:: MATRIX15 semantic presentation
begin
theorem Th1: :: MATRIX15:1
for K being Field
for a being Element of K
for A, B being Matrix of K st width A = len B holds
(a * A) * B = a * (A * B)
proof
let K be Field; ::_thesis: for a being Element of K
for A, B being Matrix of K st width A = len B holds
(a * A) * B = a * (A * B)
let a be Element of K; ::_thesis: for A, B being Matrix of K st width A = len B holds
(a * A) * B = a * (A * B)
let A, B be Matrix of K; ::_thesis: ( width A = len B implies (a * A) * B = a * (A * B) )
set aA = a * A;
set AB = A * B;
set aAB = a * (A * B);
assume A1: width A = len B ; ::_thesis: (a * A) * B = a * (A * B)
then A2: width (A * B) = width B by MATRIX_3:def_4;
A3: len (a * (A * B)) = len (A * B) by MATRIX_3:def_5;
then A4: len (a * (A * B)) = len A by A1, MATRIX_3:def_4;
A5: width (a * A) = width A by MATRIX_3:def_5;
then A6: ( len (a * A) = len A & len (a * A) = len ((a * A) * B) ) by A1, MATRIX_3:def_4, MATRIX_3:def_5;
A7: now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_(a_*_(A_*_B))_holds_
(a_*_(A_*_B))_*_(i,j)_=_((a_*_A)_*_B)_*_(i,j)
let i, j be Nat; ::_thesis: ( [i,j] in Indices (a * (A * B)) implies (a * (A * B)) * (i,j) = ((a * A) * B) * (i,j) )
assume [i,j] in Indices (a * (A * B)) ; ::_thesis: (a * (A * B)) * (i,j) = ((a * A) * B) * (i,j)
then A8: [i,j] in Indices (A * B) by MATRIXR1:18;
then i in dom (A * B) by ZFMISC_1:87;
then i in Seg (len A) by A3, A4, FINSEQ_1:def_3;
then A9: ( 1 <= i & i <= len A ) by FINSEQ_1:1;
dom (A * B) = Seg (len (A * B)) by FINSEQ_1:def_3
.= dom ((a * A) * B) by A3, A4, A6, FINSEQ_1:def_3 ;
then A10: [i,j] in Indices ((a * A) * B) by A1, A5, A2, A8, MATRIX_3:def_4;
thus (a * (A * B)) * (i,j) = a * ((A * B) * (i,j)) by A8, MATRIX_3:def_5
.= a * ((Line (A,i)) "*" (Col (B,j))) by A1, A8, MATRIX_3:def_4
.= Sum (a * (mlt ((Line (A,i)),(Col (B,j))))) by FVSUM_1:73
.= Sum (mlt ((a * (Line (A,i))),(Col (B,j)))) by A1, FVSUM_1:68
.= (Line ((a * A),i)) "*" (Col (B,j)) by A9, MATRIXR1:20
.= ((a * A) * B) * (i,j) by A1, A5, A10, MATRIX_3:def_4 ; ::_thesis: verum
end;
( width (a * (A * B)) = width (A * B) & width B = width ((a * A) * B) ) by A1, A5, MATRIX_3:def_4, MATRIX_3:def_5;
hence (a * A) * B = a * (A * B) by A2, A4, A6, A7, MATRIX_1:21; ::_thesis: verum
end;
theorem Th2: :: MATRIX15:2
for K being Field
for a, b being Element of K
for A being Matrix of K holds
( (1_ K) * A = A & a * (b * A) = (a * b) * A )
proof
let K be Field; ::_thesis: for a, b being Element of K
for A being Matrix of K holds
( (1_ K) * A = A & a * (b * A) = (a * b) * A )
let a, b be Element of K; ::_thesis: for A being Matrix of K holds
( (1_ K) * A = A & a * (b * A) = (a * b) * A )
let A be Matrix of K; ::_thesis: ( (1_ K) * A = A & a * (b * A) = (a * b) * A )
set 1A = (1_ K) * A;
set bA = b * A;
set ab = a * b;
set abA = (a * b) * A;
A1: now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_A_holds_
A_*_(i,j)_=_((1__K)_*_A)_*_(i,j)
let i, j be Nat; ::_thesis: ( [i,j] in Indices A implies A * (i,j) = ((1_ K) * A) * (i,j) )
assume A2: [i,j] in Indices A ; ::_thesis: A * (i,j) = ((1_ K) * A) * (i,j)
thus A * (i,j) = (1_ K) * (A * (i,j)) by VECTSP_1:def_6
.= ((1_ K) * A) * (i,j) by A2, MATRIX_3:def_5 ; ::_thesis: verum
end;
A3: len (a * (b * A)) = len (b * A) by MATRIX_3:def_5
.= len A by MATRIX_3:def_5
.= len ((a * b) * A) by MATRIX_3:def_5 ;
A4: now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_(a_*_(b_*_A))_holds_
(a_*_(b_*_A))_*_(i,j)_=_((a_*_b)_*_A)_*_(i,j)
A5: Indices (b * A) = Indices A by MATRIXR1:18;
A6: Indices (a * (b * A)) = Indices (b * A) by MATRIXR1:18;
let i, j be Nat; ::_thesis: ( [i,j] in Indices (a * (b * A)) implies (a * (b * A)) * (i,j) = ((a * b) * A) * (i,j) )
assume A7: [i,j] in Indices (a * (b * A)) ; ::_thesis: (a * (b * A)) * (i,j) = ((a * b) * A) * (i,j)
thus (a * (b * A)) * (i,j) = a * ((b * A) * (i,j)) by A7, A6, MATRIX_3:def_5
.= a * (b * (A * (i,j))) by A7, A6, A5, MATRIX_3:def_5
.= (a * b) * (A * (i,j)) by GROUP_1:def_3
.= ((a * b) * A) * (i,j) by A7, A6, A5, MATRIX_3:def_5 ; ::_thesis: verum
end;
A8: width (a * (b * A)) = width (b * A) by MATRIX_3:def_5
.= width A by MATRIX_3:def_5
.= width ((a * b) * A) by MATRIX_3:def_5 ;
( len ((1_ K) * A) = len A & width ((1_ K) * A) = width A ) by MATRIX_3:def_5;
hence ( (1_ K) * A = A & a * (b * A) = (a * b) * A ) by A1, A3, A8, A4, MATRIX_1:21; ::_thesis: verum
end;
Lm1: for K being Field
for A being Matrix of K holds Indices A = Indices (- A)
proof
let K be Field; ::_thesis: for A being Matrix of K holds Indices A = Indices (- A)
let A be Matrix of K; ::_thesis: Indices A = Indices (- A)
dom A = Seg (len A) by FINSEQ_1:def_3
.= Seg (len (- A)) by MATRIX_3:def_2
.= dom (- A) by FINSEQ_1:def_3 ;
hence Indices A = Indices (- A) by MATRIX_3:def_2; ::_thesis: verum
end;
Lm2: for n being Nat
for K being Field
for a being Element of K st a <> 0. K holds
(power K) . (a,n) <> 0. K
proof
let n be Nat; ::_thesis: for K being Field
for a being Element of K st a <> 0. K holds
(power K) . (a,n) <> 0. K
let K be Field; ::_thesis: for a being Element of K st a <> 0. K holds
(power K) . (a,n) <> 0. K
let a be Element of K; ::_thesis: ( a <> 0. K implies (power K) . (a,n) <> 0. K )
defpred S1[ Nat] means for n being Nat st n = $1 holds
(power K) . (a,n) <> 0. K;
assume A1: a <> 0. K ; ::_thesis: (power K) . (a,n) <> 0. K
A2: for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] )
assume S1[k] ; ::_thesis: S1[k + 1]
then A3: (power K) . (a,k) <> 0. K ;
A4: k in NAT by ORDINAL1:def_12;
let n be Nat; ::_thesis: ( n = k + 1 implies (power K) . (a,n) <> 0. K )
assume n = k + 1 ; ::_thesis: (power K) . (a,n) <> 0. K
then (power K) . (a,n) = ((power K) . (a,k)) * a by A4, GROUP_1:def_7;
hence (power K) . (a,n) <> 0. K by A1, A3, VECTSP_1:12; ::_thesis: verum
end;
A5: S1[ 0 ]
proof
A6: 1_ K <> 0. K ;
let n be Nat; ::_thesis: ( n = 0 implies (power K) . (a,n) <> 0. K )
assume n = 0 ; ::_thesis: (power K) . (a,n) <> 0. K
hence (power K) . (a,n) <> 0. K by A6, GROUP_1:def_7; ::_thesis: verum
end;
for k being Nat holds S1[k] from NAT_1:sch_2(A5, A2);
hence (power K) . (a,n) <> 0. K ; ::_thesis: verum
end;
theorem Th3: :: MATRIX15:3
for K being non empty addLoopStr
for f, g, h, w being FinSequence of K st len f = len g & len h = len w holds
(f ^ h) + (g ^ w) = (f + g) ^ (h + w)
proof
let K be non empty addLoopStr ; ::_thesis: for f, g, h, w being FinSequence of K st len f = len g & len h = len w holds
(f ^ h) + (g ^ w) = (f + g) ^ (h + w)
let f, g, h, w be FinSequence of K; ::_thesis: ( len f = len g & len h = len w implies (f ^ h) + (g ^ w) = (f + g) ^ (h + w) )
assume that
A1: len f = len g and
A2: len h = len w ; ::_thesis: (f ^ h) + (g ^ w) = (f + g) ^ (h + w)
set KK = the carrier of K;
reconsider H = h, W = w as Element of (len h) -tuples_on the carrier of K by A2, FINSEQ_2:92;
reconsider F = f, G = g as Element of (len f) -tuples_on the carrier of K by A1, FINSEQ_2:92;
reconsider FH = F ^ H, GW = G ^ W, Th36W = (F + G) ^ (H + W) as Tuple of (len f) + (len h), the carrier of K ;
reconsider FH = FH, GW = GW, Th36W = Th36W as Element of ((len f) + (len h)) -tuples_on the carrier of K by FINSEQ_2:131;
now__::_thesis:_for_i_being_Nat_st_i_in_Seg_((len_f)_+_(len_h))_holds_
(FH_+_GW)_._i_=_Th36W_._i
let i be Nat; ::_thesis: ( i in Seg ((len f) + (len h)) implies (FH + GW) . i = Th36W . i )
assume A3: i in Seg ((len f) + (len h)) ; ::_thesis: (FH + GW) . i = Th36W . i
A4: i in dom FH by A3, FINSEQ_2:124;
now__::_thesis:_(FH_+_GW)_._i_=_Th36W_._i
percases ( i in dom f or ex n being Nat st
( n in dom h & i = (len f) + n ) ) by A4, FINSEQ_1:25;
supposeA5: i in dom f ; ::_thesis: (FH + GW) . i = Th36W . i
A6: ( rng f c= the carrier of K & rng g c= the carrier of K ) by RELAT_1:def_19;
A7: dom (F + G) = Seg (len f) by FINSEQ_2:124;
A8: f . i in rng f by A5, FUNCT_1:def_3;
A9: dom F = Seg (len f) by FINSEQ_2:124;
A10: dom G = Seg (len f) by FINSEQ_2:124;
then g . i in rng g by A5, A9, FUNCT_1:def_3;
then reconsider fi = f . i, gi = g . i as Element of K by A8, A6;
A11: FH . i = fi by A5, FINSEQ_1:def_7;
GW . i = gi by A5, A9, A10, FINSEQ_1:def_7;
hence (FH + GW) . i = fi + gi by A3, A11, FVSUM_1:18
.= (F + G) . i by A5, A9, FVSUM_1:18
.= Th36W . i by A5, A9, A7, FINSEQ_1:def_7 ;
::_thesis: verum
end;
suppose ex n being Nat st
( n in dom h & i = (len f) + n ) ; ::_thesis: (FH + GW) . i = Th36W . i
then consider n being Nat such that
A12: n in dom h and
A13: i = (len f) + n ;
A14: h . n in rng h by A12, FUNCT_1:def_3;
A15: ( rng h c= the carrier of K & rng w c= the carrier of K ) by RELAT_1:def_19;
A16: dom H = Seg (len h) by FINSEQ_2:124;
A17: dom W = Seg (len h) by FINSEQ_2:124;
then w . n in rng w by A12, A16, FUNCT_1:def_3;
then reconsider hn = h . n, wn = w . n as Element of K by A14, A15;
A18: FH . i = hn by A12, A13, FINSEQ_1:def_7;
A19: ( dom (H + W) = Seg (len h) & len (F + G) = len f ) by CARD_1:def_7, FINSEQ_2:124;
GW . i = wn by A1, A12, A13, A16, A17, FINSEQ_1:def_7;
hence (FH + GW) . i = hn + wn by A3, A18, FVSUM_1:18
.= (H + W) . n by A12, A16, FVSUM_1:18
.= Th36W . i by A12, A13, A16, A19, FINSEQ_1:def_7 ;
::_thesis: verum
end;
end;
end;
hence (FH + GW) . i = Th36W . i ; ::_thesis: verum
end;
hence (f ^ h) + (g ^ w) = (f + g) ^ (h + w) by FINSEQ_2:119; ::_thesis: verum
end;
theorem Th4: :: MATRIX15:4
for K being non empty multMagma
for f, g being FinSequence of K
for a being Element of K holds a * (f ^ g) = (a * f) ^ (a * g)
proof
let K be non empty multMagma ; ::_thesis: for f, g being FinSequence of K
for a being Element of K holds a * (f ^ g) = (a * f) ^ (a * g)
let f, g be FinSequence of K; ::_thesis: for a being Element of K holds a * (f ^ g) = (a * f) ^ (a * g)
let a be Element of K; ::_thesis: a * (f ^ g) = (a * f) ^ (a * g)
set KK = the carrier of K;
reconsider F = f as Element of (len f) -tuples_on the carrier of K by FINSEQ_2:92;
reconsider G = g as Element of (len g) -tuples_on the carrier of K by FINSEQ_2:92;
reconsider FG = F ^ G, aFaG = (a * F) ^ (a * G) as Element of ((len f) + (len g)) -tuples_on the carrier of K by FINSEQ_2:131;
now__::_thesis:_for_i_being_Nat_st_i_in_Seg_((len_f)_+_(len_g))_holds_
(a_*_FG)_._i_=_aFaG_._i
let i be Nat; ::_thesis: ( i in Seg ((len f) + (len g)) implies (a * FG) . i = aFaG . i )
assume A1: i in Seg ((len f) + (len g)) ; ::_thesis: (a * FG) . i = aFaG . i
A2: i in dom FG by A1, FINSEQ_2:124;
now__::_thesis:_(a_*_FG)_._i_=_aFaG_._i
percases ( i in dom f or ex n being Nat st
( n in dom g & i = (len f) + n ) ) by A2, FINSEQ_1:25;
supposeA3: i in dom f ; ::_thesis: (a * FG) . i = aFaG . i
A4: rng f c= the carrier of K by RELAT_1:def_19;
f . i in rng f by A3, FUNCT_1:def_3;
then reconsider fi = f . i as Element of K by A4;
A5: dom F = Seg (len f) by FINSEQ_2:124;
A6: dom (a * F) = Seg (len f) by FINSEQ_2:124;
FG . i = fi by A3, FINSEQ_1:def_7;
hence (a * FG) . i = a * fi by A1, FVSUM_1:51
.= (a * F) . i by A3, A5, FVSUM_1:51
.= aFaG . i by A3, A5, A6, FINSEQ_1:def_7 ;
::_thesis: verum
end;
supposeA7: ex n being Nat st
( n in dom g & i = (len f) + n ) ; ::_thesis: (a * FG) . i = aFaG . i
A8: rng g c= the carrier of K by RELAT_1:def_19;
A9: ( dom (a * G) = Seg (len g) & len (a * F) = len f ) by CARD_1:def_7, FINSEQ_2:124;
consider n being Nat such that
A10: n in dom g and
A11: i = (len f) + n by A7;
g . n in rng g by A10, FUNCT_1:def_3;
then reconsider gn = g . n as Element of K by A8;
A12: dom G = Seg (len g) by FINSEQ_2:124;
FG . i = gn by A10, A11, FINSEQ_1:def_7;
hence (a * FG) . i = a * gn by A1, FVSUM_1:51
.= (a * G) . n by A10, A12, FVSUM_1:51
.= aFaG . i by A10, A11, A12, A9, FINSEQ_1:def_7 ;
::_thesis: verum
end;
end;
end;
hence (a * FG) . i = aFaG . i ; ::_thesis: verum
end;
hence a * (f ^ g) = (a * f) ^ (a * g) by FINSEQ_2:119; ::_thesis: verum
end;
theorem Th5: :: MATRIX15:5
for f being Function
for p1, p2, f1, f2 being FinSequence st rng p1 c= dom f & rng p2 c= dom f & f1 = f * p1 & f2 = f * p2 holds
f * (p1 ^ p2) = f1 ^ f2
proof
let f be Function; ::_thesis: for p1, p2, f1, f2 being FinSequence st rng p1 c= dom f & rng p2 c= dom f & f1 = f * p1 & f2 = f * p2 holds
f * (p1 ^ p2) = f1 ^ f2
let p1, p2, f1, f2 be FinSequence; ::_thesis: ( rng p1 c= dom f & rng p2 c= dom f & f1 = f * p1 & f2 = f * p2 implies f * (p1 ^ p2) = f1 ^ f2 )
assume that
A1: rng p1 c= dom f and
A2: rng p2 c= dom f and
A3: f1 = f * p1 and
A4: f2 = f * p2 ; ::_thesis: f * (p1 ^ p2) = f1 ^ f2
A5: dom (p1 ^ p2) = Seg ((len p1) + (len p2)) by FINSEQ_1:def_7;
rng (p1 ^ p2) = (rng p1) \/ (rng p2) by FINSEQ_1:31;
then A6: dom (f * (p1 ^ p2)) = dom (p1 ^ p2) by A1, A2, RELAT_1:27, XBOOLE_1:8;
A7: dom f1 = dom p1 by A1, A3, RELAT_1:27;
then A8: len f1 = len p1 by FINSEQ_3:29;
A9: dom f2 = dom p2 by A2, A4, RELAT_1:27;
then len f2 = len p2 by FINSEQ_3:29;
then A10: dom (f1 ^ f2) = dom (f * (p1 ^ p2)) by A6, A8, A5, FINSEQ_1:def_7;
now__::_thesis:_for_x_being_set_st_x_in_dom_(f1_^_f2)_holds_
(f1_^_f2)_._x_=_(f_*_(p1_^_p2))_._x
let x be set ; ::_thesis: ( x in dom (f1 ^ f2) implies (f1 ^ f2) . x = (f * (p1 ^ p2)) . x )
assume A11: x in dom (f1 ^ f2) ; ::_thesis: (f1 ^ f2) . x = (f * (p1 ^ p2)) . x
reconsider i = x as Element of NAT by A11;
now__::_thesis:_(f1_^_f2)_._i_=_(f_*_(p1_^_p2))_._i
percases ( i in dom p1 or not i in dom p1 ) ;
supposeA12: i in dom p1 ; ::_thesis: (f1 ^ f2) . i = (f * (p1 ^ p2)) . i
hence (f1 ^ f2) . i = f1 . i by A7, FINSEQ_1:def_7
.= f . (p1 . i) by A3, A7, A12, FUNCT_1:12
.= f . ((p1 ^ p2) . i) by A12, FINSEQ_1:def_7
.= (f * (p1 ^ p2)) . i by A10, A11, FUNCT_1:12 ;
::_thesis: verum
end;
suppose not i in dom p1 ; ::_thesis: (f1 ^ f2) . i = (f * (p1 ^ p2)) . i
then consider n being Nat such that
A13: n in dom p2 and
A14: i = (len p1) + n by A7, A9, A8, A11, FINSEQ_1:25;
thus (f1 ^ f2) . i = f2 . n by A9, A8, A13, A14, FINSEQ_1:def_7
.= f . (p2 . n) by A4, A9, A13, FUNCT_1:12
.= f . ((p1 ^ p2) . i) by A13, A14, FINSEQ_1:def_7
.= (f * (p1 ^ p2)) . i by A10, A11, FUNCT_1:12 ; ::_thesis: verum
end;
end;
end;
hence (f1 ^ f2) . x = (f * (p1 ^ p2)) . x ; ::_thesis: verum
end;
hence f * (p1 ^ p2) = f1 ^ f2 by A10, FUNCT_1:2; ::_thesis: verum
end;
theorem Th6: :: MATRIX15:6
for f being FinSequence of NAT
for n being Nat st f is one-to-one & rng f c= Seg n & ( for i, j being Nat st i in dom f & j in dom f & i < j holds
f . i < f . j ) holds
Sgm (rng f) = f
proof
defpred S1[ Nat] means for f being FinSequence of NAT
for n being Nat st len f = $1 & rng f c= Seg n & f is one-to-one & ( for i, j being Nat st i in dom f & j in dom f & i < j holds
f . i < f . j ) holds
Sgm (rng f) = f;
A1: S1[ 0 ]
proof
let f be FinSequence of NAT ; ::_thesis: for n being Nat st len f = 0 & rng f c= Seg n & f is one-to-one & ( for i, j being Nat st i in dom f & j in dom f & i < j holds
f . i < f . j ) holds
Sgm (rng f) = f
let n be Nat; ::_thesis: ( len f = 0 & rng f c= Seg n & f is one-to-one & ( for i, j being Nat st i in dom f & j in dom f & i < j holds
f . i < f . j ) implies Sgm (rng f) = f )
assume that
A2: len f = 0 and
rng f c= Seg n and
f is one-to-one and
for i, j being Nat st i in dom f & j in dom f & i < j holds
f . i < f . j ; ::_thesis: Sgm (rng f) = f
f = {} by A2;
hence Sgm (rng f) = f by FINSEQ_3:43; ::_thesis: verum
end;
A3: for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A4: S1[n] ; ::_thesis: S1[n + 1]
set n1 = n + 1;
let f be FinSequence of NAT ; ::_thesis: for n being Nat st len f = n + 1 & rng f c= Seg n & f is one-to-one & ( for i, j being Nat st i in dom f & j in dom f & i < j holds
f . i < f . j ) holds
Sgm (rng f) = f
let k be Nat; ::_thesis: ( len f = n + 1 & rng f c= Seg k & f is one-to-one & ( for i, j being Nat st i in dom f & j in dom f & i < j holds
f . i < f . j ) implies Sgm (rng f) = f )
assume that
A5: len f = n + 1 and
A6: rng f c= Seg k and
A7: f is one-to-one and
A8: for i, j being Nat st i in dom f & j in dom f & i < j holds
f . i < f . j ; ::_thesis: Sgm (rng f) = f
set fn = f | n;
A9: f = (f | n) ^ <*(f . (n + 1))*> by A5, FINSEQ_3:55;
then A10: rng (f | n) c= rng f by FINSEQ_1:29;
A11: dom (f | n) c= dom f by A9, FINSEQ_1:26;
A12: for i, j being Nat st i in dom (f | n) & j in dom (f | n) & i < j holds
(f | n) . i < (f | n) . j
proof
let i, j be Nat; ::_thesis: ( i in dom (f | n) & j in dom (f | n) & i < j implies (f | n) . i < (f | n) . j )
assume that
A13: ( i in dom (f | n) & j in dom (f | n) ) and
A14: i < j ; ::_thesis: (f | n) . i < (f | n) . j
( (f | n) . i = f . i & (f | n) . j = f . j ) by A9, A13, FINSEQ_1:def_7;
hence (f | n) . i < (f | n) . j by A8, A11, A13, A14; ::_thesis: verum
end;
A15: len (f | n) = n by A5, FINSEQ_3:53;
A16: now__::_thesis:_for_m9,_n9_being_Element_of_NAT_st_m9_in_rng_(f_|_n)_&_n9_in_{(f_._(n_+_1))}_holds_
m9_<_n9
A17: ( n + 1 in Seg (n + 1) & dom f = Seg (n + 1) ) by A5, FINSEQ_1:4, FINSEQ_1:def_3;
let m9, n9 be Element of NAT ; ::_thesis: ( m9 in rng (f | n) & n9 in {(f . (n + 1))} implies m9 < n9 )
assume that
A18: m9 in rng (f | n) and
A19: n9 in {(f . (n + 1))} ; ::_thesis: m9 < n9
consider x being set such that
A20: x in dom (f | n) and
A21: (f | n) . x = m9 by A18, FUNCT_1:def_3;
reconsider x = x as Element of NAT by A20;
A22: f . x = (f | n) . x by A9, A20, FINSEQ_1:def_7;
dom (f | n) = Seg n by A15, FINSEQ_1:def_3;
then x <= n by A20, FINSEQ_1:1;
then x < n + 1 by NAT_1:13;
then f . x < f . (n + 1) by A8, A11, A20, A17;
hence m9 < n9 by A19, A21, A22, TARSKI:def_1; ::_thesis: verum
end;
f | n is one-to-one by A7, FUNCT_1:52;
then A23: Sgm (rng (f | n)) = f | n by A4, A6, A15, A10, A12, XBOOLE_1:1;
A24: rng <*(f . (n + 1))*> = {(f . (n + 1))} by FINSEQ_1:39;
rng <*(f . (n + 1))*> c= rng f by A9, FINSEQ_1:30;
then A25: {(f . (n + 1))} c= Seg k by A6, A24, XBOOLE_1:1;
A26: rng f = (rng (f | n)) \/ (rng <*(f . (n + 1))*>) by A9, FINSEQ_1:31;
A27: f . (n + 1) in {(f . (n + 1))} by TARSKI:def_1;
rng (f | n) c= Seg k by A6, A10, XBOOLE_1:1;
hence Sgm (rng f) = (f | n) ^ (Sgm {(f . (n + 1))}) by A26, A24, A25, A23, A16, FINSEQ_3:42
.= f by A9, A25, A27, FINSEQ_3:44 ;
::_thesis: verum
end;
let f be FinSequence of NAT ; ::_thesis: for n being Nat st f is one-to-one & rng f c= Seg n & ( for i, j being Nat st i in dom f & j in dom f & i < j holds
f . i < f . j ) holds
Sgm (rng f) = f
let n be Nat; ::_thesis: ( f is one-to-one & rng f c= Seg n & ( for i, j being Nat st i in dom f & j in dom f & i < j holds
f . i < f . j ) implies Sgm (rng f) = f )
assume A28: ( f is one-to-one & rng f c= Seg n & ( for i, j being Nat st i in dom f & j in dom f & i < j holds
f . i < f . j ) ) ; ::_thesis: Sgm (rng f) = f
for n being Nat holds S1[n] from NAT_1:sch_2(A1, A3);
then for g being FinSequence of NAT
for n being Nat st len g = len f & rng g c= Seg n & g is one-to-one & ( for i, j being Nat st i in dom g & j in dom g & i < j holds
g . i < g . j ) holds
Sgm (rng g) = g ;
hence Sgm (rng f) = f by A28; ::_thesis: verum
end;
theorem Th7: :: MATRIX15:7
for K being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for p being FinSequence of K
for i, j being Nat st i in dom p & j in dom p & i <> j & ( for k being Nat st k in dom p & k <> i & k <> j holds
p . k = 0. K ) holds
Sum p = (p /. i) + (p /. j)
proof
let K be non empty right_complementable Abelian add-associative right_zeroed addLoopStr ; ::_thesis: for p being FinSequence of K
for i, j being Nat st i in dom p & j in dom p & i <> j & ( for k being Nat st k in dom p & k <> i & k <> j holds
p . k = 0. K ) holds
Sum p = (p /. i) + (p /. j)
let p be FinSequence of K; ::_thesis: for i, j being Nat st i in dom p & j in dom p & i <> j & ( for k being Nat st k in dom p & k <> i & k <> j holds
p . k = 0. K ) holds
Sum p = (p /. i) + (p /. j)
A1: now__::_thesis:_for_i,_j_being_Nat_st_i_in_dom_p_&_j_in_dom_p_&_i_<_j_&_(_for_k_being_Nat_st_k_in_dom_p_&_k_<>_i_&_k_<>_j_holds_
p_._k_=_0._K_)_holds_
Sum_p_=_(p_/._i)_+_(p_/._j)
let i, j be Nat; ::_thesis: ( i in dom p & j in dom p & i < j & ( for k being Nat st k in dom p & k <> i & k <> j holds
p . k = 0. K ) implies Sum p = (p /. i) + (p /. j) )
assume that
A2: i in dom p and
A3: j in dom p and
A4: i < j and
A5: for k being Nat st k in dom p & k <> i & k <> j holds
p . k = 0. K ; ::_thesis: Sum p = (p /. i) + (p /. j)
A6: dom p = Seg (len p) by FINSEQ_1:def_3;
then ( i in NAT & 1 <= i ) by A2, FINSEQ_1:1;
then A7: i in Seg i ;
set pI = p | i;
consider q being FinSequence such that
A8: p = (p | i) ^ q by FINSEQ_1:80;
reconsider q = q as FinSequence of K by A8, FINSEQ_1:36;
A9: i <= len p by A2, A6, FINSEQ_1:1;
then A10: len (p | i) = i by FINSEQ_1:17;
A11: dom (p | i) = Seg i by A9, FINSEQ_1:17;
then not j in dom (p | i) by A4, FINSEQ_1:1;
then consider ji being Nat such that
A12: ji in dom q and
A13: j = i + ji by A3, A8, A10, FINSEQ_1:25;
now__::_thesis:_for_k_being_Nat_st_k_in_dom_q_&_k_<>_ji_holds_
q_._k_=_0._K
let k be Nat; ::_thesis: ( k in dom q & k <> ji implies q . k = 0. K )
assume that
A14: k in dom q and
A15: k <> ji ; ::_thesis: q . k = 0. K
reconsider kk = k as Element of NAT by ORDINAL1:def_12;
A16: i + kk <> i + ji by A15;
dom q = Seg (len q) by FINSEQ_1:def_3;
then k >= 1 by A14, FINSEQ_1:1;
then k + i >= i + 1 by XREAL_1:7;
then A17: i + kk <> i by NAT_1:13;
thus q . k = p . (i + kk) by A8, A10, A14, FINSEQ_1:def_7
.= 0. K by A5, A8, A10, A13, A14, A17, A16, FINSEQ_1:28 ; ::_thesis: verum
end;
then A18: Sum q = q . ji by A12, MATRIX_3:12
.= p . j by A8, A10, A12, A13, FINSEQ_1:def_7
.= p /. j by A3, PARTFUN1:def_6 ;
A19: Seg i c= Seg (len p) by A9, FINSEQ_1:5;
now__::_thesis:_for_k_being_Nat_st_k_in_dom_(p_|_i)_&_k_<>_i_holds_
(p_|_i)_._k_=_0._K
let k be Nat; ::_thesis: ( k in dom (p | i) & k <> i implies (p | i) . k = 0. K )
assume that
A20: k in dom (p | i) and
A21: k <> i ; ::_thesis: (p | i) . k = 0. K
reconsider kk = k as Element of NAT by ORDINAL1:def_12;
A22: k <> j by A4, A11, A20, FINSEQ_1:1;
thus (p | i) . k = p . kk by A8, A20, FINSEQ_1:def_7
.= 0. K by A5, A6, A11, A19, A20, A21, A22 ; ::_thesis: verum
end;
then Sum (p | i) = (p | i) . i by A7, A11, MATRIX_3:12
.= p . i by A8, A7, A11, FINSEQ_1:def_7
.= p /. i by A6, A7, A19, PARTFUN1:def_6 ;
hence Sum p = (p /. i) + (p /. j) by A8, A18, RLVECT_1:41; ::_thesis: verum
end;
let i, j be Nat; ::_thesis: ( i in dom p & j in dom p & i <> j & ( for k being Nat st k in dom p & k <> i & k <> j holds
p . k = 0. K ) implies Sum p = (p /. i) + (p /. j) )
assume that
A23: ( i in dom p & j in dom p ) and
A24: i <> j and
A25: for k being Nat st k in dom p & k <> i & k <> j holds
p . k = 0. K ; ::_thesis: Sum p = (p /. i) + (p /. j)
A26: ( i < j or j < i ) by A24, XXREAL_0:1;
for k being Nat st k in dom p & k <> j & k <> i holds
p . k = 0. K by A25;
hence Sum p = (p /. i) + (p /. j) by A1, A23, A25, A26; ::_thesis: verum
end;
theorem Th8: :: MATRIX15:8
for i, m, n being Nat st i in Seg m holds
(Sgm ((Seg (n + m)) \ (Seg n))) . i = n + i
proof
let i, m, n be Nat; ::_thesis: ( i in Seg m implies (Sgm ((Seg (n + m)) \ (Seg n))) . i = n + i )
assume A1: i in Seg m ; ::_thesis: (Sgm ((Seg (n + m)) \ (Seg n))) . i = n + i
reconsider N = n as Element of NAT by ORDINAL1:def_12;
set I = idseq m;
A2: dom (idseq m) = Seg (len (idseq m)) by FINSEQ_1:def_3;
A3: len (idseq m) = m by CARD_1:def_7;
A4: dom (N Shift (idseq m)) = { (N + k) where k is Element of NAT : k in dom (idseq m) } by PNPROC_1:def_14;
A5: (Seg (n + m)) \ (Seg n) c= dom (N Shift (idseq m))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (Seg (n + m)) \ (Seg n) or x in dom (N Shift (idseq m)) )
assume A6: x in (Seg (n + m)) \ (Seg n) ; ::_thesis: x in dom (N Shift (idseq m))
reconsider i = x as Element of NAT by A6;
A7: i in Seg (n + m) by A6, XBOOLE_0:def_5;
not i in Seg n by A6, XBOOLE_0:def_5;
then A8: ( i < 1 or i > n ) ;
then reconsider IN = i - n as Element of NAT by A7, FINSEQ_1:1, NAT_1:21;
A9: n + IN = i ;
i <= n + m by A7, FINSEQ_1:1;
then A10: IN <= m by A9, XREAL_1:8;
IN >= 1 by A7, A8, A9, FINSEQ_1:1, NAT_1:19;
then IN in dom (idseq m) by A2, A3, A10;
then n + IN in dom (N Shift (idseq m)) by A4;
hence x in dom (N Shift (idseq m)) ; ::_thesis: verum
end;
dom (N Shift (idseq m)) c= (Seg (n + m)) \ (Seg n)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom (N Shift (idseq m)) or x in (Seg (n + m)) \ (Seg n) )
assume x in dom (N Shift (idseq m)) ; ::_thesis: x in (Seg (n + m)) \ (Seg n)
then consider k being Element of NAT such that
A11: n + k = x and
A12: k in dom (idseq m) by A4;
k <= m by A2, A3, A12, FINSEQ_1:1;
then A13: n + k <= n + m by XREAL_1:7;
1 <= k by A2, A12, FINSEQ_1:1;
then A14: n + 1 <= n + k by XREAL_1:7;
then n + k > n by NAT_1:13;
then A15: not k + n in Seg n by FINSEQ_1:1;
1 <= n + 1 by NAT_1:11;
then 1 <= n + k by A14, XXREAL_0:2;
then n + k in Seg (n + m) by A13;
hence x in (Seg (n + m)) \ (Seg n) by A11, A15, XBOOLE_0:def_5; ::_thesis: verum
end;
then (Seg (n + m)) \ (Seg n) = dom (N Shift (idseq m)) by A5, XBOOLE_0:def_10;
hence (Sgm ((Seg (n + m)) \ (Seg n))) . i = n + i by A1, A2, A3, PNPROC_1:56; ::_thesis: verum
end;
theorem Th9: :: MATRIX15:9
for D being non empty set
for A being Matrix of D
for Bx, By, Cx, Cy being finite without_zero Subset of NAT st [:Bx,By:] c= Indices A & [:Cx,Cy:] c= Indices A holds
for B being Matrix of card Bx, card By,D
for C being Matrix of card Cx, card Cy,D st ( for i, j, bi, bj, ci, cj being Nat st [i,j] in [:Bx,By:] /\ [:Cx,Cy:] & bi = ((Sgm Bx) ") . i & bj = ((Sgm By) ") . j & ci = ((Sgm Cx) ") . i & cj = ((Sgm Cy) ") . j holds
B * (bi,bj) = C * (ci,cj) ) holds
ex M being Matrix of len A, width A,D st
( Segm (M,Bx,By) = B & Segm (M,Cx,Cy) = C & ( for i, j being Nat st [i,j] in (Indices M) \ ([:Bx,By:] \/ [:Cx,Cy:]) holds
M * (i,j) = A * (i,j) ) )
proof
let D be non empty set ; ::_thesis: for A being Matrix of D
for Bx, By, Cx, Cy being finite without_zero Subset of NAT st [:Bx,By:] c= Indices A & [:Cx,Cy:] c= Indices A holds
for B being Matrix of card Bx, card By,D
for C being Matrix of card Cx, card Cy,D st ( for i, j, bi, bj, ci, cj being Nat st [i,j] in [:Bx,By:] /\ [:Cx,Cy:] & bi = ((Sgm Bx) ") . i & bj = ((Sgm By) ") . j & ci = ((Sgm Cx) ") . i & cj = ((Sgm Cy) ") . j holds
B * (bi,bj) = C * (ci,cj) ) holds
ex M being Matrix of len A, width A,D st
( Segm (M,Bx,By) = B & Segm (M,Cx,Cy) = C & ( for i, j being Nat st [i,j] in (Indices M) \ ([:Bx,By:] \/ [:Cx,Cy:]) holds
M * (i,j) = A * (i,j) ) )
let A be Matrix of D; ::_thesis: for Bx, By, Cx, Cy being finite without_zero Subset of NAT st [:Bx,By:] c= Indices A & [:Cx,Cy:] c= Indices A holds
for B being Matrix of card Bx, card By,D
for C being Matrix of card Cx, card Cy,D st ( for i, j, bi, bj, ci, cj being Nat st [i,j] in [:Bx,By:] /\ [:Cx,Cy:] & bi = ((Sgm Bx) ") . i & bj = ((Sgm By) ") . j & ci = ((Sgm Cx) ") . i & cj = ((Sgm Cy) ") . j holds
B * (bi,bj) = C * (ci,cj) ) holds
ex M being Matrix of len A, width A,D st
( Segm (M,Bx,By) = B & Segm (M,Cx,Cy) = C & ( for i, j being Nat st [i,j] in (Indices M) \ ([:Bx,By:] \/ [:Cx,Cy:]) holds
M * (i,j) = A * (i,j) ) )
let Bx, By, Cx, Cy be finite without_zero Subset of NAT; ::_thesis: ( [:Bx,By:] c= Indices A & [:Cx,Cy:] c= Indices A implies for B being Matrix of card Bx, card By,D
for C being Matrix of card Cx, card Cy,D st ( for i, j, bi, bj, ci, cj being Nat st [i,j] in [:Bx,By:] /\ [:Cx,Cy:] & bi = ((Sgm Bx) ") . i & bj = ((Sgm By) ") . j & ci = ((Sgm Cx) ") . i & cj = ((Sgm Cy) ") . j holds
B * (bi,bj) = C * (ci,cj) ) holds
ex M being Matrix of len A, width A,D st
( Segm (M,Bx,By) = B & Segm (M,Cx,Cy) = C & ( for i, j being Nat st [i,j] in (Indices M) \ ([:Bx,By:] \/ [:Cx,Cy:]) holds
M * (i,j) = A * (i,j) ) ) )
assume that
A1: [:Bx,By:] c= Indices A and
A2: [:Cx,Cy:] c= Indices A ; ::_thesis: for B being Matrix of card Bx, card By,D
for C being Matrix of card Cx, card Cy,D st ( for i, j, bi, bj, ci, cj being Nat st [i,j] in [:Bx,By:] /\ [:Cx,Cy:] & bi = ((Sgm Bx) ") . i & bj = ((Sgm By) ") . j & ci = ((Sgm Cx) ") . i & cj = ((Sgm Cy) ") . j holds
B * (bi,bj) = C * (ci,cj) ) holds
ex M being Matrix of len A, width A,D st
( Segm (M,Bx,By) = B & Segm (M,Cx,Cy) = C & ( for i, j being Nat st [i,j] in (Indices M) \ ([:Bx,By:] \/ [:Cx,Cy:]) holds
M * (i,j) = A * (i,j) ) )
set w = width A;
set l = len A;
set cy = card Cy;
set cx = card Cx;
set bY = card By;
set bx = card Bx;
let B be Matrix of card Bx, card By,D; ::_thesis: for C being Matrix of card Cx, card Cy,D st ( for i, j, bi, bj, ci, cj being Nat st [i,j] in [:Bx,By:] /\ [:Cx,Cy:] & bi = ((Sgm Bx) ") . i & bj = ((Sgm By) ") . j & ci = ((Sgm Cx) ") . i & cj = ((Sgm Cy) ") . j holds
B * (bi,bj) = C * (ci,cj) ) holds
ex M being Matrix of len A, width A,D st
( Segm (M,Bx,By) = B & Segm (M,Cx,Cy) = C & ( for i, j being Nat st [i,j] in (Indices M) \ ([:Bx,By:] \/ [:Cx,Cy:]) holds
M * (i,j) = A * (i,j) ) )
let C be Matrix of card Cx, card Cy,D; ::_thesis: ( ( for i, j, bi, bj, ci, cj being Nat st [i,j] in [:Bx,By:] /\ [:Cx,Cy:] & bi = ((Sgm Bx) ") . i & bj = ((Sgm By) ") . j & ci = ((Sgm Cx) ") . i & cj = ((Sgm Cy) ") . j holds
B * (bi,bj) = C * (ci,cj) ) implies ex M being Matrix of len A, width A,D st
( Segm (M,Bx,By) = B & Segm (M,Cx,Cy) = C & ( for i, j being Nat st [i,j] in (Indices M) \ ([:Bx,By:] \/ [:Cx,Cy:]) holds
M * (i,j) = A * (i,j) ) ) )
assume A3: for i, j, bi, bj, ci, cj being Nat st [i,j] in [:Bx,By:] /\ [:Cx,Cy:] & bi = ((Sgm Bx) ") . i & bj = ((Sgm By) ") . j & ci = ((Sgm Cx) ") . i & cj = ((Sgm Cy) ") . j holds
B * (bi,bj) = C * (ci,cj) ; ::_thesis: ex M being Matrix of len A, width A,D st
( Segm (M,Bx,By) = B & Segm (M,Cx,Cy) = C & ( for i, j being Nat st [i,j] in (Indices M) \ ([:Bx,By:] \/ [:Cx,Cy:]) holds
M * (i,j) = A * (i,j) ) )
A4: ex kBy being Nat st By c= Seg kBy by MATRIX13:43;
then A5: rng (Sgm By) = By by FINSEQ_1:def_13;
defpred S1[ set , set , set ] means for i, j being Nat st $1 = i & $2 = j holds
( ( [i,j] in [:Bx,By:] implies ( ex m, n being Nat st
( m in dom (Sgm Bx) & n in dom (Sgm By) & (Sgm Bx) . m = i & (Sgm By) . n = j ) & ( for m, n being Nat st m in dom (Sgm Bx) & n in dom (Sgm By) & (Sgm Bx) . m = i & (Sgm By) . n = j holds
$3 = B * (m,n) ) ) ) & ( [i,j] in [:Cx,Cy:] implies ( ex m, n being Nat st
( m in dom (Sgm Cx) & n in dom (Sgm Cy) & (Sgm Cx) . m = i & (Sgm Cy) . n = j ) & ( for m, n being Nat st m in dom (Sgm Cx) & n in dom (Sgm Cy) & (Sgm Cx) . m = i & (Sgm Cy) . n = j holds
$3 = C * (m,n) ) ) ) & ( not [i,j] in [:Bx,By:] \/ [:Cx,Cy:] implies $3 = A * (i,j) ) );
A6: ex kBx being Nat st Bx c= Seg kBx by MATRIX13:43;
then A7: rng (Sgm Bx) = Bx by FINSEQ_1:def_13;
A8: dom (Sgm By) = Seg (card By) by A4, FINSEQ_3:40;
A9: Sgm By is one-to-one by A4, FINSEQ_3:92;
A10: Sgm Bx is one-to-one by A6, FINSEQ_3:92;
A11: ex kCy being Nat st Cy c= Seg kCy by MATRIX13:43;
then A12: rng (Sgm Cy) = Cy by FINSEQ_1:def_13;
A13: Sgm Cy is one-to-one by A11, FINSEQ_3:92;
A14: ex kCx being Nat st Cx c= Seg kCx by MATRIX13:43;
then A15: rng (Sgm Cx) = Cx by FINSEQ_1:def_13;
A16: Sgm Cx is one-to-one by A14, FINSEQ_3:92;
A17: for i, j being Nat st [i,j] in [:(Seg (len A)),(Seg (width A)):] holds
ex x being Element of D st S1[i,j,x]
proof
let i, j be Nat; ::_thesis: ( [i,j] in [:(Seg (len A)),(Seg (width A)):] implies ex x being Element of D st S1[i,j,x] )
assume [i,j] in [:(Seg (len A)),(Seg (width A)):] ; ::_thesis: ex x being Element of D st S1[i,j,x]
percases ( ( [i,j] in [:Bx,By:] & [i,j] in [:Cx,Cy:] ) or ( [i,j] in [:Bx,By:] & not [i,j] in [:Cx,Cy:] ) or ( not [i,j] in [:Bx,By:] & [i,j] in [:Cx,Cy:] ) or ( not [i,j] in [:Bx,By:] & not [i,j] in [:Cx,Cy:] ) ) ;
supposeA18: ( [i,j] in [:Bx,By:] & [i,j] in [:Cx,Cy:] ) ; ::_thesis: ex x being Element of D st S1[i,j,x]
then j in Cy by ZFMISC_1:87;
then consider yC being set such that
A19: yC in dom (Sgm Cy) and
A20: (Sgm Cy) . yC = j by A12, FUNCT_1:def_3;
j in By by A18, ZFMISC_1:87;
then consider y being set such that
A21: y in dom (Sgm By) and
A22: (Sgm By) . y = j by A5, FUNCT_1:def_3;
i in Cx by A18, ZFMISC_1:87;
then consider xC being set such that
A23: xC in dom (Sgm Cx) and
A24: (Sgm Cx) . xC = i by A15, FUNCT_1:def_3;
i in Bx by A18, ZFMISC_1:87;
then consider x being set such that
A25: x in dom (Sgm Bx) and
A26: (Sgm Bx) . x = i by A7, FUNCT_1:def_3;
reconsider x = x, y = y as Element of NAT by A25, A21;
take BB = B * (x,y); ::_thesis: S1[i,j,BB]
A27: now__::_thesis:_for_m,_n_being_Nat_st_m_in_dom_(Sgm_Cx)_&_n_in_dom_(Sgm_Cy)_&_(Sgm_Cx)_._m_=_i_&_(Sgm_Cy)_._n_=_j_holds_
BB_=_C_*_(m,n)
let m, n be Nat; ::_thesis: ( m in dom (Sgm Cx) & n in dom (Sgm Cy) & (Sgm Cx) . m = i & (Sgm Cy) . n = j implies BB = C * (m,n) )
assume ( m in dom (Sgm Cx) & n in dom (Sgm Cy) & (Sgm Cx) . m = i & (Sgm Cy) . n = j ) ; ::_thesis: BB = C * (m,n)
then A28: ( ((Sgm Cx) ") . i = m & ((Sgm Cy) ") . j = n ) by A16, A13, FUNCT_1:32;
A29: ((Sgm By) ") . j = y by A9, A21, A22, FUNCT_1:32;
( [i,j] in [:Bx,By:] /\ [:Cx,Cy:] & ((Sgm Bx) ") . i = x ) by A10, A18, A25, A26, FUNCT_1:32, XBOOLE_0:def_4;
hence BB = C * (m,n) by A3, A29, A28; ::_thesis: verum
end;
now__::_thesis:_for_m,_n_being_Nat_st_m_in_dom_(Sgm_Bx)_&_n_in_dom_(Sgm_By)_&_(Sgm_Bx)_._m_=_i_&_(Sgm_By)_._n_=_j_holds_
BB_=_B_*_(m,n)
let m, n be Nat; ::_thesis: ( m in dom (Sgm Bx) & n in dom (Sgm By) & (Sgm Bx) . m = i & (Sgm By) . n = j implies BB = B * (m,n) )
assume that
A30: m in dom (Sgm Bx) and
A31: n in dom (Sgm By) and
A32: (Sgm Bx) . m = i and
A33: (Sgm By) . n = j ; ::_thesis: BB = B * (m,n)
x = m by A10, A25, A26, A30, A32, FUNCT_1:def_4;
hence BB = B * (m,n) by A9, A21, A22, A31, A33, FUNCT_1:def_4; ::_thesis: verum
end;
hence S1[i,j,BB] by A18, A25, A26, A21, A22, A23, A24, A19, A20, A27, XBOOLE_0:def_3; ::_thesis: verum
end;
supposeA34: ( [i,j] in [:Bx,By:] & not [i,j] in [:Cx,Cy:] ) ; ::_thesis: ex x being Element of D st S1[i,j,x]
then j in By by ZFMISC_1:87;
then consider y being set such that
A35: y in dom (Sgm By) and
A36: (Sgm By) . y = j by A5, FUNCT_1:def_3;
i in Bx by A34, ZFMISC_1:87;
then consider x being set such that
A37: x in dom (Sgm Bx) and
A38: (Sgm Bx) . x = i by A7, FUNCT_1:def_3;
reconsider x = x, y = y as Element of NAT by A37, A35;
take BB = B * (x,y); ::_thesis: S1[i,j,BB]
now__::_thesis:_for_m,_n_being_Nat_st_m_in_dom_(Sgm_Bx)_&_n_in_dom_(Sgm_By)_&_(Sgm_Bx)_._m_=_i_&_(Sgm_By)_._n_=_j_holds_
BB_=_B_*_(m,n)
let m, n be Nat; ::_thesis: ( m in dom (Sgm Bx) & n in dom (Sgm By) & (Sgm Bx) . m = i & (Sgm By) . n = j implies BB = B * (m,n) )
assume that
A39: m in dom (Sgm Bx) and
A40: n in dom (Sgm By) and
A41: (Sgm Bx) . m = i and
A42: (Sgm By) . n = j ; ::_thesis: BB = B * (m,n)
x = m by A10, A37, A38, A39, A41, FUNCT_1:def_4;
hence BB = B * (m,n) by A9, A35, A36, A40, A42, FUNCT_1:def_4; ::_thesis: verum
end;
hence S1[i,j,BB] by A34, A37, A38, A35, A36, XBOOLE_0:def_3; ::_thesis: verum
end;
supposeA43: ( not [i,j] in [:Bx,By:] & [i,j] in [:Cx,Cy:] ) ; ::_thesis: ex x being Element of D st S1[i,j,x]
then j in Cy by ZFMISC_1:87;
then consider y being set such that
A44: y in dom (Sgm Cy) and
A45: (Sgm Cy) . y = j by A12, FUNCT_1:def_3;
i in Cx by A43, ZFMISC_1:87;
then consider x being set such that
A46: x in dom (Sgm Cx) and
A47: (Sgm Cx) . x = i by A15, FUNCT_1:def_3;
reconsider x = x, y = y as Element of NAT by A46, A44;
take CC = C * (x,y); ::_thesis: S1[i,j,CC]
now__::_thesis:_for_m,_n_being_Nat_st_m_in_dom_(Sgm_Cx)_&_n_in_dom_(Sgm_Cy)_&_(Sgm_Cx)_._m_=_i_&_(Sgm_Cy)_._n_=_j_holds_
CC_=_C_*_(m,n)
let m, n be Nat; ::_thesis: ( m in dom (Sgm Cx) & n in dom (Sgm Cy) & (Sgm Cx) . m = i & (Sgm Cy) . n = j implies CC = C * (m,n) )
assume that
A48: m in dom (Sgm Cx) and
A49: n in dom (Sgm Cy) and
A50: (Sgm Cx) . m = i and
A51: (Sgm Cy) . n = j ; ::_thesis: CC = C * (m,n)
x = m by A16, A46, A47, A48, A50, FUNCT_1:def_4;
hence CC = C * (m,n) by A13, A44, A45, A49, A51, FUNCT_1:def_4; ::_thesis: verum
end;
hence S1[i,j,CC] by A43, A46, A47, A44, A45, XBOOLE_0:def_3; ::_thesis: verum
end;
supposeA52: ( not [i,j] in [:Bx,By:] & not [i,j] in [:Cx,Cy:] ) ; ::_thesis: ex x being Element of D st S1[i,j,x]
take A * (i,j) ; ::_thesis: S1[i,j,A * (i,j)]
thus S1[i,j,A * (i,j)] by A52; ::_thesis: verum
end;
end;
end;
consider M being Matrix of len A, width A,D such that
A53: for i, j being Nat st [i,j] in Indices M holds
S1[i,j,M * (i,j)] from MATRIX_1:sch_2(A17);
set MB = Segm (M,Bx,By);
take M ; ::_thesis: ( Segm (M,Bx,By) = B & Segm (M,Cx,Cy) = C & ( for i, j being Nat st [i,j] in (Indices M) \ ([:Bx,By:] \/ [:Cx,Cy:]) holds
M * (i,j) = A * (i,j) ) )
A is Matrix of len A, width A,D by MATRIX_2:7;
then A54: Indices A = Indices M by MATRIX_1:26;
A55: dom (Sgm Bx) = Seg (card Bx) by A6, FINSEQ_3:40;
now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_(Segm_(M,Bx,By))_holds_
B_*_(i,j)_=_(Segm_(M,Bx,By))_*_(i,j)
let i, j be Nat; ::_thesis: ( [i,j] in Indices (Segm (M,Bx,By)) implies B * (i,j) = (Segm (M,Bx,By)) * (i,j) )
assume A56: [i,j] in Indices (Segm (M,Bx,By)) ; ::_thesis: B * (i,j) = (Segm (M,Bx,By)) * (i,j)
card Bx <> 0 by A56, MATRIX_1:22;
then A57: Indices (Segm (M,Bx,By)) = [:(Seg (card Bx)),(Seg (card By)):] by MATRIX_1:23;
then A58: j in Seg (card By) by A56, ZFMISC_1:87;
then A59: (Sgm By) . j in By by A5, A8, FUNCT_1:def_3;
A60: i in Seg (card Bx) by A56, A57, ZFMISC_1:87;
then (Sgm Bx) . i in Bx by A7, A55, FUNCT_1:def_3;
then [((Sgm Bx) . i),((Sgm By) . j)] in [:Bx,By:] by A59, ZFMISC_1:87;
hence B * (i,j) = M * (((Sgm Bx) . i),((Sgm By) . j)) by A1, A55, A8, A53, A54, A60, A58
.= (Segm (M,Bx,By)) * (i,j) by A56, MATRIX13:def_1 ;
::_thesis: verum
end;
hence Segm (M,Bx,By) = B by MATRIX_1:27; ::_thesis: ( Segm (M,Cx,Cy) = C & ( for i, j being Nat st [i,j] in (Indices M) \ ([:Bx,By:] \/ [:Cx,Cy:]) holds
M * (i,j) = A * (i,j) ) )
set MC = Segm (M,Cx,Cy);
A61: dom (Sgm Cy) = Seg (card Cy) by A11, FINSEQ_3:40;
A62: dom (Sgm Cx) = Seg (card Cx) by A14, FINSEQ_3:40;
now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_(Segm_(M,Cx,Cy))_holds_
C_*_(i,j)_=_(Segm_(M,Cx,Cy))_*_(i,j)
let i, j be Nat; ::_thesis: ( [i,j] in Indices (Segm (M,Cx,Cy)) implies C * (i,j) = (Segm (M,Cx,Cy)) * (i,j) )
assume A63: [i,j] in Indices (Segm (M,Cx,Cy)) ; ::_thesis: C * (i,j) = (Segm (M,Cx,Cy)) * (i,j)
card Cx <> 0 by A63, MATRIX_1:22;
then A64: Indices (Segm (M,Cx,Cy)) = [:(Seg (card Cx)),(Seg (card Cy)):] by MATRIX_1:23;
then A65: j in Seg (card Cy) by A63, ZFMISC_1:87;
then A66: (Sgm Cy) . j in Cy by A12, A61, FUNCT_1:def_3;
A67: i in Seg (card Cx) by A63, A64, ZFMISC_1:87;
then (Sgm Cx) . i in Cx by A15, A62, FUNCT_1:def_3;
then [((Sgm Cx) . i),((Sgm Cy) . j)] in [:Cx,Cy:] by A66, ZFMISC_1:87;
hence C * (i,j) = M * (((Sgm Cx) . i),((Sgm Cy) . j)) by A2, A62, A61, A53, A54, A67, A65
.= (Segm (M,Cx,Cy)) * (i,j) by A63, MATRIX13:def_1 ;
::_thesis: verum
end;
hence Segm (M,Cx,Cy) = C by MATRIX_1:27; ::_thesis: for i, j being Nat st [i,j] in (Indices M) \ ([:Bx,By:] \/ [:Cx,Cy:]) holds
M * (i,j) = A * (i,j)
let i, j be Nat; ::_thesis: ( [i,j] in (Indices M) \ ([:Bx,By:] \/ [:Cx,Cy:]) implies M * (i,j) = A * (i,j) )
assume A68: [i,j] in (Indices M) \ ([:Bx,By:] \/ [:Cx,Cy:]) ; ::_thesis: M * (i,j) = A * (i,j)
not [i,j] in [:Bx,By:] \/ [:Cx,Cy:] by A68, XBOOLE_0:def_5;
hence M * (i,j) = A * (i,j) by A53, A68; ::_thesis: verum
end;
theorem Th10: :: MATRIX15:10
for K being Field
for A being Matrix of K
for P, Q, Q9 being finite without_zero Subset of NAT st [:P,Q9:] c= Indices A holds
for i, j being Nat st i in (dom A) \ P & j in (Seg (width A)) \ Q & A * (i,j) <> 0. K & Q c= Q9 & (Line (A,i)) * (Sgm Q9) = (card Q9) |-> (0. K) holds
the_rank_of A > the_rank_of (Segm (A,P,Q))
proof
let K be Field; ::_thesis: for A being Matrix of K
for P, Q, Q9 being finite without_zero Subset of NAT st [:P,Q9:] c= Indices A holds
for i, j being Nat st i in (dom A) \ P & j in (Seg (width A)) \ Q & A * (i,j) <> 0. K & Q c= Q9 & (Line (A,i)) * (Sgm Q9) = (card Q9) |-> (0. K) holds
the_rank_of A > the_rank_of (Segm (A,P,Q))
let A be Matrix of K; ::_thesis: for P, Q, Q9 being finite without_zero Subset of NAT st [:P,Q9:] c= Indices A holds
for i, j being Nat st i in (dom A) \ P & j in (Seg (width A)) \ Q & A * (i,j) <> 0. K & Q c= Q9 & (Line (A,i)) * (Sgm Q9) = (card Q9) |-> (0. K) holds
the_rank_of A > the_rank_of (Segm (A,P,Q))
let P, Q, R be finite without_zero Subset of NAT; ::_thesis: ( [:P,R:] c= Indices A implies for i, j being Nat st i in (dom A) \ P & j in (Seg (width A)) \ Q & A * (i,j) <> 0. K & Q c= R & (Line (A,i)) * (Sgm R) = (card R) |-> (0. K) holds
the_rank_of A > the_rank_of (Segm (A,P,Q)) )
assume A1: [:P,R:] c= Indices A ; ::_thesis: for i, j being Nat st i in (dom A) \ P & j in (Seg (width A)) \ Q & A * (i,j) <> 0. K & Q c= R & (Line (A,i)) * (Sgm R) = (card R) |-> (0. K) holds
the_rank_of A > the_rank_of (Segm (A,P,Q))
let i, j be Nat; ::_thesis: ( i in (dom A) \ P & j in (Seg (width A)) \ Q & A * (i,j) <> 0. K & Q c= R & (Line (A,i)) * (Sgm R) = (card R) |-> (0. K) implies the_rank_of A > the_rank_of (Segm (A,P,Q)) )
assume that
A2: i in (dom A) \ P and
A3: j in (Seg (width A)) \ Q and
A4: A * (i,j) <> 0. K and
A5: Q c= R and
A6: (Line (A,i)) * (Sgm R) = (card R) |-> (0. K) ; ::_thesis: the_rank_of A > the_rank_of (Segm (A,P,Q))
A7: dom A = Seg (len A) by FINSEQ_1:def_3;
then A8: i in Seg (len A) by A2, XBOOLE_0:def_5;
A9: [:P,Q:] c= [:P,R:] by A5, ZFMISC_1:95;
then A10: [:P,Q:] c= Indices A by A1, XBOOLE_1:1;
reconsider i0 = i, j0 = j as non zero Element of NAT by A2, A3, A7;
A11: j in Seg (width A) by A3, XBOOLE_0:def_5;
set S = Segm (A,P,Q);
consider P9, Q9 being finite without_zero Subset of NAT such that
A12: [:P9,Q9:] c= Indices (Segm (A,P,Q)) and
A13: card P9 = card Q9 and
A14: card P9 = the_rank_of (Segm (A,P,Q)) and
A15: Det (EqSegm ((Segm (A,P,Q)),P9,Q9)) <> 0. K by MATRIX13:def_4;
( P9 = {} iff Q9 = {} ) by A13;
then consider P2, Q2 being finite without_zero Subset of NAT such that
A16: P2 c= P and
A17: Q2 c= Q and
P2 = (Sgm P) .: P9 and
Q2 = (Sgm Q) .: Q9 and
A18: card P2 = card P9 and
A19: card Q2 = card Q9 and
A20: Segm ((Segm (A,P,Q)),P9,Q9) = Segm (A,P2,Q2) by A12, MATRIX13:57;
set Q2j = Q2 \/ {j0};
set P2i = P2 \/ {i0};
set ESS = EqSegm (A,(P2 \/ {i0}),(Q2 \/ {j0}));
set SS = Segm (A,(P2 \/ {i0}),(Q2 \/ {j0}));
percases ( [:P,Q:] = {} or [:P,Q:] <> {} ) ;
suppose [:P,Q:] = {} ; ::_thesis: the_rank_of A > the_rank_of (Segm (A,P,Q))
then ( card P = 0 or card Q = 0 ) by CARD_1:27, ZFMISC_1:90;
then A21: the_rank_of (Segm (A,P,Q)) = 0 by MATRIX13:77;
[i,j] in Indices A by A7, A8, A11, ZFMISC_1:87;
hence the_rank_of A > the_rank_of (Segm (A,P,Q)) by A4, A21, MATRIX13:94; ::_thesis: verum
end;
supposeA22: [:P,Q:] <> {} ; ::_thesis: the_rank_of A > the_rank_of (Segm (A,P,Q))
then P c= dom A by A10, ZFMISC_1:114;
then A23: P2 c= dom A by A16, XBOOLE_1:1;
[:P,R:] <> {} by A9, A22, XBOOLE_1:3;
then A24: R c= Seg (width A) by A1, ZFMISC_1:114;
then A25: dom (Sgm R) = Seg (card R) by FINSEQ_3:40;
Q c= Seg (width A) by A10, A22, ZFMISC_1:114;
then A26: Q2 c= Seg (width A) by A17, XBOOLE_1:1;
A27: {j0} c= Seg (width A) by A11, ZFMISC_1:31;
then A28: Sgm (Q2 \/ {j0}) is one-to-one by A26, FINSEQ_3:92, XBOOLE_1:8;
A29: Q2 \/ {j0} c= Seg (width A) by A26, A27, XBOOLE_1:8;
then A30: rng (Sgm (Q2 \/ {j0})) = Q2 \/ {j0} by FINSEQ_1:def_13;
A31: {i0} c= dom A by A7, A8, ZFMISC_1:31;
then A32: P2 \/ {i0} c= dom A by A23, XBOOLE_1:8;
then A33: [:(P2 \/ {i0}),(Q2 \/ {j0}):] c= Indices A by A29, ZFMISC_1:96;
A34: dom (Sgm (P2 \/ {i0})) = Seg (card (P2 \/ {i0})) by A7, A23, A31, FINSEQ_3:40, XBOOLE_1:8;
i in {i} by TARSKI:def_1;
then A35: i in P2 \/ {i0} by XBOOLE_0:def_3;
A36: not i in P2 by A2, A16, XBOOLE_0:def_5;
then A37: card (P2 \/ {i0}) = (card P2) + 1 by CARD_2:41;
then A38: (card (P2 \/ {i0})) -' 1 = card P9 by A18, NAT_D:34;
A39: not j in Q2 by A3, A17, XBOOLE_0:def_5;
then A40: card (Q2 \/ {j0}) = (card Q2) + 1 by CARD_2:41;
then A41: EqSegm (A,(P2 \/ {i0}),(Q2 \/ {j0})) = Segm (A,(P2 \/ {i0}),(Q2 \/ {j0})) by A13, A18, A19, A36, CARD_2:41, MATRIX13:def_3;
j in {j} by TARSKI:def_1;
then j in Q2 \/ {j0} by XBOOLE_0:def_3;
then consider y being set such that
A42: y in dom (Sgm (Q2 \/ {j0})) and
A43: (Sgm (Q2 \/ {j0})) . y = j by A30, FUNCT_1:def_3;
rng (Sgm (P2 \/ {i0})) = P2 \/ {i0} by A7, A32, FINSEQ_1:def_13;
then consider x being set such that
A44: x in dom (Sgm (P2 \/ {i0})) and
A45: (Sgm (P2 \/ {i0})) . x = i by A35, FUNCT_1:def_3;
reconsider x = x, y = y as Element of NAT by A44, A42;
- (1_ K) <> 0. K by VECTSP_1:28;
then A46: (power K) . ((- (1_ K)),(x + y)) <> 0. K by Lm2;
set L = LaplaceExpL ((EqSegm (A,(P2 \/ {i0}),(Q2 \/ {j0}))),x);
A47: dom (LaplaceExpL ((EqSegm (A,(P2 \/ {i0}),(Q2 \/ {j0}))),x)) = Seg (len (LaplaceExpL ((EqSegm (A,(P2 \/ {i0}),(Q2 \/ {j0}))),x))) by FINSEQ_1:def_3
.= Seg (card (P2 \/ {i0})) by LAPLACE:def_7 ;
then A48: y in dom (LaplaceExpL ((EqSegm (A,(P2 \/ {i0}),(Q2 \/ {j0}))),x)) by A13, A18, A19, A26, A27, A37, A40, A42, FINSEQ_3:40, XBOOLE_1:8;
A49: dom (Sgm (Q2 \/ {j0})) = Seg (card (Q2 \/ {j0})) by A26, A27, FINSEQ_3:40, XBOOLE_1:8;
then Delete ((EqSegm (A,(P2 \/ {i0}),(Q2 \/ {j0}))),x,y) = EqSegm (A,((P2 \/ {i0}) \ {i}),((Q2 \/ {j0}) \ {j})) by A13, A18, A19, A37, A40, A34, A44, A45, A42, A43, MATRIX13:64
.= EqSegm (A,P2,((Q2 \/ {j0}) \ {j})) by A36, ZFMISC_1:117
.= EqSegm (A,P2,Q2) by A39, ZFMISC_1:117
.= Segm (A,P2,Q2) by A13, A18, A19, MATRIX13:def_3
.= EqSegm ((Segm (A,P,Q)),P9,Q9) by A13, A20, MATRIX13:def_3 ;
then A50: ((power K) . ((- (1_ K)),(x + y))) * (Det (Delete ((EqSegm (A,(P2 \/ {i0}),(Q2 \/ {j0}))),x,y))) <> 0. K by A15, A38, A46, VECTSP_1:12;
A51: Indices (EqSegm (A,(P2 \/ {i0}),(Q2 \/ {j0}))) = [:(Seg (card (P2 \/ {i0}))),(Seg (card (P2 \/ {i0}))):] by MATRIX_1:24;
then A52: [x,y] in Indices (EqSegm (A,(P2 \/ {i0}),(Q2 \/ {j0}))) by A13, A18, A19, A37, A40, A34, A49, A44, A42, ZFMISC_1:87;
A53: rng (Sgm R) = R by A24, FINSEQ_1:def_13;
now__::_thesis:_for_k_being_Nat_st_k_in_dom_(LaplaceExpL_((EqSegm_(A,(P2_\/_{i0}),(Q2_\/_{j0}))),x))_&_k_<>_y_holds_
(LaplaceExpL_((EqSegm_(A,(P2_\/_{i0}),(Q2_\/_{j0}))),x))_._k_=_0._K
let k be Nat; ::_thesis: ( k in dom (LaplaceExpL ((EqSegm (A,(P2 \/ {i0}),(Q2 \/ {j0}))),x)) & k <> y implies (LaplaceExpL ((EqSegm (A,(P2 \/ {i0}),(Q2 \/ {j0}))),x)) . k = 0. K )
assume that
A54: k in dom (LaplaceExpL ((EqSegm (A,(P2 \/ {i0}),(Q2 \/ {j0}))),x)) and
A55: k <> y ; ::_thesis: (LaplaceExpL ((EqSegm (A,(P2 \/ {i0}),(Q2 \/ {j0}))),x)) . k = 0. K
(Sgm (Q2 \/ {j0})) . k <> j by A13, A18, A19, A37, A40, A49, A28, A42, A43, A47, A54, A55, FUNCT_1:def_4;
then A56: not (Sgm (Q2 \/ {j0})) . k in {j} by TARSKI:def_1;
(Sgm (Q2 \/ {j0})) . k in Q2 \/ {j0} by A13, A18, A19, A37, A40, A30, A49, A47, A54, FUNCT_1:def_3;
then (Sgm (Q2 \/ {j0})) . k in Q2 by A56, XBOOLE_0:def_3;
then A57: (Sgm (Q2 \/ {j0})) . k in Q by A17;
then A58: (Sgm (Q2 \/ {j0})) . k in R by A5;
consider z being set such that
A59: z in dom (Sgm R) and
A60: (Sgm R) . z = (Sgm (Q2 \/ {j0})) . k by A5, A53, A57, FUNCT_1:def_3;
reconsider z = z as Element of NAT by A59;
[x,k] in Indices (EqSegm (A,(P2 \/ {i0}),(Q2 \/ {j0}))) by A34, A44, A47, A51, A54, ZFMISC_1:87;
then (EqSegm (A,(P2 \/ {i0}),(Q2 \/ {j0}))) * (x,k) = A * (i,((Sgm (Q2 \/ {j0})) . k)) by A45, A41, MATRIX13:def_1
.= (Line (A,i)) . ((Sgm R) . z) by A24, A60, A58, MATRIX_1:def_7
.= ((card R) |-> (0. K)) . z by A6, A59, FUNCT_1:13
.= 0. K by A25, A59, FINSEQ_2:57 ;
hence (LaplaceExpL ((EqSegm (A,(P2 \/ {i0}),(Q2 \/ {j0}))),x)) . k = (0. K) * (Cofactor ((EqSegm (A,(P2 \/ {i0}),(Q2 \/ {j0}))),x,k)) by A54, LAPLACE:def_7
.= 0. K by VECTSP_1:7 ;
::_thesis: verum
end;
then A61: (LaplaceExpL ((EqSegm (A,(P2 \/ {i0}),(Q2 \/ {j0}))),x)) . y = Sum (LaplaceExpL ((EqSegm (A,(P2 \/ {i0}),(Q2 \/ {j0}))),x)) by A48, MATRIX_3:12
.= Det (EqSegm (A,(P2 \/ {i0}),(Q2 \/ {j0}))) by A34, A44, LAPLACE:25 ;
(LaplaceExpL ((EqSegm (A,(P2 \/ {i0}),(Q2 \/ {j0}))),x)) . y = ((Segm (A,(P2 \/ {i0}),(Q2 \/ {j0}))) * (x,y)) * (Cofactor ((EqSegm (A,(P2 \/ {i0}),(Q2 \/ {j0}))),x,y)) by A48, A41, LAPLACE:def_7
.= (A * (i,j)) * (((power K) . ((- (1_ K)),(x + y))) * (Det (Delete ((EqSegm (A,(P2 \/ {i0}),(Q2 \/ {j0}))),x,y)))) by A45, A43, A41, A52, MATRIX13:def_1 ;
then Det (EqSegm (A,(P2 \/ {i0}),(Q2 \/ {j0}))) <> 0. K by A4, A61, A50, VECTSP_1:12;
then the_rank_of A >= card (P2 \/ {i0}) by A13, A18, A19, A37, A40, A33, MATRIX13:def_4;
hence the_rank_of A > the_rank_of (Segm (A,P,Q)) by A14, A18, A37, NAT_1:13; ::_thesis: verum
end;
end;
end;
theorem Th11: :: MATRIX15:11
for K being Field
for A being Matrix of K
for N being finite without_zero Subset of NAT st N c= dom A & ( for i being Nat st i in (dom A) \ N holds
Line (A,i) = (width A) |-> (0. K) ) holds
the_rank_of A = the_rank_of (Segm (A,N,(Seg (width A))))
proof
let K be Field; ::_thesis: for A being Matrix of K
for N being finite without_zero Subset of NAT st N c= dom A & ( for i being Nat st i in (dom A) \ N holds
Line (A,i) = (width A) |-> (0. K) ) holds
the_rank_of A = the_rank_of (Segm (A,N,(Seg (width A))))
let A be Matrix of K; ::_thesis: for N being finite without_zero Subset of NAT st N c= dom A & ( for i being Nat st i in (dom A) \ N holds
Line (A,i) = (width A) |-> (0. K) ) holds
the_rank_of A = the_rank_of (Segm (A,N,(Seg (width A))))
let N be finite without_zero Subset of NAT; ::_thesis: ( N c= dom A & ( for i being Nat st i in (dom A) \ N holds
Line (A,i) = (width A) |-> (0. K) ) implies the_rank_of A = the_rank_of (Segm (A,N,(Seg (width A)))) )
assume that
A1: N c= dom A and
A2: for i being Nat st i in (dom A) \ N holds
Line (A,i) = (width A) |-> (0. K) ; ::_thesis: the_rank_of A = the_rank_of (Segm (A,N,(Seg (width A))))
set w = width A;
set l = len A;
reconsider A9 = A as Matrix of len A, width A,K by MATRIX_2:7;
set S = Segm (A9,N,(Seg (width A9)));
consider U being finite Subset of ((width A) -VectSp_over K) such that
A3: U is linearly-independent and
A4: U c= lines A9 and
A5: card U = the_rank_of A9 by MATRIX13:123;
A6: U c= lines (Segm (A9,N,(Seg (width A9))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in U or x in lines (Segm (A9,N,(Seg (width A9)))) )
assume A7: x in U ; ::_thesis: x in lines (Segm (A9,N,(Seg (width A9))))
consider Ni being Nat such that
A8: Ni in Seg (len A) and
A9: x = Line (A9,Ni) by A4, A7, MATRIX13:103;
A10: dom A = Seg (len A) by FINSEQ_1:def_3;
A11: Ni in N
proof
assume not Ni in N ; ::_thesis: contradiction
then Ni in (dom A) \ N by A8, A10, XBOOLE_0:def_5;
then x = (width A) |-> (0. K) by A2, A9
.= 0. ((width A) -VectSp_over K) by MATRIX13:102 ;
hence contradiction by A3, A7, VECTSP_7:2; ::_thesis: verum
end;
rng (Sgm N) = N by A1, A10, FINSEQ_1:def_13;
then consider i being set such that
A12: i in dom (Sgm N) and
A13: (Sgm N) . i = Ni by A11, FUNCT_1:def_3;
reconsider i = i as Element of NAT by A12;
A14: dom (Sgm N) = Seg (card N) by A1, A10, FINSEQ_3:40;
then Line ((Segm (A9,N,(Seg (width A9)))),i) = x by A9, A12, A13, MATRIX13:48;
hence x in lines (Segm (A9,N,(Seg (width A9)))) by A12, A14, MATRIX13:103; ::_thesis: verum
end;
A15: now__::_thesis:_for_W_being_finite_Subset_of_((width_A)_-VectSp_over_K)_st_W_is_linearly-independent_&_W_c=_lines_(Segm_(A9,N,(Seg_(width_A9))))_holds_
card_W_<=_the_rank_of_A9
let W be finite Subset of ((width A) -VectSp_over K); ::_thesis: ( W is linearly-independent & W c= lines (Segm (A9,N,(Seg (width A9)))) implies card W <= the_rank_of A9 )
assume that
A16: W is linearly-independent and
A17: W c= lines (Segm (A9,N,(Seg (width A9)))) ; ::_thesis: card W <= the_rank_of A9
dom A = Seg (len A) by FINSEQ_1:def_3;
then lines (Segm (A9,N,(Seg (width A9)))) c= lines A9 by A1, MATRIX13:118;
then W c= lines A9 by A17, XBOOLE_1:1;
hence card W <= the_rank_of A9 by A16, MATRIX13:123; ::_thesis: verum
end;
width A = card (Seg (width A)) by FINSEQ_1:57;
hence the_rank_of A = the_rank_of (Segm (A,N,(Seg (width A)))) by A3, A5, A6, A15, MATRIX13:123; ::_thesis: verum
end;
theorem Th12: :: MATRIX15:12
for K being Field
for A being Matrix of K
for N being finite without_zero Subset of NAT st N c= Seg (width A) & ( for i being Nat st i in (Seg (width A)) \ N holds
Col (A,i) = (len A) |-> (0. K) ) holds
the_rank_of A = the_rank_of (Segm (A,(Seg (len A)),N))
proof
let K be Field; ::_thesis: for A being Matrix of K
for N being finite without_zero Subset of NAT st N c= Seg (width A) & ( for i being Nat st i in (Seg (width A)) \ N holds
Col (A,i) = (len A) |-> (0. K) ) holds
the_rank_of A = the_rank_of (Segm (A,(Seg (len A)),N))
let A be Matrix of K; ::_thesis: for N being finite without_zero Subset of NAT st N c= Seg (width A) & ( for i being Nat st i in (Seg (width A)) \ N holds
Col (A,i) = (len A) |-> (0. K) ) holds
the_rank_of A = the_rank_of (Segm (A,(Seg (len A)),N))
let N be finite without_zero Subset of NAT; ::_thesis: ( N c= Seg (width A) & ( for i being Nat st i in (Seg (width A)) \ N holds
Col (A,i) = (len A) |-> (0. K) ) implies the_rank_of A = the_rank_of (Segm (A,(Seg (len A)),N)) )
assume that
A1: N c= Seg (width A) and
A2: for i being Nat st i in (Seg (width A)) \ N holds
Col (A,i) = (len A) |-> (0. K) ; ::_thesis: the_rank_of A = the_rank_of (Segm (A,(Seg (len A)),N))
A3: dom A = Seg (len A) by FINSEQ_1:def_3;
percases ( card N = 0 or card N > 0 ) ;
supposeA4: card N = 0 ; ::_thesis: the_rank_of A = the_rank_of (Segm (A,(Seg (len A)),N))
the_rank_of A = 0
proof
A5: N = {} by A4;
assume the_rank_of A <> 0 ; ::_thesis: contradiction
then consider i, j being Nat such that
A6: [i,j] in Indices A and
A7: A * (i,j) <> 0. K by MATRIX13:94;
A8: j in Seg (width A) by A6, ZFMISC_1:87;
A9: i in dom A by A6, ZFMISC_1:87;
then 0. K = ((len A) |-> (0. K)) . i by A3, FINSEQ_2:57
.= (Col (A,j)) . i by A2, A8, A5
.= A * (i,j) by A9, MATRIX_1:def_8 ;
hence contradiction by A7; ::_thesis: verum
end;
hence the_rank_of A = the_rank_of (Segm (A,(Seg (len A)),N)) by A4, MATRIX13:77; ::_thesis: verum
end;
supposeA10: card N > 0 ; ::_thesis: the_rank_of A = the_rank_of (Segm (A,(Seg (len A)),N))
then N <> {} ;
then Seg (width A) <> {} by A1, XBOOLE_1:3;
then A11: width A <> 0 ;
then A12: len A <> 0 by MATRIX_1:def_3;
set AT = A @ ;
A13: [:(dom A),N:] c= Indices A by A1, ZFMISC_1:95;
A14: width (A @) = len A by A11, MATRIX_2:10;
A15: len (A @) = width A by A11, MATRIX_2:10;
then A16: N c= dom (A @) by A1, FINSEQ_1:def_3;
A17: dom (A @) = Seg (len (A @)) by FINSEQ_1:def_3;
A18: now__::_thesis:_for_i_being_Nat_st_i_in_(dom_(A_@))_\_N_holds_
Line_((A_@),i)_=_(width_(A_@))_|->_(0._K)
let i be Nat; ::_thesis: ( i in (dom (A @)) \ N implies Line ((A @),i) = (width (A @)) |-> (0. K) )
assume A19: i in (dom (A @)) \ N ; ::_thesis: Line ((A @),i) = (width (A @)) |-> (0. K)
i in dom (A @) by A19, XBOOLE_0:def_5;
hence Line ((A @),i) = Col (A,i) by A15, A17, MATRIX_2:15
.= (width (A @)) |-> (0. K) by A2, A15, A14, A17, A19 ;
::_thesis: verum
end;
thus the_rank_of A = the_rank_of (A @) by MATRIX13:84
.= the_rank_of (Segm ((A @),N,(Seg (len A)))) by A14, A16, A18, Th11
.= the_rank_of ((Segm (A,(Seg (len A)),N)) @) by A3, A10, A12, A13, MATRIX13:61
.= the_rank_of (Segm (A,(Seg (len A)),N)) by MATRIX13:84 ; ::_thesis: verum
end;
end;
end;
theorem Th13: :: MATRIX15:13
for K being Field
for V being VectSp of K
for U being finite Subset of V
for u, v being Vector of V
for a being Element of K st u in U & v in U holds
Lin ((U \ {u}) \/ {(u + (a * v))}) is Subspace of Lin U
proof
let K be Field; ::_thesis: for V being VectSp of K
for U being finite Subset of V
for u, v being Vector of V
for a being Element of K st u in U & v in U holds
Lin ((U \ {u}) \/ {(u + (a * v))}) is Subspace of Lin U
let V be VectSp of K; ::_thesis: for U being finite Subset of V
for u, v being Vector of V
for a being Element of K st u in U & v in U holds
Lin ((U \ {u}) \/ {(u + (a * v))}) is Subspace of Lin U
let U be finite Subset of V; ::_thesis: for u, v being Vector of V
for a being Element of K st u in U & v in U holds
Lin ((U \ {u}) \/ {(u + (a * v))}) is Subspace of Lin U
let u, v be Vector of V; ::_thesis: for a being Element of K st u in U & v in U holds
Lin ((U \ {u}) \/ {(u + (a * v))}) is Subspace of Lin U
let a be Element of K; ::_thesis: ( u in U & v in U implies Lin ((U \ {u}) \/ {(u + (a * v))}) is Subspace of Lin U )
assume A1: ( u in U & v in U ) ; ::_thesis: Lin ((U \ {u}) \/ {(u + (a * v))}) is Subspace of Lin U
set ua = u + (a * v);
set UU = U \ {u};
(U \ {u}) \/ {(u + (a * v))} c= the carrier of (Lin U)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (U \ {u}) \/ {(u + (a * v))} or x in the carrier of (Lin U) )
assume A2: x in (U \ {u}) \/ {(u + (a * v))} ; ::_thesis: x in the carrier of (Lin U)
percases ( x in U \ {u} or x in {(u + (a * v))} ) by A2, XBOOLE_0:def_3;
suppose x in U \ {u} ; ::_thesis: x in the carrier of (Lin U)
then x in U by XBOOLE_0:def_5;
then x in Lin U by VECTSP_7:8;
hence x in the carrier of (Lin U) by STRUCT_0:def_5; ::_thesis: verum
end;
supposeA3: x in {(u + (a * v))} ; ::_thesis: x in the carrier of (Lin U)
A4: ( u in Lin U & a * v in Lin U ) by A1, VECTSP_4:21, VECTSP_7:8;
x = u + (a * v) by A3, TARSKI:def_1;
then x in Lin U by A4, VECTSP_4:20;
hence x in the carrier of (Lin U) by STRUCT_0:def_5; ::_thesis: verum
end;
end;
end;
hence Lin ((U \ {u}) \/ {(u + (a * v))}) is Subspace of Lin U by VECTSP_9:16; ::_thesis: verum
end;
theorem Th14: :: MATRIX15:14
for K being Field
for V being VectSp of K
for U being finite Subset of V
for u, v being Vector of V
for a being Element of K st u in U & v in U & ( not u = v or a <> - (1_ K) or u = 0. V ) holds
Lin ((U \ {u}) \/ {(u + (a * v))}) = Lin U
proof
let K be Field; ::_thesis: for V being VectSp of K
for U being finite Subset of V
for u, v being Vector of V
for a being Element of K st u in U & v in U & ( not u = v or a <> - (1_ K) or u = 0. V ) holds
Lin ((U \ {u}) \/ {(u + (a * v))}) = Lin U
let V be VectSp of K; ::_thesis: for U being finite Subset of V
for u, v being Vector of V
for a being Element of K st u in U & v in U & ( not u = v or a <> - (1_ K) or u = 0. V ) holds
Lin ((U \ {u}) \/ {(u + (a * v))}) = Lin U
let U be finite Subset of V; ::_thesis: for u, v being Vector of V
for a being Element of K st u in U & v in U & ( not u = v or a <> - (1_ K) or u = 0. V ) holds
Lin ((U \ {u}) \/ {(u + (a * v))}) = Lin U
let u, v be Vector of V; ::_thesis: for a being Element of K st u in U & v in U & ( not u = v or a <> - (1_ K) or u = 0. V ) holds
Lin ((U \ {u}) \/ {(u + (a * v))}) = Lin U
let a be Element of K; ::_thesis: ( u in U & v in U & ( not u = v or a <> - (1_ K) or u = 0. V ) implies Lin ((U \ {u}) \/ {(u + (a * v))}) = Lin U )
assume that
A1: u in U and
A2: v in U and
A3: ( not u = v or a <> - (1_ K) or u = 0. V ) ; ::_thesis: Lin ((U \ {u}) \/ {(u + (a * v))}) = Lin U
set ua = u + (a * v);
set UU = U \ {u};
U c= the carrier of (Lin ((U \ {u}) \/ {(u + (a * v))}))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in U or x in the carrier of (Lin ((U \ {u}) \/ {(u + (a * v))})) )
assume A4: x in U ; ::_thesis: x in the carrier of (Lin ((U \ {u}) \/ {(u + (a * v))}))
percases ( x = u or x <> u ) ;
supposeA5: x = u ; ::_thesis: x in the carrier of (Lin ((U \ {u}) \/ {(u + (a * v))}))
percases ( u <> v or u = v ) ;
supposeA6: u <> v ; ::_thesis: x in the carrier of (Lin ((U \ {u}) \/ {(u + (a * v))}))
A7: (u + (a * v)) + ((- a) * v) = (u + (a * v)) - (a * v) by VECTSP_1:21
.= u + ((a * v) - (a * v)) by RLVECT_1:def_3
.= u + (0. V) by VECTSP_1:16
.= u by RLVECT_1:def_4 ;
u + (a * v) in {(u + (a * v))} by TARSKI:def_1;
then u + (a * v) in (U \ {u}) \/ {(u + (a * v))} by XBOOLE_0:def_3;
then A8: u + (a * v) in Lin ((U \ {u}) \/ {(u + (a * v))}) by VECTSP_7:8;
v in U \ {u} by A2, A6, ZFMISC_1:56;
then v in (U \ {u}) \/ {(u + (a * v))} by XBOOLE_0:def_3;
then (- a) * v in Lin ((U \ {u}) \/ {(u + (a * v))}) by VECTSP_4:21, VECTSP_7:8;
then (u + (a * v)) + ((- a) * v) in Lin ((U \ {u}) \/ {(u + (a * v))}) by A8, VECTSP_4:20;
hence x in the carrier of (Lin ((U \ {u}) \/ {(u + (a * v))})) by A5, A7, STRUCT_0:def_5; ::_thesis: verum
end;
supposeA9: u = v ; ::_thesis: x in the carrier of (Lin ((U \ {u}) \/ {(u + (a * v))}))
percases ( a <> - (1_ K) or u = 0. V ) by A3, A9;
suppose a <> - (1_ K) ; ::_thesis: x in the carrier of (Lin ((U \ {u}) \/ {(u + (a * v))}))
then 0. K <> (- (1_ K)) - a by VECTSP_1:19;
then 0. K <> - ((- (1_ K)) - a) by VECTSP_1:28;
then A10: 0. K <> a + (1_ K) by VECTSP_1:31;
u + (a * v) in {(u + (a * v))} by TARSKI:def_1;
then A11: u + (a * v) in (U \ {u}) \/ {(u + (a * v))} by XBOOLE_0:def_3;
u + (a * v) = ((1_ K) * u) + (a * u) by A9, VECTSP_1:def_17
.= ((1. K) + a) * u by VECTSP_1:def_15 ;
then (((1. K) + a) ") * (u + (a * v)) = u by A10, VECTSP_1:20;
then u in Lin ((U \ {u}) \/ {(u + (a * v))}) by A11, VECTSP_4:21, VECTSP_7:8;
hence x in the carrier of (Lin ((U \ {u}) \/ {(u + (a * v))})) by A5, STRUCT_0:def_5; ::_thesis: verum
end;
suppose u = 0. V ; ::_thesis: x in the carrier of (Lin ((U \ {u}) \/ {(u + (a * v))}))
then x in Lin ((U \ {u}) \/ {(u + (a * v))}) by A5, VECTSP_4:17;
hence x in the carrier of (Lin ((U \ {u}) \/ {(u + (a * v))})) by STRUCT_0:def_5; ::_thesis: verum
end;
end;
end;
end;
end;
suppose x <> u ; ::_thesis: x in the carrier of (Lin ((U \ {u}) \/ {(u + (a * v))}))
then x in U \ {u} by A4, ZFMISC_1:56;
then x in (U \ {u}) \/ {(u + (a * v))} by XBOOLE_0:def_3;
then x in Lin ((U \ {u}) \/ {(u + (a * v))}) by VECTSP_7:8;
hence x in the carrier of (Lin ((U \ {u}) \/ {(u + (a * v))})) by STRUCT_0:def_5; ::_thesis: verum
end;
end;
end;
then Lin U is Subspace of Lin ((U \ {u}) \/ {(u + (a * v))}) by VECTSP_9:16;
then A12: the carrier of (Lin U) c= the carrier of (Lin ((U \ {u}) \/ {(u + (a * v))})) by VECTSP_4:def_2;
Lin ((U \ {u}) \/ {(u + (a * v))}) is Subspace of Lin U by A1, A2, Th13;
then the carrier of (Lin ((U \ {u}) \/ {(u + (a * v))})) c= the carrier of (Lin U) by VECTSP_4:def_2;
then the carrier of (Lin U) = the carrier of (Lin ((U \ {u}) \/ {(u + (a * v))})) by A12, XBOOLE_0:def_10;
hence Lin ((U \ {u}) \/ {(u + (a * v))}) = Lin U by VECTSP_4:29; ::_thesis: verum
end;
begin
definition
let D be non empty set ;
let n, m, k be Nat;
let A be Matrix of n,m,D;
let B be Matrix of n,k,D;
:: original: ^^
redefine funcA ^^ B -> Matrix of n,(width A) + (width B),D;
coherence
A ^^ B is Matrix of n,(width A) + (width B),D
proof
reconsider N = n as Element of NAT by ORDINAL1:def_12;
set AB = A ^^ B;
A1: dom B = Seg (len B) by FINSEQ_1:def_3
.= Seg n by MATRIX_1:def_2 ;
dom A = Seg (len A) by FINSEQ_1:def_3
.= Seg n by MATRIX_1:def_2 ;
then A2: dom (A ^^ B) = (Seg n) /\ (Seg n) by A1, PRE_POLY:def_4
.= Seg n ;
then A3: len (A ^^ B) = N by FINSEQ_1:def_3;
percases ( N = 0 or N > 0 ) ;
supposeA4: N = 0 ; ::_thesis: A ^^ B is Matrix of n,(width A) + (width B),D
then A ^^ B = {} by A2;
hence A ^^ B is Matrix of n,(width A) + (width B),D by A4, MATRIX_1:13; ::_thesis: verum
end;
supposeA5: N > 0 ; ::_thesis: A ^^ B is Matrix of n,(width A) + (width B),D
then A6: N in Seg N by FINSEQ_1:3;
then ( A . N = Line (A,N) & B . N = Line (B,N) ) by MATRIX_2:8;
then A7: (A ^^ B) . N = (Line (A,N)) ^ (Line (B,N)) by A2, A6, PRE_POLY:def_4;
A8: len ((Line (A,N)) ^ (Line (B,N))) = (width A) + (width B) by CARD_1:def_7;
(A ^^ B) . N in rng (A ^^ B) by A2, A6, FUNCT_1:def_3;
then width (A ^^ B) = (width A) + (width B) by A3, A5, A7, A8, MATRIX_1:def_3;
hence A ^^ B is Matrix of n,(width A) + (width B),D by A3, MATRIX_2:7; ::_thesis: verum
end;
end;
end;
end;
theorem Th15: :: MATRIX15:15
for n, m, k being Nat
for D being non empty set
for A being Matrix of n,m,D
for B being Matrix of n,k,D
for i being Nat st i in Seg n holds
Line ((A ^^ B),i) = (Line (A,i)) ^ (Line (B,i))
proof
let n, m, k be Nat; ::_thesis: for D being non empty set
for A being Matrix of n,m,D
for B being Matrix of n,k,D
for i being Nat st i in Seg n holds
Line ((A ^^ B),i) = (Line (A,i)) ^ (Line (B,i))
let D be non empty set ; ::_thesis: for A being Matrix of n,m,D
for B being Matrix of n,k,D
for i being Nat st i in Seg n holds
Line ((A ^^ B),i) = (Line (A,i)) ^ (Line (B,i))
let A be Matrix of n,m,D; ::_thesis: for B being Matrix of n,k,D
for i being Nat st i in Seg n holds
Line ((A ^^ B),i) = (Line (A,i)) ^ (Line (B,i))
let B be Matrix of n,k,D; ::_thesis: for i being Nat st i in Seg n holds
Line ((A ^^ B),i) = (Line (A,i)) ^ (Line (B,i))
set AB = A ^^ B;
A1: ( len (A ^^ B) = n & dom (A ^^ B) = Seg (len (A ^^ B)) ) by FINSEQ_1:def_3, MATRIX_1:def_2;
let i be Nat; ::_thesis: ( i in Seg n implies Line ((A ^^ B),i) = (Line (A,i)) ^ (Line (B,i)) )
assume A2: i in Seg n ; ::_thesis: Line ((A ^^ B),i) = (Line (A,i)) ^ (Line (B,i))
( Line (A,i) = A . i & Line (B,i) = B . i ) by A2, MATRIX_2:8;
hence (Line (A,i)) ^ (Line (B,i)) = (A ^^ B) . i by A2, A1, PRE_POLY:def_4
.= Line ((A ^^ B),i) by A2, MATRIX_2:8 ;
::_thesis: verum
end;
theorem Th16: :: MATRIX15:16
for n, m, k being Nat
for D being non empty set
for A being Matrix of n,m,D
for B being Matrix of n,k,D
for i being Nat st i in Seg (width A) holds
Col ((A ^^ B),i) = Col (A,i)
proof
let n, m, k be Nat; ::_thesis: for D being non empty set
for A being Matrix of n,m,D
for B being Matrix of n,k,D
for i being Nat st i in Seg (width A) holds
Col ((A ^^ B),i) = Col (A,i)
let D be non empty set ; ::_thesis: for A being Matrix of n,m,D
for B being Matrix of n,k,D
for i being Nat st i in Seg (width A) holds
Col ((A ^^ B),i) = Col (A,i)
let A be Matrix of n,m,D; ::_thesis: for B being Matrix of n,k,D
for i being Nat st i in Seg (width A) holds
Col ((A ^^ B),i) = Col (A,i)
let B be Matrix of n,k,D; ::_thesis: for i being Nat st i in Seg (width A) holds
Col ((A ^^ B),i) = Col (A,i)
let i be Nat; ::_thesis: ( i in Seg (width A) implies Col ((A ^^ B),i) = Col (A,i) )
assume A1: i in Seg (width A) ; ::_thesis: Col ((A ^^ B),i) = Col (A,i)
set AB = A ^^ B;
A2: len (A ^^ B) = n by MATRIX_1:def_2;
A3: len A = n by MATRIX_1:def_2;
now__::_thesis:_for_j_being_Nat_st_j_in_Seg_n_holds_
(Col_((A_^^_B),i))_._j_=_(Col_(A,i))_._j
let j be Nat; ::_thesis: ( j in Seg n implies (Col ((A ^^ B),i)) . j = (Col (A,i)) . j )
assume A4: j in Seg n ; ::_thesis: (Col ((A ^^ B),i)) . j = (Col (A,i)) . j
n <> 0 by A4;
then width (A ^^ B) = (width A) + (width B) by MATRIX_1:23;
then width A <= width (A ^^ B) by NAT_1:11;
then A5: Seg (width A) c= Seg (width (A ^^ B)) by FINSEQ_1:5;
A6: dom A = Seg n by A3, FINSEQ_1:def_3;
A7: dom (Line (A,j)) = Seg (width A) by FINSEQ_2:124;
dom (A ^^ B) = Seg n by A2, FINSEQ_1:def_3;
hence (Col ((A ^^ B),i)) . j = (A ^^ B) * (j,i) by A4, MATRIX_1:def_8
.= (Line ((A ^^ B),j)) . i by A1, A5, MATRIX_1:def_7
.= ((Line (A,j)) ^ (Line (B,j))) . i by A4, Th15
.= (Line (A,j)) . i by A1, A7, FINSEQ_1:def_7
.= A * (j,i) by A1, MATRIX_1:def_7
.= (Col (A,i)) . j by A4, A6, MATRIX_1:def_8 ;
::_thesis: verum
end;
hence Col ((A ^^ B),i) = Col (A,i) by A3, A2, FINSEQ_2:119; ::_thesis: verum
end;
theorem Th17: :: MATRIX15:17
for n, m, k being Nat
for D being non empty set
for A being Matrix of n,m,D
for B being Matrix of n,k,D
for i being Nat st i in Seg (width B) holds
Col ((A ^^ B),((width A) + i)) = Col (B,i)
proof
let n, m, k be Nat; ::_thesis: for D being non empty set
for A being Matrix of n,m,D
for B being Matrix of n,k,D
for i being Nat st i in Seg (width B) holds
Col ((A ^^ B),((width A) + i)) = Col (B,i)
let D be non empty set ; ::_thesis: for A being Matrix of n,m,D
for B being Matrix of n,k,D
for i being Nat st i in Seg (width B) holds
Col ((A ^^ B),((width A) + i)) = Col (B,i)
let A be Matrix of n,m,D; ::_thesis: for B being Matrix of n,k,D
for i being Nat st i in Seg (width B) holds
Col ((A ^^ B),((width A) + i)) = Col (B,i)
let B be Matrix of n,k,D; ::_thesis: for i being Nat st i in Seg (width B) holds
Col ((A ^^ B),((width A) + i)) = Col (B,i)
let i be Nat; ::_thesis: ( i in Seg (width B) implies Col ((A ^^ B),((width A) + i)) = Col (B,i) )
assume A1: i in Seg (width B) ; ::_thesis: Col ((A ^^ B),((width A) + i)) = Col (B,i)
set AB = A ^^ B;
A2: len (A ^^ B) = n by MATRIX_1:def_2;
A3: len B = n by MATRIX_1:def_2;
now__::_thesis:_for_j_being_Nat_st_j_in_Seg_n_holds_
(Col_((A_^^_B),((width_A)_+_i)))_._j_=_(Col_(B,i))_._j
A4: dom B = Seg n by A3, FINSEQ_1:def_3;
let j be Nat; ::_thesis: ( j in Seg n implies (Col ((A ^^ B),((width A) + i))) . j = (Col (B,i)) . j )
assume A5: j in Seg n ; ::_thesis: (Col ((A ^^ B),((width A) + i))) . j = (Col (B,i)) . j
n <> 0 by A5;
then width (A ^^ B) = (width A) + (width B) by MATRIX_1:23;
then A6: (width A) + i in Seg (width (A ^^ B)) by A1, FINSEQ_1:60;
A7: ( dom (Line (B,j)) = Seg (width B) & len (Line (A,j)) = width A ) by CARD_1:def_7, FINSEQ_2:124;
dom (A ^^ B) = Seg n by A2, FINSEQ_1:def_3;
hence (Col ((A ^^ B),((width A) + i))) . j = (A ^^ B) * (j,((width A) + i)) by A5, MATRIX_1:def_8
.= (Line ((A ^^ B),j)) . ((width A) + i) by A6, MATRIX_1:def_7
.= ((Line (A,j)) ^ (Line (B,j))) . ((width A) + i) by A5, Th15
.= (Line (B,j)) . i by A1, A7, FINSEQ_1:def_7
.= B * (j,i) by A1, MATRIX_1:def_7
.= (Col (B,i)) . j by A5, A4, MATRIX_1:def_8 ;
::_thesis: verum
end;
hence Col ((A ^^ B),((width A) + i)) = Col (B,i) by A3, A2, FINSEQ_2:119; ::_thesis: verum
end;
theorem Th18: :: MATRIX15:18
for n, m, k, i being Nat
for D being non empty set
for A being Matrix of n,m,D
for B being Matrix of n,k,D
for pA, pB being FinSequence of D st len pA = width A & len pB = width B holds
ReplaceLine ((A ^^ B),i,(pA ^ pB)) = (ReplaceLine (A,i,pA)) ^^ (ReplaceLine (B,i,pB))
proof
let n, m, k, i be Nat; ::_thesis: for D being non empty set
for A being Matrix of n,m,D
for B being Matrix of n,k,D
for pA, pB being FinSequence of D st len pA = width A & len pB = width B holds
ReplaceLine ((A ^^ B),i,(pA ^ pB)) = (ReplaceLine (A,i,pA)) ^^ (ReplaceLine (B,i,pB))
let D be non empty set ; ::_thesis: for A being Matrix of n,m,D
for B being Matrix of n,k,D
for pA, pB being FinSequence of D st len pA = width A & len pB = width B holds
ReplaceLine ((A ^^ B),i,(pA ^ pB)) = (ReplaceLine (A,i,pA)) ^^ (ReplaceLine (B,i,pB))
let A be Matrix of n,m,D; ::_thesis: for B being Matrix of n,k,D
for pA, pB being FinSequence of D st len pA = width A & len pB = width B holds
ReplaceLine ((A ^^ B),i,(pA ^ pB)) = (ReplaceLine (A,i,pA)) ^^ (ReplaceLine (B,i,pB))
let B be Matrix of n,k,D; ::_thesis: for pA, pB being FinSequence of D st len pA = width A & len pB = width B holds
ReplaceLine ((A ^^ B),i,(pA ^ pB)) = (ReplaceLine (A,i,pA)) ^^ (ReplaceLine (B,i,pB))
let pA, pB be FinSequence of D; ::_thesis: ( len pA = width A & len pB = width B implies ReplaceLine ((A ^^ B),i,(pA ^ pB)) = (ReplaceLine (A,i,pA)) ^^ (ReplaceLine (B,i,pB)) )
assume that
A1: len pA = width A and
A2: len pB = width B ; ::_thesis: ReplaceLine ((A ^^ B),i,(pA ^ pB)) = (ReplaceLine (A,i,pA)) ^^ (ReplaceLine (B,i,pB))
set RB = RLine (B,i,pB);
set RA = RLine (A,i,pA);
set AB = A ^^ B;
set RAB = RLine ((A ^^ B),i,(pA ^ pB));
set Rab = (RLine (A,i,pA)) ^^ (RLine (B,i,pB));
A3: now__::_thesis:_for_j_being_Nat_st_1_<=_j_&_j_<=_n_holds_
(RLine_((A_^^_B),i,(pA_^_pB)))_._j_=_((RLine_(A,i,pA))_^^_(RLine_(B,i,pB)))_._j
( pA is Element of (width A) -tuples_on D & pB is Element of (width B) -tuples_on D ) by A1, A2, FINSEQ_2:92;
then pA ^ pB is Tuple of (width A) + (width B),D by FINSEQ_2:107;
then pA ^ pB is Element of ((width A) + (width B)) -tuples_on D by FINSEQ_2:131;
then A4: len (pA ^ pB) = (width A) + (width B) by CARD_1:def_7;
let j be Nat; ::_thesis: ( 1 <= j & j <= n implies (RLine ((A ^^ B),i,(pA ^ pB))) . j = ((RLine (A,i,pA)) ^^ (RLine (B,i,pB))) . j )
assume A5: ( 1 <= j & j <= n ) ; ::_thesis: (RLine ((A ^^ B),i,(pA ^ pB))) . j = ((RLine (A,i,pA)) ^^ (RLine (B,i,pB))) . j
j in NAT by ORDINAL1:def_12;
then A6: j in Seg n by A5;
A7: width (A ^^ B) = (width A) + (width B) by A5, MATRIX_1:23;
A8: now__::_thesis:_Line_((RLine_((A_^^_B),i,(pA_^_pB))),j)_=_Line_(((RLine_(A,i,pA))_^^_(RLine_(B,i,pB))),j)
percases ( i = j or i <> j ) ;
supposeA9: i = j ; ::_thesis: Line ((RLine ((A ^^ B),i,(pA ^ pB))),j) = Line (((RLine (A,i,pA)) ^^ (RLine (B,i,pB))),j)
then A10: Line ((RLine (B,i,pB)),j) = pB by A2, A6, MATRIX11:28;
( Line ((RLine ((A ^^ B),i,(pA ^ pB))),j) = pA ^ pB & Line ((RLine (A,i,pA)),j) = pA ) by A1, A6, A4, A7, A9, MATRIX11:28;
hence Line ((RLine ((A ^^ B),i,(pA ^ pB))),j) = Line (((RLine (A,i,pA)) ^^ (RLine (B,i,pB))),j) by A6, A10, Th15; ::_thesis: verum
end;
supposeA11: i <> j ; ::_thesis: Line ((RLine ((A ^^ B),i,(pA ^ pB))),j) = Line (((RLine (A,i,pA)) ^^ (RLine (B,i,pB))),j)
then A12: Line ((RLine (B,i,pB)),j) = Line (B,j) by A6, MATRIX11:28;
( Line ((RLine ((A ^^ B),i,(pA ^ pB))),j) = Line ((A ^^ B),j) & Line ((RLine (A,i,pA)),j) = Line (A,j) ) by A6, A11, MATRIX11:28;
hence Line ((RLine ((A ^^ B),i,(pA ^ pB))),j) = (Line ((RLine (A,i,pA)),j)) ^ (Line ((RLine (B,i,pB)),j)) by A6, A12, Th15
.= Line (((RLine (A,i,pA)) ^^ (RLine (B,i,pB))),j) by A6, Th15 ;
::_thesis: verum
end;
end;
end;
thus (RLine ((A ^^ B),i,(pA ^ pB))) . j = Line ((RLine ((A ^^ B),i,(pA ^ pB))),j) by A6, MATRIX_2:8
.= ((RLine (A,i,pA)) ^^ (RLine (B,i,pB))) . j by A6, A8, MATRIX_2:8 ; ::_thesis: verum
end;
( len (RLine ((A ^^ B),i,(pA ^ pB))) = n & len ((RLine (A,i,pA)) ^^ (RLine (B,i,pB))) = n ) by MATRIX_1:def_2;
hence ReplaceLine ((A ^^ B),i,(pA ^ pB)) = (ReplaceLine (A,i,pA)) ^^ (ReplaceLine (B,i,pB)) by A3, FINSEQ_1:14; ::_thesis: verum
end;
theorem Th19: :: MATRIX15:19
for n, m, k being Nat
for D being non empty set
for A being Matrix of n,m,D
for B being Matrix of n,k,D holds
( Segm ((A ^^ B),(Seg n),(Seg (width A))) = A & Segm ((A ^^ B),(Seg n),((Seg ((width A) + (width B))) \ (Seg (width A)))) = B )
proof
let n, m, k be Nat; ::_thesis: for D being non empty set
for A being Matrix of n,m,D
for B being Matrix of n,k,D holds
( Segm ((A ^^ B),(Seg n),(Seg (width A))) = A & Segm ((A ^^ B),(Seg n),((Seg ((width A) + (width B))) \ (Seg (width A)))) = B )
let D be non empty set ; ::_thesis: for A being Matrix of n,m,D
for B being Matrix of n,k,D holds
( Segm ((A ^^ B),(Seg n),(Seg (width A))) = A & Segm ((A ^^ B),(Seg n),((Seg ((width A) + (width B))) \ (Seg (width A)))) = B )
let A be Matrix of n,m,D; ::_thesis: for B being Matrix of n,k,D holds
( Segm ((A ^^ B),(Seg n),(Seg (width A))) = A & Segm ((A ^^ B),(Seg n),((Seg ((width A) + (width B))) \ (Seg (width A)))) = B )
let B be Matrix of n,k,D; ::_thesis: ( Segm ((A ^^ B),(Seg n),(Seg (width A))) = A & Segm ((A ^^ B),(Seg n),((Seg ((width A) + (width B))) \ (Seg (width A)))) = B )
set AB = A ^^ B;
A1: card (Seg n) = n by FINSEQ_1:57;
A2: len A = n by MATRIX_1:def_2;
then reconsider A9 = A as Matrix of n, width A,D by MATRIX_2:7;
set S1 = Segm ((A ^^ B),(Seg n),(Seg (width A)));
A3: card (Seg (width A)) = width A by FINSEQ_1:57;
A4: len (A ^^ B) = n by MATRIX_1:def_2;
now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_A9_holds_
(Segm_((A_^^_B),(Seg_n),(Seg_(width_A))))_*_(i,j)_=_A_*_(i,j)
let i, j be Nat; ::_thesis: ( [i,j] in Indices A9 implies (Segm ((A ^^ B),(Seg n),(Seg (width A)))) * (i,j) = A * (i,j) )
assume A5: [i,j] in Indices A9 ; ::_thesis: (Segm ((A ^^ B),(Seg n),(Seg (width A)))) * (i,j) = A * (i,j)
reconsider I = i, J = j as Element of NAT by ORDINAL1:def_12;
A6: dom A = Seg n by A2, FINSEQ_1:def_3;
n <> 0 by A5, MATRIX_1:22;
then Indices A9 = [:(Seg n),(Seg (width A)):] by MATRIX_1:23;
then A7: I in Seg n by A5, ZFMISC_1:87;
A8: J in Seg (width A) by A5, ZFMISC_1:87;
then A9: J = (idseq (width A)) . J by FINSEQ_2:49
.= (Sgm (Seg (width A))) . J by FINSEQ_3:48 ;
dom (Segm ((A ^^ B),(Seg n),(Seg (width A)))) = Seg (len (Segm ((A ^^ B),(Seg n),(Seg (width A))))) by FINSEQ_1:def_3
.= Seg n by A1, MATRIX_1:def_2 ;
hence (Segm ((A ^^ B),(Seg n),(Seg (width A)))) * (i,j) = (Col ((Segm ((A ^^ B),(Seg n),(Seg (width A)))),J)) . I by A7, MATRIX_1:def_8
.= (Col ((A ^^ B),((Sgm (Seg (width A))) . J))) . I by A4, A3, A8, MATRIX13:50
.= (Col (A,J)) . I by A8, A9, Th16
.= A * (i,j) by A7, A6, MATRIX_1:def_8 ;
::_thesis: verum
end;
hence A = Segm ((A ^^ B),(Seg n),(Seg (width A))) by A1, A3, MATRIX_1:27; ::_thesis: Segm ((A ^^ B),(Seg n),((Seg ((width A) + (width B))) \ (Seg (width A)))) = B
set w = (width A) + (width B);
set SS = (Seg ((width A) + (width B))) \ (Seg (width A));
set S2 = Segm ((A ^^ B),(Seg n),((Seg ((width A) + (width B))) \ (Seg (width A))));
A10: len B = n by MATRIX_1:def_2;
then reconsider B9 = B as Matrix of n, width B,D by MATRIX_2:7;
width A <= (width A) + (width B) by NAT_1:11;
then ( card (Seg ((width A) + (width B))) = (width A) + (width B) & Seg (width A) c= Seg ((width A) + (width B)) ) by FINSEQ_1:5, FINSEQ_1:57;
then A11: card ((Seg ((width A) + (width B))) \ (Seg (width A))) = ((width A) + (width B)) - (width A) by A3, CARD_2:44
.= width B ;
now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_B9_holds_
(Segm_((A_^^_B),(Seg_n),((Seg_((width_A)_+_(width_B)))_\_(Seg_(width_A)))))_*_(i,j)_=_B9_*_(i,j)
A12: dom B = Seg n by A10, FINSEQ_1:def_3;
let i, j be Nat; ::_thesis: ( [i,j] in Indices B9 implies (Segm ((A ^^ B),(Seg n),((Seg ((width A) + (width B))) \ (Seg (width A))))) * (i,j) = B9 * (i,j) )
assume A13: [i,j] in Indices B9 ; ::_thesis: (Segm ((A ^^ B),(Seg n),((Seg ((width A) + (width B))) \ (Seg (width A))))) * (i,j) = B9 * (i,j)
reconsider I = i, J = j as Element of NAT by ORDINAL1:def_12;
A14: J in Seg (width B) by A13, ZFMISC_1:87;
n <> 0 by A13, MATRIX_1:22;
then Indices B9 = [:(Seg n),(Seg (width B)):] by MATRIX_1:23;
then A15: I in Seg n by A13, ZFMISC_1:87;
dom (Segm ((A ^^ B),(Seg n),((Seg ((width A) + (width B))) \ (Seg (width A))))) = Seg (len (Segm ((A ^^ B),(Seg n),((Seg ((width A) + (width B))) \ (Seg (width A)))))) by FINSEQ_1:def_3
.= Seg n by A1, MATRIX_1:def_2 ;
hence (Segm ((A ^^ B),(Seg n),((Seg ((width A) + (width B))) \ (Seg (width A))))) * (i,j) = (Col ((Segm ((A ^^ B),(Seg n),((Seg ((width A) + (width B))) \ (Seg (width A))))),J)) . I by A15, MATRIX_1:def_8
.= (Col ((A ^^ B),((Sgm ((Seg ((width A) + (width B))) \ (Seg (width A)))) . J))) . I by A4, A11, A14, MATRIX13:50
.= (Col ((A ^^ B),((width A) + J))) . I by A14, Th8
.= (Col (B,J)) . I by A14, Th17
.= B9 * (i,j) by A15, A12, MATRIX_1:def_8 ;
::_thesis: verum
end;
hence Segm ((A ^^ B),(Seg n),((Seg ((width A) + (width B))) \ (Seg (width A)))) = B by A1, A11, MATRIX_1:27; ::_thesis: verum
end;
theorem Th20: :: MATRIX15:20
for K being Field
for A, B being Matrix of K st len A = len B holds
( the_rank_of A <= the_rank_of (A ^^ B) & the_rank_of B <= the_rank_of (A ^^ B) )
proof
let K be Field; ::_thesis: for A, B being Matrix of K st len A = len B holds
( the_rank_of A <= the_rank_of (A ^^ B) & the_rank_of B <= the_rank_of (A ^^ B) )
let A, B be Matrix of K; ::_thesis: ( len A = len B implies ( the_rank_of A <= the_rank_of (A ^^ B) & the_rank_of B <= the_rank_of (A ^^ B) ) )
assume A1: len A = len B ; ::_thesis: ( the_rank_of A <= the_rank_of (A ^^ B) & the_rank_of B <= the_rank_of (A ^^ B) )
set L = len A;
reconsider B9 = B as Matrix of len A, width B,K by A1, MATRIX_2:7;
reconsider A9 = A as Matrix of len A, width A,K by MATRIX_2:7;
set AB = A9 ^^ B9;
percases ( len A = 0 or len A > 0 ) ;
suppose len A = 0 ; ::_thesis: ( the_rank_of A <= the_rank_of (A ^^ B) & the_rank_of B <= the_rank_of (A ^^ B) )
hence ( the_rank_of A <= the_rank_of (A ^^ B) & the_rank_of B <= the_rank_of (A ^^ B) ) by A1, MATRIX13:74; ::_thesis: verum
end;
supposeA2: len A > 0 ; ::_thesis: ( the_rank_of A <= the_rank_of (A ^^ B) & the_rank_of B <= the_rank_of (A ^^ B) )
A3: Segm ((A9 ^^ B9),(Seg (len A)),(Seg (width A))) = A by Th19;
A4: Indices (A9 ^^ B9) = [:(Seg (len A)),(Seg ((width A) + (width B))):] by A2, MATRIX_1:23;
A5: width (A9 ^^ B9) = (width A) + (width B) by A2, MATRIX_1:23;
then width A <= width (A9 ^^ B9) by NAT_1:11;
then Seg (width A) c= Seg (width (A9 ^^ B9)) by FINSEQ_1:5;
then A6: [:(Seg (len A)),(Seg (width A)):] c= Indices (A9 ^^ B9) by A5, A4, ZFMISC_1:95;
(Seg (width (A9 ^^ B9))) \ (Seg (width A)) c= Seg (width (A9 ^^ B9)) by XBOOLE_1:36;
then A7: [:(Seg (len A)),((Seg (width (A9 ^^ B9))) \ (Seg (width A))):] c= Indices (A9 ^^ B9) by A5, A4, ZFMISC_1:95;
Segm ((A9 ^^ B9),(Seg (len A)),((Seg (width (A9 ^^ B9))) \ (Seg (width A)))) = B by A5, Th19;
hence ( the_rank_of A <= the_rank_of (A ^^ B) & the_rank_of B <= the_rank_of (A ^^ B) ) by A3, A6, A7, MATRIX13:79; ::_thesis: verum
end;
end;
end;
theorem :: MATRIX15:21
for K being Field
for A, B being Matrix of K st len A = len B & len A = the_rank_of A holds
the_rank_of A = the_rank_of (A ^^ B)
proof
let K be Field; ::_thesis: for A, B being Matrix of K st len A = len B & len A = the_rank_of A holds
the_rank_of A = the_rank_of (A ^^ B)
let A, B be Matrix of K; ::_thesis: ( len A = len B & len A = the_rank_of A implies the_rank_of A = the_rank_of (A ^^ B) )
assume that
A1: len A = len B and
A2: len A = the_rank_of A ; ::_thesis: the_rank_of A = the_rank_of (A ^^ B)
set L = len A;
reconsider B9 = B as Matrix of len A, width B,K by A1, MATRIX_2:7;
reconsider A9 = A as Matrix of len A, width A,K by MATRIX_2:7;
A3: ( the_rank_of (A9 ^^ B9) <= len (A9 ^^ B9) & len (A9 ^^ B9) = len A ) by MATRIX13:74, MATRIX_1:def_2;
the_rank_of (A9 ^^ B9) >= len A by A1, A2, Th20;
hence the_rank_of A = the_rank_of (A ^^ B) by A2, A3, XXREAL_0:1; ::_thesis: verum
end;
theorem Th22: :: MATRIX15:22
for K being Field
for A, B being Matrix of K st len A = len B & width A = 0 holds
( A ^^ B = B & B ^^ A = B )
proof
let K be Field; ::_thesis: for A, B being Matrix of K st len A = len B & width A = 0 holds
( A ^^ B = B & B ^^ A = B )
let A, B be Matrix of K; ::_thesis: ( len A = len B & width A = 0 implies ( A ^^ B = B & B ^^ A = B ) )
assume that
A1: len A = len B and
A2: width A = 0 ; ::_thesis: ( A ^^ B = B & B ^^ A = B )
A3: Seg (width A) = {} by A2;
set L = len A;
reconsider B9 = B as Matrix of len A, width B,K by A1, MATRIX_2:7;
reconsider A9 = A as Matrix of len A, width A,K by MATRIX_2:7;
set AB = A9 ^^ B9;
set BA = B9 ^^ A9;
percases ( len A = 0 or len A > 0 ) ;
supposeA4: len A = 0 ; ::_thesis: ( A ^^ B = B & B ^^ A = B )
then len (B9 ^^ A9) = 0 by MATRIX_1:def_2;
then A5: B9 ^^ A9 = {} ;
len (A9 ^^ B9) = 0 by A4, MATRIX_1:def_2;
then A9 ^^ B9 = {} ;
hence ( A ^^ B = B & B ^^ A = B ) by A1, A4, A5; ::_thesis: verum
end;
supposeA6: len A > 0 ; ::_thesis: ( A ^^ B = B & B ^^ A = B )
then ( width (A9 ^^ B9) = width B & len (A9 ^^ B9) = len A ) by A2, MATRIX_1:23;
hence A ^^ B = Segm ((A9 ^^ B9),(Seg (len A)),((Seg (width B)) \ (Seg (width A)))) by A3, MATRIX13:46
.= B by A2, Th19 ;
::_thesis: B ^^ A = B
( width (B9 ^^ A9) = width B & len (B9 ^^ A9) = len A ) by A2, A6, MATRIX_1:23;
hence B ^^ A = Segm ((B9 ^^ A9),(Seg (len A)),(Seg (width B))) by MATRIX13:46
.= B by Th19 ;
::_thesis: verum
end;
end;
end;
theorem Th23: :: MATRIX15:23
for m being Nat
for K being Field
for A, B being Matrix of K st B = 0. (K,(len A),m) holds
the_rank_of A = the_rank_of (A ^^ B)
proof
let m be Nat; ::_thesis: for K being Field
for A, B being Matrix of K st B = 0. (K,(len A),m) holds
the_rank_of A = the_rank_of (A ^^ B)
let K be Field; ::_thesis: for A, B being Matrix of K st B = 0. (K,(len A),m) holds
the_rank_of A = the_rank_of (A ^^ B)
let A, B be Matrix of K; ::_thesis: ( B = 0. (K,(len A),m) implies the_rank_of A = the_rank_of (A ^^ B) )
assume A1: B = 0. (K,(len A),m) ; ::_thesis: the_rank_of A = the_rank_of (A ^^ B)
A2: len B = len A by A1, MATRIX_1:def_2;
set L = len A;
reconsider B9 = B as Matrix of len A, width B,K by A2, MATRIX_2:7;
reconsider A9 = A as Matrix of len A, width A,K by MATRIX_2:7;
set AB = A9 ^^ B9;
percases ( width B = 0 or width B > 0 ) ;
suppose width B = 0 ; ::_thesis: the_rank_of A = the_rank_of (A ^^ B)
hence the_rank_of A = the_rank_of (A ^^ B) by A2, Th22; ::_thesis: verum
end;
suppose width B > 0 ; ::_thesis: the_rank_of A = the_rank_of (A ^^ B)
then A3: len A > 0 by A2, MATRIX_1:def_3;
then A4: width (A9 ^^ B9) = (width A) + (width B) by MATRIX_1:23;
A5: len (A9 ^^ B9) = len A by A3, MATRIX_1:23;
A6: now__::_thesis:_for_i_being_Nat_st_i_in_(Seg_(width_(A9_^^_B9)))_\_(Seg_(width_A))_holds_
Col_((A9_^^_B9),i)_=_(len_(A9_^^_B9))_|->_(0._K)
set L0 = (len (A9 ^^ B9)) |-> (0. K);
let i be Nat; ::_thesis: ( i in (Seg (width (A9 ^^ B9))) \ (Seg (width A)) implies Col ((A9 ^^ B9),i) = (len (A9 ^^ B9)) |-> (0. K) )
assume A7: i in (Seg (width (A9 ^^ B9))) \ (Seg (width A)) ; ::_thesis: Col ((A9 ^^ B9),i) = (len (A9 ^^ B9)) |-> (0. K)
A8: i in Seg (width (A9 ^^ B9)) by A7, XBOOLE_0:def_5;
not i in Seg (width A) by A7, XBOOLE_0:def_5;
then A9: ( i < 1 or i > width A ) by A7;
then reconsider n = i - (width A) as Element of NAT by A8, FINSEQ_1:1, NAT_1:21;
A10: i = n + (width A) ;
n <> 0 by A8, A9, FINSEQ_1:1;
then A11: n in Seg (width B) by A4, A8, A10, FINSEQ_1:61;
A12: now__::_thesis:_for_i_being_Nat_st_1_<=_i_&_i_<=_len_A_holds_
(Col_(B9,n))_._i_=_((len_(A9_^^_B9))_|->_(0._K))_._i
let i be Nat; ::_thesis: ( 1 <= i & i <= len A implies (Col (B9,n)) . i = ((len (A9 ^^ B9)) |-> (0. K)) . i )
assume A13: ( 1 <= i & i <= len A ) ; ::_thesis: (Col (B9,n)) . i = ((len (A9 ^^ B9)) |-> (0. K)) . i
i in NAT by ORDINAL1:def_12;
then A14: i in Seg (len A) by A13;
then A15: ((len (A9 ^^ B9)) |-> (0. K)) . i = 0. K by A5, FINSEQ_2:57;
Seg (len A) = dom B by A2, FINSEQ_1:def_3;
then ( (Col (B9,n)) . i = B9 * (i,n) & [i,n] in Indices B ) by A11, A14, MATRIX_1:def_8, ZFMISC_1:87;
hence (Col (B9,n)) . i = ((len (A9 ^^ B9)) |-> (0. K)) . i by A1, A15, MATRIX_3:1; ::_thesis: verum
end;
A16: ( len (Col (B,n)) = len A & len ((len (A9 ^^ B9)) |-> (0. K)) = len A ) by A2, A5, CARD_1:def_7;
Col ((A9 ^^ B9),i) = Col (B9,n) by A10, A11, Th17;
hence Col ((A9 ^^ B9),i) = (len (A9 ^^ B9)) |-> (0. K) by A16, A12, FINSEQ_1:14; ::_thesis: verum
end;
width A <= width (A9 ^^ B9) by A4, NAT_1:11;
then Seg (width A) c= Seg (width (A9 ^^ B9)) by FINSEQ_1:5;
hence the_rank_of (A ^^ B) = the_rank_of (Segm ((A9 ^^ B9),(Seg (len (A9 ^^ B9))),(Seg (width A)))) by A6, Th12
.= the_rank_of A by A5, Th19 ;
::_thesis: verum
end;
end;
end;
theorem Th24: :: MATRIX15:24
for K being Field
for A, B being Matrix of K st the_rank_of A = the_rank_of (A ^^ B) & len A = len B holds
for N being finite without_zero Subset of NAT st N c= dom A & ( for i being Nat st i in N holds
Line (A,i) = (width A) |-> (0. K) ) holds
for i being Nat st i in N holds
Line (B,i) = (width B) |-> (0. K)
proof
let K be Field; ::_thesis: for A, B being Matrix of K st the_rank_of A = the_rank_of (A ^^ B) & len A = len B holds
for N being finite without_zero Subset of NAT st N c= dom A & ( for i being Nat st i in N holds
Line (A,i) = (width A) |-> (0. K) ) holds
for i being Nat st i in N holds
Line (B,i) = (width B) |-> (0. K)
let A, B be Matrix of K; ::_thesis: ( the_rank_of A = the_rank_of (A ^^ B) & len A = len B implies for N being finite without_zero Subset of NAT st N c= dom A & ( for i being Nat st i in N holds
Line (A,i) = (width A) |-> (0. K) ) holds
for i being Nat st i in N holds
Line (B,i) = (width B) |-> (0. K) )
assume that
A1: the_rank_of A = the_rank_of (A ^^ B) and
A2: len A = len B ; ::_thesis: for N being finite without_zero Subset of NAT st N c= dom A & ( for i being Nat st i in N holds
Line (A,i) = (width A) |-> (0. K) ) holds
for i being Nat st i in N holds
Line (B,i) = (width B) |-> (0. K)
reconsider B9 = B as Matrix of len A, width B,K by A2, MATRIX_2:7;
reconsider A9 = A as Matrix of len A, width A,K by MATRIX_2:7;
set AB = A9 ^^ B9;
let N be finite without_zero Subset of NAT; ::_thesis: ( N c= dom A & ( for i being Nat st i in N holds
Line (A,i) = (width A) |-> (0. K) ) implies for i being Nat st i in N holds
Line (B,i) = (width B) |-> (0. K) )
assume that
A3: N c= dom A and
A4: for i being Nat st i in N holds
Line (A,i) = (width A) |-> (0. K) ; ::_thesis: for i being Nat st i in N holds
Line (B,i) = (width B) |-> (0. K)
let i be Nat; ::_thesis: ( i in N implies Line (B,i) = (width B) |-> (0. K) )
assume A5: i in N ; ::_thesis: Line (B,i) = (width B) |-> (0. K)
dom A <> {} by A3, A5;
then Seg (len A) <> {} by FINSEQ_1:def_3;
then A6: len A <> 0 ;
then A7: width (A9 ^^ B9) = (width A) + (width B) by MATRIX_1:23;
then width A <= width (A9 ^^ B9) by NAT_1:11;
then A8: Seg (width A) c= Seg (width (A9 ^^ B9)) by FINSEQ_1:5;
A9: card (Seg (len A)) = len A by FINSEQ_1:57;
A10: Segm ((A9 ^^ B9),(Seg (len A)),(Seg (width A))) = A by Th19;
A11: dom A = Seg (len A) by FINSEQ_1:def_3;
A12: (Sgm (Seg (len A))) . i = (idseq (len A)) . i by FINSEQ_3:48
.= i by A3, A5, A11, FINSEQ_2:49 ;
card (Seg (width A)) = width A by FINSEQ_1:57;
then A13: (card (Seg (width A))) |-> (0. K) = Line (A,i) by A4, A5
.= (Line ((A9 ^^ B9),i)) * (Sgm (Seg (width A))) by A3, A5, A11, A10, A8, A9, A12, MATRIX13:47 ;
assume Line (B,i) <> (width B) |-> (0. K) ; ::_thesis: contradiction
then consider j being Nat such that
A14: j in Seg (width B) and
A15: (Line (B,i)) . j <> ((width B) |-> (0. K)) . j by FINSEQ_2:119;
A16: ( len (Line (A9,i)) = width A9 & 1 <= j ) by A14, FINSEQ_1:1, MATRIX_1:def_7;
len (Line (B9,i)) = width B9 by MATRIX_1:def_7;
then A17: j <= len (Line (B9,i)) by A14, FINSEQ_1:1;
A18: j + (width A) in Seg (width (A9 ^^ B9)) by A14, A7, FINSEQ_1:60;
then (A9 ^^ B9) * (i,(j + (width A))) = (Line ((A9 ^^ B9),i)) . (j + (width A)) by MATRIX_1:def_7
.= ((Line (A9,i)) ^ (Line (B9,i))) . (j + (width A)) by A3, A5, A11, Th15
.= (Line (B9,i)) . j by A16, A17, FINSEQ_1:65 ;
then A19: (A9 ^^ B9) * (i,(j + (width A))) <> 0. K by A14, A15, FINSEQ_2:57;
consider P, Q being finite without_zero Subset of NAT such that
A20: [:P,Q:] c= Indices A9 and
A21: card P = card Q and
A22: card P = the_rank_of A9 and
A23: Det (EqSegm (A9,P,Q)) <> 0. K by MATRIX13:def_4;
( P = {} iff Q = {} ) by A21;
then consider P2, Q2 being finite without_zero Subset of NAT such that
A24: P2 c= Seg (len A) and
A25: Q2 c= Seg (width A) and
A26: P2 = (Sgm (Seg (len A))) .: P and
Q2 = (Sgm (Seg (width A))) .: Q and
card P2 = card P and
card Q2 = card Q and
A27: Segm (A,P,Q) = Segm ((A9 ^^ B9),P2,Q2) by A20, A10, MATRIX13:57;
A28: Segm ((A9 ^^ B9),P2,Q2) = EqSegm (A,P,Q) by A21, A27, MATRIX13:def_3;
A29: ( dom (A9 ^^ B9) = Seg (len (A9 ^^ B9)) & len (A9 ^^ B9) = len A ) by A6, FINSEQ_1:def_3, MATRIX_1:23;
then A30: [:P2,(Seg (width A)):] c= Indices (A9 ^^ B9) by A24, A8, ZFMISC_1:96;
j >= 1 by A14, FINSEQ_1:1;
then j + (width A) >= 1 + (width A) by XREAL_1:6;
then j + (width A) > width A by NAT_1:13;
then not j + (width A) in Q2 by A25, FINSEQ_1:1;
then A31: j + (width A) in (Seg (width (A9 ^^ B9))) \ Q2 by A18, XBOOLE_0:def_5;
not i in P2
proof
A32: Line (A,i) = (width A) |-> (0. K) by A4, A5
.= (0. K) * (Line (A,i)) by FVSUM_1:58 ;
A33: Sgm (Seg (len A)) = idseq (len A) by FINSEQ_3:48
.= id (Seg (len A)) ;
A34: P c= Seg (len A) by A20, A21, MATRIX13:67;
then A35: rng (Sgm P) = P by FINSEQ_1:def_13;
assume i in P2 ; ::_thesis: contradiction
then i in P by A26, A33, A34, FUNCT_1:92;
then consider x being set such that
A36: x in dom (Sgm P) and
A37: (Sgm P) . x = i by A35, FUNCT_1:def_3;
reconsider x = x as Element of NAT by A36;
A38: Segm (A,P,Q) = EqSegm (A,P,Q) by A21, MATRIX13:def_3;
A39: Q c= Seg (width A) by A20, A21, MATRIX13:67;
then A40: rng (Sgm Q) = Q by FINSEQ_1:def_13;
A41: dom (Sgm P) = Seg (card P) by A34, FINSEQ_3:40;
then ( dom (Line (A,i)) = Seg (width A) & Line ((Segm (A,P,Q)),x) = (Line (A,i)) * (Sgm Q) ) by A39, A36, A37, FINSEQ_2:124, MATRIX13:47;
then Line ((Segm (A,P,Q)),x) = (0. K) * (Line ((Segm (A,P,Q)),x)) by A39, A40, A32, MATRIX13:87;
then (0. K) * (Det (EqSegm (A,P,Q))) = Det (RLine ((EqSegm (A,P,Q)),x,(Line ((EqSegm (A,P,Q)),x)))) by A41, A36, A38, MATRIX11:35
.= Det (EqSegm (A,P,Q)) by MATRIX11:30 ;
hence contradiction by A23, VECTSP_1:6; ::_thesis: verum
end;
then i in (dom (A9 ^^ B9)) \ P2 by A3, A5, A11, A29, XBOOLE_0:def_5;
then card P > the_rank_of (Segm ((A9 ^^ B9),P2,Q2)) by A1, A19, A22, A25, A30, A31, A13, Th10;
hence contradiction by A23, A28, MATRIX13:83; ::_thesis: verum
end;
begin
definition
let D be non empty set ;
let b be FinSequence of D;
func LineVec2Mx b -> Matrix of 1, len b,D equals :: MATRIX15:def 1
<*b*>;
coherence
<*b*> is Matrix of 1, len b,D ;
func ColVec2Mx b -> Matrix of len b,1,D equals :: MATRIX15:def 2
<*b*> @ ;
coherence
<*b*> @ is Matrix of len b,1,D
proof
set B = <*b*>;
A1: len <*b*> = 1 by MATRIX_1:23;
A2: width <*b*> = len b by MATRIX_1:23;
A3: len (<*b*> @) = width <*b*> by MATRIX_1:def_6;
percases ( len b = 0 or len b > 0 ) ;
supposeA4: len b = 0 ; ::_thesis: <*b*> @ is Matrix of len b,1,D
then <*b*> @ = {} by A3, MATRIX_1:23;
hence <*b*> @ is Matrix of len b,1,D by A4, MATRIX_1:13; ::_thesis: verum
end;
suppose len b > 0 ; ::_thesis: <*b*> @ is Matrix of len b,1,D
then width (<*b*> @) = len <*b*> by A2, MATRIX_2:10;
hence <*b*> @ is Matrix of len b,1,D by A1, A2, A3, MATRIX_2:7; ::_thesis: verum
end;
end;
end;
end;
:: deftheorem defines LineVec2Mx MATRIX15:def_1_:_
for D being non empty set
for b being FinSequence of D holds LineVec2Mx b = <*b*>;
:: deftheorem defines ColVec2Mx MATRIX15:def_2_:_
for D being non empty set
for b being FinSequence of D holds ColVec2Mx b = <*b*> @ ;
theorem Th25: :: MATRIX15:25
for D being non empty set
for bD being FinSequence of D
for MD being Matrix of D holds
( MD = LineVec2Mx bD iff ( Line (MD,1) = bD & len MD = 1 ) )
proof
let D be non empty set ; ::_thesis: for bD being FinSequence of D
for MD being Matrix of D holds
( MD = LineVec2Mx bD iff ( Line (MD,1) = bD & len MD = 1 ) )
let bD be FinSequence of D; ::_thesis: for MD being Matrix of D holds
( MD = LineVec2Mx bD iff ( Line (MD,1) = bD & len MD = 1 ) )
let MD be Matrix of D; ::_thesis: ( MD = LineVec2Mx bD iff ( Line (MD,1) = bD & len MD = 1 ) )
thus ( MD = LineVec2Mx bD implies ( Line (MD,1) = bD & len MD = 1 ) ) ::_thesis: ( Line (MD,1) = bD & len MD = 1 implies MD = LineVec2Mx bD )
proof
1 in Seg 1 ;
then A1: Line ((LineVec2Mx bD),1) = (LineVec2Mx bD) . 1 by MATRIX_2:8;
assume MD = LineVec2Mx bD ; ::_thesis: ( Line (MD,1) = bD & len MD = 1 )
hence ( Line (MD,1) = bD & len MD = 1 ) by A1, FINSEQ_1:40; ::_thesis: verum
end;
assume that
A2: Line (MD,1) = bD and
A3: len MD = 1 ; ::_thesis: MD = LineVec2Mx bD
reconsider md = MD as Matrix of 1, width MD,D by A3, MATRIX_2:7;
1 in Seg 1 ;
then md . 1 = bD by A2, MATRIX_2:8;
hence MD = LineVec2Mx bD by A3, FINSEQ_1:40; ::_thesis: verum
end;
theorem Th26: :: MATRIX15:26
for D being non empty set
for bD being FinSequence of D
for MD being Matrix of D st ( len MD <> 0 or len bD <> 0 ) holds
( MD = ColVec2Mx bD iff ( Col (MD,1) = bD & width MD = 1 ) )
proof
let D be non empty set ; ::_thesis: for bD being FinSequence of D
for MD being Matrix of D st ( len MD <> 0 or len bD <> 0 ) holds
( MD = ColVec2Mx bD iff ( Col (MD,1) = bD & width MD = 1 ) )
let bD be FinSequence of D; ::_thesis: for MD being Matrix of D st ( len MD <> 0 or len bD <> 0 ) holds
( MD = ColVec2Mx bD iff ( Col (MD,1) = bD & width MD = 1 ) )
let MD be Matrix of D; ::_thesis: ( ( len MD <> 0 or len bD <> 0 ) implies ( MD = ColVec2Mx bD iff ( Col (MD,1) = bD & width MD = 1 ) ) )
assume A1: ( len MD <> 0 or len bD <> 0 ) ; ::_thesis: ( MD = ColVec2Mx bD iff ( Col (MD,1) = bD & width MD = 1 ) )
thus ( MD = ColVec2Mx bD implies ( Col (MD,1) = bD & width MD = 1 ) ) ::_thesis: ( Col (MD,1) = bD & width MD = 1 implies MD = ColVec2Mx bD )
proof
len (LineVec2Mx bD) = 1 by Th25;
then A2: dom (LineVec2Mx bD) = Seg 1 by FINSEQ_1:def_3;
assume A3: MD = ColVec2Mx bD ; ::_thesis: ( Col (MD,1) = bD & width MD = 1 )
1 in Seg 1 ;
hence Col (MD,1) = Line ((LineVec2Mx bD),1) by A3, A2, MATRIX_2:14
.= bD by Th25 ;
::_thesis: width MD = 1
len MD = len bD by A3, MATRIX_1:def_2;
hence width MD = 1 by A1, A3, MATRIX_1:23; ::_thesis: verum
end;
assume that
A4: Col (MD,1) = bD and
A5: width MD = 1 ; ::_thesis: MD = ColVec2Mx bD
A6: len MD > 0 by A1, A4, MATRIX_1:def_8;
A7: len (MD @) = 1 by A5, MATRIX_1:def_6;
1 in Seg 1 ;
then Line ((MD @),1) = bD by A4, A5, MATRIX_2:15;
then (LineVec2Mx bD) @ = (MD @) @ by A7, Th25
.= MD by A5, A6, MATRIX_2:13 ;
hence MD = ColVec2Mx bD ; ::_thesis: verum
end;
theorem :: MATRIX15:27
for K being Field
for f, g being FinSequence of K st len f = len g holds
(LineVec2Mx f) + (LineVec2Mx g) = LineVec2Mx (f + g)
proof
let K be Field; ::_thesis: for f, g being FinSequence of K st len f = len g holds
(LineVec2Mx f) + (LineVec2Mx g) = LineVec2Mx (f + g)
let f, g be FinSequence of K; ::_thesis: ( len f = len g implies (LineVec2Mx f) + (LineVec2Mx g) = LineVec2Mx (f + g) )
set Lf = LineVec2Mx f;
set Lg = LineVec2Mx g;
A1: len (LineVec2Mx f) = 1 by CARD_1:def_7;
assume A2: len f = len g ; ::_thesis: (LineVec2Mx f) + (LineVec2Mx g) = LineVec2Mx (f + g)
then reconsider F = f, G = g as Element of (len f) -tuples_on the carrier of K by FINSEQ_2:92;
A3: width (LineVec2Mx g) = len f by A2, MATRIX_1:23;
set FG = F + G;
set Lfg = LineVec2Mx (F + G);
A4: ( len (F + G) = len f & len (LineVec2Mx (F + G)) = 1 ) by CARD_1:def_7;
A5: width (LineVec2Mx (F + G)) = len (F + G) by MATRIX_1:23;
A6: len ((LineVec2Mx f) + (LineVec2Mx g)) = len (LineVec2Mx f) by MATRIX_3:def_3;
A7: width ((LineVec2Mx f) + (LineVec2Mx g)) = width (LineVec2Mx f) by MATRIX_3:def_3;
A8: width (LineVec2Mx f) = len f by MATRIX_1:23;
percases ( len f = 0 or len f > 0 ) ;
suppose len f = 0 ; ::_thesis: (LineVec2Mx f) + (LineVec2Mx g) = LineVec2Mx (f + g)
then Seg (len f) = {} ;
then for i, j being Nat st [i,j] in Indices ((LineVec2Mx f) + (LineVec2Mx g)) holds
((LineVec2Mx f) + (LineVec2Mx g)) * (i,j) = (LineVec2Mx (F + G)) * (i,j) by A7, A8, ZFMISC_1:90;
hence (LineVec2Mx f) + (LineVec2Mx g) = LineVec2Mx (f + g) by A4, A1, A6, A7, A8, A5, MATRIX_1:21; ::_thesis: verum
end;
supposeA9: len f > 0 ; ::_thesis: (LineVec2Mx f) + (LineVec2Mx g) = LineVec2Mx (f + g)
A10: ( dom (LineVec2Mx f) = Seg 1 & 1 in Seg 1 ) by A1, FINSEQ_1:def_3;
len f in Seg (len f) by A9, FINSEQ_1:3;
then [1,(len f)] in Indices (LineVec2Mx f) by A8, A10, ZFMISC_1:87;
then Line (((LineVec2Mx f) + (LineVec2Mx g)),1) = (Line ((LineVec2Mx f),1)) + (Line ((LineVec2Mx g),1)) by A3, MATRIX_1:23, MATRIX_4:59
.= f + (Line ((LineVec2Mx g),1)) by Th25
.= f + g by Th25 ;
hence (LineVec2Mx f) + (LineVec2Mx g) = LineVec2Mx (f + g) by A1, A6, Th25; ::_thesis: verum
end;
end;
end;
theorem Th28: :: MATRIX15:28
for K being Field
for f, g being FinSequence of K st len f = len g holds
(ColVec2Mx f) + (ColVec2Mx g) = ColVec2Mx (f + g)
proof
let K be Field; ::_thesis: for f, g being FinSequence of K st len f = len g holds
(ColVec2Mx f) + (ColVec2Mx g) = ColVec2Mx (f + g)
let f, g be FinSequence of K; ::_thesis: ( len f = len g implies (ColVec2Mx f) + (ColVec2Mx g) = ColVec2Mx (f + g) )
set Cf = ColVec2Mx f;
set Cg = ColVec2Mx g;
A1: len (ColVec2Mx f) = len f by MATRIX_1:def_2;
assume A2: len f = len g ; ::_thesis: (ColVec2Mx f) + (ColVec2Mx g) = ColVec2Mx (f + g)
then reconsider F = f, G = g as Element of (len f) -tuples_on the carrier of K by FINSEQ_2:92;
A3: len (ColVec2Mx g) = len f by A2, MATRIX_1:def_2;
set FG = F + G;
set Cfg = ColVec2Mx (F + G);
A4: len (ColVec2Mx (F + G)) = len (F + G) by MATRIX_1:def_2;
A5: ( len (F + G) = len f & width ((ColVec2Mx f) + (ColVec2Mx g)) = width (ColVec2Mx f) ) by CARD_1:def_7, MATRIX_3:def_3;
A6: len ((ColVec2Mx f) + (ColVec2Mx g)) = len (ColVec2Mx f) by MATRIX_3:def_3;
percases ( len f = 0 or len f > 0 ) ;
supposeA7: len f = 0 ; ::_thesis: (ColVec2Mx f) + (ColVec2Mx g) = ColVec2Mx (f + g)
then (ColVec2Mx f) + (ColVec2Mx g) = {} by A6, MATRIX_1:def_2;
hence (ColVec2Mx f) + (ColVec2Mx g) = ColVec2Mx (f + g) by A4, A7; ::_thesis: verum
end;
supposeA8: len f > 0 ; ::_thesis: (ColVec2Mx f) + (ColVec2Mx g) = ColVec2Mx (f + g)
A9: ( dom (ColVec2Mx f) = Seg (len f) & 1 in Seg 1 ) by A1, FINSEQ_1:def_3;
A10: width (ColVec2Mx f) = 1 by A8, MATRIX_1:23;
len f in Seg (len f) by A8, FINSEQ_1:3;
then [(len f),1] in Indices (ColVec2Mx f) by A9, A10, ZFMISC_1:87;
then Col (((ColVec2Mx f) + (ColVec2Mx g)),1) = (Col ((ColVec2Mx f),1)) + (Col ((ColVec2Mx g),1)) by A1, A3, MATRIX_4:60
.= f + (Col ((ColVec2Mx g),1)) by A8, Th26
.= f + g by A2, A8, Th26 ;
hence (ColVec2Mx f) + (ColVec2Mx g) = ColVec2Mx (f + g) by A5, A8, A10, Th26; ::_thesis: verum
end;
end;
end;
theorem :: MATRIX15:29
for K being Field
for a being Element of K
for f being FinSequence of K holds a * (LineVec2Mx f) = LineVec2Mx (a * f)
proof
let K be Field; ::_thesis: for a being Element of K
for f being FinSequence of K holds a * (LineVec2Mx f) = LineVec2Mx (a * f)
let a be Element of K; ::_thesis: for f being FinSequence of K holds a * (LineVec2Mx f) = LineVec2Mx (a * f)
let f be FinSequence of K; ::_thesis: a * (LineVec2Mx f) = LineVec2Mx (a * f)
A1: len (a * (LineVec2Mx f)) = len (LineVec2Mx f) by MATRIX_3:def_5;
A2: len (LineVec2Mx f) = 1 by MATRIX_1:def_2;
then Line ((a * (LineVec2Mx f)),1) = a * (Line ((LineVec2Mx f),1)) by MATRIXR1:20
.= a * f by Th25 ;
hence a * (LineVec2Mx f) = LineVec2Mx (a * f) by A2, A1, Th25; ::_thesis: verum
end;
theorem Th30: :: MATRIX15:30
for K being Field
for a being Element of K
for f being FinSequence of K holds a * (ColVec2Mx f) = ColVec2Mx (a * f)
proof
let K be Field; ::_thesis: for a being Element of K
for f being FinSequence of K holds a * (ColVec2Mx f) = ColVec2Mx (a * f)
let a be Element of K; ::_thesis: for f being FinSequence of K holds a * (ColVec2Mx f) = ColVec2Mx (a * f)
let f be FinSequence of K; ::_thesis: a * (ColVec2Mx f) = ColVec2Mx (a * f)
A1: len f = len (a * f) by MATRIXR1:16;
percases ( len f = 0 or len f > 0 ) ;
supposeA2: len f = 0 ; ::_thesis: a * (ColVec2Mx f) = ColVec2Mx (a * f)
len (ColVec2Mx f) = len (a * (ColVec2Mx f)) by MATRIX_3:def_5;
then A3: a * (ColVec2Mx f) = {} by A2, MATRIX_1:def_2;
len (ColVec2Mx (a * f)) = 0 by A1, A2, MATRIX_1:def_2;
hence a * (ColVec2Mx f) = ColVec2Mx (a * f) by A3; ::_thesis: verum
end;
supposeA4: len f > 0 ; ::_thesis: a * (ColVec2Mx f) = ColVec2Mx (a * f)
A5: width (a * (ColVec2Mx f)) = width (ColVec2Mx f) by MATRIX_3:def_5;
A6: width (ColVec2Mx f) = 1 by A4, MATRIX_1:23;
then Col ((a * (ColVec2Mx f)),1) = a * (Col ((ColVec2Mx f),1)) by MATRIXR1:19
.= a * f by A4, Th26 ;
hence a * (ColVec2Mx f) = ColVec2Mx (a * f) by A1, A4, A6, A5, Th26; ::_thesis: verum
end;
end;
end;
theorem :: MATRIX15:31
for k being Nat
for K being Field holds LineVec2Mx (k |-> (0. K)) = 0. (K,1,k)
proof
let k be Nat; ::_thesis: for K being Field holds LineVec2Mx (k |-> (0. K)) = 0. (K,1,k)
let K be Field; ::_thesis: LineVec2Mx (k |-> (0. K)) = 0. (K,1,k)
reconsider L = LineVec2Mx (k |-> (0. K)) as Matrix of 1,k,K by CARD_1:def_7;
set Z = 0. (K,1,k);
now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_L_holds_
(0._(K,1,k))_*_(i,j)_=_L_*_(i,j)
A1: width L = k by MATRIX_1:23;
A2: ( dom L = Seg (len L) & len L = 1 ) by FINSEQ_1:def_3, MATRIX_1:def_2;
let i, j be Nat; ::_thesis: ( [i,j] in Indices L implies (0. (K,1,k)) * (i,j) = L * (i,j) )
assume A3: [i,j] in Indices L ; ::_thesis: (0. (K,1,k)) * (i,j) = L * (i,j)
A4: j in Seg (width L) by A3, ZFMISC_1:87;
i in dom L by A3, ZFMISC_1:87;
then A5: i = 1 by A2, FINSEQ_1:2, TARSKI:def_1;
Indices (0. (K,1,k)) = Indices L by MATRIX_1:26;
hence (0. (K,1,k)) * (i,j) = 0. K by A3, MATRIX_3:1
.= (k |-> (0. K)) . j by A4, A1, FINSEQ_2:57
.= (Line (L,i)) . j by A5, Th25
.= L * (i,j) by A4, MATRIX_1:def_7 ;
::_thesis: verum
end;
hence LineVec2Mx (k |-> (0. K)) = 0. (K,1,k) by MATRIX_1:27; ::_thesis: verum
end;
theorem Th32: :: MATRIX15:32
for k being Nat
for K being Field holds ColVec2Mx (k |-> (0. K)) = 0. (K,k,1)
proof
let k be Nat; ::_thesis: for K being Field holds ColVec2Mx (k |-> (0. K)) = 0. (K,k,1)
let K be Field; ::_thesis: ColVec2Mx (k |-> (0. K)) = 0. (K,k,1)
reconsider C = ColVec2Mx (k |-> (0. K)) as Matrix of k,1,K by CARD_1:def_7;
set Z = 0. (K,k,1);
now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_C_holds_
(0._(K,k,1))_*_(i,j)_=_C_*_(i,j)
A1: len (k |-> (0. K)) = k by CARD_1:def_7;
let i, j be Nat; ::_thesis: ( [i,j] in Indices C implies (0. (K,k,1)) * (i,j) = C * (i,j) )
assume A2: [i,j] in Indices C ; ::_thesis: (0. (K,k,1)) * (i,j) = C * (i,j)
A3: i in dom C by A2, ZFMISC_1:87;
A4: j in Seg (width C) by A2, ZFMISC_1:87;
A5: ( dom C = Seg (len C) & len C = len (k |-> (0. K)) ) by FINSEQ_1:def_3, MATRIX_1:def_2;
then width C = 1 by A3, A1, Th26;
then A6: j = 1 by A4, FINSEQ_1:2, TARSKI:def_1;
Indices (0. (K,k,1)) = Indices C by MATRIX_1:26;
hence (0. (K,k,1)) * (i,j) = 0. K by A2, MATRIX_3:1
.= (k |-> (0. K)) . i by A3, A5, A1, FINSEQ_2:57
.= (Col (C,j)) . i by A3, A5, A1, A6, Th26
.= C * (i,j) by A3, MATRIX_1:def_8 ;
::_thesis: verum
end;
hence ColVec2Mx (k |-> (0. K)) = 0. (K,k,1) by MATRIX_1:27; ::_thesis: verum
end;
begin
definition
let K be Field;
let A, B be Matrix of K;
func Solutions_of (A,B) -> set equals :: MATRIX15:def 3
{ X where X is Matrix of K : ( len X = width A & width X = width B & A * X = B ) } ;
coherence
{ X where X is Matrix of K : ( len X = width A & width X = width B & A * X = B ) } is set ;
end;
:: deftheorem defines Solutions_of MATRIX15:def_3_:_
for K being Field
for A, B being Matrix of K holds Solutions_of (A,B) = { X where X is Matrix of K : ( len X = width A & width X = width B & A * X = B ) } ;
theorem Th33: :: MATRIX15:33
for K being Field
for A, B being Matrix of K st not Solutions_of (A,B) is empty holds
len A = len B
proof
let K be Field; ::_thesis: for A, B being Matrix of K st not Solutions_of (A,B) is empty holds
len A = len B
let A, B be Matrix of K; ::_thesis: ( not Solutions_of (A,B) is empty implies len A = len B )
assume not Solutions_of (A,B) is empty ; ::_thesis: len A = len B
then consider x being set such that
A1: x in Solutions_of (A,B) by XBOOLE_0:def_1;
ex X being Matrix of K st
( X = x & len X = width A & width X = width B & A * X = B ) by A1;
hence len A = len B by MATRIX_3:def_4; ::_thesis: verum
end;
theorem :: MATRIX15:34
for i being Nat
for K being Field
for X, A, B being Matrix of K st X in Solutions_of (A,B) & i in Seg (width X) & Col (X,i) = (len X) |-> (0. K) holds
Col (B,i) = (len B) |-> (0. K)
proof
let i be Nat; ::_thesis: for K being Field
for X, A, B being Matrix of K st X in Solutions_of (A,B) & i in Seg (width X) & Col (X,i) = (len X) |-> (0. K) holds
Col (B,i) = (len B) |-> (0. K)
let K be Field; ::_thesis: for X, A, B being Matrix of K st X in Solutions_of (A,B) & i in Seg (width X) & Col (X,i) = (len X) |-> (0. K) holds
Col (B,i) = (len B) |-> (0. K)
let X, A, B be Matrix of K; ::_thesis: ( X in Solutions_of (A,B) & i in Seg (width X) & Col (X,i) = (len X) |-> (0. K) implies Col (B,i) = (len B) |-> (0. K) )
assume that
A1: X in Solutions_of (A,B) and
A2: i in Seg (width X) and
A3: Col (X,i) = (len X) |-> (0. K) ; ::_thesis: Col (B,i) = (len B) |-> (0. K)
set LB0 = (len B) |-> (0. K);
consider X1 being Matrix of K such that
A4: X = X1 and
A5: len X1 = width A and
A6: width X1 = width B and
A7: A * X1 = B by A1;
A8: now__::_thesis:_for_k_being_Nat_st_1_<=_k_&_k_<=_len_B_holds_
(Col_(B,i))_._k_=_((len_B)_|->_(0._K))_._k
let k be Nat; ::_thesis: ( 1 <= k & k <= len B implies (Col (B,i)) . k = ((len B) |-> (0. K)) . k )
assume A9: ( 1 <= k & k <= len B ) ; ::_thesis: (Col (B,i)) . k = ((len B) |-> (0. K)) . k
k in NAT by ORDINAL1:def_12;
then A10: k in Seg (len B) by A9;
Indices B = [:(Seg (len B)),(Seg (width B)):] by FINSEQ_1:def_3;
then [k,i] in Indices B by A2, A4, A6, A10, ZFMISC_1:87;
then A11: B * (k,i) = (Line (A,k)) "*" (Col (X1,i)) by A5, A7, MATRIX_3:def_4
.= Sum ((0. K) * (Line (A,k))) by A3, A4, A5, FVSUM_1:66
.= (0. K) * (Sum (Line (A,k))) by FVSUM_1:73
.= 0. K by VECTSP_1:6
.= ((len B) |-> (0. K)) . k by A10, FINSEQ_2:57 ;
k in dom B by A10, FINSEQ_1:def_3;
hence (Col (B,i)) . k = ((len B) |-> (0. K)) . k by A11, MATRIX_1:def_8; ::_thesis: verum
end;
( len (Col (B,i)) = len B & len ((len B) |-> (0. K)) = len B ) by CARD_1:def_7;
hence Col (B,i) = (len B) |-> (0. K) by A8, FINSEQ_1:14; ::_thesis: verum
end;
theorem Th35: :: MATRIX15:35
for K being Field
for a being Element of K
for X, A, B being Matrix of K st X in Solutions_of (A,B) holds
( a * X in Solutions_of (A,(a * B)) & X in Solutions_of ((a * A),(a * B)) )
proof
let K be Field; ::_thesis: for a being Element of K
for X, A, B being Matrix of K st X in Solutions_of (A,B) holds
( a * X in Solutions_of (A,(a * B)) & X in Solutions_of ((a * A),(a * B)) )
let a be Element of K; ::_thesis: for X, A, B being Matrix of K st X in Solutions_of (A,B) holds
( a * X in Solutions_of (A,(a * B)) & X in Solutions_of ((a * A),(a * B)) )
let X, A, B be Matrix of K; ::_thesis: ( X in Solutions_of (A,B) implies ( a * X in Solutions_of (A,(a * B)) & X in Solutions_of ((a * A),(a * B)) ) )
A1: ( width (a * B) = width B & width (a * A) = width A ) by MATRIX_3:def_5;
assume X in Solutions_of (A,B) ; ::_thesis: ( a * X in Solutions_of (A,(a * B)) & X in Solutions_of ((a * A),(a * B)) )
then consider X1 being Matrix of K such that
A2: ( X = X1 & len X1 = width A ) and
A3: ( width X1 = width B & A * X1 = B ) ;
A4: ( len (a * X) = width A & width (a * X) = width X1 ) by A2, MATRIX_3:def_5;
( A * (a * X) = a * (A * X) & (a * A) * X = a * (A * X) ) by A2, Th1, MATRIXR1:22;
hence ( a * X in Solutions_of (A,(a * B)) & X in Solutions_of ((a * A),(a * B)) ) by A2, A3, A4, A1; ::_thesis: verum
end;
theorem :: MATRIX15:36
for K being Field
for a being Element of K
for A, B being Matrix of K st a <> 0. K holds
Solutions_of (A,B) = Solutions_of ((a * A),(a * B))
proof
let K be Field; ::_thesis: for a being Element of K
for A, B being Matrix of K st a <> 0. K holds
Solutions_of (A,B) = Solutions_of ((a * A),(a * B))
let a be Element of K; ::_thesis: for A, B being Matrix of K st a <> 0. K holds
Solutions_of (A,B) = Solutions_of ((a * A),(a * B))
let A, B be Matrix of K; ::_thesis: ( a <> 0. K implies Solutions_of (A,B) = Solutions_of ((a * A),(a * B)) )
assume A1: a <> 0. K ; ::_thesis: Solutions_of (A,B) = Solutions_of ((a * A),(a * B))
thus Solutions_of (A,B) c= Solutions_of ((a * A),(a * B)) :: according to XBOOLE_0:def_10 ::_thesis: Solutions_of ((a * A),(a * B)) c= Solutions_of (A,B)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Solutions_of (A,B) or x in Solutions_of ((a * A),(a * B)) )
assume A2: x in Solutions_of (A,B) ; ::_thesis: x in Solutions_of ((a * A),(a * B))
ex X being Matrix of K st
( x = X & len X = width A & width X = width B & A * X = B ) by A2;
hence x in Solutions_of ((a * A),(a * B)) by A2, Th35; ::_thesis: verum
end;
A3: (a ") * (a * A) = ((a ") * a) * A by Th2
.= (1_ K) * A by A1, VECTSP_1:def_10
.= A by Th2 ;
A4: (a ") * (a * B) = ((a ") * a) * B by Th2
.= (1_ K) * B by A1, VECTSP_1:def_10
.= B by Th2 ;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Solutions_of ((a * A),(a * B)) or x in Solutions_of (A,B) )
assume A5: x in Solutions_of ((a * A),(a * B)) ; ::_thesis: x in Solutions_of (A,B)
ex X being Matrix of K st
( x = X & len X = width (a * A) & width X = width (a * B) & (a * A) * X = a * B ) by A5;
hence x in Solutions_of (A,B) by A5, A3, A4, Th35; ::_thesis: verum
end;
Lm3: for D being non empty set
for A, B being Matrix of D st len A = len B & width A = 0 & width B = 0 holds
A = B
proof
let D be non empty set ; ::_thesis: for A, B being Matrix of D st len A = len B & width A = 0 & width B = 0 holds
A = B
let A, B be Matrix of D; ::_thesis: ( len A = len B & width A = 0 & width B = 0 implies A = B )
assume that
A1: len A = len B and
A2: width A = 0 and
A3: width B = 0 ; ::_thesis: A = B
Seg (width A) = {} by A2;
then Indices A = {} by ZFMISC_1:90;
then for i, j being Nat st [i,j] in Indices A holds
A * (i,j) = B * (i,j) ;
hence A = B by A1, A2, A3, MATRIX_1:21; ::_thesis: verum
end;
theorem Th37: :: MATRIX15:37
for K being Field
for X1, A, B1, X2, B2 being Matrix of K st X1 in Solutions_of (A,B1) & X2 in Solutions_of (A,B2) & width B1 = width B2 holds
X1 + X2 in Solutions_of (A,(B1 + B2))
proof
let K be Field; ::_thesis: for X1, A, B1, X2, B2 being Matrix of K st X1 in Solutions_of (A,B1) & X2 in Solutions_of (A,B2) & width B1 = width B2 holds
X1 + X2 in Solutions_of (A,(B1 + B2))
let X1, A, B1, X2, B2 be Matrix of K; ::_thesis: ( X1 in Solutions_of (A,B1) & X2 in Solutions_of (A,B2) & width B1 = width B2 implies X1 + X2 in Solutions_of (A,(B1 + B2)) )
assume that
A1: X1 in Solutions_of (A,B1) and
A2: X2 in Solutions_of (A,B2) and
A3: width B1 = width B2 ; ::_thesis: X1 + X2 in Solutions_of (A,(B1 + B2))
A4: ex Y1 being Matrix of K st
( Y1 = X1 & len Y1 = width A & width Y1 = width B1 & A * Y1 = B1 ) by A1;
A5: width (X1 + X2) = width X1 by MATRIX_3:def_3;
A6: len (X1 + X2) = len X1 by MATRIX_3:def_3;
A7: ex Y2 being Matrix of K st
( Y2 = X2 & len Y2 = width A & width Y2 = width B2 & A * Y2 = B2 ) by A2;
A8: now__::_thesis:_A_*_(X1_+_X2)_=_(A_*_X1)_+_(A_*_X2)
percases ( len A = 0 or len X1 = 0 or ( len A > 0 & len X1 > 0 ) ) ;
supposeA9: len A = 0 ; ::_thesis: A * (X1 + X2) = (A * X1) + (A * X2)
then len (A * X1) = 0 by A4, MATRIX_3:def_4;
then A10: len ((A * X1) + (A * X2)) = 0 by MATRIX_3:def_3;
len (A * (X1 + X2)) = 0 by A4, A6, A9, MATRIX_3:def_4;
hence A * (X1 + X2) = (A * X1) + (A * X2) by A10, CARD_2:64; ::_thesis: verum
end;
suppose len X1 = 0 ; ::_thesis: A * (X1 + X2) = (A * X1) + (A * X2)
then width (A * X1) = 0 by A4, MATRIX_1:def_3;
then A11: ( width ((A * X1) + (A * X2)) = 0 & width (A * (X1 + X2)) = 0 ) by A4, A6, A5, MATRIX_3:def_3, MATRIX_3:def_4;
len (A * X1) = len A by A4, MATRIX_3:def_4;
then A12: len ((A * X1) + (A * X2)) = len A by MATRIX_3:def_3;
len (A * (X1 + X2)) = len A by A4, A6, MATRIX_3:def_4;
hence A * (X1 + X2) = (A * X1) + (A * X2) by A12, A11, Lm3; ::_thesis: verum
end;
suppose ( len A > 0 & len X1 > 0 ) ; ::_thesis: A * (X1 + X2) = (A * X1) + (A * X2)
hence A * (X1 + X2) = (A * X1) + (A * X2) by A3, A4, A7, MATRIX_4:62; ::_thesis: verum
end;
end;
end;
width (B1 + B2) = width B1 by MATRIX_3:def_3;
hence X1 + X2 in Solutions_of (A,(B1 + B2)) by A4, A7, A6, A5, A8; ::_thesis: verum
end;
theorem Th38: :: MATRIX15:38
for n, m, k, i being Nat
for K being Field
for a being Element of K
for X being Matrix of K
for A9 being Matrix of m,n,K
for B9 being Matrix of m,k,K st X in Solutions_of (A9,B9) holds
X in Solutions_of ((RLine (A9,i,(a * (Line (A9,i))))),(RLine (B9,i,(a * (Line (B9,i))))))
proof
let n, m, k, i be Nat; ::_thesis: for K being Field
for a being Element of K
for X being Matrix of K
for A9 being Matrix of m,n,K
for B9 being Matrix of m,k,K st X in Solutions_of (A9,B9) holds
X in Solutions_of ((RLine (A9,i,(a * (Line (A9,i))))),(RLine (B9,i,(a * (Line (B9,i))))))
let K be Field; ::_thesis: for a being Element of K
for X being Matrix of K
for A9 being Matrix of m,n,K
for B9 being Matrix of m,k,K st X in Solutions_of (A9,B9) holds
X in Solutions_of ((RLine (A9,i,(a * (Line (A9,i))))),(RLine (B9,i,(a * (Line (B9,i))))))
let a be Element of K; ::_thesis: for X being Matrix of K
for A9 being Matrix of m,n,K
for B9 being Matrix of m,k,K st X in Solutions_of (A9,B9) holds
X in Solutions_of ((RLine (A9,i,(a * (Line (A9,i))))),(RLine (B9,i,(a * (Line (B9,i))))))
let X be Matrix of K; ::_thesis: for A9 being Matrix of m,n,K
for B9 being Matrix of m,k,K st X in Solutions_of (A9,B9) holds
X in Solutions_of ((RLine (A9,i,(a * (Line (A9,i))))),(RLine (B9,i,(a * (Line (B9,i))))))
let A9 be Matrix of m,n,K; ::_thesis: for B9 being Matrix of m,k,K st X in Solutions_of (A9,B9) holds
X in Solutions_of ((RLine (A9,i,(a * (Line (A9,i))))),(RLine (B9,i,(a * (Line (B9,i))))))
let B9 be Matrix of m,k,K; ::_thesis: ( X in Solutions_of (A9,B9) implies X in Solutions_of ((RLine (A9,i,(a * (Line (A9,i))))),(RLine (B9,i,(a * (Line (B9,i)))))) )
set LA = Line (A9,i);
set LB = Line (B9,i);
set RA = RLine (A9,i,(a * (Line (A9,i))));
set RB = RLine (B9,i,(a * (Line (B9,i))));
A1: Indices (RLine (B9,i,(a * (Line (B9,i))))) = Indices B9 by MATRIX_1:26;
A2: ( len (a * (Line (B9,i))) = len (Line (B9,i)) & len (Line (B9,i)) = width B9 ) by CARD_1:def_7, MATRIXR1:16;
then A3: width (RLine (B9,i,(a * (Line (B9,i))))) = width B9 by MATRIX11:def_3;
A4: ( len (a * (Line (A9,i))) = len (Line (A9,i)) & len (Line (A9,i)) = width A9 ) by CARD_1:def_7, MATRIXR1:16;
then A5: len (RLine (A9,i,(a * (Line (A9,i))))) = len A9 by MATRIX11:def_3;
assume A6: X in Solutions_of (A9,B9) ; ::_thesis: X in Solutions_of ((RLine (A9,i,(a * (Line (A9,i))))),(RLine (B9,i,(a * (Line (B9,i))))))
then consider X1 being Matrix of K such that
A7: X = X1 and
A8: len X1 = width A9 and
A9: width X1 = width B9 and
A10: A9 * X1 = B9 ;
set RX = (RLine (A9,i,(a * (Line (A9,i))))) * X1;
A11: width (RLine (A9,i,(a * (Line (A9,i))))) = width A9 by A4, MATRIX11:def_3;
then A12: ( len ((RLine (A9,i,(a * (Line (A9,i))))) * X1) = len (RLine (A9,i,(a * (Line (A9,i))))) & width ((RLine (A9,i,(a * (Line (A9,i))))) * X1) = width X1 ) by A8, MATRIX_3:def_4;
A13: len A9 = len B9 by A6, Th33;
then dom B9 = Seg (len (RLine (A9,i,(a * (Line (A9,i)))))) by A5, FINSEQ_1:def_3;
then A14: Indices ((RLine (A9,i,(a * (Line (A9,i))))) * X1) = Indices B9 by A9, A12, FINSEQ_1:def_3;
A15: now__::_thesis:_for_j,_k_being_Nat_st_[j,k]_in_Indices_(RLine_(B9,i,(a_*_(Line_(B9,i)))))_holds_
(RLine_(B9,i,(a_*_(Line_(B9,i)))))_*_(j,k)_=_((RLine_(A9,i,(a_*_(Line_(A9,i)))))_*_X1)_*_(j,k)
len B9 = m by MATRIX_1:def_2;
then A16: dom B9 = Seg m by FINSEQ_1:def_3;
let j, k be Nat; ::_thesis: ( [j,k] in Indices (RLine (B9,i,(a * (Line (B9,i))))) implies (RLine (B9,i,(a * (Line (B9,i))))) * (j,k) = ((RLine (A9,i,(a * (Line (A9,i))))) * X1) * (j,k) )
assume A17: [j,k] in Indices (RLine (B9,i,(a * (Line (B9,i))))) ; ::_thesis: (RLine (B9,i,(a * (Line (B9,i))))) * (j,k) = ((RLine (A9,i,(a * (Line (A9,i))))) * X1) * (j,k)
A18: j in dom B9 by A1, A17, ZFMISC_1:87;
A19: k in Seg (width B9) by A1, A17, ZFMISC_1:87;
then B9 * (i,k) = (Line (B9,i)) . k by MATRIX_1:def_7;
then reconsider LBk = (Line (B9,i)) . k as Element of K ;
A20: B9 * (j,k) = (Line (A9,j)) "*" (Col (X1,k)) by A8, A10, A1, A17, MATRIX_3:def_4;
now__::_thesis:_((RLine_(A9,i,(a_*_(Line_(A9,i)))))_*_X1)_*_(j,k)_=_(RLine_(B9,i,(a_*_(Line_(B9,i)))))_*_(j,k)
percases ( j = i or j <> i ) ;
supposeA21: j = i ; ::_thesis: ((RLine (A9,i,(a * (Line (A9,i))))) * X1) * (j,k) = (RLine (B9,i,(a * (Line (B9,i))))) * (j,k)
then Line ((RLine (A9,i,(a * (Line (A9,i))))),i) = a * (Line (A9,i)) by A4, A18, A16, MATRIX11:28;
hence ((RLine (A9,i,(a * (Line (A9,i))))) * X1) * (j,k) = (a * (Line (A9,i))) "*" (Col (X1,k)) by A8, A11, A14, A1, A17, A21, MATRIX_3:def_4
.= Sum (a * (mlt ((Line (A9,i)),(Col (X1,k))))) by A8, FVSUM_1:68
.= a * (Sum (mlt ((Line (A9,i)),(Col (X1,k))))) by FVSUM_1:73
.= a * LBk by A19, A20, A21, MATRIX_1:def_7
.= (a * (Line (B9,i))) . k by A19, FVSUM_1:51
.= (RLine (B9,i,(a * (Line (B9,i))))) * (j,k) by A2, A1, A17, A21, MATRIX11:def_3 ;
::_thesis: verum
end;
supposeA22: j <> i ; ::_thesis: ((RLine (A9,i,(a * (Line (A9,i))))) * X1) * (j,k) = (RLine (B9,i,(a * (Line (B9,i))))) * (j,k)
then Line ((RLine (A9,i,(a * (Line (A9,i))))),j) = Line (A9,j) by A18, A16, MATRIX11:28;
hence ((RLine (A9,i,(a * (Line (A9,i))))) * X1) * (j,k) = (Line (A9,j)) "*" (Col (X1,k)) by A8, A11, A14, A1, A17, MATRIX_3:def_4
.= B9 * (j,k) by A8, A10, A1, A17, MATRIX_3:def_4
.= (RLine (B9,i,(a * (Line (B9,i))))) * (j,k) by A2, A1, A17, A22, MATRIX11:def_3 ;
::_thesis: verum
end;
end;
end;
hence (RLine (B9,i,(a * (Line (B9,i))))) * (j,k) = ((RLine (A9,i,(a * (Line (A9,i))))) * X1) * (j,k) ; ::_thesis: verum
end;
len (RLine (B9,i,(a * (Line (B9,i))))) = len B9 by A2, MATRIX11:def_3;
then (RLine (A9,i,(a * (Line (A9,i))))) * X1 = RLine (B9,i,(a * (Line (B9,i)))) by A9, A13, A5, A3, A12, A15, MATRIX_1:21;
hence X in Solutions_of ((RLine (A9,i,(a * (Line (A9,i))))),(RLine (B9,i,(a * (Line (B9,i)))))) by A7, A8, A9, A11, A3; ::_thesis: verum
end;
Lm4: for n, m, k, i being Nat
for K being Field
for a being Element of K
for A9 being Matrix of m,n,K
for B9 being Matrix of m,k,K st a <> 0. K holds
Solutions_of (A9,B9) = Solutions_of ((RLine (A9,i,(a * (Line (A9,i))))),(RLine (B9,i,(a * (Line (B9,i))))))
proof
let n, m, k, i be Nat; ::_thesis: for K being Field
for a being Element of K
for A9 being Matrix of m,n,K
for B9 being Matrix of m,k,K st a <> 0. K holds
Solutions_of (A9,B9) = Solutions_of ((RLine (A9,i,(a * (Line (A9,i))))),(RLine (B9,i,(a * (Line (B9,i))))))
let K be Field; ::_thesis: for a being Element of K
for A9 being Matrix of m,n,K
for B9 being Matrix of m,k,K st a <> 0. K holds
Solutions_of (A9,B9) = Solutions_of ((RLine (A9,i,(a * (Line (A9,i))))),(RLine (B9,i,(a * (Line (B9,i))))))
let a be Element of K; ::_thesis: for A9 being Matrix of m,n,K
for B9 being Matrix of m,k,K st a <> 0. K holds
Solutions_of (A9,B9) = Solutions_of ((RLine (A9,i,(a * (Line (A9,i))))),(RLine (B9,i,(a * (Line (B9,i))))))
let A9 be Matrix of m,n,K; ::_thesis: for B9 being Matrix of m,k,K st a <> 0. K holds
Solutions_of (A9,B9) = Solutions_of ((RLine (A9,i,(a * (Line (A9,i))))),(RLine (B9,i,(a * (Line (B9,i))))))
let B9 be Matrix of m,k,K; ::_thesis: ( a <> 0. K implies Solutions_of (A9,B9) = Solutions_of ((RLine (A9,i,(a * (Line (A9,i))))),(RLine (B9,i,(a * (Line (B9,i)))))) )
assume A1: a <> 0. K ; ::_thesis: Solutions_of (A9,B9) = Solutions_of ((RLine (A9,i,(a * (Line (A9,i))))),(RLine (B9,i,(a * (Line (B9,i))))))
set RB = RLine (B9,i,(a * (Line (B9,i))));
set RA = RLine (A9,i,(a * (Line (A9,i))));
thus Solutions_of (A9,B9) c= Solutions_of ((RLine (A9,i,(a * (Line (A9,i))))),(RLine (B9,i,(a * (Line (B9,i)))))) :: according to XBOOLE_0:def_10 ::_thesis: Solutions_of ((RLine (A9,i,(a * (Line (A9,i))))),(RLine (B9,i,(a * (Line (B9,i)))))) c= Solutions_of (A9,B9)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Solutions_of (A9,B9) or x in Solutions_of ((RLine (A9,i,(a * (Line (A9,i))))),(RLine (B9,i,(a * (Line (B9,i)))))) )
assume A2: x in Solutions_of (A9,B9) ; ::_thesis: x in Solutions_of ((RLine (A9,i,(a * (Line (A9,i))))),(RLine (B9,i,(a * (Line (B9,i))))))
ex X being Matrix of K st
( x = X & len X = width A9 & width X = width B9 & A9 * X = B9 ) by A2;
hence x in Solutions_of ((RLine (A9,i,(a * (Line (A9,i))))),(RLine (B9,i,(a * (Line (B9,i)))))) by A2, Th38; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Solutions_of ((RLine (A9,i,(a * (Line (A9,i))))),(RLine (B9,i,(a * (Line (B9,i)))))) or x in Solutions_of (A9,B9) )
assume A3: x in Solutions_of ((RLine (A9,i,(a * (Line (A9,i))))),(RLine (B9,i,(a * (Line (B9,i)))))) ; ::_thesis: x in Solutions_of (A9,B9)
percases ( not i in Seg m or i in Seg m ) ;
supposeA4: not i in Seg m ; ::_thesis: x in Solutions_of (A9,B9)
len A9 = m by MATRIX_1:def_2;
then ( len B9 = m & RLine (A9,i,(a * (Line (A9,i)))) = A9 ) by A4, MATRIX13:40, MATRIX_1:def_2;
hence x in Solutions_of (A9,B9) by A3, A4, MATRIX13:40; ::_thesis: verum
end;
supposeA5: i in Seg m ; ::_thesis: x in Solutions_of (A9,B9)
reconsider aLA = a * (Line (A9,i)), aLB = a * (Line (B9,i)), aLAR = (a ") * (Line ((RLine (A9,i,(a * (Line (A9,i))))),i)), aLBR = (a ") * (Line ((RLine (B9,i,(a * (Line (B9,i))))),i)) as Element of the carrier of K * by FINSEQ_1:def_11;
set RRB = RLine ((RLine (B9,i,(a * (Line (B9,i))))),i,((a ") * (Line ((RLine (B9,i,(a * (Line (B9,i))))),i))));
set RRA = RLine ((RLine (A9,i,(a * (Line (A9,i))))),i,((a ") * (Line ((RLine (A9,i,(a * (Line (A9,i))))),i))));
A6: ex X being Matrix of K st
( x = X & len X = width (RLine (A9,i,(a * (Line (A9,i))))) & width X = width (RLine (B9,i,(a * (Line (B9,i))))) & (RLine (A9,i,(a * (Line (A9,i))))) * X = RLine (B9,i,(a * (Line (B9,i)))) ) by A3;
A7: len (a * (Line (A9,i))) = width A9 by CARD_1:def_7;
then A8: (a ") * (Line ((RLine (A9,i,(a * (Line (A9,i))))),i)) = (a ") * (a * (Line (A9,i))) by A5, MATRIX11:28
.= ((a ") * a) * (Line (A9,i)) by FVSUM_1:54
.= (1_ K) * (Line (A9,i)) by A1, VECTSP_1:def_10
.= Line (A9,i) by FVSUM_1:57 ;
A9: len (a * (Line (B9,i))) = width B9 by CARD_1:def_7;
then A10: (a ") * (Line ((RLine (B9,i,(a * (Line (B9,i))))),i)) = (a ") * (a * (Line (B9,i))) by A5, MATRIX11:28
.= ((a ") * a) * (Line (B9,i)) by FVSUM_1:54
.= (1_ K) * (Line (B9,i)) by A1, VECTSP_1:def_10
.= Line (B9,i) by FVSUM_1:57 ;
A11: width (RLine (B9,i,(a * (Line (B9,i))))) = width B9 by A9, MATRIX11:def_3;
A12: len ((a ") * (Line ((RLine (B9,i,(a * (Line (B9,i))))),i))) = width (RLine (B9,i,(a * (Line (B9,i))))) by CARD_1:def_7;
then A13: RLine ((RLine (B9,i,(a * (Line (B9,i))))),i,((a ") * (Line ((RLine (B9,i,(a * (Line (B9,i))))),i)))) = Replace ((RLine (B9,i,(a * (Line (B9,i))))),i,aLBR) by MATRIX11:29
.= Replace ((Replace (B9,i,aLB)),i,aLBR) by A9, MATRIX11:29
.= Replace (B9,i,aLBR) by FUNCT_7:34
.= RLine (B9,i,(Line (B9,i))) by A12, A11, A10, MATRIX11:29
.= B9 by MATRIX11:30 ;
A14: width (RLine (A9,i,(a * (Line (A9,i))))) = width A9 by A7, MATRIX11:def_3;
A15: len ((a ") * (Line ((RLine (A9,i,(a * (Line (A9,i))))),i))) = width (RLine (A9,i,(a * (Line (A9,i))))) by CARD_1:def_7;
then RLine ((RLine (A9,i,(a * (Line (A9,i))))),i,((a ") * (Line ((RLine (A9,i,(a * (Line (A9,i))))),i)))) = Replace ((RLine (A9,i,(a * (Line (A9,i))))),i,aLAR) by MATRIX11:29
.= Replace ((Replace (A9,i,aLA)),i,aLAR) by A7, MATRIX11:29
.= Replace (A9,i,aLAR) by FUNCT_7:34
.= RLine (A9,i,(Line (A9,i))) by A15, A14, A8, MATRIX11:29
.= A9 by MATRIX11:30 ;
hence x in Solutions_of (A9,B9) by A3, A6, A13, Th38; ::_thesis: verum
end;
end;
end;
theorem Th39: :: MATRIX15:39
for n, k, j, m, i being Nat
for K being Field
for a being Element of K
for X being Matrix of K
for A9 being Matrix of m,n,K
for B9 being Matrix of m,k,K st X in Solutions_of (A9,B9) & j in Seg m holds
X in Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),(RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))))
proof
let n, k, j, m, i be Nat; ::_thesis: for K being Field
for a being Element of K
for X being Matrix of K
for A9 being Matrix of m,n,K
for B9 being Matrix of m,k,K st X in Solutions_of (A9,B9) & j in Seg m holds
X in Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),(RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))))
let K be Field; ::_thesis: for a being Element of K
for X being Matrix of K
for A9 being Matrix of m,n,K
for B9 being Matrix of m,k,K st X in Solutions_of (A9,B9) & j in Seg m holds
X in Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),(RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))))
let a be Element of K; ::_thesis: for X being Matrix of K
for A9 being Matrix of m,n,K
for B9 being Matrix of m,k,K st X in Solutions_of (A9,B9) & j in Seg m holds
X in Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),(RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))))
let X be Matrix of K; ::_thesis: for A9 being Matrix of m,n,K
for B9 being Matrix of m,k,K st X in Solutions_of (A9,B9) & j in Seg m holds
X in Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),(RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))))
let A9 be Matrix of m,n,K; ::_thesis: for B9 being Matrix of m,k,K st X in Solutions_of (A9,B9) & j in Seg m holds
X in Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),(RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))))
let B9 be Matrix of m,k,K; ::_thesis: ( X in Solutions_of (A9,B9) & j in Seg m implies X in Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),(RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j))))))) )
assume that
A1: X in Solutions_of (A9,B9) and
A2: j in Seg m ; ::_thesis: X in Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),(RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))))
consider X1 being Matrix of K such that
A3: X = X1 and
A4: len X1 = width A9 and
A5: width X1 = width B9 and
A6: A9 * X1 = B9 by A1;
set LAj = Line (A9,j);
set LAi = Line (A9,i);
set RA = RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))));
A7: len ((Line (A9,i)) + (a * (Line (A9,j)))) = width A9 by CARD_1:def_7;
then A8: len (RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))) = len A9 by MATRIX11:def_3;
set RX = (RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))) * X1;
A9: width (RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))) = width A9 by A7, MATRIX11:def_3;
then A10: ( len ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))) * X1) = len (RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))) & width ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))) * X1) = width X1 ) by A4, MATRIX_3:def_4;
A11: len A9 = len B9 by A1, Th33;
then dom B9 = Seg (len (RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j))))))) by A8, FINSEQ_1:def_3;
then A12: Indices ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))) * X1) = Indices B9 by A5, A10, FINSEQ_1:def_3;
set LBj = Line (B9,j);
set LBi = Line (B9,i);
set RB = RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))));
A13: Indices (RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))) = Indices B9 by MATRIX_1:26;
A14: len ((Line (B9,i)) + (a * (Line (B9,j)))) = width B9 by CARD_1:def_7;
then A15: width (RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))) = width B9 by MATRIX11:def_3;
A16: ( len (a * (Line (A9,j))) = width A9 & len (Line (A9,i)) = width A9 ) by CARD_1:def_7;
A17: now__::_thesis:_for_o,_p_being_Nat_st_[o,p]_in_Indices_(RLine_(B9,i,((Line_(B9,i))_+_(a_*_(Line_(B9,j))))))_holds_
(RLine_(B9,i,((Line_(B9,i))_+_(a_*_(Line_(B9,j))))))_*_(o,p)_=_((RLine_(A9,i,((Line_(A9,i))_+_(a_*_(Line_(A9,j))))))_*_X1)_*_(o,p)
A18: rng (a * (Line (B9,j))) c= the carrier of K by FINSEQ_1:def_4;
let o, p be Nat; ::_thesis: ( [o,p] in Indices (RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))) implies (RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))) * (o,p) = ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))) * X1) * (o,p) )
assume A19: [o,p] in Indices (RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))) ; ::_thesis: (RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))) * (o,p) = ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))) * X1) * (o,p)
A20: o in dom B9 by A13, A19, ZFMISC_1:87;
A21: B9 * (o,p) = (Line (A9,o)) "*" (Col (X1,p)) by A4, A6, A13, A19, MATRIX_3:def_4;
reconsider CX = Col (X1,p) as Element of (width A9) -tuples_on the carrier of K by A4;
A22: len (Col (X1,p)) = width A9 by A4, MATRIX_1:def_8;
A23: p in Seg (width B9) by A13, A19, ZFMISC_1:87;
then ( B9 * (o,p) = (Line (B9,o)) . p & B9 * (j,p) = (Line (B9,j)) . p ) by MATRIX_1:def_7;
then reconsider LBop = (Line (B9,o)) . p, LBjp = (Line (B9,j)) . p as Element of the carrier of K ;
p in dom (a * (Line (B9,j))) by A23, FINSEQ_2:124;
then (a * (Line (B9,j))) . p in rng (a * (Line (B9,j))) by FUNCT_1:def_3;
then reconsider aLBjp = (a * (Line (B9,j))) . p as Element of K by A18;
len B9 = m by MATRIX_1:def_2;
then A24: dom B9 = Seg m by FINSEQ_1:def_3;
then [j,p] in Indices B9 by A2, A23, ZFMISC_1:87;
then A25: B9 * (j,p) = (Line (A9,j)) "*" (Col (X1,p)) by A4, A6, MATRIX_3:def_4;
now__::_thesis:_((RLine_(A9,i,((Line_(A9,i))_+_(a_*_(Line_(A9,j))))))_*_X1)_*_(o,p)_=_(RLine_(B9,i,((Line_(B9,i))_+_(a_*_(Line_(B9,j))))))_*_(o,p)
percases ( o = i or o <> i ) ;
supposeA26: o = i ; ::_thesis: ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))) * X1) * (o,p) = (RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))) * (o,p)
then Line ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),o) = (Line (A9,i)) + (a * (Line (A9,j))) by A7, A20, A24, MATRIX11:28;
hence ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))) * X1) * (o,p) = ((Line (A9,i)) + (a * (Line (A9,j)))) "*" CX by A4, A9, A12, A13, A19, MATRIX_3:def_4
.= Sum ((mlt ((Line (A9,i)),CX)) + (mlt ((a * (Line (A9,j))),CX))) by A16, A22, MATRIX_4:56
.= Sum ((mlt ((Line (A9,i)),CX)) + (a * (mlt ((Line (A9,j)),CX)))) by FVSUM_1:68
.= (Sum (mlt ((Line (A9,i)),CX))) + (Sum (a * (mlt ((Line (A9,j)),CX)))) by FVSUM_1:76
.= (B9 * (o,p)) + (a * (B9 * (j,p))) by A21, A25, A26, FVSUM_1:73
.= LBop + (a * (B9 * (j,p))) by A23, MATRIX_1:def_7
.= LBop + (a * LBjp) by A23, MATRIX_1:def_7
.= LBop + aLBjp by A23, FVSUM_1:51
.= ((Line (B9,i)) + (a * (Line (B9,j)))) . p by A23, A26, FVSUM_1:18
.= (RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))) * (o,p) by A14, A13, A19, A26, MATRIX11:def_3 ;
::_thesis: verum
end;
supposeA27: o <> i ; ::_thesis: ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))) * X1) * (o,p) = (RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))) * (o,p)
then Line ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),o) = Line (A9,o) by A20, A24, MATRIX11:28;
hence ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))) * X1) * (o,p) = (Line (A9,o)) "*" (Col (X1,p)) by A4, A9, A12, A13, A19, MATRIX_3:def_4
.= B9 * (o,p) by A4, A6, A13, A19, MATRIX_3:def_4
.= (RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))) * (o,p) by A14, A13, A19, A27, MATRIX11:def_3 ;
::_thesis: verum
end;
end;
end;
hence (RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))) * (o,p) = ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))) * X1) * (o,p) ; ::_thesis: verum
end;
len (RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))) = len B9 by A14, MATRIX11:def_3;
then (RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))) * X1 = RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j))))) by A5, A11, A8, A15, A10, A17, MATRIX_1:21;
hence X in Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),(RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j))))))) by A3, A4, A5, A9, A15; ::_thesis: verum
end;
Lm5: for n, k, j, m, i being Nat
for K being Field
for a being Element of K
for A9 being Matrix of m,n,K
for B9 being Matrix of m,k,K st j in Seg m & i <> j holds
Solutions_of (A9,B9) = Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),(RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))))
proof
let n, k, j, m, i be Nat; ::_thesis: for K being Field
for a being Element of K
for A9 being Matrix of m,n,K
for B9 being Matrix of m,k,K st j in Seg m & i <> j holds
Solutions_of (A9,B9) = Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),(RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))))
let K be Field; ::_thesis: for a being Element of K
for A9 being Matrix of m,n,K
for B9 being Matrix of m,k,K st j in Seg m & i <> j holds
Solutions_of (A9,B9) = Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),(RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))))
let a be Element of K; ::_thesis: for A9 being Matrix of m,n,K
for B9 being Matrix of m,k,K st j in Seg m & i <> j holds
Solutions_of (A9,B9) = Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),(RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))))
let A9 be Matrix of m,n,K; ::_thesis: for B9 being Matrix of m,k,K st j in Seg m & i <> j holds
Solutions_of (A9,B9) = Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),(RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))))
let B9 be Matrix of m,k,K; ::_thesis: ( j in Seg m & i <> j implies Solutions_of (A9,B9) = Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),(RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j))))))) )
assume that
A1: j in Seg m and
A2: i <> j ; ::_thesis: Solutions_of (A9,B9) = Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),(RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))))
set LB = Line (B9,j);
set LA = Line (A9,j);
set RA = RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))));
set RB = RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))));
thus Solutions_of (A9,B9) c= Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),(RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j))))))) :: according to XBOOLE_0:def_10 ::_thesis: Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),(RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j))))))) c= Solutions_of (A9,B9)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Solutions_of (A9,B9) or x in Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),(RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j))))))) )
assume A3: x in Solutions_of (A9,B9) ; ::_thesis: x in Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),(RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))))
ex X being Matrix of K st
( x = X & len X = width A9 & width X = width B9 & A9 * X = B9 ) by A3;
hence x in Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),(RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j))))))) by A1, A3, Th39; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),(RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j))))))) or x in Solutions_of (A9,B9) )
assume A4: x in Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),(RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j))))))) ; ::_thesis: x in Solutions_of (A9,B9)
percases ( not i in Seg m or i in Seg m ) ;
supposeA5: not i in Seg m ; ::_thesis: x in Solutions_of (A9,B9)
len A9 = m by MATRIX_1:def_2;
then ( len B9 = m & RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j))))) = A9 ) by A5, MATRIX13:40, MATRIX_1:def_2;
hence x in Solutions_of (A9,B9) by A4, A5, MATRIX13:40; ::_thesis: verum
end;
supposeA6: i in Seg m ; ::_thesis: x in Solutions_of (A9,B9)
reconsider LLA = (Line (A9,i)) + (a * (Line (A9,j))), LLB = (Line (B9,i)) + (a * (Line (B9,j))), LLRA = (Line ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),i)) + ((- a) * (Line ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),j))), LLRB = (Line ((RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))),i)) + ((- a) * (Line ((RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))),j))) as Element of the carrier of K * by FINSEQ_1:def_11;
set RRB = RLine ((RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))),i,((Line ((RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))),i)) + ((- a) * (Line ((RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))),j)))));
set RRA = RLine ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),i,((Line ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),i)) + ((- a) * (Line ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),j)))));
A7: ex X being Matrix of K st
( x = X & len X = width (RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))) & width X = width (RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))) & (RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))) * X = RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j))))) ) by A4;
A8: Line ((RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))),j) = Line (B9,j) by A1, A2, MATRIX11:28;
A9: len ((Line (B9,i)) + (a * (Line (B9,j)))) = width B9 by CARD_1:def_7;
then A10: width (RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))) = width B9 by MATRIX11:def_3;
Line ((RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))),i) = (Line (B9,i)) + (a * (Line (B9,j))) by A6, A9, MATRIX11:28;
then A11: (Line ((RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))),i)) + ((- a) * (Line ((RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))),j))) = ((Line (B9,i)) + (a * (Line (B9,j)))) + ((- ((1_ K) * a)) * (Line (B9,j))) by A8, VECTSP_1:def_8
.= ((Line (B9,i)) + (a * (Line (B9,j)))) + (((- (1_ K)) * a) * (Line (B9,j))) by VECTSP_1:9
.= ((Line (B9,i)) + (a * (Line (B9,j)))) + ((- (1_ K)) * (a * (Line (B9,j)))) by FVSUM_1:54
.= ((Line (B9,i)) + (a * (Line (B9,j)))) + (- (a * (Line (B9,j)))) by FVSUM_1:59
.= (Line (B9,i)) + ((a * (Line (B9,j))) + (- (a * (Line (B9,j))))) by FINSEQOP:28
.= (Line (B9,i)) + ((width B9) |-> (0. K)) by FVSUM_1:26
.= Line (B9,i) by FVSUM_1:21 ;
A12: len ((Line ((RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))),i)) + ((- a) * (Line ((RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))),j)))) = width (RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))) by CARD_1:def_7;
then A13: RLine ((RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))),i,((Line ((RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))),i)) + ((- a) * (Line ((RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))),j))))) = Replace ((RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))),i,LLRB) by MATRIX11:29
.= Replace ((Replace (B9,i,LLB)),i,LLRB) by A9, MATRIX11:29
.= Replace (B9,i,LLRB) by FUNCT_7:34
.= RLine (B9,i,((Line ((RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))),i)) + ((- a) * (Line ((RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))),j))))) by A12, A10, MATRIX11:29
.= B9 by A11, MATRIX11:30 ;
A14: Line ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),j) = Line (A9,j) by A1, A2, MATRIX11:28;
A15: len ((Line (A9,i)) + (a * (Line (A9,j)))) = width A9 by CARD_1:def_7;
then A16: width (RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))) = width A9 by MATRIX11:def_3;
Line ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),i) = (Line (A9,i)) + (a * (Line (A9,j))) by A6, A15, MATRIX11:28;
then A17: (Line ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),i)) + ((- a) * (Line ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),j))) = ((Line (A9,i)) + (a * (Line (A9,j)))) + ((- ((1_ K) * a)) * (Line (A9,j))) by A14, VECTSP_1:def_8
.= ((Line (A9,i)) + (a * (Line (A9,j)))) + (((- (1_ K)) * a) * (Line (A9,j))) by VECTSP_1:9
.= ((Line (A9,i)) + (a * (Line (A9,j)))) + ((- (1_ K)) * (a * (Line (A9,j)))) by FVSUM_1:54
.= ((Line (A9,i)) + (a * (Line (A9,j)))) + (- (a * (Line (A9,j)))) by FVSUM_1:59
.= (Line (A9,i)) + ((a * (Line (A9,j))) + (- (a * (Line (A9,j))))) by FINSEQOP:28
.= (Line (A9,i)) + ((width A9) |-> (0. K)) by FVSUM_1:26
.= Line (A9,i) by FVSUM_1:21 ;
A18: len ((Line ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),i)) + ((- a) * (Line ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),j)))) = width (RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))) by CARD_1:def_7;
then RLine ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),i,((Line ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),i)) + ((- a) * (Line ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),j))))) = Replace ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),i,LLRA) by MATRIX11:29
.= Replace ((Replace (A9,i,LLA)),i,LLRA) by A15, MATRIX11:29
.= Replace (A9,i,LLRA) by FUNCT_7:34
.= RLine (A9,i,((Line ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),i)) + ((- a) * (Line ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),j))))) by A18, A16, MATRIX11:29
.= A9 by A17, MATRIX11:30 ;
hence x in Solutions_of (A9,B9) by A1, A4, A7, A13, Th39; ::_thesis: verum
end;
end;
end;
theorem Th40: :: MATRIX15:40
for n, k, j, m, i being Nat
for K being Field
for a being Element of K
for A9 being Matrix of m,n,K
for B9 being Matrix of m,k,K st j in Seg m & ( i = j implies a <> - (1_ K) ) holds
Solutions_of (A9,B9) = Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),(RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))))
proof
let n, k, j, m, i be Nat; ::_thesis: for K being Field
for a being Element of K
for A9 being Matrix of m,n,K
for B9 being Matrix of m,k,K st j in Seg m & ( i = j implies a <> - (1_ K) ) holds
Solutions_of (A9,B9) = Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),(RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))))
let K be Field; ::_thesis: for a being Element of K
for A9 being Matrix of m,n,K
for B9 being Matrix of m,k,K st j in Seg m & ( i = j implies a <> - (1_ K) ) holds
Solutions_of (A9,B9) = Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),(RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))))
let a be Element of K; ::_thesis: for A9 being Matrix of m,n,K
for B9 being Matrix of m,k,K st j in Seg m & ( i = j implies a <> - (1_ K) ) holds
Solutions_of (A9,B9) = Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),(RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))))
let A9 be Matrix of m,n,K; ::_thesis: for B9 being Matrix of m,k,K st j in Seg m & ( i = j implies a <> - (1_ K) ) holds
Solutions_of (A9,B9) = Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),(RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))))
let B9 be Matrix of m,k,K; ::_thesis: ( j in Seg m & ( i = j implies a <> - (1_ K) ) implies Solutions_of (A9,B9) = Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),(RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j))))))) )
assume that
A1: j in Seg m and
A2: ( i = j implies a <> - (1_ K) ) ; ::_thesis: Solutions_of (A9,B9) = Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),(RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))))
percases ( i <> j or i = j ) ;
suppose i <> j ; ::_thesis: Solutions_of (A9,B9) = Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),(RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))))
hence Solutions_of (A9,B9) = Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),(RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j))))))) by A1, Lm5; ::_thesis: verum
end;
supposeA3: i = j ; ::_thesis: Solutions_of (A9,B9) = Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),(RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j)))))))
A4: (1_ K) + a <> 0. K
proof
assume (1_ K) + a = 0. K ; ::_thesis: contradiction
then - (1. K) = (- (1. K)) + ((1_ K) + a) by RLVECT_1:def_4
.= ((- (1. K)) + (1_ K)) + a by RLVECT_1:def_3
.= (0. K) + a by VECTSP_1:19
.= a by RLVECT_1:def_4 ;
hence contradiction by A2, A3; ::_thesis: verum
end;
set LB = Line (B9,i);
set LA = Line (A9,i);
A5: (Line (B9,i)) + (a * (Line (B9,i))) = ((1_ K) * (Line (B9,i))) + (a * (Line (B9,i))) by FVSUM_1:57
.= ((1_ K) + a) * (Line (B9,i)) by FVSUM_1:55 ;
(Line (A9,i)) + (a * (Line (A9,i))) = ((1_ K) * (Line (A9,i))) + (a * (Line (A9,i))) by FVSUM_1:57
.= ((1_ K) + a) * (Line (A9,i)) by FVSUM_1:55 ;
hence Solutions_of (A9,B9) = Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),(RLine (B9,i,((Line (B9,i)) + (a * (Line (B9,j))))))) by A3, A4, A5, Lm4; ::_thesis: verum
end;
end;
end;
theorem Th41: :: MATRIX15:41
for i being Nat
for K being Field
for X, A, B being Matrix of K st X in Solutions_of (A,B) & i in dom A & Line (A,i) = (width A) |-> (0. K) holds
Line (B,i) = (width B) |-> (0. K)
proof
let i be Nat; ::_thesis: for K being Field
for X, A, B being Matrix of K st X in Solutions_of (A,B) & i in dom A & Line (A,i) = (width A) |-> (0. K) holds
Line (B,i) = (width B) |-> (0. K)
let K be Field; ::_thesis: for X, A, B being Matrix of K st X in Solutions_of (A,B) & i in dom A & Line (A,i) = (width A) |-> (0. K) holds
Line (B,i) = (width B) |-> (0. K)
let X, A, B be Matrix of K; ::_thesis: ( X in Solutions_of (A,B) & i in dom A & Line (A,i) = (width A) |-> (0. K) implies Line (B,i) = (width B) |-> (0. K) )
assume that
A1: X in Solutions_of (A,B) and
A2: i in dom A and
A3: Line (A,i) = (width A) |-> (0. K) ; ::_thesis: Line (B,i) = (width B) |-> (0. K)
set wB0 = (width B) |-> (0. K);
set LB = Line (B,i);
A4: len (Line (B,i)) = width B by CARD_1:def_7;
A5: ex X1 being Matrix of K st
( X = X1 & len X1 = width A & width X1 = width B & A * X1 = B ) by A1;
A6: now__::_thesis:_for_k_being_Nat_st_1_<=_k_&_k_<=_len_(Line_(B,i))_holds_
((width_B)_|->_(0._K))_._k_=_(Line_(B,i))_._k
let k be Nat; ::_thesis: ( 1 <= k & k <= len (Line (B,i)) implies ((width B) |-> (0. K)) . k = (Line (B,i)) . k )
assume A7: ( 1 <= k & k <= len (Line (B,i)) ) ; ::_thesis: ((width B) |-> (0. K)) . k = (Line (B,i)) . k
k in NAT by ORDINAL1:def_12;
then A8: k in Seg (width B) by A4, A7;
len A = len B by A1, Th33;
then dom A = Seg (len B) by FINSEQ_1:def_3;
then i in dom B by A2, FINSEQ_1:def_3;
then [i,k] in Indices B by A8, ZFMISC_1:87;
then B * (i,k) = (Line (A,i)) "*" (Col (X,k)) by A5, MATRIX_3:def_4
.= Sum ((0. K) * (Col (X,k))) by A3, A5, FVSUM_1:66
.= (0. K) * (Sum (Col (X,k))) by FVSUM_1:73
.= 0. K by VECTSP_1:6
.= ((width B) |-> (0. K)) . k by A8, FINSEQ_2:57 ;
hence ((width B) |-> (0. K)) . k = (Line (B,i)) . k by A8, MATRIX_1:def_7; ::_thesis: verum
end;
len ((width B) |-> (0. K)) = width B by CARD_1:def_7;
hence Line (B,i) = (width B) |-> (0. K) by A4, A6, FINSEQ_1:14; ::_thesis: verum
end;
Lm6: for n, i being Nat
for K being Field
for A being Matrix of K
for nt being Element of n -tuples_on NAT st i in Seg n holds
Line ((Segm (A,nt,(Sgm (Seg (width A))))),i) = Line (A,(nt . i))
proof
let n, i be Nat; ::_thesis: for K being Field
for A being Matrix of K
for nt being Element of n -tuples_on NAT st i in Seg n holds
Line ((Segm (A,nt,(Sgm (Seg (width A))))),i) = Line (A,(nt . i))
let K be Field; ::_thesis: for A being Matrix of K
for nt being Element of n -tuples_on NAT st i in Seg n holds
Line ((Segm (A,nt,(Sgm (Seg (width A))))),i) = Line (A,(nt . i))
let A be Matrix of K; ::_thesis: for nt being Element of n -tuples_on NAT st i in Seg n holds
Line ((Segm (A,nt,(Sgm (Seg (width A))))),i) = Line (A,(nt . i))
let nt be Element of n -tuples_on NAT; ::_thesis: ( i in Seg n implies Line ((Segm (A,nt,(Sgm (Seg (width A))))),i) = Line (A,(nt . i)) )
set S = Seg (width A);
A1: rng (Sgm (Seg (width A))) = Seg (width A) by FINSEQ_1:def_13;
len (Line (A,(nt . i))) = width A by MATRIX_1:def_7;
then A2: dom (Line (A,(nt . i))) = Seg (width A) by FINSEQ_1:def_3;
Sgm (Seg (width A)) = idseq (width A) by FINSEQ_3:48;
then A3: (Line (A,(nt . i))) * (Sgm (Seg (width A))) = Line (A,(nt . i)) by A2, RELAT_1:52;
assume i in Seg n ; ::_thesis: Line ((Segm (A,nt,(Sgm (Seg (width A))))),i) = Line (A,(nt . i))
hence Line ((Segm (A,nt,(Sgm (Seg (width A))))),i) = Line (A,(nt . i)) by A3, A1, MATRIX13:24; ::_thesis: verum
end;
theorem Th42: :: MATRIX15:42
for n being Nat
for K being Field
for A, B being Matrix of K
for nt being Element of n -tuples_on NAT st rng nt c= dom A & n > 0 holds
Solutions_of (A,B) c= Solutions_of ((Segm (A,nt,(Sgm (Seg (width A))))),(Segm (B,nt,(Sgm (Seg (width B))))))
proof
let n be Nat; ::_thesis: for K being Field
for A, B being Matrix of K
for nt being Element of n -tuples_on NAT st rng nt c= dom A & n > 0 holds
Solutions_of (A,B) c= Solutions_of ((Segm (A,nt,(Sgm (Seg (width A))))),(Segm (B,nt,(Sgm (Seg (width B))))))
let K be Field; ::_thesis: for A, B being Matrix of K
for nt being Element of n -tuples_on NAT st rng nt c= dom A & n > 0 holds
Solutions_of (A,B) c= Solutions_of ((Segm (A,nt,(Sgm (Seg (width A))))),(Segm (B,nt,(Sgm (Seg (width B))))))
let A, B be Matrix of K; ::_thesis: for nt being Element of n -tuples_on NAT st rng nt c= dom A & n > 0 holds
Solutions_of (A,B) c= Solutions_of ((Segm (A,nt,(Sgm (Seg (width A))))),(Segm (B,nt,(Sgm (Seg (width B))))))
let nt be Element of n -tuples_on NAT; ::_thesis: ( rng nt c= dom A & n > 0 implies Solutions_of (A,B) c= Solutions_of ((Segm (A,nt,(Sgm (Seg (width A))))),(Segm (B,nt,(Sgm (Seg (width B)))))) )
assume that
A1: rng nt c= dom A and
A2: n > 0 ; ::_thesis: Solutions_of (A,B) c= Solutions_of ((Segm (A,nt,(Sgm (Seg (width A))))),(Segm (B,nt,(Sgm (Seg (width B))))))
set SA = Segm (A,nt,(Sgm (Seg (width A))));
A3: len (Segm (A,nt,(Sgm (Seg (width A))))) = n by A2, MATRIX_1:23;
width (Segm (A,nt,(Sgm (Seg (width A))))) = card (Seg (width A)) by A2, MATRIX_1:23;
then A4: width (Segm (A,nt,(Sgm (Seg (width A))))) = width A by FINSEQ_1:57;
set SB = Segm (B,nt,(Sgm (Seg (width B))));
A5: len (Segm (B,nt,(Sgm (Seg (width B))))) = n by A2, MATRIX_1:23;
width (Segm (B,nt,(Sgm (Seg (width B))))) = card (Seg (width B)) by A2, MATRIX_1:23;
then A6: width (Segm (B,nt,(Sgm (Seg (width B))))) = width B by FINSEQ_1:57;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Solutions_of (A,B) or x in Solutions_of ((Segm (A,nt,(Sgm (Seg (width A))))),(Segm (B,nt,(Sgm (Seg (width B)))))) )
assume A7: x in Solutions_of (A,B) ; ::_thesis: x in Solutions_of ((Segm (A,nt,(Sgm (Seg (width A))))),(Segm (B,nt,(Sgm (Seg (width B))))))
consider X being Matrix of K such that
A8: x = X and
A9: len X = width A and
A10: width X = width B and
A11: A * X = B by A7;
set SX = (Segm (A,nt,(Sgm (Seg (width A))))) * X;
A12: len A = len B by A7, Th33;
A13: now__::_thesis:_for_j,_k_being_Nat_st_[j,k]_in_Indices_((Segm_(A,nt,(Sgm_(Seg_(width_A)))))_*_X)_holds_
((Segm_(A,nt,(Sgm_(Seg_(width_A)))))_*_X)_*_(j,k)_=_(Segm_(B,nt,(Sgm_(Seg_(width_B)))))_*_(j,k)
A14: len ((Segm (A,nt,(Sgm (Seg (width A))))) * X) = len (Segm (A,nt,(Sgm (Seg (width A))))) by A9, A4, MATRIX_3:def_4;
let j, k be Nat; ::_thesis: ( [j,k] in Indices ((Segm (A,nt,(Sgm (Seg (width A))))) * X) implies ((Segm (A,nt,(Sgm (Seg (width A))))) * X) * (j,k) = (Segm (B,nt,(Sgm (Seg (width B))))) * (j,k) )
assume A15: [j,k] in Indices ((Segm (A,nt,(Sgm (Seg (width A))))) * X) ; ::_thesis: ((Segm (A,nt,(Sgm (Seg (width A))))) * X) * (j,k) = (Segm (B,nt,(Sgm (Seg (width B))))) * (j,k)
j in dom ((Segm (A,nt,(Sgm (Seg (width A))))) * X) by A15, ZFMISC_1:87;
then A16: j in Seg n by A3, A14, FINSEQ_1:def_3;
width ((Segm (A,nt,(Sgm (Seg (width A))))) * X) = width X by A9, A4, MATRIX_3:def_4;
then A17: k in Seg (width B) by A10, A15, ZFMISC_1:87;
dom nt = Seg n by FINSEQ_2:124;
then nt . j in rng nt by A16, FUNCT_1:def_3;
then A18: nt . j in dom A by A1;
dom A = Seg (len B) by A12, FINSEQ_1:def_3;
then nt . j in dom B by A18, FINSEQ_1:def_3;
then A19: [(nt . j),k] in Indices B by A17, ZFMISC_1:87;
reconsider j9 = j, k9 = k as Element of NAT by ORDINAL1:def_12;
Sgm (Seg (width B)) = idseq (width B) by FINSEQ_3:48;
then A20: (Sgm (Seg (width B))) . k9 = k by A17, FINSEQ_2:49;
j in dom (Segm (B,nt,(Sgm (Seg (width B))))) by A5, A16, FINSEQ_1:def_3;
then A21: [j,k] in Indices (Segm (B,nt,(Sgm (Seg (width B))))) by A6, A17, ZFMISC_1:87;
Line ((Segm (A,nt,(Sgm (Seg (width A))))),j) = Line (A,(nt . j)) by A16, Lm6;
hence ((Segm (A,nt,(Sgm (Seg (width A))))) * X) * (j,k) = (Line (A,(nt . j))) "*" (Col (X,k)) by A9, A4, A15, MATRIX_3:def_4
.= B * ((nt . j9),k) by A9, A11, A19, MATRIX_3:def_4
.= (Segm (B,nt,(Sgm (Seg (width B))))) * (j,k) by A21, A20, MATRIX13:def_1 ;
::_thesis: verum
end;
( len ((Segm (A,nt,(Sgm (Seg (width A))))) * X) = len (Segm (A,nt,(Sgm (Seg (width A))))) & width ((Segm (A,nt,(Sgm (Seg (width A))))) * X) = width X ) by A9, A4, MATRIX_3:def_4;
then (Segm (A,nt,(Sgm (Seg (width A))))) * X = Segm (B,nt,(Sgm (Seg (width B)))) by A10, A3, A5, A6, A13, MATRIX_1:21;
hence x in Solutions_of ((Segm (A,nt,(Sgm (Seg (width A))))),(Segm (B,nt,(Sgm (Seg (width B)))))) by A8, A9, A10, A4, A6; ::_thesis: verum
end;
theorem Th43: :: MATRIX15:43
for n being Nat
for K being Field
for A, B being Matrix of K
for nt being Element of n -tuples_on NAT st rng nt c= dom A & dom A = dom B & n > 0 & ( for i being Nat st i in (dom A) \ (rng nt) holds
( Line (A,i) = (width A) |-> (0. K) & Line (B,i) = (width B) |-> (0. K) ) ) holds
Solutions_of (A,B) = Solutions_of ((Segm (A,nt,(Sgm (Seg (width A))))),(Segm (B,nt,(Sgm (Seg (width B))))))
proof
let n be Nat; ::_thesis: for K being Field
for A, B being Matrix of K
for nt being Element of n -tuples_on NAT st rng nt c= dom A & dom A = dom B & n > 0 & ( for i being Nat st i in (dom A) \ (rng nt) holds
( Line (A,i) = (width A) |-> (0. K) & Line (B,i) = (width B) |-> (0. K) ) ) holds
Solutions_of (A,B) = Solutions_of ((Segm (A,nt,(Sgm (Seg (width A))))),(Segm (B,nt,(Sgm (Seg (width B))))))
let K be Field; ::_thesis: for A, B being Matrix of K
for nt being Element of n -tuples_on NAT st rng nt c= dom A & dom A = dom B & n > 0 & ( for i being Nat st i in (dom A) \ (rng nt) holds
( Line (A,i) = (width A) |-> (0. K) & Line (B,i) = (width B) |-> (0. K) ) ) holds
Solutions_of (A,B) = Solutions_of ((Segm (A,nt,(Sgm (Seg (width A))))),(Segm (B,nt,(Sgm (Seg (width B))))))
let A, B be Matrix of K; ::_thesis: for nt being Element of n -tuples_on NAT st rng nt c= dom A & dom A = dom B & n > 0 & ( for i being Nat st i in (dom A) \ (rng nt) holds
( Line (A,i) = (width A) |-> (0. K) & Line (B,i) = (width B) |-> (0. K) ) ) holds
Solutions_of (A,B) = Solutions_of ((Segm (A,nt,(Sgm (Seg (width A))))),(Segm (B,nt,(Sgm (Seg (width B))))))
let nt be Element of n -tuples_on NAT; ::_thesis: ( rng nt c= dom A & dom A = dom B & n > 0 & ( for i being Nat st i in (dom A) \ (rng nt) holds
( Line (A,i) = (width A) |-> (0. K) & Line (B,i) = (width B) |-> (0. K) ) ) implies Solutions_of (A,B) = Solutions_of ((Segm (A,nt,(Sgm (Seg (width A))))),(Segm (B,nt,(Sgm (Seg (width B)))))) )
assume that
A1: rng nt c= dom A and
A2: dom A = dom B and
A3: n > 0 and
A4: for i being Nat st i in (dom A) \ (rng nt) holds
( Line (A,i) = (width A) |-> (0. K) & Line (B,i) = (width B) |-> (0. K) ) ; ::_thesis: Solutions_of (A,B) = Solutions_of ((Segm (A,nt,(Sgm (Seg (width A))))),(Segm (B,nt,(Sgm (Seg (width B))))))
set SB = Segm (B,nt,(Sgm (Seg (width B))));
set SA = Segm (A,nt,(Sgm (Seg (width A))));
A5: Solutions_of ((Segm (A,nt,(Sgm (Seg (width A))))),(Segm (B,nt,(Sgm (Seg (width B)))))) c= Solutions_of (A,B)
proof
A6: Seg (len A) = dom B by A2, FINSEQ_1:def_3;
A7: width (Segm (B,nt,(Sgm (Seg (width B))))) = card (Seg (width B)) by A3, MATRIX_1:23;
then A8: width (Segm (B,nt,(Sgm (Seg (width B))))) = width B by FINSEQ_1:57;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Solutions_of ((Segm (A,nt,(Sgm (Seg (width A))))),(Segm (B,nt,(Sgm (Seg (width B)))))) or x in Solutions_of (A,B) )
assume x in Solutions_of ((Segm (A,nt,(Sgm (Seg (width A))))),(Segm (B,nt,(Sgm (Seg (width B)))))) ; ::_thesis: x in Solutions_of (A,B)
then consider X being Matrix of K such that
A9: x = X and
A10: len X = width (Segm (A,nt,(Sgm (Seg (width A))))) and
A11: width X = width (Segm (B,nt,(Sgm (Seg (width B))))) and
A12: (Segm (A,nt,(Sgm (Seg (width A))))) * X = Segm (B,nt,(Sgm (Seg (width B)))) ;
set AX = A * X;
width (Segm (A,nt,(Sgm (Seg (width A))))) = card (Seg (width A)) by A3, MATRIX_1:23;
then A13: width (Segm (A,nt,(Sgm (Seg (width A))))) = width A by FINSEQ_1:57;
then A14: width (A * X) = width X by A10, MATRIX_3:def_4;
A15: len (A * X) = len A by A10, A13, MATRIX_3:def_4;
A16: now__::_thesis:_for_j,_k_being_Nat_st_[j,k]_in_Indices_(A_*_X)_holds_
(A_*_X)_*_(j,k)_=_B_*_(j,k)
A17: dom (A * X) = Seg (len A) by A15, FINSEQ_1:def_3;
let j, k be Nat; ::_thesis: ( [j,k] in Indices (A * X) implies (A * X) * (j,k) = B * (j,k) )
assume A18: [j,k] in Indices (A * X) ; ::_thesis: (A * X) * (j,k) = B * (j,k)
A19: k in Seg (width (A * X)) by A18, ZFMISC_1:87;
reconsider j9 = j, k9 = k as Element of NAT by ORDINAL1:def_12;
A20: j in dom (A * X) by A18, ZFMISC_1:87;
now__::_thesis:_(A_*_X)_*_(j,k)_=_B_*_(j,k)
percases ( j9 in rng nt or not j9 in rng nt ) ;
supposeA21: j9 in rng nt ; ::_thesis: (A * X) * (j,k) = B * (j,k)
A22: dom nt = Seg n by FINSEQ_2:124;
Sgm (Seg (width B)) = idseq (width B) by FINSEQ_3:48;
then A23: (Sgm (Seg (width B))) . k9 = k by A11, A8, A14, A19, FINSEQ_2:49;
consider p being set such that
A24: p in dom nt and
A25: nt . p = j9 by A21, FUNCT_1:def_3;
reconsider p = p as Element of NAT by A24;
Indices (Segm (B,nt,(Sgm (Seg (width B))))) = [:(Seg n),(Seg (card (Seg (width B)))):] by A3, MATRIX_1:23;
then A26: [p,k] in Indices (Segm (B,nt,(Sgm (Seg (width B))))) by A11, A7, A14, A19, A24, A22, ZFMISC_1:87;
Line ((Segm (A,nt,(Sgm (Seg (width A))))),p) = Line (A,j9) by A24, A25, A22, Lm6;
hence (A * X) * (j,k) = (Line ((Segm (A,nt,(Sgm (Seg (width A))))),p)) "*" (Col (X,k)) by A10, A13, A18, MATRIX_3:def_4
.= (Segm (B,nt,(Sgm (Seg (width B))))) * (p,k9) by A10, A12, A26, MATRIX_3:def_4
.= B * (j,k) by A25, A26, A23, MATRIX13:def_1 ;
::_thesis: verum
end;
suppose not j9 in rng nt ; ::_thesis: (A * X) * (j,k) = B * (j,k)
then A27: j9 in (dom A) \ (rng nt) by A2, A6, A20, A17, XBOOLE_0:def_5;
then A28: Line (B,j) = (width B) |-> (0. K) by A4;
Line (A,j) = (width A) |-> (0. K) by A4, A27;
hence (A * X) * (j,k) = ((width A) |-> (0. K)) "*" (Col (X,k)) by A10, A13, A18, MATRIX_3:def_4
.= Sum ((0. K) * (Col (X,k))) by A10, A13, FVSUM_1:66
.= (0. K) * (Sum (Col (X,k))) by FVSUM_1:73
.= 0. K by VECTSP_1:6
.= (Line (B,j)) . k by A11, A8, A14, A19, A28, FINSEQ_2:57
.= B * (j,k) by A11, A8, A14, A19, MATRIX_1:def_7 ;
::_thesis: verum
end;
end;
end;
hence (A * X) * (j,k) = B * (j,k) ; ::_thesis: verum
end;
len (A * X) = len B by A15, A6, FINSEQ_1:def_3;
then A * X = B by A11, A7, A14, A16, FINSEQ_1:57, MATRIX_1:21;
hence x in Solutions_of (A,B) by A9, A10, A11, A13, A8; ::_thesis: verum
end;
Solutions_of (A,B) c= Solutions_of ((Segm (A,nt,(Sgm (Seg (width A))))),(Segm (B,nt,(Sgm (Seg (width B)))))) by A1, A3, Th42;
hence Solutions_of (A,B) = Solutions_of ((Segm (A,nt,(Sgm (Seg (width A))))),(Segm (B,nt,(Sgm (Seg (width B)))))) by A5, XBOOLE_0:def_10; ::_thesis: verum
end;
theorem Th44: :: MATRIX15:44
for K being Field
for A, B being Matrix of K
for N being finite without_zero Subset of NAT st N c= dom A & not N is empty holds
Solutions_of (A,B) c= Solutions_of ((Segm (A,N,(Seg (width A)))),(Segm (B,N,(Seg (width B)))))
proof
let K be Field; ::_thesis: for A, B being Matrix of K
for N being finite without_zero Subset of NAT st N c= dom A & not N is empty holds
Solutions_of (A,B) c= Solutions_of ((Segm (A,N,(Seg (width A)))),(Segm (B,N,(Seg (width B)))))
let A, B be Matrix of K; ::_thesis: for N being finite without_zero Subset of NAT st N c= dom A & not N is empty holds
Solutions_of (A,B) c= Solutions_of ((Segm (A,N,(Seg (width A)))),(Segm (B,N,(Seg (width B)))))
let N be finite without_zero Subset of NAT; ::_thesis: ( N c= dom A & not N is empty implies Solutions_of (A,B) c= Solutions_of ((Segm (A,N,(Seg (width A)))),(Segm (B,N,(Seg (width B))))) )
assume that
A1: N c= dom A and
A2: not N is empty ; ::_thesis: Solutions_of (A,B) c= Solutions_of ((Segm (A,N,(Seg (width A)))),(Segm (B,N,(Seg (width B)))))
dom A = Seg (len A) by FINSEQ_1:def_3;
then rng (Sgm N) = N by A1, FINSEQ_1:def_13;
hence Solutions_of (A,B) c= Solutions_of ((Segm (A,N,(Seg (width A)))),(Segm (B,N,(Seg (width B))))) by A1, A2, Th42; ::_thesis: verum
end;
theorem Th45: :: MATRIX15:45
for K being Field
for A, B being Matrix of K
for N being finite without_zero Subset of NAT st N c= dom A & not N is empty & dom A = dom B & ( for i being Nat st i in (dom A) \ N holds
( Line (A,i) = (width A) |-> (0. K) & Line (B,i) = (width B) |-> (0. K) ) ) holds
Solutions_of (A,B) = Solutions_of ((Segm (A,N,(Seg (width A)))),(Segm (B,N,(Seg (width B)))))
proof
let K be Field; ::_thesis: for A, B being Matrix of K
for N being finite without_zero Subset of NAT st N c= dom A & not N is empty & dom A = dom B & ( for i being Nat st i in (dom A) \ N holds
( Line (A,i) = (width A) |-> (0. K) & Line (B,i) = (width B) |-> (0. K) ) ) holds
Solutions_of (A,B) = Solutions_of ((Segm (A,N,(Seg (width A)))),(Segm (B,N,(Seg (width B)))))
let A, B be Matrix of K; ::_thesis: for N being finite without_zero Subset of NAT st N c= dom A & not N is empty & dom A = dom B & ( for i being Nat st i in (dom A) \ N holds
( Line (A,i) = (width A) |-> (0. K) & Line (B,i) = (width B) |-> (0. K) ) ) holds
Solutions_of (A,B) = Solutions_of ((Segm (A,N,(Seg (width A)))),(Segm (B,N,(Seg (width B)))))
let N be finite without_zero Subset of NAT; ::_thesis: ( N c= dom A & not N is empty & dom A = dom B & ( for i being Nat st i in (dom A) \ N holds
( Line (A,i) = (width A) |-> (0. K) & Line (B,i) = (width B) |-> (0. K) ) ) implies Solutions_of (A,B) = Solutions_of ((Segm (A,N,(Seg (width A)))),(Segm (B,N,(Seg (width B))))) )
assume that
A1: N c= dom A and
A2: ( not N is empty & dom A = dom B & ( for i being Nat st i in (dom A) \ N holds
( Line (A,i) = (width A) |-> (0. K) & Line (B,i) = (width B) |-> (0. K) ) ) ) ; ::_thesis: Solutions_of (A,B) = Solutions_of ((Segm (A,N,(Seg (width A)))),(Segm (B,N,(Seg (width B)))))
dom A = Seg (len A) by FINSEQ_1:def_3;
then rng (Sgm N) = N by A1, FINSEQ_1:def_13;
hence Solutions_of (A,B) = Solutions_of ((Segm (A,N,(Seg (width A)))),(Segm (B,N,(Seg (width B))))) by A1, A2, Th43; ::_thesis: verum
end;
theorem Th46: :: MATRIX15:46
for i being Nat
for K being Field
for A, B being Matrix of K st i in dom A & len A > 1 holds
Solutions_of (A,B) c= Solutions_of ((DelLine (A,i)),(DelLine (B,i)))
proof
let i be Nat; ::_thesis: for K being Field
for A, B being Matrix of K st i in dom A & len A > 1 holds
Solutions_of (A,B) c= Solutions_of ((DelLine (A,i)),(DelLine (B,i)))
let K be Field; ::_thesis: for A, B being Matrix of K st i in dom A & len A > 1 holds
Solutions_of (A,B) c= Solutions_of ((DelLine (A,i)),(DelLine (B,i)))
let A, B be Matrix of K; ::_thesis: ( i in dom A & len A > 1 implies Solutions_of (A,B) c= Solutions_of ((DelLine (A,i)),(DelLine (B,i))) )
assume that
A1: i in dom A and
A2: len A > 1 ; ::_thesis: Solutions_of (A,B) c= Solutions_of ((DelLine (A,i)),(DelLine (B,i)))
reconsider l1 = (len A) - 1 as Element of NAT by A2, NAT_1:20;
A3: l1 > 1 - 1 by A2, XREAL_1:9;
A4: Seg (len A) = dom A by FINSEQ_1:def_3;
card (Seg (len A)) = l1 + 1 by FINSEQ_1:57;
then card ((Seg (len A)) \ {i}) = l1 by A1, A4, STIRL2_1:55;
then A5: Solutions_of (A,B) c= Solutions_of ((Segm (A,((Seg (len A)) \ {i}),(Seg (width A)))),(Segm (B,((Seg (len A)) \ {i}),(Seg (width B))))) by A4, A3, Th44, CARD_1:27, XBOOLE_1:36;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Solutions_of (A,B) or x in Solutions_of ((DelLine (A,i)),(DelLine (B,i))) )
assume A6: x in Solutions_of (A,B) ; ::_thesis: x in Solutions_of ((DelLine (A,i)),(DelLine (B,i)))
len A = len B by A6, Th33;
then ( Segm (A,((Seg (len A)) \ {i}),(Seg (width A))) = Del (A,i) & Segm (B,((Seg (len A)) \ {i}),(Seg (width B))) = Del (B,i) ) by MATRIX13:51;
hence x in Solutions_of ((DelLine (A,i)),(DelLine (B,i))) by A5, A6; ::_thesis: verum
end;
theorem :: MATRIX15:47
for K being Field
for A, B being Matrix of K
for i being Nat st i in dom A & len A > 1 & Line (A,i) = (width A) |-> (0. K) & i in dom B & Line (B,i) = (width B) |-> (0. K) holds
Solutions_of (A,B) = Solutions_of ((DelLine (A,i)),(DelLine (B,i)))
proof
let K be Field; ::_thesis: for A, B being Matrix of K
for i being Nat st i in dom A & len A > 1 & Line (A,i) = (width A) |-> (0. K) & i in dom B & Line (B,i) = (width B) |-> (0. K) holds
Solutions_of (A,B) = Solutions_of ((DelLine (A,i)),(DelLine (B,i)))
let A, B be Matrix of K; ::_thesis: for i being Nat st i in dom A & len A > 1 & Line (A,i) = (width A) |-> (0. K) & i in dom B & Line (B,i) = (width B) |-> (0. K) holds
Solutions_of (A,B) = Solutions_of ((DelLine (A,i)),(DelLine (B,i)))
let i be Nat; ::_thesis: ( i in dom A & len A > 1 & Line (A,i) = (width A) |-> (0. K) & i in dom B & Line (B,i) = (width B) |-> (0. K) implies Solutions_of (A,B) = Solutions_of ((DelLine (A,i)),(DelLine (B,i))) )
assume that
A1: i in dom A and
A2: len A > 1 and
A3: Line (A,i) = (width A) |-> (0. K) and
A4: i in dom B and
A5: Line (B,i) = (width B) |-> (0. K) ; ::_thesis: Solutions_of (A,B) = Solutions_of ((DelLine (A,i)),(DelLine (B,i)))
reconsider l1 = (len A) - 1 as Element of NAT by A2, NAT_1:20;
A6: l1 > 1 - 1 by A2, XREAL_1:9;
thus Solutions_of (A,B) c= Solutions_of ((DelLine (A,i)),(DelLine (B,i))) by A1, A2, Th46; :: according to XBOOLE_0:def_10 ::_thesis: Solutions_of ((DelLine (A,i)),(DelLine (B,i))) c= Solutions_of (A,B)
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Solutions_of ((DelLine (A,i)),(DelLine (B,i))) or x in Solutions_of (A,B) )
assume A7: x in Solutions_of ((DelLine (A,i)),(DelLine (B,i))) ; ::_thesis: x in Solutions_of (A,B)
set S = Seg (len A);
A8: dom A = Seg (len A) by FINSEQ_1:def_3;
A9: now__::_thesis:_for_j_being_Nat_st_j_in_(dom_A)_\_((Seg_(len_A))_\_{i})_holds_
(_Line_(A,j)_=_(width_A)_|->_(0._K)_&_Line_(B,j)_=_(width_B)_|->_(0._K)_)
let j be Nat; ::_thesis: ( j in (dom A) \ ((Seg (len A)) \ {i}) implies ( Line (A,j) = (width A) |-> (0. K) & Line (B,j) = (width B) |-> (0. K) ) )
assume j in (dom A) \ ((Seg (len A)) \ {i}) ; ::_thesis: ( Line (A,j) = (width A) |-> (0. K) & Line (B,j) = (width B) |-> (0. K) )
then j in (dom A) /\ {i} by A8, XBOOLE_1:48;
then j in {i} by XBOOLE_0:def_4;
hence ( Line (A,j) = (width A) |-> (0. K) & Line (B,j) = (width B) |-> (0. K) ) by A3, A5, TARSKI:def_1; ::_thesis: verum
end;
card (Seg (len A)) = l1 + 1 by FINSEQ_1:57;
then A10: card ((Seg (len A)) \ {i}) = l1 by A1, A8, STIRL2_1:55;
( ex mA being Nat st
( len A = mA + 1 & len (Del (A,i)) = mA ) & ex mB being Nat st
( len B = mB + 1 & len (Del (B,i)) = mB ) ) by A1, A4, FINSEQ_3:104;
then A11: len B = len A by A7, Th33;
then dom A = dom B by A8, FINSEQ_1:def_3;
then Solutions_of (A,B) = Solutions_of ((Segm (A,((Seg (len A)) \ {i}),(Seg (width A)))),(Segm (B,((Seg (len A)) \ {i}),(Seg (width B))))) by A8, A10, A6, A9, Th45, CARD_1:27, XBOOLE_1:36
.= Solutions_of ((DelLine (A,i)),(Segm (B,((Seg (len A)) \ {i}),(Seg (width B))))) by MATRIX13:51
.= Solutions_of ((DelLine (A,i)),(DelLine (B,i))) by A11, MATRIX13:51 ;
hence x in Solutions_of (A,B) by A7; ::_thesis: verum
end;
theorem :: MATRIX15:48
for n, m, k being Nat
for K being Field
for A being Matrix of n,m,K
for B being Matrix of n,k,K
for P being Function of (Seg n),(Seg n) holds
( Solutions_of (A,B) c= Solutions_of ((A * P),(B * P)) & ( P is one-to-one implies Solutions_of (A,B) = Solutions_of ((A * P),(B * P)) ) )
proof
let n, m, k be Nat; ::_thesis: for K being Field
for A being Matrix of n,m,K
for B being Matrix of n,k,K
for P being Function of (Seg n),(Seg n) holds
( Solutions_of (A,B) c= Solutions_of ((A * P),(B * P)) & ( P is one-to-one implies Solutions_of (A,B) = Solutions_of ((A * P),(B * P)) ) )
let K be Field; ::_thesis: for A being Matrix of n,m,K
for B being Matrix of n,k,K
for P being Function of (Seg n),(Seg n) holds
( Solutions_of (A,B) c= Solutions_of ((A * P),(B * P)) & ( P is one-to-one implies Solutions_of (A,B) = Solutions_of ((A * P),(B * P)) ) )
set IDn = idseq n;
( len (idseq n) = n & idseq n is FinSequence of NAT ) by CARD_1:def_7, FINSEQ_2:48;
then reconsider IDn = idseq n as Element of n -tuples_on NAT by FINSEQ_2:92;
let A be Matrix of n,m,K; ::_thesis: for B being Matrix of n,k,K
for P being Function of (Seg n),(Seg n) holds
( Solutions_of (A,B) c= Solutions_of ((A * P),(B * P)) & ( P is one-to-one implies Solutions_of (A,B) = Solutions_of ((A * P),(B * P)) ) )
let B be Matrix of n,k,K; ::_thesis: for P being Function of (Seg n),(Seg n) holds
( Solutions_of (A,B) c= Solutions_of ((A * P),(B * P)) & ( P is one-to-one implies Solutions_of (A,B) = Solutions_of ((A * P),(B * P)) ) )
let P be Function of (Seg n),(Seg n); ::_thesis: ( Solutions_of (A,B) c= Solutions_of ((A * P),(B * P)) & ( P is one-to-one implies Solutions_of (A,B) = Solutions_of ((A * P),(B * P)) ) )
A1: rng P c= Seg n by RELAT_1:def_19;
dom IDn = Seg n by FUNCT_2:52;
then reconsider IDnP = IDn * P as FinSequence of NAT by FINSEQ_2:47;
dom P = Seg n by FUNCT_2:52;
then ( n in NAT & dom IDnP = Seg n ) by A1, ORDINAL1:def_12, RELAT_1:53;
then len IDnP = n by FINSEQ_1:def_3;
then reconsider IDnP = IDnP as Element of n -tuples_on NAT by FINSEQ_2:92;
A2: n = len A by MATRIX_1:def_2;
A3: (idseq n) * P = P by A1, RELAT_1:53;
then A4: rng IDnP c= dom A by A1, A2, FINSEQ_1:def_3;
A5: ( IDn = Sgm (Seg n) & card (Seg n) = n ) by FINSEQ_1:57, FINSEQ_3:48;
then A6: Segm (A,IDnP,(Sgm (Seg (width A)))) = (Segm (A,(Seg (len A)),(Seg (width A)))) * P by A2, MATRIX13:33
.= A * P by MATRIX13:46 ;
A7: len B = n by MATRIX_1:def_2;
then A8: Segm (B,IDnP,(Sgm (Seg (width B)))) = (Segm (B,(Seg (len B)),(Seg (width B)))) * P by A5, MATRIX13:33
.= B * P by MATRIX13:46 ;
percases ( n > 0 or n = 0 ) ;
supposeA9: n > 0 ; ::_thesis: ( Solutions_of (A,B) c= Solutions_of ((A * P),(B * P)) & ( P is one-to-one implies Solutions_of (A,B) = Solutions_of ((A * P),(B * P)) ) )
hence Solutions_of (A,B) c= Solutions_of ((A * P),(B * P)) by A6, A8, A4, Th42; ::_thesis: ( P is one-to-one implies Solutions_of (A,B) = Solutions_of ((A * P),(B * P)) )
A10: card (Seg n) = card (Seg n) ;
A11: dom A = Seg n by A2, FINSEQ_1:def_3;
A12: dom B = Seg n by A7, FINSEQ_1:def_3;
assume P is one-to-one ; ::_thesis: Solutions_of (A,B) = Solutions_of ((A * P),(B * P))
then P is onto by A10, STIRL2_1:60;
then rng P = Seg n by FUNCT_2:def_3;
then for i being Nat st i in (dom A) \ (rng IDnP) holds
( Line (A,i) = (width A) |-> (0. K) & Line (B,i) = (width B) |-> (0. K) ) by A3, A11, XBOOLE_1:37;
hence Solutions_of (A,B) = Solutions_of ((A * P),(B * P)) by A1, A3, A6, A8, A9, A11, A12, Th43; ::_thesis: verum
end;
supposeA13: n = 0 ; ::_thesis: ( Solutions_of (A,B) c= Solutions_of ((A * P),(B * P)) & ( P is one-to-one implies Solutions_of (A,B) = Solutions_of ((A * P),(B * P)) ) )
then len B = 0 by MATRIX_1:22;
then A14: B = {} ;
len A = 0 by A13, MATRIX_1:22;
then A15: A = {} ;
A * P = {} by A13;
hence ( Solutions_of (A,B) c= Solutions_of ((A * P),(B * P)) & ( P is one-to-one implies Solutions_of (A,B) = Solutions_of ((A * P),(B * P)) ) ) by A15, A14; ::_thesis: verum
end;
end;
end;
theorem Th49: :: MATRIX15:49
for n, m being Nat
for K being Field
for A being Matrix of n,m,K
for N being finite without_zero Subset of NAT st card N = n & N c= Seg m & Segm (A,(Seg n),N) = 1. (K,n) & n > 0 holds
ex MVectors being Matrix of m -' n,m,K st
( Segm (MVectors,(Seg (m -' n)),((Seg m) \ N)) = 1. (K,(m -' n)) & Segm (MVectors,(Seg (m -' n)),N) = - ((Segm (A,(Seg n),((Seg m) \ N))) @) & ( for l being Nat
for M being Matrix of m,l,K st ( for i being Nat holds
( not i in Seg l or ex j being Nat st
( j in Seg (m -' n) & Col (M,i) = Line (MVectors,j) ) or Col (M,i) = m |-> (0. K) ) ) holds
M in Solutions_of (A,(0. (K,n,l))) ) )
proof
let n, m be Nat; ::_thesis: for K being Field
for A being Matrix of n,m,K
for N being finite without_zero Subset of NAT st card N = n & N c= Seg m & Segm (A,(Seg n),N) = 1. (K,n) & n > 0 holds
ex MVectors being Matrix of m -' n,m,K st
( Segm (MVectors,(Seg (m -' n)),((Seg m) \ N)) = 1. (K,(m -' n)) & Segm (MVectors,(Seg (m -' n)),N) = - ((Segm (A,(Seg n),((Seg m) \ N))) @) & ( for l being Nat
for M being Matrix of m,l,K st ( for i being Nat holds
( not i in Seg l or ex j being Nat st
( j in Seg (m -' n) & Col (M,i) = Line (MVectors,j) ) or Col (M,i) = m |-> (0. K) ) ) holds
M in Solutions_of (A,(0. (K,n,l))) ) )
let K be Field; ::_thesis: for A being Matrix of n,m,K
for N being finite without_zero Subset of NAT st card N = n & N c= Seg m & Segm (A,(Seg n),N) = 1. (K,n) & n > 0 holds
ex MVectors being Matrix of m -' n,m,K st
( Segm (MVectors,(Seg (m -' n)),((Seg m) \ N)) = 1. (K,(m -' n)) & Segm (MVectors,(Seg (m -' n)),N) = - ((Segm (A,(Seg n),((Seg m) \ N))) @) & ( for l being Nat
for M being Matrix of m,l,K st ( for i being Nat holds
( not i in Seg l or ex j being Nat st
( j in Seg (m -' n) & Col (M,i) = Line (MVectors,j) ) or Col (M,i) = m |-> (0. K) ) ) holds
M in Solutions_of (A,(0. (K,n,l))) ) )
let A be Matrix of n,m,K; ::_thesis: for N being finite without_zero Subset of NAT st card N = n & N c= Seg m & Segm (A,(Seg n),N) = 1. (K,n) & n > 0 holds
ex MVectors being Matrix of m -' n,m,K st
( Segm (MVectors,(Seg (m -' n)),((Seg m) \ N)) = 1. (K,(m -' n)) & Segm (MVectors,(Seg (m -' n)),N) = - ((Segm (A,(Seg n),((Seg m) \ N))) @) & ( for l being Nat
for M being Matrix of m,l,K st ( for i being Nat holds
( not i in Seg l or ex j being Nat st
( j in Seg (m -' n) & Col (M,i) = Line (MVectors,j) ) or Col (M,i) = m |-> (0. K) ) ) holds
M in Solutions_of (A,(0. (K,n,l))) ) )
let N be finite without_zero Subset of NAT; ::_thesis: ( card N = n & N c= Seg m & Segm (A,(Seg n),N) = 1. (K,n) & n > 0 implies ex MVectors being Matrix of m -' n,m,K st
( Segm (MVectors,(Seg (m -' n)),((Seg m) \ N)) = 1. (K,(m -' n)) & Segm (MVectors,(Seg (m -' n)),N) = - ((Segm (A,(Seg n),((Seg m) \ N))) @) & ( for l being Nat
for M being Matrix of m,l,K st ( for i being Nat holds
( not i in Seg l or ex j being Nat st
( j in Seg (m -' n) & Col (M,i) = Line (MVectors,j) ) or Col (M,i) = m |-> (0. K) ) ) holds
M in Solutions_of (A,(0. (K,n,l))) ) ) )
assume that
A1: card N = n and
A2: N c= Seg m and
A3: Segm (A,(Seg n),N) = 1. (K,n) and
A4: n > 0 ; ::_thesis: ex MVectors being Matrix of m -' n,m,K st
( Segm (MVectors,(Seg (m -' n)),((Seg m) \ N)) = 1. (K,(m -' n)) & Segm (MVectors,(Seg (m -' n)),N) = - ((Segm (A,(Seg n),((Seg m) \ N))) @) & ( for l being Nat
for M being Matrix of m,l,K st ( for i being Nat holds
( not i in Seg l or ex j being Nat st
( j in Seg (m -' n) & Col (M,i) = Line (MVectors,j) ) or Col (M,i) = m |-> (0. K) ) ) holds
M in Solutions_of (A,(0. (K,n,l))) ) )
set SN = (Seg m) \ N;
A5: ( m -' n = 0 or m -' n > 0 ) ;
A6: card (Seg m) = m by FINSEQ_1:57;
then A7: card ((Seg m) \ N) = m - n by A1, A2, CARD_2:44;
set ZERO = 0. (K,(m -' n),m);
A8: (Seg m) \ N c= Seg m by XBOOLE_1:36;
A9: now__::_thesis:_(_[:(Seg_(m_-'_n)),N:]_c=_Indices_(0._(K,(m_-'_n),m))_&_[:(Seg_(m_-'_n)),((Seg_m)_\_N):]_c=_Indices_(0._(K,(m_-'_n),m))_)
percases ( m -' n = 0 or m -' n > 0 ) ;
suppose m -' n = 0 ; ::_thesis: ( [:(Seg (m -' n)),N:] c= Indices (0. (K,(m -' n),m)) & [:(Seg (m -' n)),((Seg m) \ N):] c= Indices (0. (K,(m -' n),m)) )
then Seg (m -' n) = {} ;
then ( [:(Seg (m -' n)),N:] = {} & [:(Seg (m -' n)),((Seg m) \ N):] = {} ) by ZFMISC_1:90;
hence ( [:(Seg (m -' n)),N:] c= Indices (0. (K,(m -' n),m)) & [:(Seg (m -' n)),((Seg m) \ N):] c= Indices (0. (K,(m -' n),m)) ) by XBOOLE_1:2; ::_thesis: verum
end;
suppose m -' n > 0 ; ::_thesis: ( [:(Seg (m -' n)),N:] c= Indices (0. (K,(m -' n),m)) & [:(Seg (m -' n)),((Seg m) \ N):] c= Indices (0. (K,(m -' n),m)) )
then Indices (0. (K,(m -' n),m)) = [:(Seg (m -' n)),(Seg m):] by MATRIX_1:23;
hence ( [:(Seg (m -' n)),N:] c= Indices (0. (K,(m -' n),m)) & [:(Seg (m -' n)),((Seg m) \ N):] c= Indices (0. (K,(m -' n),m)) ) by A2, A8, ZFMISC_1:96; ::_thesis: verum
end;
end;
end;
set SA = Segm (A,(Seg n),((Seg m) \ N));
card (Seg n) = n by FINSEQ_1:57;
then A10: len (Segm (A,(Seg n),((Seg m) \ N))) = n by A4, MATRIX_1:23;
A11: ( len ((Segm (A,(Seg n),((Seg m) \ N))) @) = len (- ((Segm (A,(Seg n),((Seg m) \ N))) @)) & width ((Segm (A,(Seg n),((Seg m) \ N))) @) = width (- ((Segm (A,(Seg n),((Seg m) \ N))) @)) ) by MATRIX_3:def_2;
A12: width A = m by A4, MATRIX_1:23;
n c= card (Seg m) by A1, A2, CARD_1:11;
then A13: n <= m by A6, NAT_1:39;
then A14: m -' n = m - n by XREAL_1:233;
then width (Segm (A,(Seg n),((Seg m) \ N))) = m -' n by A4, A7, MATRIX_1:23;
then ( ( len ((Segm (A,(Seg n),((Seg m) \ N))) @) = 0 & m -' n = 0 ) or ( len ((Segm (A,(Seg n),((Seg m) \ N))) @) = m -' n & width ((Segm (A,(Seg n),((Seg m) \ N))) @) = n ) ) by A10, A5, MATRIX_1:def_6, MATRIX_2:10;
then ( ( - ((Segm (A,(Seg n),((Seg m) \ N))) @) = {} & m -' n = 0 ) or - ((Segm (A,(Seg n),((Seg m) \ N))) @) is Matrix of m -' n,n,K ) by A11, MATRIX_2:7;
then reconsider SAT = - ((Segm (A,(Seg n),((Seg m) \ N))) @) as Matrix of m -' n,n,K by MATRIX_1:13;
set ONE = 1. (K,(m -' n));
A15: N misses (Seg m) \ N by XBOOLE_1:79;
[:(Seg (m -' n)),N:] /\ [:(Seg (m -' n)),((Seg m) \ N):] = [:(Seg (m -' n)),(N /\ ((Seg m) \ N)):] by ZFMISC_1:99
.= [:(Seg (m -' n)),{}:] by A15, XBOOLE_0:def_7
.= {} by ZFMISC_1:90 ;
then ( card (Seg (m -' n)) = m -' n & ( for i, j, bi, bj, ci, cj being Nat st [i,j] in [:(Seg (m -' n)),N:] /\ [:(Seg (m -' n)),((Seg m) \ N):] & bi = ((Sgm (Seg (m -' n))) ") . i & bj = ((Sgm N) ") . j & ci = ((Sgm (Seg (m -' n))) ") . i & cj = ((Sgm ((Seg m) \ N)) ") . j holds
SAT * (bi,bj) = (1. (K,(m -' n))) * (ci,cj) ) ) by FINSEQ_1:57;
then consider V being Matrix of len (0. (K,(m -' n),m)), width (0. (K,(m -' n),m)),K such that
A16: Segm (V,(Seg (m -' n)),N) = SAT and
A17: Segm (V,(Seg (m -' n)),((Seg m) \ N)) = 1. (K,(m -' n)) and
for i, j being Nat st [i,j] in (Indices V) \ ([:(Seg (m -' n)),N:] \/ [:(Seg (m -' n)),((Seg m) \ N):]) holds
V * (i,j) = (0. (K,(m -' n),m)) * (i,j) by A1, A9, A14, A7, Th9;
( m -' n = 0 or m -' n > 0 ) ;
then ( ( len (0. (K,(m -' n),m)) = 0 & m -' n = 0 & len V = len (0. (K,(m -' n),m)) ) or ( len (0. (K,(m -' n),m)) = m -' n & width (0. (K,(m -' n),m)) = m ) ) by MATRIX_1:23, MATRIX_1:def_2;
then ( ( V = {} & m -' n = 0 ) or V is Matrix of m -' n,m,K ) ;
then reconsider V = V as Matrix of m -' n,m,K ;
take V ; ::_thesis: ( Segm (V,(Seg (m -' n)),((Seg m) \ N)) = 1. (K,(m -' n)) & Segm (V,(Seg (m -' n)),N) = - ((Segm (A,(Seg n),((Seg m) \ N))) @) & ( for l being Nat
for M being Matrix of m,l,K st ( for i being Nat holds
( not i in Seg l or ex j being Nat st
( j in Seg (m -' n) & Col (M,i) = Line (V,j) ) or Col (M,i) = m |-> (0. K) ) ) holds
M in Solutions_of (A,(0. (K,n,l))) ) )
thus ( Segm (V,(Seg (m -' n)),((Seg m) \ N)) = 1. (K,(m -' n)) & Segm (V,(Seg (m -' n)),N) = - ((Segm (A,(Seg n),((Seg m) \ N))) @) ) by A16, A17; ::_thesis: for l being Nat
for M being Matrix of m,l,K st ( for i being Nat holds
( not i in Seg l or ex j being Nat st
( j in Seg (m -' n) & Col (M,i) = Line (V,j) ) or Col (M,i) = m |-> (0. K) ) ) holds
M in Solutions_of (A,(0. (K,n,l)))
let l be Nat; ::_thesis: for M being Matrix of m,l,K st ( for i being Nat holds
( not i in Seg l or ex j being Nat st
( j in Seg (m -' n) & Col (M,i) = Line (V,j) ) or Col (M,i) = m |-> (0. K) ) ) holds
M in Solutions_of (A,(0. (K,n,l)))
let M be Matrix of m,l,K; ::_thesis: ( ( for i being Nat holds
( not i in Seg l or ex j being Nat st
( j in Seg (m -' n) & Col (M,i) = Line (V,j) ) or Col (M,i) = m |-> (0. K) ) ) implies M in Solutions_of (A,(0. (K,n,l))) )
assume A18: for i being Nat holds
( not i in Seg l or ex j being Nat st
( j in Seg (m -' n) & Col (M,i) = Line (V,j) ) or Col (M,i) = m |-> (0. K) ) ; ::_thesis: M in Solutions_of (A,(0. (K,n,l)))
set Z = 0. (K,n,l);
A19: len M = m by A4, A13, MATRIX_1:23;
A20: width M = l by A4, A13, MATRIX_1:23;
then A21: width (A * M) = l by A12, A19, MATRIX_3:def_4;
len A = n by A4, MATRIX_1:23;
then len (A * M) = n by A12, A19, MATRIX_3:def_4;
then reconsider AM = A * M as Matrix of n,l,K by A21, MATRIX_2:7;
A22: Indices A = [:(Seg n),(Seg m):] by A4, MATRIX_1:23;
then A23: [:(Seg n),N:] c= Indices A by A2, ZFMISC_1:96;
now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_AM_holds_
AM_*_(i,j)_=_(0._(K,n,l))_*_(i,j)
A24: Indices AM = Indices (0. (K,n,l)) by MATRIX_1:26;
let i, j be Nat; ::_thesis: ( [i,j] in Indices AM implies AM * (i,j) = (0. (K,n,l)) * (i,j) )
assume A25: [i,j] in Indices AM ; ::_thesis: AM * (i,j) = (0. (K,n,l)) * (i,j)
reconsider I = i, J = j as Element of NAT by ORDINAL1:def_12;
A26: Indices AM = [:(Seg n),(Seg l):] by A4, MATRIX_1:23;
then A27: I in Seg n by A25, ZFMISC_1:87;
A28: J in Seg l by A25, A26, ZFMISC_1:87;
now__::_thesis:_(0._(K,n,l))_*_(i,j)_=_AM_*_(i,j)
percases ( ex jj being Nat st
( jj in Seg (m -' n) & Col (M,J) = Line (V,jj) ) or Col (M,J) = m |-> (0. K) ) by A18, A28;
suppose ex jj being Nat st
( jj in Seg (m -' n) & Col (M,J) = Line (V,jj) ) ; ::_thesis: (0. (K,n,l)) * (i,j) = AM * (i,j)
then consider jj being Nat such that
A29: jj in Seg (m -' n) and
A30: Col (M,J) = Line (V,jj) ;
A31: jj = (idseq (m -' n)) . jj by A29, FINSEQ_2:49
.= (Sgm (Seg (m -' n))) . jj by FINSEQ_3:48 ;
A32: Indices (1. (K,(m -' n))) = [:(Seg (m -' n)),(Seg (m -' n)):] by MATRIX_1:24;
then A33: [jj,jj] in Indices (1. (K,(m -' n))) by A29, ZFMISC_1:87;
A34: rng (Sgm ((Seg m) \ N)) = (Seg m) \ N by A8, FINSEQ_1:def_13;
A35: dom (Sgm ((Seg m) \ N)) = Seg (m -' n) by A14, A7, FINSEQ_3:40, XBOOLE_1:36;
then A36: (Sgm ((Seg m) \ N)) . jj in (Seg m) \ N by A29, A34, FUNCT_1:def_3;
then A37: (Line (A,I)) . ((Sgm ((Seg m) \ N)) . jj) = A * (I,((Sgm ((Seg m) \ N)) . jj)) by A12, A8, MATRIX_1:def_7;
A38: m -' n <> 0 by A29;
then A39: width V = m by MATRIX_1:23;
then A40: (Line (V,jj)) . ((Sgm ((Seg m) \ N)) . jj) = V * (jj,((Sgm ((Seg m) \ N)) . jj)) by A8, A36, MATRIX_1:def_7
.= (1. (K,(m -' n))) * (jj,jj) by A17, A31, A33, MATRIX13:def_1
.= 1_ K by A33, MATRIX_1:def_11 ;
A41: len (Line (A,I)) = m by A12, MATRIX_1:def_7;
A42: I = (idseq n) . I by A27, FINSEQ_2:49
.= (Sgm (Seg n)) . I by FINSEQ_3:48 ;
len (Line (V,jj)) = m by A39, MATRIX_1:def_7;
then len (mlt ((Line (A,I)),(Line (V,jj)))) = m by A41, MATRIX_3:6;
then A43: dom (mlt ((Line (A,I)),(Line (V,jj)))) = Seg m by FINSEQ_1:def_3;
then A44: (mlt ((Line (A,I)),(Line (V,jj)))) /. ((Sgm ((Seg m) \ N)) . jj) = (mlt ((Line (A,I)),(Line (V,jj)))) . ((Sgm ((Seg m) \ N)) . jj) by A8, A36, PARTFUN1:def_6
.= (A * (I,((Sgm ((Seg m) \ N)) . jj))) * (1_ K) by A12, A8, A39, A36, A37, A40, FVSUM_1:61
.= A * (I,((Sgm ((Seg m) \ N)) . jj)) by VECTSP_1:def_6 ;
A45: ( Indices V = Indices (0. (K,(m -' n),m)) & rng (Sgm (Seg (m -' n))) = Seg (m -' n) ) by FINSEQ_1:def_13, MATRIX_1:26;
A46: rng (Sgm N) = N by A2, FINSEQ_1:def_13;
A47: rng (Sgm (Seg n)) = Seg n by FINSEQ_1:def_13;
A48: now__::_thesis:_for_kk_being_Nat_st_kk_in_Seg_m_&_kk_<>_(Sgm_((Seg_m)_\_N))_._jj_&_kk_<>_(Sgm_N)_._I_holds_
(mlt_((Line_(A,I)),(Line_(V,jj))))_._kk_=_0._K
let kk be Nat; ::_thesis: ( kk in Seg m & kk <> (Sgm ((Seg m) \ N)) . jj & kk <> (Sgm N) . I implies (mlt ((Line (A,I)),(Line (V,jj)))) . kk = 0. K )
assume that
A49: kk in Seg m and
A50: kk <> (Sgm ((Seg m) \ N)) . jj and
A51: kk <> (Sgm N) . I ; ::_thesis: (mlt ((Line (A,I)),(Line (V,jj)))) . kk = 0. K
now__::_thesis:_(mlt_((Line_(A,I)),(Line_(V,jj))))_._kk_=_0._K
percases ( kk in N or kk in (Seg m) \ N ) by A49, XBOOLE_0:def_5;
suppose kk in N ; ::_thesis: (mlt ((Line (A,I)),(Line (V,jj)))) . kk = 0. K
then consider x being set such that
A52: x in dom (Sgm N) and
A53: (Sgm N) . x = kk by A46, FUNCT_1:def_3;
reconsider x = x as Element of NAT by A52;
A54: (Line (V,jj)) . ((Sgm N) . x) = V * (jj,((Sgm N) . x)) by A39, A49, A53, MATRIX_1:def_7;
[((Sgm (Seg n)) . I),((Sgm N) . x)] in Indices A by A22, A27, A42, A49, A53, ZFMISC_1:87;
then A55: [I,x] in Indices (Segm (A,(Seg n),N)) by A23, A46, A47, MATRIX13:17;
(Line (A,I)) . ((Sgm N) . x) = A * (I,((Sgm N) . x)) by A12, A49, A53, MATRIX_1:def_7
.= (Segm (A,(Seg n),N)) * (I,x) by A42, A55, MATRIX13:def_1
.= 0. K by A3, A51, A53, A55, MATRIX_1:def_11 ;
hence (mlt ((Line (A,I)),(Line (V,jj)))) . kk = (0. K) * (V * (jj,((Sgm N) . x))) by A12, A39, A49, A53, A54, FVSUM_1:61
.= 0. K by VECTSP_1:7 ;
::_thesis: verum
end;
suppose kk in (Seg m) \ N ; ::_thesis: (mlt ((Line (A,I)),(Line (V,jj)))) . kk = 0. K
then consider x being set such that
A56: x in dom (Sgm ((Seg m) \ N)) and
A57: (Sgm ((Seg m) \ N)) . x = kk by A34, FUNCT_1:def_3;
reconsider x = x as Element of NAT by A56;
A58: (Line (A,I)) . ((Sgm ((Seg m) \ N)) . x) = A * (I,((Sgm ((Seg m) \ N)) . x)) by A12, A49, A57, MATRIX_1:def_7;
A59: [jj,x] in Indices (1. (K,(m -' n))) by A29, A35, A32, A56, ZFMISC_1:87;
(Line (V,jj)) . ((Sgm ((Seg m) \ N)) . x) = V * (jj,((Sgm ((Seg m) \ N)) . x)) by A39, A49, A57, MATRIX_1:def_7
.= (1. (K,(m -' n))) * (jj,x) by A17, A31, A59, MATRIX13:def_1
.= 0. K by A50, A57, A59, MATRIX_1:def_11 ;
hence (mlt ((Line (A,I)),(Line (V,jj)))) . kk = (A * (I,((Sgm ((Seg m) \ N)) . x))) * (0. K) by A12, A39, A49, A57, A58, FVSUM_1:61
.= 0. K by VECTSP_1:7 ;
::_thesis: verum
end;
end;
end;
hence (mlt ((Line (A,I)),(Line (V,jj)))) . kk = 0. K ; ::_thesis: verum
end;
dom (Sgm N) = Seg n by A1, A2, FINSEQ_3:40;
then A60: (Sgm N) . I in N by A27, A46, FUNCT_1:def_3;
then A61: (Sgm ((Seg m) \ N)) . jj <> (Sgm N) . I by A15, A36, XBOOLE_0:3;
[((Sgm (Seg n)) . I),((Sgm N) . I)] in Indices A by A2, A22, A27, A60, A42, ZFMISC_1:87;
then A62: [I,I] in Indices (Segm (A,(Seg n),N)) by A23, A46, A47, MATRIX13:17;
Indices V = [:(Seg (m -' n)),(Seg m):] by A38, MATRIX_1:23;
then [((Sgm (Seg (m -' n))) . jj),((Sgm N) . I)] in Indices V by A2, A29, A60, A31, ZFMISC_1:87;
then A63: [jj,I] in Indices (Segm (V,(Seg (m -' n)),N)) by A9, A46, A45, MATRIX13:17;
A64: Indices SAT = Indices ((Segm (A,(Seg n),((Seg m) \ N))) @) by Lm1;
then A65: [I,jj] in Indices (Segm (A,(Seg n),((Seg m) \ N))) by A16, A63, MATRIX_1:def_6;
A66: (Line (V,jj)) . ((Sgm N) . I) = V * (jj,((Sgm N) . I)) by A2, A39, A60, MATRIX_1:def_7
.= (Segm (V,(Seg (m -' n)),N)) * (jj,I) by A31, A63, MATRIX13:def_1
.= - (((Segm (A,(Seg n),((Seg m) \ N))) @) * (jj,I)) by A16, A63, A64, MATRIX_3:def_2
.= - ((Segm (A,(Seg n),((Seg m) \ N))) * (I,jj)) by A65, MATRIX_1:def_6
.= - (A * (I,((Sgm ((Seg m) \ N)) . jj))) by A42, A65, MATRIX13:def_1 ;
A67: (Line (A,I)) . ((Sgm N) . I) = A * (I,((Sgm N) . I)) by A2, A12, A60, MATRIX_1:def_7
.= (Segm (A,(Seg n),N)) * (I,I) by A42, A62, MATRIX13:def_1
.= 1_ K by A3, A62, MATRIX_1:def_11 ;
(mlt ((Line (A,I)),(Line (V,jj)))) /. ((Sgm N) . I) = (mlt ((Line (A,I)),(Line (V,jj)))) . ((Sgm N) . I) by A2, A43, A60, PARTFUN1:def_6
.= (1_ K) * (- (A * (I,((Sgm ((Seg m) \ N)) . jj)))) by A2, A12, A39, A60, A67, A66, FVSUM_1:61
.= - (A * (I,((Sgm ((Seg m) \ N)) . jj))) by VECTSP_1:def_6 ;
then Sum (mlt ((Line (A,I)),(Line (V,jj)))) = (A * (I,((Sgm ((Seg m) \ N)) . jj))) + (- (A * (I,((Sgm ((Seg m) \ N)) . jj)))) by A2, A8, A43, A60, A36, A44, A61, A48, Th7
.= 0. K by VECTSP_1:16 ;
hence (0. (K,n,l)) * (i,j) = (Line (A,I)) "*" (Line (V,jj)) by A25, A24, MATRIX_3:1
.= AM * (i,j) by A12, A19, A25, A30, MATRIX_3:def_4 ;
::_thesis: verum
end;
suppose Col (M,J) = m |-> (0. K) ; ::_thesis: AM * (i,j) = (0. (K,n,l)) * (i,j)
hence AM * (i,j) = (Line (A,I)) "*" (m |-> (0. K)) by A12, A19, A25, MATRIX_3:def_4
.= Sum ((0. K) * (Line (A,I))) by A12, FVSUM_1:66
.= (0. K) * (Sum (Line (A,I))) by FVSUM_1:73
.= 0. K by VECTSP_1:6
.= (0. (K,n,l)) * (i,j) by A25, A24, MATRIX_3:1 ;
::_thesis: verum
end;
end;
end;
hence AM * (i,j) = (0. (K,n,l)) * (i,j) ; ::_thesis: verum
end;
then AM = 0. (K,n,l) by MATRIX_1:27;
hence M in Solutions_of (A,(0. (K,n,l))) by A12, A19, A20, A21; ::_thesis: verum
end;
theorem Th50: :: MATRIX15:50
for n, m, l being Nat
for K being Field
for A being Matrix of n,m,K
for B being Matrix of n,l,K
for N being finite without_zero Subset of NAT st card N = n & N c= Seg m & n > 0 & Segm (A,(Seg n),N) = 1. (K,n) holds
ex X being Matrix of m,l,K st
( Segm (X,((Seg m) \ N),(Seg l)) = 0. (K,(m -' n),l) & Segm (X,N,(Seg l)) = B & X in Solutions_of (A,B) )
proof
let n, m, l be Nat; ::_thesis: for K being Field
for A being Matrix of n,m,K
for B being Matrix of n,l,K
for N being finite without_zero Subset of NAT st card N = n & N c= Seg m & n > 0 & Segm (A,(Seg n),N) = 1. (K,n) holds
ex X being Matrix of m,l,K st
( Segm (X,((Seg m) \ N),(Seg l)) = 0. (K,(m -' n),l) & Segm (X,N,(Seg l)) = B & X in Solutions_of (A,B) )
let K be Field; ::_thesis: for A being Matrix of n,m,K
for B being Matrix of n,l,K
for N being finite without_zero Subset of NAT st card N = n & N c= Seg m & n > 0 & Segm (A,(Seg n),N) = 1. (K,n) holds
ex X being Matrix of m,l,K st
( Segm (X,((Seg m) \ N),(Seg l)) = 0. (K,(m -' n),l) & Segm (X,N,(Seg l)) = B & X in Solutions_of (A,B) )
let A be Matrix of n,m,K; ::_thesis: for B being Matrix of n,l,K
for N being finite without_zero Subset of NAT st card N = n & N c= Seg m & n > 0 & Segm (A,(Seg n),N) = 1. (K,n) holds
ex X being Matrix of m,l,K st
( Segm (X,((Seg m) \ N),(Seg l)) = 0. (K,(m -' n),l) & Segm (X,N,(Seg l)) = B & X in Solutions_of (A,B) )
let B be Matrix of n,l,K; ::_thesis: for N being finite without_zero Subset of NAT st card N = n & N c= Seg m & n > 0 & Segm (A,(Seg n),N) = 1. (K,n) holds
ex X being Matrix of m,l,K st
( Segm (X,((Seg m) \ N),(Seg l)) = 0. (K,(m -' n),l) & Segm (X,N,(Seg l)) = B & X in Solutions_of (A,B) )
let N be finite without_zero Subset of NAT; ::_thesis: ( card N = n & N c= Seg m & n > 0 & Segm (A,(Seg n),N) = 1. (K,n) implies ex X being Matrix of m,l,K st
( Segm (X,((Seg m) \ N),(Seg l)) = 0. (K,(m -' n),l) & Segm (X,N,(Seg l)) = B & X in Solutions_of (A,B) ) )
assume that
A1: card N = n and
A2: N c= Seg m and
A3: n > 0 and
A4: Segm (A,(Seg n),N) = 1. (K,n) ; ::_thesis: ex X being Matrix of m,l,K st
( Segm (X,((Seg m) \ N),(Seg l)) = 0. (K,(m -' n),l) & Segm (X,N,(Seg l)) = B & X in Solutions_of (A,B) )
A5: width A = m by A3, MATRIX_1:23;
set Z = 0. (K,m,l);
set SN = (Seg m) \ N;
A6: card (Seg m) = m by FINSEQ_1:57;
then A7: ( m -' n = m - n & card ((Seg m) \ N) = m - n ) by A1, A2, CARD_2:44, NAT_1:43, XREAL_1:233;
set ZERO = 0. (K,(m -' n),l);
A8: N misses (Seg m) \ N by XBOOLE_1:79;
[:N,(Seg l):] /\ [:((Seg m) \ N),(Seg l):] = [:(N /\ ((Seg m) \ N)),(Seg l):] by ZFMISC_1:99
.= [:{},(Seg l):] by A8, XBOOLE_0:def_7
.= {} by ZFMISC_1:90 ;
then A9: for i, j, bi, bj, ci, cj being Nat st [i,j] in [:N,(Seg l):] /\ [:((Seg m) \ N),(Seg l):] & bi = ((Sgm N) ") . i & bj = ((Sgm (Seg l)) ") . j & ci = ((Sgm ((Seg m) \ N)) ") . i & cj = ((Sgm (Seg l)) ") . j holds
B * (bi,bj) = (0. (K,(m -' n),l)) * (ci,cj) ;
A10: Indices A = [:(Seg n),(Seg m):] by A3, MATRIX_1:23;
A11: n <= card (Seg m) by A1, A2, NAT_1:43;
then A12: ( len (0. (K,m,l)) = m & width (0. (K,m,l)) = l ) by A3, A6, MATRIX_1:23;
A13: Indices (0. (K,m,l)) = [:(Seg m),(Seg l):] by A3, A11, A6, MATRIX_1:23;
then A14: [:N,(Seg l):] c= Indices (0. (K,m,l)) by A2, ZFMISC_1:95;
A15: (Seg m) \ N c= Seg m by XBOOLE_1:36;
then ( card (Seg l) = l & [:((Seg m) \ N),(Seg l):] c= Indices (0. (K,m,l)) ) by A13, FINSEQ_1:57, ZFMISC_1:95;
then consider X being Matrix of m,l,K such that
A16: Segm (X,N,(Seg l)) = B and
A17: Segm (X,((Seg m) \ N),(Seg l)) = 0. (K,(m -' n),l) and
for i, j being Nat st [i,j] in (Indices X) \ ([:N,(Seg l):] \/ [:((Seg m) \ N),(Seg l):]) holds
X * (i,j) = (0. (K,m,l)) * (i,j) by A1, A7, A12, A14, A9, Th9;
set AX = A * X;
A18: len X = m by A3, A11, A6, MATRIX_1:23;
then A19: dom X = Seg m by FINSEQ_1:def_3;
len A = n by A3, MATRIX_1:23;
then A20: len (A * X) = n by A5, A18, MATRIX_3:def_4;
take X ; ::_thesis: ( Segm (X,((Seg m) \ N),(Seg l)) = 0. (K,(m -' n),l) & Segm (X,N,(Seg l)) = B & X in Solutions_of (A,B) )
thus ( Segm (X,((Seg m) \ N),(Seg l)) = 0. (K,(m -' n),l) & Segm (X,N,(Seg l)) = B ) by A16, A17; ::_thesis: X in Solutions_of (A,B)
A21: Indices X = [:(Seg m),(Seg l):] by A3, A11, A6, MATRIX_1:23;
A22: width B = l by A3, MATRIX_1:23;
A23: width X = l by A3, A11, A6, MATRIX_1:23;
then width (A * X) = l by A5, A18, MATRIX_3:def_4;
then reconsider AX = A * X as Matrix of n,l,K by A20, MATRIX_2:7;
A24: Indices B = [:(Seg n),(Seg l):] by A3, MATRIX_1:23;
now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_AX_holds_
AX_*_(i,j)_=_B_*_(i,j)
A25: [:N,(Seg l):] c= Indices X by A2, A21, ZFMISC_1:95;
let i, j be Nat; ::_thesis: ( [i,j] in Indices AX implies AX * (i,j) = B * (i,j) )
assume A26: [i,j] in Indices AX ; ::_thesis: AX * (i,j) = B * (i,j)
reconsider I = i, J = j as Element of NAT by ORDINAL1:def_12;
A27: Indices AX = Indices B by MATRIX_1:26;
then A28: i in Seg n by A24, A26, ZFMISC_1:87;
( len (Line (A,i)) = m & len (Col (X,j)) = m ) by A5, A18, CARD_1:def_7;
then len (mlt ((Line (A,i)),(Col (X,j)))) = m by MATRIX_3:6;
then A29: dom (mlt ((Line (A,i)),(Col (X,j)))) = Seg m by FINSEQ_1:def_3;
A30: rng (Sgm (Seg l)) = Seg l by FINSEQ_1:def_13;
A31: ( rng (Sgm (Seg n)) = Seg n & [:(Seg n),N:] c= Indices A ) by A2, A10, FINSEQ_1:def_13, ZFMISC_1:95;
A32: rng (Sgm N) = N by A2, FINSEQ_1:def_13;
dom (Sgm N) = Seg n by A1, A2, FINSEQ_3:40;
then A33: (Sgm N) . i in N by A28, A32, FUNCT_1:def_3;
A34: j in Seg l by A22, A26, A27, ZFMISC_1:87;
then A35: j = (idseq l) . j by FINSEQ_2:49
.= (Sgm (Seg l)) . j by FINSEQ_3:48 ;
then [((Sgm N) . I),((Sgm (Seg l)) . J)] in Indices X by A2, A21, A34, A33, ZFMISC_1:87;
then A36: [I,J] in Indices B by A16, A32, A30, A25, MATRIX13:17;
A37: rng (Sgm ((Seg m) \ N)) = (Seg m) \ N by A15, FINSEQ_1:def_13;
A38: i = (idseq n) . i by A28, FINSEQ_2:49
.= (Sgm (Seg n)) . i by FINSEQ_3:48 ;
then [((Sgm (Seg n)) . i),((Sgm N) . i)] in Indices A by A2, A10, A28, A33, ZFMISC_1:87;
then A39: [I,I] in Indices (1. (K,n)) by A4, A32, A31, MATRIX13:17;
A40: [:((Seg m) \ N),(Seg l):] c= Indices X by A15, A21, ZFMISC_1:95;
A41: now__::_thesis:_for_kk_being_Nat_st_kk_in_Seg_m_&_kk_<>_(Sgm_N)_._I_holds_
(mlt_((Line_(A,i)),(Col_(X,j))))_._kk_=_0._K
let kk be Nat; ::_thesis: ( kk in Seg m & kk <> (Sgm N) . I implies (mlt ((Line (A,i)),(Col (X,j)))) . b1 = 0. K )
assume that
A42: kk in Seg m and
A43: kk <> (Sgm N) . I ; ::_thesis: (mlt ((Line (A,i)),(Col (X,j)))) . b1 = 0. K
percases ( kk in N or kk in (Seg m) \ N ) by A42, XBOOLE_0:def_5;
supposeA44: kk in N ; ::_thesis: (mlt ((Line (A,i)),(Col (X,j)))) . b1 = 0. K
then consider x being set such that
A45: x in dom (Sgm N) and
A46: (Sgm N) . x = kk by A32, FUNCT_1:def_3;
reconsider x = x as Element of NAT by A45;
[((Sgm (Seg n)) . i),((Sgm N) . x)] in Indices A by A2, A10, A28, A38, A44, A46, ZFMISC_1:87;
then A47: [I,x] in Indices (1. (K,n)) by A4, A32, A31, MATRIX13:17;
A48: (Col (X,j)) . kk = X * (kk,j) by A2, A19, A44, MATRIX_1:def_8;
(Line (A,i)) . ((Sgm N) . x) = A * (I,((Sgm N) . x)) by A2, A5, A44, A46, MATRIX_1:def_7
.= (Segm (A,(Seg n),N)) * (I,x) by A4, A38, A47, MATRIX13:def_1
.= 0. K by A4, A43, A46, A47, MATRIX_1:def_11 ;
hence (mlt ((Line (A,i)),(Col (X,j)))) . kk = (0. K) * (X * (kk,j)) by A2, A5, A18, A44, A46, A48, FVSUM_1:61
.= 0. K by VECTSP_1:6 ;
::_thesis: verum
end;
supposeA49: kk in (Seg m) \ N ; ::_thesis: (mlt ((Line (A,i)),(Col (X,j)))) . b1 = 0. K
then consider x being set such that
A50: x in dom (Sgm ((Seg m) \ N)) and
A51: (Sgm ((Seg m) \ N)) . x = kk by A37, FUNCT_1:def_3;
reconsider x = x as Element of NAT by A50;
A52: (Line (A,i)) . kk = A * (I,((Sgm ((Seg m) \ N)) . x)) by A5, A42, A51, MATRIX_1:def_7;
[((Sgm ((Seg m) \ N)) . x),((Sgm (Seg l)) . J)] in Indices X by A15, A21, A34, A35, A49, A51, ZFMISC_1:87;
then A53: [x,J] in Indices (0. (K,(m -' n),l)) by A17, A30, A37, A40, MATRIX13:17;
(Col (X,j)) . kk = X * (((Sgm ((Seg m) \ N)) . x),((Sgm (Seg l)) . j)) by A15, A19, A35, A49, A51, MATRIX_1:def_8
.= (0. (K,(m -' n),l)) * (x,J) by A17, A53, MATRIX13:def_1
.= 0. K by A53, MATRIX_3:1 ;
hence (mlt ((Line (A,i)),(Col (X,j)))) . kk = (A * (I,((Sgm ((Seg m) \ N)) . x))) * (0. K) by A5, A18, A42, A52, FVSUM_1:61
.= 0. K by VECTSP_1:6 ;
::_thesis: verum
end;
end;
end;
A54: (Col (X,j)) . ((Sgm N) . i) = X * (((Sgm N) . i),j) by A2, A19, A33, MATRIX_1:def_8
.= B * (I,J) by A16, A35, A36, MATRIX13:def_1 ;
(Line (A,i)) . ((Sgm N) . i) = A * (I,((Sgm N) . I)) by A2, A5, A33, MATRIX_1:def_7
.= (Segm (A,(Seg n),N)) * (I,I) by A4, A38, A39, MATRIX13:def_1
.= 1_ K by A4, A39, MATRIX_1:def_11 ;
then A55: (mlt ((Line (A,i)),(Col (X,j)))) . ((Sgm N) . i) = (1_ K) * (B * (I,J)) by A2, A5, A18, A33, A54, FVSUM_1:61
.= B * (I,J) by VECTSP_1:def_4 ;
AX * (i,j) = (Line (A,i)) "*" (Col (X,j)) by A5, A18, A26, MATRIX_3:def_4
.= Sum (mlt ((Line (A,i)),(Col (X,j)))) ;
hence AX * (i,j) = B * (i,j) by A2, A33, A55, A29, A41, MATRIX_3:12; ::_thesis: verum
end;
then AX = B by MATRIX_1:27;
hence X in Solutions_of (A,B) by A5, A22, A18, A23; ::_thesis: verum
end;
theorem Th51: :: MATRIX15:51
for n, m being Nat
for K being Field
for A being Matrix of 0 ,n,K
for B being Matrix of 0 ,m,K holds Solutions_of (A,B) = {{}}
proof
let n, m be Nat; ::_thesis: for K being Field
for A being Matrix of 0 ,n,K
for B being Matrix of 0 ,m,K holds Solutions_of (A,B) = {{}}
let K be Field; ::_thesis: for A being Matrix of 0 ,n,K
for B being Matrix of 0 ,m,K holds Solutions_of (A,B) = {{}}
let A be Matrix of 0 ,n,K; ::_thesis: for B being Matrix of 0 ,m,K holds Solutions_of (A,B) = {{}}
let B be Matrix of 0 ,m,K; ::_thesis: Solutions_of (A,B) = {{}}
A1: len A = 0 by MATRIX_1:def_2;
A2: Solutions_of (A,B) c= {{}}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Solutions_of (A,B) or x in {{}} )
assume x in Solutions_of (A,B) ; ::_thesis: x in {{}}
then ex X being Matrix of K st
( X = x & len X = width A & width X = width B & A * X = B ) ;
then x = {} by A1, MATRIX_1:def_3;
hence x in {{}} by TARSKI:def_1; ::_thesis: verum
end;
len B = 0 by MATRIX_1:def_2;
then A3: ( B = {} & width B = 0 ) by MATRIX_1:def_3;
A4: width A = 0 by A1, MATRIX_1:def_3;
then len (A * A) = 0 by A1, MATRIX_3:def_4;
then A * A = {} ;
then A in Solutions_of (A,B) by A1, A4, A3;
hence Solutions_of (A,B) = {{}} by A2, ZFMISC_1:33; ::_thesis: verum
end;
theorem Th52: :: MATRIX15:52
for n, k being Nat
for K being Field
for B being Matrix of K st not Solutions_of ((0. (K,n,k)),B) is empty holds
B = 0. (K,n,(width B))
proof
let n, k be Nat; ::_thesis: for K being Field
for B being Matrix of K st not Solutions_of ((0. (K,n,k)),B) is empty holds
B = 0. (K,n,(width B))
let K be Field; ::_thesis: for B being Matrix of K st not Solutions_of ((0. (K,n,k)),B) is empty holds
B = 0. (K,n,(width B))
let B be Matrix of K; ::_thesis: ( not Solutions_of ((0. (K,n,k)),B) is empty implies B = 0. (K,n,(width B)) )
set A = 0. (K,n,k);
set ZERO = 0. (K,n,(width B));
assume not Solutions_of ((0. (K,n,k)),B) is empty ; ::_thesis: B = 0. (K,n,(width B))
then consider x being set such that
A1: x in Solutions_of ((0. (K,n,k)),B) by XBOOLE_0:def_1;
A2: len (0. (K,n,k)) = n by MATRIX_1:def_2;
then A3: dom (0. (K,n,k)) = Seg n by FINSEQ_1:def_3;
A4: len (0. (K,n,(width B))) = n by MATRIX_1:def_2;
then A5: len B = len (0. (K,n,(width B))) by A1, A2, Th33;
then reconsider B9 = B as Matrix of n, width B,K by A4, MATRIX_2:7;
A6: ex X being Matrix of K st
( X = x & len X = width (0. (K,n,k)) & width X = width B & (0. (K,n,k)) * X = B ) by A1;
now__::_thesis:_for_i_being_Nat_st_1_<=_i_&_i_<=_n_holds_
B_._i_=_(0._(K,n,(width_B)))_._i
let i be Nat; ::_thesis: ( 1 <= i & i <= n implies B . i = (0. (K,n,(width B))) . i )
assume A7: ( 1 <= i & i <= n ) ; ::_thesis: B . i = (0. (K,n,(width B))) . i
A8: width (0. (K,n,k)) = k by A7, MATRIX_1:23;
i in NAT by ORDINAL1:def_12;
then A9: i in Seg n by A7;
then Line ((0. (K,n,k)),i) = (0. (K,n,k)) . i by MATRIX_2:8
.= (width (0. (K,n,k))) |-> (0. K) by A9, A8, FINSEQ_2:57 ;
then (width B) |-> (0. K) = Line (B,i) by A1, A6, A3, A9, Th41
.= B9 . i by A9, MATRIX_2:8 ;
hence B . i = (0. (K,n,(width B))) . i by A9, FINSEQ_2:57; ::_thesis: verum
end;
hence B = 0. (K,n,(width B)) by A4, A5, FINSEQ_1:14; ::_thesis: verum
end;
theorem Th53: :: MATRIX15:53
for x being set
for n, k, m being Nat
for K being Field
for A being Matrix of n,k,K
for B being Matrix of n,m,K st n > 0 & x in Solutions_of (A,B) holds
x is Matrix of k,m,K
proof
let x be set ; ::_thesis: for n, k, m being Nat
for K being Field
for A being Matrix of n,k,K
for B being Matrix of n,m,K st n > 0 & x in Solutions_of (A,B) holds
x is Matrix of k,m,K
let n, k, m be Nat; ::_thesis: for K being Field
for A being Matrix of n,k,K
for B being Matrix of n,m,K st n > 0 & x in Solutions_of (A,B) holds
x is Matrix of k,m,K
let K be Field; ::_thesis: for A being Matrix of n,k,K
for B being Matrix of n,m,K st n > 0 & x in Solutions_of (A,B) holds
x is Matrix of k,m,K
let A be Matrix of n,k,K; ::_thesis: for B being Matrix of n,m,K st n > 0 & x in Solutions_of (A,B) holds
x is Matrix of k,m,K
let B be Matrix of n,m,K; ::_thesis: ( n > 0 & x in Solutions_of (A,B) implies x is Matrix of k,m,K )
assume n > 0 ; ::_thesis: ( not x in Solutions_of (A,B) or x is Matrix of k,m,K )
then A1: ( width A = k & width B = m ) by MATRIX_1:23;
assume x in Solutions_of (A,B) ; ::_thesis: x is Matrix of k,m,K
then ex X being Matrix of K st
( X = x & len X = k & width X = m & A * X = B ) by A1;
hence x is Matrix of k,m,K by MATRIX_2:7; ::_thesis: verum
end;
theorem Th54: :: MATRIX15:54
for n, k, m being Nat
for K being Field st n > 0 & k > 0 holds
Solutions_of ((0. (K,n,k)),(0. (K,n,m))) = { X where X is Matrix of k,m,K : verum }
proof
let n, k, m be Nat; ::_thesis: for K being Field st n > 0 & k > 0 holds
Solutions_of ((0. (K,n,k)),(0. (K,n,m))) = { X where X is Matrix of k,m,K : verum }
let K be Field; ::_thesis: ( n > 0 & k > 0 implies Solutions_of ((0. (K,n,k)),(0. (K,n,m))) = { X where X is Matrix of k,m,K : verum } )
assume that
A1: n > 0 and
A2: k > 0 ; ::_thesis: Solutions_of ((0. (K,n,k)),(0. (K,n,m))) = { X where X is Matrix of k,m,K : verum }
set B = 0. (K,n,m);
A3: width (0. (K,n,m)) = m by A1, MATRIX_1:23;
set XX = { X where X is Matrix of k,m,K : verum } ;
set A = 0. (K,n,k);
thus Solutions_of ((0. (K,n,k)),(0. (K,n,m))) c= { X where X is Matrix of k,m,K : verum } :: according to XBOOLE_0:def_10 ::_thesis: { X where X is Matrix of k,m,K : verum } c= Solutions_of ((0. (K,n,k)),(0. (K,n,m)))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Solutions_of ((0. (K,n,k)),(0. (K,n,m))) or x in { X where X is Matrix of k,m,K : verum } )
assume x in Solutions_of ((0. (K,n,k)),(0. (K,n,m))) ; ::_thesis: x in { X where X is Matrix of k,m,K : verum }
then x is Matrix of k,m,K by A1, Th53;
hence x in { X where X is Matrix of k,m,K : verum } ; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { X where X is Matrix of k,m,K : verum } or x in Solutions_of ((0. (K,n,k)),(0. (K,n,m))) )
assume x in { X where X is Matrix of k,m,K : verum } ; ::_thesis: x in Solutions_of ((0. (K,n,k)),(0. (K,n,m)))
then consider X being Matrix of k,m,K such that
A4: x = X and
verum ;
A5: ( width (0. (K,n,k)) = k & len X = k ) by A1, A2, MATRIX_1:23;
A6: width X = m by A2, MATRIX_1:23;
len (0. (K,n,k)) = n by A1, MATRIX_1:23;
then (0. (K,n,k)) * X = 0. (K,n,m) by A1, A2, A5, A6, MATRIX_5:22;
hence x in Solutions_of ((0. (K,n,k)),(0. (K,n,m))) by A4, A3, A5, A6; ::_thesis: verum
end;
theorem :: MATRIX15:55
for n, m being Nat
for K being Field st n > 0 & not Solutions_of ((0. (K,n,0)),(0. (K,n,m))) is empty holds
m = 0
proof
let n, m be Nat; ::_thesis: for K being Field st n > 0 & not Solutions_of ((0. (K,n,0)),(0. (K,n,m))) is empty holds
m = 0
let K be Field; ::_thesis: ( n > 0 & not Solutions_of ((0. (K,n,0)),(0. (K,n,m))) is empty implies m = 0 )
assume that
A1: n > 0 and
A2: not Solutions_of ((0. (K,n,0)),(0. (K,n,m))) is empty ; ::_thesis: m = 0
consider x being set such that
A3: x in Solutions_of ((0. (K,n,0)),(0. (K,n,m))) by A2, XBOOLE_0:def_1;
A4: width (0. (K,n,0)) = 0 by A1, MATRIX_1:23;
ex X being Matrix of K st
( X = x & len X = width (0. (K,n,0)) & width X = width (0. (K,n,m)) & (0. (K,n,0)) * X = 0. (K,n,m) ) by A3;
hence 0 = width (0. (K,n,m)) by A4, MATRIX_1:def_3
.= m by A1, MATRIX_1:23 ;
::_thesis: verum
end;
theorem Th56: :: MATRIX15:56
for n being Nat
for K being Field holds Solutions_of ((0. (K,n,0)),(0. (K,n,0))) = {{}}
proof
let n be Nat; ::_thesis: for K being Field holds Solutions_of ((0. (K,n,0)),(0. (K,n,0))) = {{}}
let K be Field; ::_thesis: Solutions_of ((0. (K,n,0)),(0. (K,n,0))) = {{}}
percases ( n = 0 or n > 0 ) ;
suppose n = 0 ; ::_thesis: Solutions_of ((0. (K,n,0)),(0. (K,n,0))) = {{}}
hence Solutions_of ((0. (K,n,0)),(0. (K,n,0))) = {{}} by Th51; ::_thesis: verum
end;
supposeA1: n > 0 ; ::_thesis: Solutions_of ((0. (K,n,0)),(0. (K,n,0))) = {{}}
set B = 0. (K,n,0);
set A = 0. (K,n,0);
reconsider E = {} as Matrix of 0 , 0 ,K by MATRIX_1:13;
A2: width (0. (K,n,0)) = 0 by A1, MATRIX_1:23;
then A3: for i, j being Nat st [i,j] in Indices (0. (K,n,0)) holds
(0. (K,n,0)) * (i,j) = ((0. (K,n,0)) * E) * (i,j) by ZFMISC_1:90;
A4: Solutions_of ((0. (K,n,0)),(0. (K,n,0))) c= {{}}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Solutions_of ((0. (K,n,0)),(0. (K,n,0))) or x in {{}} )
assume x in Solutions_of ((0. (K,n,0)),(0. (K,n,0))) ; ::_thesis: x in {{}}
then reconsider X = x as Matrix of 0 , 0 ,K by A1, Th53;
len X = 0 by MATRIX_1:def_2;
then X = {} ;
hence x in {{}} by TARSKI:def_1; ::_thesis: verum
end;
A5: len E = 0 ;
A6: width E = 0 by MATRIX_1:24;
then A7: width ((0. (K,n,0)) * E) = 0 by A2, A5, MATRIX_3:def_4;
A8: len (0. (K,n,0)) = n by A1, MATRIX_1:23;
then len ((0. (K,n,0)) * E) = n by A2, A5, MATRIX_3:def_4;
then (0. (K,n,0)) * E = 0. (K,n,0) by A8, A2, A7, A3, MATRIX_1:21;
then E in Solutions_of ((0. (K,n,0)),(0. (K,n,0))) by A2, A5, A6;
hence Solutions_of ((0. (K,n,0)),(0. (K,n,0))) = {{}} by A4, ZFMISC_1:33; ::_thesis: verum
end;
end;
end;
begin
scheme :: MATRIX15:sch 1
GAUSS1{ F1() -> Field, F2() -> Nat, F3() -> Nat, F4() -> Nat, F5() -> Matrix of F2(),F3(),F1(), F6() -> Matrix of F2(),F4(),F1(), F7( Matrix of F2(),F4(),F1(), Nat, Nat, Element of F1()) -> Matrix of F2(),F4(),F1(), P1[ set , set ] } :
ex A9 being Matrix of F2(),F3(),F1() ex B9 being Matrix of F2(),F4(),F1() ex N being finite without_zero Subset of NAT st
( N c= Seg F3() & the_rank_of F5() = the_rank_of A9 & the_rank_of F5() = card N & P1[A9,B9] & Segm (A9,(Seg (card N)),N) is diagonal & ( for i being Nat st i in Seg (card N) holds
A9 * (i,((Sgm N) /. i)) <> 0. F1() ) & ( for i being Nat st i in dom A9 & i > card N holds
Line (A9,i) = F3() |-> (0. F1()) ) & ( for i, j being Nat st i in Seg (card N) & j in Seg (width A9) & j < (Sgm N) . i holds
A9 * (i,j) = 0. F1() ) )
provided
A1: P1[F5(),F6()] and
A2: for A9 being Matrix of F2(),F3(),F1()
for B9 being Matrix of F2(),F4(),F1() st P1[A9,B9] holds
for i, j being Nat st i <> j & j in dom A9 holds
for a being Element of F1() holds P1[ RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j))))),F7(B9,i,j,a)]
proof
defpred S1[ FinSequence of NAT , Nat, Nat, Matrix of F2(),F3(),F1()] means ( $4 * ($2,($1 /. $2)) <> 0. F1() & ( $3 in dom $1 & $2 < $3 implies $1 /. $2 < $1 /. $3 ) & ( $3 in (dom $1) \ {$2} implies $4 * ($3,($1 /. $2)) = 0. F1() ) & ( $3 in Seg (width $4) & $3 < $1 /. $2 implies $4 * ($2,$3) = 0. F1() ) );
set r = the_rank_of F5();
ex A9 being Matrix of F2(),F3(),F1() ex B9 being Matrix of F2(),F4(),F1() st
( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 & ( for i being Nat st i in dom A9 & i > the_rank_of F5() holds
Line (A9,i) = F3() |-> (0. F1()) ) & ex f being FinSequence of NAT st
( len f = the_rank_of A9 & f is one-to-one & rng f c= Seg (width A9) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) ) )
proof
percases ( F2() = 0 or F2() > 0 ) ;
supposeA3: F2() = 0 ; ::_thesis: ex A9 being Matrix of F2(),F3(),F1() ex B9 being Matrix of F2(),F4(),F1() st
( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 & ( for i being Nat st i in dom A9 & i > the_rank_of F5() holds
Line (A9,i) = F3() |-> (0. F1()) ) & ex f being FinSequence of NAT st
( len f = the_rank_of A9 & f is one-to-one & rng f c= Seg (width A9) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) ) )
take A9 = F5(); ::_thesis: ex B9 being Matrix of F2(),F4(),F1() st
( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 & ( for i being Nat st i in dom A9 & i > the_rank_of F5() holds
Line (A9,i) = F3() |-> (0. F1()) ) & ex f being FinSequence of NAT st
( len f = the_rank_of A9 & f is one-to-one & rng f c= Seg (width A9) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) ) )
take B9 = F6(); ::_thesis: ( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 & ( for i being Nat st i in dom A9 & i > the_rank_of F5() holds
Line (A9,i) = F3() |-> (0. F1()) ) & ex f being FinSequence of NAT st
( len f = the_rank_of A9 & f is one-to-one & rng f c= Seg (width A9) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) ) )
( dom A9 = Seg (len A9) & len A9 = 0 ) by A3, FINSEQ_1:def_3, MATRIX_1:def_2;
hence ( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 & ( for i being Nat st i in dom A9 & i > the_rank_of F5() holds
Line (A9,i) = F3() |-> (0. F1()) ) ) by A1; ::_thesis: ex f being FinSequence of NAT st
( len f = the_rank_of A9 & f is one-to-one & rng f c= Seg (width A9) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) )
take <*> NAT ; ::_thesis: ( len (<*> NAT) = the_rank_of A9 & <*> NAT is one-to-one & rng (<*> NAT) c= Seg (width A9) & ( for i, j being Nat st i in dom (<*> NAT) holds
S1[ <*> NAT,i,j,A9] ) )
len A9 = 0 by A3, MATRIX_1:22;
hence ( len (<*> NAT) = the_rank_of A9 & <*> NAT is one-to-one & rng (<*> NAT) c= Seg (width A9) & ( for i, j being Nat st i in dom (<*> NAT) holds
S1[ <*> NAT,i,j,A9] ) ) by MATRIX13:74, XBOOLE_1:2; ::_thesis: verum
end;
supposeA4: F2() > 0 ; ::_thesis: ex A9 being Matrix of F2(),F3(),F1() ex B9 being Matrix of F2(),F4(),F1() st
( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 & ( for i being Nat st i in dom A9 & i > the_rank_of F5() holds
Line (A9,i) = F3() |-> (0. F1()) ) & ex f being FinSequence of NAT st
( len f = the_rank_of A9 & f is one-to-one & rng f c= Seg (width A9) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) ) )
defpred S2[ Nat] means ( $1 <= F3() implies ex A9 being Matrix of F2(),F3(),F1() ex B9 being Matrix of F2(),F4(),F1() st
( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 & ex f being FinSequence of NAT st
( ( for i, j being Nat st [i,j] in Indices A9 & i > len f & j <= $1 holds
A9 * (i,j) = 0. F1() ) & f is one-to-one & len f <= $1 & len f <= F2() & rng f c= Seg $1 & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) ) ) );
A5: for n being Nat st S2[n] holds
S2[n + 1]
proof
let n be Nat; ::_thesis: ( S2[n] implies S2[n + 1] )
assume A6: S2[n] ; ::_thesis: S2[n + 1]
set n1 = n + 1;
assume A7: n + 1 <= F3() ; ::_thesis: ex A9 being Matrix of F2(),F3(),F1() ex B9 being Matrix of F2(),F4(),F1() st
( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 & ex f being FinSequence of NAT st
( ( for i, j being Nat st [i,j] in Indices A9 & i > len f & j <= n + 1 holds
A9 * (i,j) = 0. F1() ) & f is one-to-one & len f <= n + 1 & len f <= F2() & rng f c= Seg (n + 1) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) ) )
then consider A9 being Matrix of F2(),F3(),F1(), B9 being Matrix of F2(),F4(),F1() such that
A8: ( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 ) and
A9: ex f being FinSequence of NAT st
( ( for i, j being Nat st [i,j] in Indices A9 & i > len f & j <= n holds
A9 * (i,j) = 0. F1() ) & f is one-to-one & len f <= n & len f <= F2() & rng f c= Seg n & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) ) by A6, NAT_1:13;
consider f being FinSequence of NAT such that
A10: for i, j being Nat st [i,j] in Indices A9 & i > len f & j <= n holds
A9 * (i,j) = 0. F1() and
A11: f is one-to-one and
A12: len f <= n and
A13: len f <= F2() and
A14: rng f c= Seg n and
A15: for i, j being Nat st i in dom f holds
S1[f,i,j,A9] by A9;
percases ( for i, j being Nat st [i,j] in Indices A9 & i > len f & j = n + 1 holds
A9 * (i,j) = 0. F1() or ex i, j being Nat st
( [i,j] in Indices A9 & i > len f & j = n + 1 & A9 * (i,j) <> 0. F1() ) ) ;
supposeA16: for i, j being Nat st [i,j] in Indices A9 & i > len f & j = n + 1 holds
A9 * (i,j) = 0. F1() ; ::_thesis: ex A9 being Matrix of F2(),F3(),F1() ex B9 being Matrix of F2(),F4(),F1() st
( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 & ex f being FinSequence of NAT st
( ( for i, j being Nat st [i,j] in Indices A9 & i > len f & j <= n + 1 holds
A9 * (i,j) = 0. F1() ) & f is one-to-one & len f <= n + 1 & len f <= F2() & rng f c= Seg (n + 1) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) ) )
A17: now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_A9_&_i_>_len_f_&_j_<=_n_+_1_holds_
A9_*_(i,j)_=_0._F1()
let i, j be Nat; ::_thesis: ( [i,j] in Indices A9 & i > len f & j <= n + 1 implies A9 * (i,j) = 0. F1() )
assume that
A18: ( [i,j] in Indices A9 & i > len f ) and
A19: j <= n + 1 ; ::_thesis: A9 * (i,j) = 0. F1()
( j <= n or j = n + 1 ) by A19, NAT_1:8;
hence A9 * (i,j) = 0. F1() by A10, A16, A18; ::_thesis: verum
end;
n <= n + 1 by NAT_1:13;
then Seg n c= Seg (n + 1) by FINSEQ_1:5;
then A20: rng f c= Seg (n + 1) by A14, XBOOLE_1:1;
len f <= n + 1 by A12, NAT_1:12;
hence ex A9 being Matrix of F2(),F3(),F1() ex B9 being Matrix of F2(),F4(),F1() st
( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 & ex f being FinSequence of NAT st
( ( for i, j being Nat st [i,j] in Indices A9 & i > len f & j <= n + 1 holds
A9 * (i,j) = 0. F1() ) & f is one-to-one & len f <= n + 1 & len f <= F2() & rng f c= Seg (n + 1) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) ) ) by A8, A11, A13, A15, A17, A20; ::_thesis: verum
end;
suppose ex i, j being Nat st
( [i,j] in Indices A9 & i > len f & j = n + 1 & A9 * (i,j) <> 0. F1() ) ; ::_thesis: ex A9 being Matrix of F2(),F3(),F1() ex B9 being Matrix of F2(),F4(),F1() st
( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 & ex f being FinSequence of NAT st
( ( for i, j being Nat st [i,j] in Indices A9 & i > len f & j <= n + 1 holds
A9 * (i,j) = 0. F1() ) & f is one-to-one & len f <= n + 1 & len f <= F2() & rng f c= Seg (n + 1) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) ) )
then consider i0, j0 being Nat such that
A21: [i0,j0] in Indices A9 and
A22: i0 > len f and
A23: j0 = n + 1 and
A24: A9 * (i0,j0) <> 0. F1() ;
A25: Indices A9 = [:(Seg F2()),(Seg F3()):] by A4, MATRIX_1:23;
then A26: n + 1 in Seg F3() by A21, A23, ZFMISC_1:87;
A27: i0 in Seg F2() by A21, A25, ZFMISC_1:87;
then A28: i0 <= F2() by FINSEQ_1:1;
A29: (len f) + 1 <= i0 by A22, NAT_1:13;
then ( 0 + 1 <= (len f) + 1 & (len f) + 1 <= F2() ) by A28, XREAL_1:7, XXREAL_0:2;
then A30: (len f) + 1 in Seg F2() ;
defpred S3[ Nat] means ( $1 <= F2() implies ex A9 being Matrix of F2(),F3(),F1() ex B9 being Matrix of F2(),F4(),F1() st
( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 & A9 * (((len f) + 1),(n + 1)) <> 0. F1() & ( for i, j being Nat st [i,j] in Indices A9 & i > len f & j <= n holds
A9 * (i,j) = 0. F1() ) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) & ( for j being Nat st j in (dom A9) \ {((len f) + 1)} & j <= $1 holds
A9 * (j,(n + 1)) = 0. F1() ) ) );
A31: dom f = Seg (len f) by FINSEQ_1:def_3;
n <= F3() by A7, NAT_1:13;
then A32: Seg n c= Seg F3() by FINSEQ_1:5;
A33: Seg (len f) c= Seg F2() by A13, FINSEQ_1:5;
A34: for k being Nat st S3[k] holds
S3[k + 1]
proof
let k be Nat; ::_thesis: ( S3[k] implies S3[k + 1] )
assume A35: S3[k] ; ::_thesis: S3[k + 1]
set k1 = k + 1;
assume k + 1 <= F2() ; ::_thesis: ex A9 being Matrix of F2(),F3(),F1() ex B9 being Matrix of F2(),F4(),F1() st
( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 & A9 * (((len f) + 1),(n + 1)) <> 0. F1() & ( for i, j being Nat st [i,j] in Indices A9 & i > len f & j <= n holds
A9 * (i,j) = 0. F1() ) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) & ( for j being Nat st j in (dom A9) \ {((len f) + 1)} & j <= k + 1 holds
A9 * (j,(n + 1)) = 0. F1() ) )
then consider AA being Matrix of F2(),F3(),F1(), BB being Matrix of F2(),F4(),F1() such that
A36: P1[AA,BB] and
A37: the_rank_of F5() = the_rank_of AA and
A38: AA * (((len f) + 1),(n + 1)) <> 0. F1() and
A39: for i, j being Nat st [i,j] in Indices AA & i > len f & j <= n holds
AA * (i,j) = 0. F1() and
A40: for i, j being Nat st i in dom f holds
S1[f,i,j,AA] and
A41: for j being Nat st j in (dom AA) \ {((len f) + 1)} & j <= k holds
AA * (j,(n + 1)) = 0. F1() by A35, NAT_1:13;
now__::_thesis:_ex_RA_being_Matrix_of_F2(),F3(),F1()_ex_RB_being_Matrix_of_F2(),F4(),F1()_ex_A9_being_Matrix_of_F2(),F3(),F1()_ex_B9_being_Matrix_of_F2(),F4(),F1()_st_
(_P1[A9,B9]_&_the_rank_of_F5()_=_the_rank_of_A9_&_A9_*_(((len_f)_+_1),(n_+_1))_<>_0._F1()_&_(_for_i,_j_being_Nat_st_[i,j]_in_Indices_A9_&_i_>_len_f_&_j_<=_n_holds_
A9_*_(i,j)_=_0._F1()_)_&_(_for_i,_j_being_Nat_st_i_in_dom_f_holds_
S1[f,i,j,A9]_)_&_(_for_j_being_Nat_st_j_in_(dom_A9)_\_{((len_f)_+_1)}_&_j_<=_k_+_1_holds_
A9_*_(j,(n_+_1))_=_0._F1()_)_)
percases ( k + 1 = (len f) + 1 or k + 1 <> (len f) + 1 ) ;
supposeA42: k + 1 = (len f) + 1 ; ::_thesis: ex RA being Matrix of F2(),F3(),F1() ex RB being Matrix of F2(),F4(),F1() ex A9 being Matrix of F2(),F3(),F1() ex B9 being Matrix of F2(),F4(),F1() st
( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 & A9 * (((len f) + 1),(n + 1)) <> 0. F1() & ( for i, j being Nat st [i,j] in Indices A9 & i > len f & j <= n holds
A9 * (i,j) = 0. F1() ) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) & ( for j being Nat st j in (dom A9) \ {((len f) + 1)} & j <= k + 1 holds
A9 * (j,(n + 1)) = 0. F1() ) )
take RA = AA; ::_thesis: ex RB being Matrix of F2(),F4(),F1() ex A9 being Matrix of F2(),F3(),F1() ex B9 being Matrix of F2(),F4(),F1() st
( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 & A9 * (((len f) + 1),(n + 1)) <> 0. F1() & ( for i, j being Nat st [i,j] in Indices A9 & i > len f & j <= n holds
A9 * (i,j) = 0. F1() ) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) & ( for j being Nat st j in (dom A9) \ {((len f) + 1)} & j <= k + 1 holds
A9 * (j,(n + 1)) = 0. F1() ) )
take RB = BB; ::_thesis: ex A9 being Matrix of F2(),F3(),F1() ex B9 being Matrix of F2(),F4(),F1() st
( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 & A9 * (((len f) + 1),(n + 1)) <> 0. F1() & ( for i, j being Nat st [i,j] in Indices A9 & i > len f & j <= n holds
A9 * (i,j) = 0. F1() ) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) & ( for j being Nat st j in (dom A9) \ {((len f) + 1)} & j <= k + 1 holds
A9 * (j,(n + 1)) = 0. F1() ) )
now__::_thesis:_for_j_being_Nat_st_j_in_(dom_RA)_\_{((len_f)_+_1)}_&_j_<=_k_+_1_holds_
RA_*_(j,(n_+_1))_=_0._F1()
let j be Nat; ::_thesis: ( j in (dom RA) \ {((len f) + 1)} & j <= k + 1 implies RA * (j,(n + 1)) = 0. F1() )
assume that
A43: j in (dom RA) \ {((len f) + 1)} and
A44: j <= k + 1 ; ::_thesis: RA * (j,(n + 1)) = 0. F1()
j <> (len f) + 1 by A43, ZFMISC_1:56;
then j < k + 1 by A42, A44, XXREAL_0:1;
then j <= k by NAT_1:13;
hence RA * (j,(n + 1)) = 0. F1() by A41, A43; ::_thesis: verum
end;
hence ex A9 being Matrix of F2(),F3(),F1() ex B9 being Matrix of F2(),F4(),F1() st
( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 & A9 * (((len f) + 1),(n + 1)) <> 0. F1() & ( for i, j being Nat st [i,j] in Indices A9 & i > len f & j <= n holds
A9 * (i,j) = 0. F1() ) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) & ( for j being Nat st j in (dom A9) \ {((len f) + 1)} & j <= k + 1 holds
A9 * (j,(n + 1)) = 0. F1() ) ) by A36, A37, A38, A39, A40; ::_thesis: verum
end;
supposeA45: k + 1 <> (len f) + 1 ; ::_thesis: ex RA being Matrix of F2(),F3(), the carrier of F1() ex RB being Matrix of F2(),F4(),F1() ex A9 being Matrix of F2(),F3(),F1() ex B9 being Matrix of F2(),F4(),F1() st
( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 & A9 * (((len f) + 1),(n + 1)) <> 0. F1() & ( for i, j being Nat st [i,j] in Indices A9 & i > len f & j <= n holds
A9 * (i,j) = 0. F1() ) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) & ( for j being Nat st j in (dom A9) \ {((len f) + 1)} & j <= k + 1 holds
A9 * (j,(n + 1)) = 0. F1() ) )
set LA = Line (AA,(k + 1));
set LAf = Line (AA,((len f) + 1));
set a = AA * (((len f) + 1),(n + 1));
set RA = RLine (AA,(k + 1),((Line (AA,(k + 1))) + ((- ((AA * ((k + 1),(n + 1))) * ((AA * (((len f) + 1),(n + 1))) "))) * (Line (AA,((len f) + 1))))));
A46: width AA = F3() by A4, MATRIX_1:23;
then A47: len ((Line (AA,(k + 1))) + ((- ((AA * ((k + 1),(n + 1))) * ((AA * (((len f) + 1),(n + 1))) "))) * (Line (AA,((len f) + 1))))) = F3() by CARD_1:def_7;
A48: Indices A9 = Indices AA by MATRIX_1:26;
then [((len f) + 1),(n + 1)] in Indices AA by A25, A30, A26, ZFMISC_1:87;
then A49: (RLine (AA,(k + 1),((Line (AA,(k + 1))) + ((- ((AA * ((k + 1),(n + 1))) * ((AA * (((len f) + 1),(n + 1))) "))) * (Line (AA,((len f) + 1))))))) * (((len f) + 1),(n + 1)) <> 0. F1() by A38, A45, A46, A47, MATRIX11:def_3;
A50: Indices A9 = Indices (RLine (AA,(k + 1),((Line (AA,(k + 1))) + ((- ((AA * ((k + 1),(n + 1))) * ((AA * (((len f) + 1),(n + 1))) "))) * (Line (AA,((len f) + 1))))))) by MATRIX_1:26;
A51: now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_(RLine_(AA,(k_+_1),((Line_(AA,(k_+_1)))_+_((-_((AA_*_((k_+_1),(n_+_1)))_*_((AA_*_(((len_f)_+_1),(n_+_1)))_")))_*_(Line_(AA,((len_f)_+_1)))))))_&_i_>_len_f_&_j_<=_n_holds_
(RLine_(AA,(k_+_1),((Line_(AA,(k_+_1)))_+_((-_((AA_*_((k_+_1),(n_+_1)))_*_((AA_*_(((len_f)_+_1),(n_+_1)))_")))_*_(Line_(AA,((len_f)_+_1)))))))_*_(i,j)_=_0._F1()
let i, j be Nat; ::_thesis: ( [i,j] in Indices (RLine (AA,(k + 1),((Line (AA,(k + 1))) + ((- ((AA * ((k + 1),(n + 1))) * ((AA * (((len f) + 1),(n + 1))) "))) * (Line (AA,((len f) + 1))))))) & i > len f & j <= n implies (RLine (AA,(k + 1),((Line (AA,(k + 1))) + ((- ((AA * ((k + 1),(n + 1))) * ((AA * (((len f) + 1),(n + 1))) "))) * (Line (AA,((len f) + 1))))))) * (i,j) = 0. F1() )
assume that
A52: [i,j] in Indices (RLine (AA,(k + 1),((Line (AA,(k + 1))) + ((- ((AA * ((k + 1),(n + 1))) * ((AA * (((len f) + 1),(n + 1))) "))) * (Line (AA,((len f) + 1))))))) and
A53: i > len f and
A54: j <= n ; ::_thesis: (RLine (AA,(k + 1),((Line (AA,(k + 1))) + ((- ((AA * ((k + 1),(n + 1))) * ((AA * (((len f) + 1),(n + 1))) "))) * (Line (AA,((len f) + 1))))))) * (i,j) = 0. F1()
now__::_thesis:_(RLine_(AA,(k_+_1),((Line_(AA,(k_+_1)))_+_((-_((AA_*_((k_+_1),(n_+_1)))_*_((AA_*_(((len_f)_+_1),(n_+_1)))_")))_*_(Line_(AA,((len_f)_+_1)))))))_*_(i,j)_=_0._F1()
percases ( i = k + 1 or i <> k + 1 ) ;
supposeA55: i = k + 1 ; ::_thesis: (RLine (AA,(k + 1),((Line (AA,(k + 1))) + ((- ((AA * ((k + 1),(n + 1))) * ((AA * (((len f) + 1),(n + 1))) "))) * (Line (AA,((len f) + 1))))))) * (i,j) = 0. F1()
A56: j in Seg F3() by A25, A50, A52, ZFMISC_1:87;
then ( (len f) + 1 > len f & [((len f) + 1),j] in Indices A9 ) by A25, A30, NAT_1:13, ZFMISC_1:87;
then AA * (((len f) + 1),j) = 0. F1() by A39, A48, A54;
then (Line (AA,((len f) + 1))) . j = 0. F1() by A46, A56, MATRIX_1:def_7;
then A57: ((- ((AA * ((k + 1),(n + 1))) * ((AA * (((len f) + 1),(n + 1))) "))) * (Line (AA,((len f) + 1)))) . j = (- ((AA * ((k + 1),(n + 1))) * ((AA * (((len f) + 1),(n + 1))) "))) * (0. F1()) by A46, A56, FVSUM_1:51
.= 0. F1() by VECTSP_1:6 ;
(Line (AA,(k + 1))) . j = AA * ((k + 1),j) by A46, A56, MATRIX_1:def_7
.= 0. F1() by A39, A48, A50, A52, A53, A54, A55 ;
then (0. F1()) + (0. F1()) = ((Line (AA,(k + 1))) + ((- ((AA * ((k + 1),(n + 1))) * ((AA * (((len f) + 1),(n + 1))) "))) * (Line (AA,((len f) + 1))))) . j by A46, A56, A57, FVSUM_1:18
.= (RLine (AA,(k + 1),((Line (AA,(k + 1))) + ((- ((AA * ((k + 1),(n + 1))) * ((AA * (((len f) + 1),(n + 1))) "))) * (Line (AA,((len f) + 1))))))) * (i,j) by A48, A50, A46, A47, A52, A55, MATRIX11:def_3 ;
hence (RLine (AA,(k + 1),((Line (AA,(k + 1))) + ((- ((AA * ((k + 1),(n + 1))) * ((AA * (((len f) + 1),(n + 1))) "))) * (Line (AA,((len f) + 1))))))) * (i,j) = 0. F1() by RLVECT_1:def_4; ::_thesis: verum
end;
suppose i <> k + 1 ; ::_thesis: (RLine (AA,(k + 1),((Line (AA,(k + 1))) + ((- ((AA * ((k + 1),(n + 1))) * ((AA * (((len f) + 1),(n + 1))) "))) * (Line (AA,((len f) + 1))))))) * (i,j) = 0. F1()
hence (RLine (AA,(k + 1),((Line (AA,(k + 1))) + ((- ((AA * ((k + 1),(n + 1))) * ((AA * (((len f) + 1),(n + 1))) "))) * (Line (AA,((len f) + 1))))))) * (i,j) = AA * (i,j) by A48, A50, A46, A47, A52, MATRIX11:def_3
.= 0. F1() by A39, A48, A50, A52, A53, A54 ;
::_thesis: verum
end;
end;
end;
hence (RLine (AA,(k + 1),((Line (AA,(k + 1))) + ((- ((AA * ((k + 1),(n + 1))) * ((AA * (((len f) + 1),(n + 1))) "))) * (Line (AA,((len f) + 1))))))) * (i,j) = 0. F1() ; ::_thesis: verum
end;
set RB = F7(BB,(k + 1),((len f) + 1),(- ((AA * ((k + 1),(n + 1))) * ((AA * (((len f) + 1),(n + 1))) "))));
take RA = RLine (AA,(k + 1),((Line (AA,(k + 1))) + ((- ((AA * ((k + 1),(n + 1))) * ((AA * (((len f) + 1),(n + 1))) "))) * (Line (AA,((len f) + 1)))))); ::_thesis: ex RB being Matrix of F2(),F4(),F1() ex A9 being Matrix of F2(),F3(),F1() ex B9 being Matrix of F2(),F4(),F1() st
( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 & A9 * (((len f) + 1),(n + 1)) <> 0. F1() & ( for i, j being Nat st [i,j] in Indices A9 & i > len f & j <= n holds
A9 * (i,j) = 0. F1() ) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) & ( for j being Nat st j in (dom A9) \ {((len f) + 1)} & j <= k + 1 holds
A9 * (j,(n + 1)) = 0. F1() ) )
take RB = F7(BB,(k + 1),((len f) + 1),(- ((AA * ((k + 1),(n + 1))) * ((AA * (((len f) + 1),(n + 1))) ")))); ::_thesis: ex A9 being Matrix of F2(),F3(),F1() ex B9 being Matrix of F2(),F4(),F1() st
( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 & A9 * (((len f) + 1),(n + 1)) <> 0. F1() & ( for i, j being Nat st [i,j] in Indices A9 & i > len f & j <= n holds
A9 * (i,j) = 0. F1() ) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) & ( for j being Nat st j in (dom A9) \ {((len f) + 1)} & j <= k + 1 holds
A9 * (j,(n + 1)) = 0. F1() ) )
A58: width RA = F3() by A4, MATRIX_1:23;
A59: len AA = F2() by MATRIX_1:def_2;
A60: now__::_thesis:_for_j_being_Nat_st_j_in_(dom_RA)_\_{((len_f)_+_1)}_&_j_<=_k_+_1_holds_
RA_*_(j,(n_+_1))_=_0._F1()
A61: dom AA = Seg (len AA) by FINSEQ_1:def_3;
let j be Nat; ::_thesis: ( j in (dom RA) \ {((len f) + 1)} & j <= k + 1 implies RA * (j,(n + 1)) = 0. F1() )
assume that
A62: j in (dom RA) \ {((len f) + 1)} and
A63: j <= k + 1 ; ::_thesis: RA * (j,(n + 1)) = 0. F1()
j in dom RA by A62, XBOOLE_0:def_5;
then A64: [j,(n + 1)] in Indices AA by A26, A48, A50, A58, ZFMISC_1:87;
A65: ( dom RA = Seg (len RA) & len RA = F2() ) by FINSEQ_1:def_3, MATRIX_1:def_2;
now__::_thesis:_RA_*_(j,(n_+_1))_=_0._F1()
percases ( j <= k or j = k + 1 ) by A63, NAT_1:8;
supposeA66: j <= k ; ::_thesis: RA * (j,(n + 1)) = 0. F1()
then j < k + 1 by NAT_1:13;
hence RA * (j,(n + 1)) = AA * (j,(n + 1)) by A46, A47, A64, MATRIX11:def_3
.= 0. F1() by A41, A59, A62, A65, A61, A66 ;
::_thesis: verum
end;
supposeA67: j = k + 1 ; ::_thesis: RA * (j,(n + 1)) = 0. F1()
(Line (AA,((len f) + 1))) . (n + 1) = AA * (((len f) + 1),(n + 1)) by A26, A46, MATRIX_1:def_7;
then A68: ((- ((AA * ((k + 1),(n + 1))) * ((AA * (((len f) + 1),(n + 1))) "))) * (Line (AA,((len f) + 1)))) . (n + 1) = (- ((AA * ((k + 1),(n + 1))) * ((AA * (((len f) + 1),(n + 1))) "))) * (AA * (((len f) + 1),(n + 1))) by A26, A46, FVSUM_1:51
.= ((- (AA * ((k + 1),(n + 1)))) * ((AA * (((len f) + 1),(n + 1))) ")) * (AA * (((len f) + 1),(n + 1))) by VECTSP_1:9
.= (- (AA * ((k + 1),(n + 1)))) * (((AA * (((len f) + 1),(n + 1))) ") * (AA * (((len f) + 1),(n + 1)))) by GROUP_1:def_3
.= (- (AA * ((k + 1),(n + 1)))) * (1_ F1()) by A38, VECTSP_1:def_10
.= - (AA * ((k + 1),(n + 1))) by VECTSP_1:def_4 ;
(Line (AA,(k + 1))) . (n + 1) = AA * ((k + 1),(n + 1)) by A26, A46, MATRIX_1:def_7;
then ((Line (AA,(k + 1))) + ((- ((AA * ((k + 1),(n + 1))) * ((AA * (((len f) + 1),(n + 1))) "))) * (Line (AA,((len f) + 1))))) . (n + 1) = (AA * ((k + 1),(n + 1))) + (- (AA * ((k + 1),(n + 1)))) by A26, A46, A68, FVSUM_1:18
.= 0. F1() by VECTSP_1:19 ;
hence RA * (j,(n + 1)) = 0. F1() by A46, A47, A64, A67, MATRIX11:def_3; ::_thesis: verum
end;
end;
end;
hence RA * (j,(n + 1)) = 0. F1() ; ::_thesis: verum
end;
A69: dom AA = Seg (len AA) by FINSEQ_1:def_3;
A70: now__::_thesis:_for_i,_j_being_Nat_st_i_in_dom_f_holds_
(_RA_*_(i,(f_/._i))_<>_0._F1()_&_(_j_in_dom_f_&_i_<_j_implies_f_/._i_<_f_/._j_)_&_(_j_in_(dom_f)_\_{i}_implies_RA_*_(j,(f_/._i))_=_0._F1()_)_&_(_j_in_Seg_(width_RA)_&_j_<_f_/._i_implies_RA_*_(i,j)_=_0._F1()_)_)
let i, j be Nat; ::_thesis: ( i in dom f implies ( RA * (i,(f /. i)) <> 0. F1() & ( j in dom f & i < j implies f /. i < f /. j ) & ( j in (dom f) \ {i} implies RA * (j,(f /. i)) = 0. F1() ) & ( j in Seg (width RA) & j < f /. i implies RA * (i,j) = 0. F1() ) ) )
assume A71: i in dom f ; ::_thesis: ( RA * (i,(f /. i)) <> 0. F1() & ( j in dom f & i < j implies f /. i < f /. j ) & ( j in (dom f) \ {i} implies RA * (j,(f /. i)) = 0. F1() ) & ( j in Seg (width RA) & j < f /. i implies RA * (i,j) = 0. F1() ) )
set fi = f /. i;
A72: ( f /. i = f . i & f . i in rng f ) by A71, FUNCT_1:def_3, PARTFUN1:def_6;
then A73: f /. i in Seg n by A14;
A74: ( (len f) + 1 > len f & f /. i <= n ) by A14, A72, FINSEQ_1:1, NAT_1:13;
( [((len f) + 1),(f /. i)] in Indices AA & (Line (AA,((len f) + 1))) . (f /. i) = AA * (((len f) + 1),(f /. i)) ) by A25, A32, A30, A48, A46, A73, MATRIX_1:def_7, ZFMISC_1:87;
then (Line (AA,((len f) + 1))) . (f /. i) = 0. F1() by A39, A74;
then A75: ((- ((AA * ((k + 1),(n + 1))) * ((AA * (((len f) + 1),(n + 1))) "))) * (Line (AA,((len f) + 1)))) . (f /. i) = (- ((AA * ((k + 1),(n + 1))) * ((AA * (((len f) + 1),(n + 1))) "))) * (0. F1()) by A32, A46, A73, FVSUM_1:51
.= 0. F1() by VECTSP_1:12 ;
(Line (AA,(k + 1))) . (f /. i) = AA * ((k + 1),(f /. i)) by A32, A46, A73, MATRIX_1:def_7;
then A76: ((Line (AA,(k + 1))) + ((- ((AA * ((k + 1),(n + 1))) * ((AA * (((len f) + 1),(n + 1))) "))) * (Line (AA,((len f) + 1))))) . (f /. i) = (AA * ((k + 1),(f /. i))) + (0. F1()) by A32, A46, A73, A75, FVSUM_1:18
.= AA * ((k + 1),(f /. i)) by RLVECT_1:def_4 ;
A77: [i,(f /. i)] in Indices AA by A25, A32, A33, A31, A48, A71, A73, ZFMISC_1:87;
now__::_thesis:_RA_*_(i,(f_/._i))_<>_0._F1()
percases ( i <> k + 1 or i = k + 1 ) ;
suppose i <> k + 1 ; ::_thesis: RA * (i,(f /. i)) <> 0. F1()
then RA * (i,(f /. i)) = AA * (i,(f /. i)) by A46, A47, A77, MATRIX11:def_3;
hence RA * (i,(f /. i)) <> 0. F1() by A40, A71; ::_thesis: verum
end;
suppose i = k + 1 ; ::_thesis: RA * (i,(f /. i)) <> 0. F1()
then RA * (i,(f /. i)) = AA * (i,(f /. i)) by A46, A47, A77, A76, MATRIX11:def_3;
hence RA * (i,(f /. i)) <> 0. F1() by A40, A71; ::_thesis: verum
end;
end;
end;
hence ( RA * (i,(f /. i)) <> 0. F1() & ( j in dom f & i < j implies f /. i < f /. j ) ) by A15, A71; ::_thesis: ( ( j in (dom f) \ {i} implies RA * (j,(f /. i)) = 0. F1() ) & ( j in Seg (width RA) & j < f /. i implies RA * (i,j) = 0. F1() ) )
thus ( j in (dom f) \ {i} implies RA * (j,(f /. i)) = 0. F1() ) ::_thesis: ( j in Seg (width RA) & j < f /. i implies RA * (i,j) = 0. F1() )
proof
assume A78: j in (dom f) \ {i} ; ::_thesis: RA * (j,(f /. i)) = 0. F1()
then j in Seg (len f) by A31, XBOOLE_0:def_5;
then A79: [j,(f /. i)] in Indices AA by A25, A32, A33, A48, A73, ZFMISC_1:87;
percases ( j <> k + 1 or j = k + 1 ) ;
suppose j <> k + 1 ; ::_thesis: RA * (j,(f /. i)) = 0. F1()
hence RA * (j,(f /. i)) = AA * (j,(f /. i)) by A46, A47, A79, MATRIX11:def_3
.= 0. F1() by A40, A71, A78 ;
::_thesis: verum
end;
supposeA80: j = k + 1 ; ::_thesis: RA * (j,(f /. i)) = 0. F1()
hence RA * (j,(f /. i)) = ((Line (AA,(k + 1))) + ((- ((AA * ((k + 1),(n + 1))) * ((AA * (((len f) + 1),(n + 1))) "))) * (Line (AA,((len f) + 1))))) . (f /. i) by A46, A47, A79, MATRIX11:def_3
.= 0. F1() by A40, A71, A76, A78, A80 ;
::_thesis: verum
end;
end;
end;
thus ( j in Seg (width RA) & j < f /. i implies RA * (i,j) = 0. F1() ) ::_thesis: verum
proof
assume that
A81: j in Seg (width RA) and
A82: j < f /. i ; ::_thesis: RA * (i,j) = 0. F1()
A83: [((len f) + 1),j] in Indices AA by A30, A59, A69, A46, A58, A81, ZFMISC_1:87;
A84: [i,j] in Indices AA by A33, A31, A59, A69, A46, A58, A71, A81, ZFMISC_1:87;
percases ( i <> k + 1 or i = k + 1 ) ;
suppose i <> k + 1 ; ::_thesis: RA * (i,j) = 0. F1()
hence RA * (i,j) = AA * (i,j) by A46, A47, A84, MATRIX11:def_3
.= 0. F1() by A40, A46, A58, A71, A81, A82 ;
::_thesis: verum
end;
supposeA85: i = k + 1 ; ::_thesis: RA * (i,j) = 0. F1()
f /. i <= n by A14, A72, FINSEQ_1:1;
then A86: j <= n by A82, XXREAL_0:2;
(len f) + 1 > len f by NAT_1:13;
then 0. F1() = AA * (((len f) + 1),j) by A39, A83, A86
.= (Line (AA,((len f) + 1))) . j by A46, A58, A81, MATRIX_1:def_7 ;
then A87: ((- ((AA * ((k + 1),(n + 1))) * ((AA * (((len f) + 1),(n + 1))) "))) * (Line (AA,((len f) + 1)))) . j = (- ((AA * ((k + 1),(n + 1))) * ((AA * (((len f) + 1),(n + 1))) "))) * (0. F1()) by A46, A58, A81, FVSUM_1:51
.= 0. F1() by VECTSP_1:6 ;
(Line (AA,(k + 1))) . j = AA * (i,j) by A46, A58, A81, A85, MATRIX_1:def_7
.= 0. F1() by A40, A46, A58, A71, A81, A82 ;
then ((Line (AA,(k + 1))) + ((- ((AA * ((k + 1),(n + 1))) * ((AA * (((len f) + 1),(n + 1))) "))) * (Line (AA,((len f) + 1))))) . j = (0. F1()) + (0. F1()) by A46, A58, A81, A87, FVSUM_1:18
.= 0. F1() by RLVECT_1:def_4 ;
hence RA * (i,j) = 0. F1() by A46, A47, A84, A85, MATRIX11:def_3; ::_thesis: verum
end;
end;
end;
end;
A88: (len f) + 1 in Seg (len AA) by A4, A30, MATRIX_1:23;
then A89: the_rank_of F5() = the_rank_of RA by A37, A45, MATRIX13:92;
P1[RA,RB] by A2, A36, A45, A88, A69;
hence ex A9 being Matrix of F2(),F3(),F1() ex B9 being Matrix of F2(),F4(),F1() st
( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 & A9 * (((len f) + 1),(n + 1)) <> 0. F1() & ( for i, j being Nat st [i,j] in Indices A9 & i > len f & j <= n holds
A9 * (i,j) = 0. F1() ) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) & ( for j being Nat st j in (dom A9) \ {((len f) + 1)} & j <= k + 1 holds
A9 * (j,(n + 1)) = 0. F1() ) ) by A89, A49, A51, A70, A60; ::_thesis: verum
end;
end;
end;
hence ex A9 being Matrix of F2(),F3(),F1() ex B9 being Matrix of F2(),F4(),F1() st
( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 & A9 * (((len f) + 1),(n + 1)) <> 0. F1() & ( for i, j being Nat st [i,j] in Indices A9 & i > len f & j <= n holds
A9 * (i,j) = 0. F1() ) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) & ( for j being Nat st j in (dom A9) \ {((len f) + 1)} & j <= k + 1 holds
A9 * (j,(n + 1)) = 0. F1() ) ) ; ::_thesis: verum
end;
A90: j0 in Seg F3() by A21, A25, ZFMISC_1:87;
ex A9 being Matrix of F2(),F3(),F1() ex B9 being Matrix of F2(),F4(),F1() st
( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 & A9 * (((len f) + 1),(n + 1)) <> 0. F1() & ( for i, j being Nat st [i,j] in Indices A9 & i > len f & j <= n holds
A9 * (i,j) = 0. F1() ) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) )
proof
percases ( A9 * (((len f) + 1),(n + 1)) <> 0. F1() or A9 * (((len f) + 1),(n + 1)) = 0. F1() ) ;
suppose A9 * (((len f) + 1),(n + 1)) <> 0. F1() ; ::_thesis: ex A9 being Matrix of F2(),F3(),F1() ex B9 being Matrix of F2(),F4(),F1() st
( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 & A9 * (((len f) + 1),(n + 1)) <> 0. F1() & ( for i, j being Nat st [i,j] in Indices A9 & i > len f & j <= n holds
A9 * (i,j) = 0. F1() ) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) )
hence ex A9 being Matrix of F2(),F3(),F1() ex B9 being Matrix of F2(),F4(),F1() st
( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 & A9 * (((len f) + 1),(n + 1)) <> 0. F1() & ( for i, j being Nat st [i,j] in Indices A9 & i > len f & j <= n holds
A9 * (i,j) = 0. F1() ) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) ) by A8, A10, A15; ::_thesis: verum
end;
supposeA91: A9 * (((len f) + 1),(n + 1)) = 0. F1() ; ::_thesis: ex A9 being Matrix of F2(),F3(),F1() ex B9 being Matrix of F2(),F4(),F1() st
( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 & A9 * (((len f) + 1),(n + 1)) <> 0. F1() & ( for i, j being Nat st [i,j] in Indices A9 & i > len f & j <= n holds
A9 * (i,j) = 0. F1() ) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) )
set RB = F7(B9,((len f) + 1),i0,(1_ F1()));
set LA = Line (A9,i0);
set LAf = Line (A9,((len f) + 1));
set RA = RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))));
take RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0))))) ; ::_thesis: ex B9 being Matrix of F2(),F4(),F1() st
( P1[ RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0))))),B9] & the_rank_of F5() = the_rank_of (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) & (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) * (((len f) + 1),(n + 1)) <> 0. F1() & ( for i, j being Nat st [i,j] in Indices (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) & i > len f & j <= n holds
(RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) * (i,j) = 0. F1() ) & ( for i, j being Nat st i in dom f holds
S1[f,i,j, RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))] ) )
take F7(B9,((len f) + 1),i0,(1_ F1())) ; ::_thesis: ( P1[ RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0))))),F7(B9,((len f) + 1),i0,(1_ F1()))] & the_rank_of F5() = the_rank_of (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) & (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) * (((len f) + 1),(n + 1)) <> 0. F1() & ( for i, j being Nat st [i,j] in Indices (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) & i > len f & j <= n holds
(RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) * (i,j) = 0. F1() ) & ( for i, j being Nat st i in dom f holds
S1[f,i,j, RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))] ) )
( i0 in dom A9 & dom A9 = Seg (len A9) ) by A21, FINSEQ_1:def_3, ZFMISC_1:87;
hence ( P1[ RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0))))),F7(B9,((len f) + 1),i0,(1_ F1()))] & the_rank_of F5() = the_rank_of (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) ) by A2, A8, A23, A24, A91, MATRIX13:92; ::_thesis: ( (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) * (((len f) + 1),(n + 1)) <> 0. F1() & ( for i, j being Nat st [i,j] in Indices (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) & i > len f & j <= n holds
(RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) * (i,j) = 0. F1() ) & ( for i, j being Nat st i in dom f holds
S1[f,i,j, RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))] ) )
A92: ( (1_ F1()) * (Line (A9,i0)) = Line (A9,i0) & len ((Line (A9,((len f) + 1))) + (Line (A9,i0))) = width A9 ) by CARD_1:def_7, FVSUM_1:57;
[((len f) + 1),j0] in Indices A9 by A25, A30, A90, ZFMISC_1:87;
then A93: (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) * (((len f) + 1),(n + 1)) = ((Line (A9,((len f) + 1))) + (Line (A9,i0))) . (n + 1) by A23, A92, MATRIX11:def_3;
A94: width A9 = F3() by A4, MATRIX_1:23;
then A95: (Line (A9,i0)) . (n + 1) = A9 * (i0,(n + 1)) by A23, A90, MATRIX_1:def_7;
(Line (A9,((len f) + 1))) . (n + 1) = 0. F1() by A23, A90, A91, A94, MATRIX_1:def_7;
then (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) * (((len f) + 1),(n + 1)) = (0. F1()) + (A9 * (i0,(n + 1))) by A26, A94, A93, A95, FVSUM_1:18
.= A9 * (i0,(n + 1)) by RLVECT_1:def_4 ;
hence (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) * (((len f) + 1),(n + 1)) <> 0. F1() by A23, A24; ::_thesis: ( ( for i, j being Nat st [i,j] in Indices (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) & i > len f & j <= n holds
(RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) * (i,j) = 0. F1() ) & ( for i, j being Nat st i in dom f holds
S1[f,i,j, RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))] ) )
A96: Indices (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) = Indices A9 by MATRIX_1:26;
now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_(RLine_(A9,((len_f)_+_1),((Line_(A9,((len_f)_+_1)))_+_((1__F1())_*_(Line_(A9,i0))))))_&_i_>_len_f_&_j_<=_n_holds_
(RLine_(A9,((len_f)_+_1),((Line_(A9,((len_f)_+_1)))_+_((1__F1())_*_(Line_(A9,i0))))))_*_(i,j)_=_0._F1()
let i, j be Nat; ::_thesis: ( [i,j] in Indices (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) & i > len f & j <= n implies (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) * (i,j) = 0. F1() )
assume that
A97: [i,j] in Indices (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) and
A98: i > len f and
A99: j <= n ; ::_thesis: (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) * (i,j) = 0. F1()
A100: j in Seg F3() by A94, A96, A97, ZFMISC_1:87;
A101: i >= (len f) + 1 by A98, NAT_1:13;
now__::_thesis:_(RLine_(A9,((len_f)_+_1),((Line_(A9,((len_f)_+_1)))_+_((1__F1())_*_(Line_(A9,i0))))))_*_(i,j)_=_0._F1()
percases ( i > (len f) + 1 or i = (len f) + 1 ) by A101, XXREAL_0:1;
suppose i > (len f) + 1 ; ::_thesis: (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) * (i,j) = 0. F1()
hence (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) * (i,j) = A9 * (i,j) by A92, A96, A97, MATRIX11:def_3
.= 0. F1() by A10, A96, A97, A98, A99 ;
::_thesis: verum
end;
supposeA102: i = (len f) + 1 ; ::_thesis: (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) * (i,j) = 0. F1()
A103: [i0,j] in Indices A9 by A25, A27, A100, ZFMISC_1:87;
A104: ( (Line (A9,((len f) + 1))) . j = A9 * (((len f) + 1),j) & (Line (A9,i0)) . j = A9 * (i0,j) ) by A94, A100, MATRIX_1:def_7;
(RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) * (i,j) = ((Line (A9,((len f) + 1))) + (Line (A9,i0))) . j by A92, A96, A97, A102, MATRIX11:def_3;
hence (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) * (i,j) = (A9 * (((len f) + 1),j)) + (A9 * (i0,j)) by A94, A100, A104, FVSUM_1:18
.= (0. F1()) + (A9 * (i0,j)) by A10, A96, A97, A98, A99, A102
.= (0. F1()) + (0. F1()) by A10, A22, A99, A103
.= 0. F1() by RLVECT_1:def_4 ;
::_thesis: verum
end;
end;
end;
hence (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) * (i,j) = 0. F1() ; ::_thesis: verum
end;
hence for i, j being Nat st [i,j] in Indices (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) & i > len f & j <= n holds
(RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) * (i,j) = 0. F1() ; ::_thesis: for i, j being Nat st i in dom f holds
S1[f,i,j, RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))]
let i, j be Nat; ::_thesis: ( i in dom f implies S1[f,i,j, RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))] )
assume A105: i in dom f ; ::_thesis: S1[f,i,j, RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))]
i <= len f by A31, A105, FINSEQ_1:1;
then A106: i < (len f) + 1 by NAT_1:13;
( f /. i = f . i & f . i in rng f ) by A105, FUNCT_1:def_3, PARTFUN1:def_6;
then A107: f /. i in Seg n by A14;
then [i,(f /. i)] in Indices A9 by A25, A32, A33, A31, A105, ZFMISC_1:87;
then (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) * (i,(f /. i)) = A9 * (i,(f /. i)) by A92, A106, MATRIX11:def_3;
hence ( (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) * (i,(f /. i)) <> 0. F1() & ( j in dom f & i < j implies f /. i < f /. j ) ) by A15, A105; ::_thesis: ( ( j in (dom f) \ {i} implies (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) * (j,(f /. i)) = 0. F1() ) & ( j in Seg (width (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0))))))) & j < f /. i implies (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) * (i,j) = 0. F1() ) )
thus ( j in (dom f) \ {i} implies (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) * (j,(f /. i)) = 0. F1() ) ::_thesis: ( j in Seg (width (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0))))))) & j < f /. i implies (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) * (i,j) = 0. F1() )
proof
assume A108: j in (dom f) \ {i} ; ::_thesis: (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) * (j,(f /. i)) = 0. F1()
then A109: j in dom f by XBOOLE_0:def_5;
then j <= len f by A31, FINSEQ_1:1;
then A110: j < (len f) + 1 by NAT_1:13;
[j,(f /. i)] in Indices A9 by A25, A32, A33, A31, A107, A109, ZFMISC_1:87;
hence (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) * (j,(f /. i)) = A9 * (j,(f /. i)) by A92, A110, MATRIX11:def_3
.= 0. F1() by A15, A105, A108 ;
::_thesis: verum
end;
assume that
A111: j in Seg (width (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0))))))) and
A112: j < f /. i ; ::_thesis: (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) * (i,j) = 0. F1()
A113: width (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) = width A9 by A92, MATRIX11:def_3;
then [i,j] in Indices A9 by A25, A33, A31, A94, A105, A111, ZFMISC_1:87;
hence (RLine (A9,((len f) + 1),((Line (A9,((len f) + 1))) + ((1_ F1()) * (Line (A9,i0)))))) * (i,j) = A9 * (i,j) by A92, A106, MATRIX11:def_3
.= 0. F1() by A15, A105, A111, A112, A113 ;
::_thesis: verum
end;
end;
end;
then consider A0 being Matrix of F2(),F3(),F1(), B0 being Matrix of F2(),F4(),F1() such that
A114: ( P1[A0,B0] & the_rank_of F5() = the_rank_of A0 & A0 * (((len f) + 1),(n + 1)) <> 0. F1() & ( for i, j being Nat st [i,j] in Indices A0 & i > len f & j <= n holds
A0 * (i,j) = 0. F1() ) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A0] ) ) ;
A115: S3[ 0 ]
proof
assume 0 <= F2() ; ::_thesis: ex A9 being Matrix of F2(),F3(),F1() ex B9 being Matrix of F2(),F4(),F1() st
( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 & A9 * (((len f) + 1),(n + 1)) <> 0. F1() & ( for i, j being Nat st [i,j] in Indices A9 & i > len f & j <= n holds
A9 * (i,j) = 0. F1() ) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) & ( for j being Nat st j in (dom A9) \ {((len f) + 1)} & j <= 0 holds
A9 * (j,(n + 1)) = 0. F1() ) )
take A0 ; ::_thesis: ex B9 being Matrix of F2(),F4(),F1() st
( P1[A0,B9] & the_rank_of F5() = the_rank_of A0 & A0 * (((len f) + 1),(n + 1)) <> 0. F1() & ( for i, j being Nat st [i,j] in Indices A0 & i > len f & j <= n holds
A0 * (i,j) = 0. F1() ) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A0] ) & ( for j being Nat st j in (dom A0) \ {((len f) + 1)} & j <= 0 holds
A0 * (j,(n + 1)) = 0. F1() ) )
take B0 ; ::_thesis: ( P1[A0,B0] & the_rank_of F5() = the_rank_of A0 & A0 * (((len f) + 1),(n + 1)) <> 0. F1() & ( for i, j being Nat st [i,j] in Indices A0 & i > len f & j <= n holds
A0 * (i,j) = 0. F1() ) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A0] ) & ( for j being Nat st j in (dom A0) \ {((len f) + 1)} & j <= 0 holds
A0 * (j,(n + 1)) = 0. F1() ) )
now__::_thesis:_for_j_being_Nat_st_j_in_(dom_A0)_\_{((len_f)_+_1)}_&_j_<=_0_holds_
A0_*_(j,(n_+_1))_=_0._F1()
A116: dom A0 = Seg (len A0) by FINSEQ_1:def_3;
let j be Nat; ::_thesis: ( j in (dom A0) \ {((len f) + 1)} & j <= 0 implies A0 * (j,(n + 1)) = 0. F1() )
assume ( j in (dom A0) \ {((len f) + 1)} & j <= 0 ) ; ::_thesis: A0 * (j,(n + 1)) = 0. F1()
hence A0 * (j,(n + 1)) = 0. F1() by A116; ::_thesis: verum
end;
hence ( P1[A0,B0] & the_rank_of F5() = the_rank_of A0 & A0 * (((len f) + 1),(n + 1)) <> 0. F1() & ( for i, j being Nat st [i,j] in Indices A0 & i > len f & j <= n holds
A0 * (i,j) = 0. F1() ) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A0] ) & ( for j being Nat st j in (dom A0) \ {((len f) + 1)} & j <= 0 holds
A0 * (j,(n + 1)) = 0. F1() ) ) by A114; ::_thesis: verum
end;
for k being Nat holds S3[k] from NAT_1:sch_2(A115, A34);
then consider Aa being Matrix of F2(),F3(),F1(), Bb being Matrix of F2(),F4(),F1() such that
A117: ( P1[Aa,Bb] & the_rank_of F5() = the_rank_of Aa ) and
A118: Aa * (((len f) + 1),(n + 1)) <> 0. F1() and
A119: for i, j being Nat st [i,j] in Indices Aa & i > len f & j <= n holds
Aa * (i,j) = 0. F1() and
A120: for i, j being Nat st i in dom f holds
S1[f,i,j,Aa] and
A121: for j being Nat st j in (dom Aa) \ {((len f) + 1)} & j <= F2() holds
Aa * (j,(n + 1)) = 0. F1() ;
take Aa ; ::_thesis: ex B9 being Matrix of F2(),F4(),F1() st
( P1[Aa,B9] & the_rank_of F5() = the_rank_of Aa & ex f being FinSequence of NAT st
( ( for i, j being Nat st [i,j] in Indices Aa & i > len f & j <= n + 1 holds
Aa * (i,j) = 0. F1() ) & f is one-to-one & len f <= n + 1 & len f <= F2() & rng f c= Seg (n + 1) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,Aa] ) ) )
take Bb ; ::_thesis: ( P1[Aa,Bb] & the_rank_of F5() = the_rank_of Aa & ex f being FinSequence of NAT st
( ( for i, j being Nat st [i,j] in Indices Aa & i > len f & j <= n + 1 holds
Aa * (i,j) = 0. F1() ) & f is one-to-one & len f <= n + 1 & len f <= F2() & rng f c= Seg (n + 1) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,Aa] ) ) )
thus ( P1[Aa,Bb] & the_rank_of F5() = the_rank_of Aa ) by A117; ::_thesis: ex f being FinSequence of NAT st
( ( for i, j being Nat st [i,j] in Indices Aa & i > len f & j <= n + 1 holds
Aa * (i,j) = 0. F1() ) & f is one-to-one & len f <= n + 1 & len f <= F2() & rng f c= Seg (n + 1) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,Aa] ) )
take f9 = f ^ <*(n + 1)*>; ::_thesis: ( ( for i, j being Nat st [i,j] in Indices Aa & i > len f9 & j <= n + 1 holds
Aa * (i,j) = 0. F1() ) & f9 is one-to-one & len f9 <= n + 1 & len f9 <= F2() & rng f9 c= Seg (n + 1) & ( for i, j being Nat st i in dom f9 holds
S1[f9,i,j,Aa] ) )
A122: len f9 = (len f) + 1 by FINSEQ_2:16;
A123: ( len Aa = F2() & dom Aa = Seg (len Aa) ) by A4, FINSEQ_1:def_3, MATRIX_1:23;
A124: now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_Aa_&_i_>_len_f9_&_j_<=_n_+_1_holds_
Aa_*_(i,j)_=_0._F1()
let i, j be Nat; ::_thesis: ( [i,j] in Indices Aa & i > len f9 & j <= n + 1 implies Aa * (b1,b2) = 0. F1() )
assume that
A125: [i,j] in Indices Aa and
A126: i > len f9 and
A127: j <= n + 1 ; ::_thesis: Aa * (b1,b2) = 0. F1()
percases ( j <= n or j = n + 1 ) by A127, NAT_1:8;
supposeA128: j <= n ; ::_thesis: Aa * (b1,b2) = 0. F1()
i > len f by A122, A126, NAT_1:13;
hence Aa * (i,j) = 0. F1() by A119, A125, A128; ::_thesis: verum
end;
supposeA129: j = n + 1 ; ::_thesis: Aa * (b1,b2) = 0. F1()
i in dom Aa by A125, ZFMISC_1:87;
then ( i in (dom Aa) \ {((len f) + 1)} & i <= F2() ) by A122, A123, A126, FINSEQ_1:1, ZFMISC_1:56;
hence Aa * (i,j) = 0. F1() by A121, A129; ::_thesis: verum
end;
end;
end;
A130: width Aa = F3() by A4, MATRIX_1:23;
A131: len f9 <= F2() by A28, A29, A122, XXREAL_0:2;
A132: now__::_thesis:_for_i,_j_being_Nat_st_i_in_dom_f9_holds_
(_Aa_*_(i,(f9_/._i))_<>_0._F1()_&_(_j_in_dom_f9_&_i_<_j_implies_f9_/._i_<_f9_/._j_)_&_(_j_in_(dom_f9)_\_{i}_implies_Aa_*_(j,(f9_/._i))_=_0._F1()_)_&_(_j_in_Seg_(width_Aa)_&_j_<_f9_/._i_implies_Aa_*_(i,j)_=_0._F1()_)_)
let i, j be Nat; ::_thesis: ( i in dom f9 implies ( Aa * (i,(f9 /. i)) <> 0. F1() & ( j in dom f9 & i < j implies f9 /. i < f9 /. j ) & ( j in (dom f9) \ {i} implies Aa * (j,(f9 /. i)) = 0. F1() ) & ( j in Seg (width Aa) & j < f9 /. i implies Aa * (i,j) = 0. F1() ) ) )
assume A133: i in dom f9 ; ::_thesis: ( Aa * (i,(f9 /. i)) <> 0. F1() & ( j in dom f9 & i < j implies f9 /. i < f9 /. j ) & ( j in (dom f9) \ {i} implies Aa * (j,(f9 /. i)) = 0. F1() ) & ( j in Seg (width Aa) & j < f9 /. i implies Aa * (i,j) = 0. F1() ) )
A134: dom f9 = Seg ((len f) + 1) by A122, FINSEQ_1:def_3
.= (dom f) \/ {((len f) + 1)} by A31, FINSEQ_1:9 ;
A135: now__::_thesis:_for_k_being_Nat_st_k_in_dom_f_holds_
f_/._k_=_f9_/._k
let k be Nat; ::_thesis: ( k in dom f implies f /. k = f9 /. k )
assume A136: k in dom f ; ::_thesis: f /. k = f9 /. k
A137: k in dom f9 by A134, A136, XBOOLE_0:def_3;
thus f /. k = f . k by A136, PARTFUN1:def_6
.= f9 . k by A136, FINSEQ_1:def_7
.= f9 /. k by A137, PARTFUN1:def_6 ; ::_thesis: verum
end;
now__::_thesis:_Aa_*_(i,(f9_/._i))_<>_0._F1()
percases ( i in dom f or i in {((len f) + 1)} ) by A133, A134, XBOOLE_0:def_3;
supposeA138: i in dom f ; ::_thesis: Aa * (i,(f9 /. i)) <> 0. F1()
then f /. i = f9 /. i by A135;
hence Aa * (i,(f9 /. i)) <> 0. F1() by A120, A138; ::_thesis: verum
end;
supposeA139: i in {((len f) + 1)} ; ::_thesis: Aa * (i,(f9 /. i)) <> 0. F1()
A140: f9 /. i = f9 . i by A133, PARTFUN1:def_6;
i = (len f) + 1 by A139, TARSKI:def_1;
hence Aa * (i,(f9 /. i)) <> 0. F1() by A118, A140, FINSEQ_1:42; ::_thesis: verum
end;
end;
end;
hence Aa * (i,(f9 /. i)) <> 0. F1() ; ::_thesis: ( ( j in dom f9 & i < j implies f9 /. i < f9 /. j ) & ( j in (dom f9) \ {i} implies Aa * (j,(f9 /. i)) = 0. F1() ) & ( j in Seg (width Aa) & j < f9 /. i implies Aa * (i,j) = 0. F1() ) )
thus ( j in dom f9 & i < j implies f9 /. i < f9 /. j ) ::_thesis: ( ( j in (dom f9) \ {i} implies Aa * (j,(f9 /. i)) = 0. F1() ) & ( j in Seg (width Aa) & j < f9 /. i implies Aa * (i,j) = 0. F1() ) )
proof
assume that
A141: j in dom f9 and
A142: i < j ; ::_thesis: f9 /. i < f9 /. j
percases ( ( j in {((len f) + 1)} & i in {((len f) + 1)} ) or ( j in {((len f) + 1)} & i in dom f ) or ( j in dom f & i in {((len f) + 1)} ) or ( j in dom f & i in dom f ) ) by A133, A134, A141, XBOOLE_0:def_3;
supposeA143: ( j in {((len f) + 1)} & i in {((len f) + 1)} ) ; ::_thesis: f9 /. i < f9 /. j
then i = (len f) + 1 by TARSKI:def_1;
hence f9 /. i < f9 /. j by A142, A143, TARSKI:def_1; ::_thesis: verum
end;
supposeA144: ( j in {((len f) + 1)} & i in dom f ) ; ::_thesis: f9 /. i < f9 /. j
then (len f) + 1 = j by TARSKI:def_1;
then A145: f9 . j = n + 1 by FINSEQ_1:42;
( f /. i = f . i & f . i in rng f ) by A144, FUNCT_1:def_3, PARTFUN1:def_6;
then A146: f /. i <= n by A14, FINSEQ_1:1;
f9 . j = f9 /. j by A141, PARTFUN1:def_6;
then f /. i < f9 /. j by A146, A145, NAT_1:13;
hence f9 /. i < f9 /. j by A135, A144; ::_thesis: verum
end;
suppose ( j in dom f & i in {((len f) + 1)} ) ; ::_thesis: f9 /. i < f9 /. j
then ( j <= len f & i = (len f) + 1 ) by A31, FINSEQ_1:1, TARSKI:def_1;
hence f9 /. i < f9 /. j by A142, NAT_1:13; ::_thesis: verum
end;
supposeA147: ( j in dom f & i in dom f ) ; ::_thesis: f9 /. i < f9 /. j
then ( f /. i = f9 /. i & f /. j = f9 /. j ) by A135;
hence f9 /. i < f9 /. j by A15, A142, A147; ::_thesis: verum
end;
end;
end;
dom f9 = Seg (len f9) by FINSEQ_1:def_3;
then A148: dom f9 c= dom Aa by A123, A131, FINSEQ_1:5;
thus ( j in (dom f9) \ {i} implies Aa * (j,(f9 /. i)) = 0. F1() ) ::_thesis: ( j in Seg (width Aa) & j < f9 /. i implies Aa * (i,j) = 0. F1() )
proof
assume A149: j in (dom f9) \ {i} ; ::_thesis: Aa * (j,(f9 /. i)) = 0. F1()
percases ( i = (len f) + 1 or i <> (len f) + 1 ) ;
supposeA150: i = (len f) + 1 ; ::_thesis: Aa * (j,(f9 /. i)) = 0. F1()
(len f) + 1 in {((len f) + 1)} by TARSKI:def_1;
then (len f) + 1 in dom f9 by A134, XBOOLE_0:def_3;
then A151: f9 . ((len f) + 1) = f9 /. i by A150, PARTFUN1:def_6;
A152: j in dom f9 by A149, ZFMISC_1:56;
j <> i by A149, ZFMISC_1:56;
then A153: j in (dom Aa) \ {i} by A148, A152, ZFMISC_1:56;
j <= F2() by A123, A148, A152, FINSEQ_1:1;
then Aa * (j,(n + 1)) = 0. F1() by A121, A150, A153;
hence Aa * (j,(f9 /. i)) = 0. F1() by A151, FINSEQ_1:42; ::_thesis: verum
end;
supposeA154: i <> (len f) + 1 ; ::_thesis: Aa * (j,(f9 /. i)) = 0. F1()
A155: j in dom f9 by A149, XBOOLE_0:def_5;
A156: ( i in dom f or i in {((len f) + 1)} ) by A133, A134, XBOOLE_0:def_3;
then A157: f . i in rng f by A154, FUNCT_1:def_3, TARSKI:def_1;
A158: ( f /. i = f9 /. i & f . i = f /. i ) by A135, A154, A156, PARTFUN1:def_6, TARSKI:def_1;
then A159: 1 <= f9 /. i by A14, A157, FINSEQ_1:1;
A160: f9 /. i <= n by A14, A158, A157, FINSEQ_1:1;
n <= F3() by A7, NAT_1:13;
then f9 /. i <= F3() by A160, XXREAL_0:2;
then f9 /. i in Seg (width Aa) by A130, A159;
then A161: [j,(f9 /. i)] in Indices Aa by A148, A155, ZFMISC_1:87;
percases ( j = (len f) + 1 or j <> (len f) + 1 ) ;
suppose j = (len f) + 1 ; ::_thesis: Aa * (j,(f9 /. i)) = 0. F1()
then j > len f by NAT_1:13;
hence Aa * (j,(f9 /. i)) = 0. F1() by A119, A160, A161; ::_thesis: verum
end;
supposeA162: j <> (len f) + 1 ; ::_thesis: Aa * (j,(f9 /. i)) = 0. F1()
j in dom f9 by A149, XBOOLE_0:def_5;
then A163: ( j in dom f or j in {((len f) + 1)} ) by A134, XBOOLE_0:def_3;
j <> i by A149, ZFMISC_1:56;
then j in (dom f) \ {i} by A162, A163, TARSKI:def_1, ZFMISC_1:56;
then Aa * (j,(f /. i)) = 0. F1() by A120, A154, A156, TARSKI:def_1;
hence Aa * (j,(f9 /. i)) = 0. F1() by A135, A154, A156, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
end;
end;
thus ( j in Seg (width Aa) & j < f9 /. i implies Aa * (i,j) = 0. F1() ) ::_thesis: verum
proof
assume that
A164: j in Seg (width Aa) and
A165: j < f9 /. i ; ::_thesis: Aa * (i,j) = 0. F1()
percases ( i in dom f or not i in dom f ) ;
supposeA166: i in dom f ; ::_thesis: Aa * (i,j) = 0. F1()
then f9 /. i = f /. i by A135;
hence Aa * (i,j) = 0. F1() by A120, A164, A165, A166; ::_thesis: verum
end;
supposeA167: not i in dom f ; ::_thesis: Aa * (i,j) = 0. F1()
( i in dom f or i in {((len f) + 1)} ) by A133, A134, XBOOLE_0:def_3;
then A168: i = (len f) + 1 by A167, TARSKI:def_1;
then A169: f9 . i = n + 1 by FINSEQ_1:42;
f9 . i = f9 /. i by A133, PARTFUN1:def_6;
then A170: j <= n by A165, A169, NAT_1:13;
A171: i > len f by A168, NAT_1:13;
[i,j] in Indices Aa by A133, A148, A164, ZFMISC_1:87;
hence Aa * (i,j) = 0. F1() by A119, A170, A171; ::_thesis: verum
end;
end;
end;
end;
A172: ( rng <*(n + 1)*> = {(n + 1)} & (rng f) \/ {(n + 1)} c= (Seg n) \/ {(n + 1)} ) by A14, FINSEQ_1:38, XBOOLE_1:9;
A173: (Seg n) \/ {(n + 1)} = Seg (n + 1) by FINSEQ_1:9;
( rng f misses {(n + 1)} & <*(n + 1)*> is one-to-one ) by A14, FINSEQ_3:14, XBOOLE_1:63;
hence ( ( for i, j being Nat st [i,j] in Indices Aa & i > len f9 & j <= n + 1 holds
Aa * (i,j) = 0. F1() ) & f9 is one-to-one & len f9 <= n + 1 & len f9 <= F2() & rng f9 c= Seg (n + 1) & ( for i, j being Nat st i in dom f9 holds
S1[f9,i,j,Aa] ) ) by A11, A12, A28, A29, A122, A124, A172, A173, A132, FINSEQ_1:31, FINSEQ_3:91, XREAL_1:6, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
A174: S2[ 0 ]
proof
assume 0 <= F3() ; ::_thesis: ex A9 being Matrix of F2(),F3(),F1() ex B9 being Matrix of F2(),F4(),F1() st
( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 & ex f being FinSequence of NAT st
( ( for i, j being Nat st [i,j] in Indices A9 & i > len f & j <= 0 holds
A9 * (i,j) = 0. F1() ) & f is one-to-one & len f <= 0 & len f <= F2() & rng f c= Seg 0 & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) ) )
take A9 = F5(); ::_thesis: ex B9 being Matrix of F2(),F4(),F1() st
( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 & ex f being FinSequence of NAT st
( ( for i, j being Nat st [i,j] in Indices A9 & i > len f & j <= 0 holds
A9 * (i,j) = 0. F1() ) & f is one-to-one & len f <= 0 & len f <= F2() & rng f c= Seg 0 & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) ) )
take B9 = F6(); ::_thesis: ( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 & ex f being FinSequence of NAT st
( ( for i, j being Nat st [i,j] in Indices A9 & i > len f & j <= 0 holds
A9 * (i,j) = 0. F1() ) & f is one-to-one & len f <= 0 & len f <= F2() & rng f c= Seg 0 & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) ) )
thus ( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 ) by A1; ::_thesis: ex f being FinSequence of NAT st
( ( for i, j being Nat st [i,j] in Indices A9 & i > len f & j <= 0 holds
A9 * (i,j) = 0. F1() ) & f is one-to-one & len f <= 0 & len f <= F2() & rng f c= Seg 0 & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) )
take f = <*> NAT; ::_thesis: ( ( for i, j being Nat st [i,j] in Indices A9 & i > len f & j <= 0 holds
A9 * (i,j) = 0. F1() ) & f is one-to-one & len f <= 0 & len f <= F2() & rng f c= Seg 0 & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) )
now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_A9_&_i_>_len_f_&_j_<=_0_holds_
A9_*_(i,j)_=_0._F1()
let i, j be Nat; ::_thesis: ( [i,j] in Indices A9 & i > len f & j <= 0 implies A9 * (i,j) = 0. F1() )
assume that
A175: [i,j] in Indices A9 and
i > len f and
A176: j <= 0 ; ::_thesis: A9 * (i,j) = 0. F1()
j in Seg (width A9) by A175, ZFMISC_1:87;
hence A9 * (i,j) = 0. F1() by A176; ::_thesis: verum
end;
hence ( ( for i, j being Nat st [i,j] in Indices A9 & i > len f & j <= 0 holds
A9 * (i,j) = 0. F1() ) & f is one-to-one & len f <= 0 & len f <= F2() & rng f c= Seg 0 & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) ) ; ::_thesis: verum
end;
for n being Nat holds S2[n] from NAT_1:sch_2(A174, A5);
then consider A9 being Matrix of F2(),F3(),F1(), B9 being Matrix of F2(),F4(),F1() such that
A177: P1[A9,B9] and
A178: the_rank_of F5() = the_rank_of A9 and
A179: ex f being FinSequence of NAT st
( ( for i, j being Nat st [i,j] in Indices A9 & i > len f & j <= F3() holds
A9 * (i,j) = 0. F1() ) & f is one-to-one & len f <= F3() & len f <= F2() & rng f c= Seg F3() & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) ) ;
consider f being FinSequence of NAT such that
A180: for i, j being Nat st [i,j] in Indices A9 & i > len f & j <= F3() holds
A9 * (i,j) = 0. F1() and
A181: f is one-to-one and
len f <= F3() and
A182: len f <= F2() and
A183: rng f c= Seg F3() and
A184: for i, j being Nat st i in dom f holds
S1[f,i,j,A9] by A179;
A185: len A9 = F2() by A4, MATRIX_1:23;
take A9 ; ::_thesis: ex B9 being Matrix of F2(),F4(),F1() st
( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 & ( for i being Nat st i in dom A9 & i > the_rank_of F5() holds
Line (A9,i) = F3() |-> (0. F1()) ) & ex f being FinSequence of NAT st
( len f = the_rank_of A9 & f is one-to-one & rng f c= Seg (width A9) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) ) )
take B9 ; ::_thesis: ( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 & ( for i being Nat st i in dom A9 & i > the_rank_of F5() holds
Line (A9,i) = F3() |-> (0. F1()) ) & ex f being FinSequence of NAT st
( len f = the_rank_of A9 & f is one-to-one & rng f c= Seg (width A9) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) ) )
thus ( P1[A9,B9] & the_rank_of F5() = the_rank_of A9 ) by A177, A178; ::_thesis: ( ( for i being Nat st i in dom A9 & i > the_rank_of F5() holds
Line (A9,i) = F3() |-> (0. F1()) ) & ex f being FinSequence of NAT st
( len f = the_rank_of A9 & f is one-to-one & rng f c= Seg (width A9) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) ) )
A186: dom A9 = Seg (len A9) by FINSEQ_1:def_3;
set L = len f;
A187: Seg (len f) c= Seg F2() by A182, FINSEQ_1:5;
( idseq (len f) is FinSequence of NAT & len (idseq (len f)) = len f ) by CARD_1:def_7, FINSEQ_2:48;
then reconsider idL = idseq (len f), F9 = f as Element of (len f) -tuples_on NAT by FINSEQ_2:92;
set S = Segm (A9,idL,F9);
A188: dom f = Seg (len f) by FINSEQ_1:def_3;
set D = diagonal_of_Matrix (Segm (A9,idL,F9));
A189: Indices (Segm (A9,idL,F9)) = [:(Seg (len f)),(Seg (len f)):] by MATRIX_1:24;
for k being Element of NAT st k in dom (diagonal_of_Matrix (Segm (A9,idL,F9))) holds
(diagonal_of_Matrix (Segm (A9,idL,F9))) . k <> 0. F1()
proof
A190: len (diagonal_of_Matrix (Segm (A9,idL,F9))) = len f by MATRIX_3:def_10;
let k be Element of NAT ; ::_thesis: ( k in dom (diagonal_of_Matrix (Segm (A9,idL,F9))) implies (diagonal_of_Matrix (Segm (A9,idL,F9))) . k <> 0. F1() )
assume k in dom (diagonal_of_Matrix (Segm (A9,idL,F9))) ; ::_thesis: (diagonal_of_Matrix (Segm (A9,idL,F9))) . k <> 0. F1()
then A191: k in Seg (len f) by A190, FINSEQ_1:def_3;
then [k,k] in Indices (Segm (A9,idL,F9)) by A189, ZFMISC_1:87;
then (Segm (A9,idL,F9)) * (k,k) = A9 * ((idL . k),(f . k)) by MATRIX13:def_1
.= A9 * (k,(f . k)) by A191, FINSEQ_2:49
.= A9 * (k,(f /. k)) by A188, A191, PARTFUN1:def_6 ;
then (Segm (A9,idL,F9)) * (k,k) <> 0. F1() by A184, A188, A191;
hence (diagonal_of_Matrix (Segm (A9,idL,F9))) . k <> 0. F1() by A191, MATRIX_3:def_10; ::_thesis: verum
end;
then A192: Product (diagonal_of_Matrix (Segm (A9,idL,F9))) <> 0. F1() by FVSUM_1:82;
now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_(Segm_(A9,idL,F9))_&_i_>_j_holds_
(Segm_(A9,idL,F9))_*_(i,j)_=_0._F1()
let i, j be Nat; ::_thesis: ( [i,j] in Indices (Segm (A9,idL,F9)) & i > j implies (Segm (A9,idL,F9)) * (i,j) = 0. F1() )
assume A193: [i,j] in Indices (Segm (A9,idL,F9)) ; ::_thesis: ( i > j implies (Segm (A9,idL,F9)) * (i,j) = 0. F1() )
A194: i in Seg (len f) by A189, A193, ZFMISC_1:87;
assume i > j ; ::_thesis: (Segm (A9,idL,F9)) * (i,j) = 0. F1()
then A195: i in (dom f) \ {j} by A188, A194, ZFMISC_1:56;
reconsider i9 = i, j9 = j as Element of NAT by ORDINAL1:def_12;
A196: j in Seg (len f) by A189, A193, ZFMISC_1:87;
thus (Segm (A9,idL,F9)) * (i,j) = A9 * ((idL . i9),(f . j9)) by A193, MATRIX13:def_1
.= A9 * (i,(f . j)) by A194, FINSEQ_2:49
.= A9 * (i,(f /. j)) by A188, A196, PARTFUN1:def_6
.= 0. F1() by A184, A188, A196, A195 ; ::_thesis: verum
end;
then Segm (A9,idL,F9) is Upper_Triangular_Matrix of len f,F1() by MATRIX_2:def_3;
then A197: Det (Segm (A9,idL,F9)) <> 0. F1() by A192, MATRIX13:7;
A198: len (Segm (A9,(Seg (len f)),(Seg (width A9)))) = card (Seg (len f)) by MATRIX_1:def_2;
A199: width A9 = F3() by A4, MATRIX_1:23;
rng idL = Seg (len f) by RELAT_1:45;
then [:(rng idL),(rng F9):] c= Indices A9 by A183, A187, A185, A186, A199, ZFMISC_1:96;
then A200: the_rank_of A9 >= len f by A197, MATRIX13:76;
A201: now__::_thesis:_for_i_being_Nat_st_i_in_(dom_A9)_\_(Seg_(len_f))_holds_
Line_(A9,i)_=_(width_A9)_|->_(0._F1())
set w0 = (width A9) |-> (0. F1());
let i be Nat; ::_thesis: ( i in (dom A9) \ (Seg (len f)) implies Line (A9,i) = (width A9) |-> (0. F1()) )
assume A202: i in (dom A9) \ (Seg (len f)) ; ::_thesis: Line (A9,i) = (width A9) |-> (0. F1())
set LA = Line (A9,i);
A203: now__::_thesis:_for_j_being_Nat_st_1_<=_j_&_j_<=_width_A9_holds_
(Line_(A9,i))_._j_=_((width_A9)_|->_(0._F1()))_._j
not i in Seg (len f) by A202, XBOOLE_0:def_5;
then A204: ( i > len f or i < 1 ) by A202;
let j be Nat; ::_thesis: ( 1 <= j & j <= width A9 implies (Line (A9,i)) . j = ((width A9) |-> (0. F1())) . j )
assume that
A205: 1 <= j and
A206: j <= width A9 ; ::_thesis: (Line (A9,i)) . j = ((width A9) |-> (0. F1())) . j
j in NAT by ORDINAL1:def_12;
then A207: j in Seg (width A9) by A205, A206;
A208: i in dom A9 by A202, XBOOLE_0:def_5;
then A209: [i,j] in Indices A9 by A207, ZFMISC_1:87;
thus (Line (A9,i)) . j = A9 * (i,j) by A207, MATRIX_1:def_7
.= 0. F1() by A180, A186, A199, A206, A208, A209, A204, FINSEQ_1:1
.= ((width A9) |-> (0. F1())) . j by A207, FINSEQ_2:57 ; ::_thesis: verum
end;
( len (Line (A9,i)) = width A9 & len ((width A9) |-> (0. F1())) = width A9 ) by CARD_1:def_7;
hence Line (A9,i) = (width A9) |-> (0. F1()) by A203, FINSEQ_1:14; ::_thesis: verum
end;
then the_rank_of A9 = the_rank_of (Segm (A9,(Seg (len f)),(Seg (width A9)))) by A187, A185, A186, Th11;
then the_rank_of A9 <= card (Seg (len f)) by A198, MATRIX13:74;
then A210: the_rank_of A9 <= len f by FINSEQ_1:57;
then A211: the_rank_of F5() = len f by A178, A200, XXREAL_0:1;
thus for i being Nat st i in dom A9 & i > the_rank_of F5() holds
Line (A9,i) = F3() |-> (0. F1()) ::_thesis: ex f being FinSequence of NAT st
( len f = the_rank_of A9 & f is one-to-one & rng f c= Seg (width A9) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) )
proof
let i be Nat; ::_thesis: ( i in dom A9 & i > the_rank_of F5() implies Line (A9,i) = F3() |-> (0. F1()) )
assume that
A212: i in dom A9 and
A213: i > the_rank_of F5() ; ::_thesis: Line (A9,i) = F3() |-> (0. F1())
not i in Seg (len f) by A211, A213, FINSEQ_1:1;
then i in (dom A9) \ (Seg (len f)) by A212, XBOOLE_0:def_5;
hence Line (A9,i) = F3() |-> (0. F1()) by A199, A201; ::_thesis: verum
end;
take f ; ::_thesis: ( len f = the_rank_of A9 & f is one-to-one & rng f c= Seg (width A9) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) )
thus ( len f = the_rank_of A9 & f is one-to-one & rng f c= Seg (width A9) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) ) by A4, A181, A183, A184, A200, A210, MATRIX_1:23, XXREAL_0:1; ::_thesis: verum
end;
end;
end;
then consider A9 being Matrix of F2(),F3(),F1(), B9 being Matrix of F2(),F4(),F1() such that
A214: P1[A9,B9] and
A215: the_rank_of F5() = the_rank_of A9 and
A216: for i being Nat st i in dom A9 & i > the_rank_of F5() holds
Line (A9,i) = F3() |-> (0. F1()) and
A217: ex f being FinSequence of NAT st
( len f = the_rank_of A9 & f is one-to-one & rng f c= Seg (width A9) & ( for i, j being Nat st i in dom f holds
S1[f,i,j,A9] ) ) ;
consider f being FinSequence of NAT such that
A218: len f = the_rank_of A9 and
A219: f is one-to-one and
A220: rng f c= Seg (width A9) and
A221: for i, j being Nat st i in dom f holds
S1[f,i,j,A9] by A217;
not 0 in rng f by A220;
then reconsider rngf = rng f as finite without_zero Subset of NAT by A220, MEASURE6:def_2, XBOOLE_1:1;
A222: ( F2() = 0 or F2() > 0 ) ;
set S = Segm (A9,(Seg (card rngf)),rngf);
A223: dom f = Seg (the_rank_of F5()) by A215, A218, FINSEQ_1:def_3;
take A9 ; ::_thesis: ex B9 being Matrix of F2(),F4(),F1() ex N being finite without_zero Subset of NAT st
( N c= Seg F3() & the_rank_of F5() = the_rank_of A9 & the_rank_of F5() = card N & P1[A9,B9] & Segm (A9,(Seg (card N)),N) is diagonal & ( for i being Nat st i in Seg (card N) holds
A9 * (i,((Sgm N) /. i)) <> 0. F1() ) & ( for i being Nat st i in dom A9 & i > card N holds
Line (A9,i) = F3() |-> (0. F1()) ) & ( for i, j being Nat st i in Seg (card N) & j in Seg (width A9) & j < (Sgm N) . i holds
A9 * (i,j) = 0. F1() ) )
take B9 ; ::_thesis: ex N being finite without_zero Subset of NAT st
( N c= Seg F3() & the_rank_of F5() = the_rank_of A9 & the_rank_of F5() = card N & P1[A9,B9] & Segm (A9,(Seg (card N)),N) is diagonal & ( for i being Nat st i in Seg (card N) holds
A9 * (i,((Sgm N) /. i)) <> 0. F1() ) & ( for i being Nat st i in dom A9 & i > card N holds
Line (A9,i) = F3() |-> (0. F1()) ) & ( for i, j being Nat st i in Seg (card N) & j in Seg (width A9) & j < (Sgm N) . i holds
A9 * (i,j) = 0. F1() ) )
take rngf ; ::_thesis: ( rngf c= Seg F3() & the_rank_of F5() = the_rank_of A9 & the_rank_of F5() = card rngf & P1[A9,B9] & Segm (A9,(Seg (card rngf)),rngf) is diagonal & ( for i being Nat st i in Seg (card rngf) holds
A9 * (i,((Sgm rngf) /. i)) <> 0. F1() ) & ( for i being Nat st i in dom A9 & i > card rngf holds
Line (A9,i) = F3() |-> (0. F1()) ) & ( for i, j being Nat st i in Seg (card rngf) & j in Seg (width A9) & j < (Sgm rngf) . i holds
A9 * (i,j) = 0. F1() ) )
len A9 = F2() by MATRIX_1:def_2;
then ( width A9 = 0 or width A9 = F3() ) by A222, MATRIX_1:23, MATRIX_1:def_3;
then Seg (width A9) c= Seg F3() by FINSEQ_1:5;
hence ( rngf c= Seg F3() & the_rank_of F5() = the_rank_of A9 ) by A215, A220, XBOOLE_1:1; ::_thesis: ( the_rank_of F5() = card rngf & P1[A9,B9] & Segm (A9,(Seg (card rngf)),rngf) is diagonal & ( for i being Nat st i in Seg (card rngf) holds
A9 * (i,((Sgm rngf) /. i)) <> 0. F1() ) & ( for i being Nat st i in dom A9 & i > card rngf holds
Line (A9,i) = F3() |-> (0. F1()) ) & ( for i, j being Nat st i in Seg (card rngf) & j in Seg (width A9) & j < (Sgm rngf) . i holds
A9 * (i,j) = 0. F1() ) )
dom f,rngf are_equipotent by A219, WELLORD2:def_4;
hence A224: card rngf = card (dom f) by CARD_1:5
.= card (Seg (the_rank_of F5())) by A215, A218, FINSEQ_1:def_3
.= the_rank_of F5() by FINSEQ_1:57 ;
::_thesis: ( P1[A9,B9] & Segm (A9,(Seg (card rngf)),rngf) is diagonal & ( for i being Nat st i in Seg (card rngf) holds
A9 * (i,((Sgm rngf) /. i)) <> 0. F1() ) & ( for i being Nat st i in dom A9 & i > card rngf holds
Line (A9,i) = F3() |-> (0. F1()) ) & ( for i, j being Nat st i in Seg (card rngf) & j in Seg (width A9) & j < (Sgm rngf) . i holds
A9 * (i,j) = 0. F1() ) )
now__::_thesis:_for_i,_j_being_Nat_st_i_in_dom_f_&_j_in_dom_f_&_i_<_j_holds_
f_._i_<_f_._j
let i, j be Nat; ::_thesis: ( i in dom f & j in dom f & i < j implies f . i < f . j )
assume that
A225: ( i in dom f & j in dom f ) and
A226: i < j ; ::_thesis: f . i < f . j
( f . i = f /. i & f . j = f /. j ) by A225, PARTFUN1:def_6;
hence f . i < f . j by A221, A225, A226; ::_thesis: verum
end;
then A227: Sgm rngf = f by A219, A220, Th6;
thus P1[A9,B9] by A214; ::_thesis: ( Segm (A9,(Seg (card rngf)),rngf) is diagonal & ( for i being Nat st i in Seg (card rngf) holds
A9 * (i,((Sgm rngf) /. i)) <> 0. F1() ) & ( for i being Nat st i in dom A9 & i > card rngf holds
Line (A9,i) = F3() |-> (0. F1()) ) & ( for i, j being Nat st i in Seg (card rngf) & j in Seg (width A9) & j < (Sgm rngf) . i holds
A9 * (i,j) = 0. F1() ) )
A228: card (Seg (card rngf)) = card rngf by FINSEQ_1:57;
then A229: Indices (Segm (A9,(Seg (card rngf)),rngf)) = [:(Seg (the_rank_of F5())),(Seg (the_rank_of F5())):] by A224, MATRIX_1:24;
now__::_thesis:_for_i,_j_being_Element_of_NAT_st_i_in_Seg_(the_rank_of_F5())_&_j_in_Seg_(the_rank_of_F5())_&_i_<>_j_holds_
(Segm_(A9,(Seg_(card_rngf)),rngf))_*_(i,j)_=_0._F1()
let i, j be Element of NAT ; ::_thesis: ( i in Seg (the_rank_of F5()) & j in Seg (the_rank_of F5()) & i <> j implies (Segm (A9,(Seg (card rngf)),rngf)) * (i,j) = 0. F1() )
assume that
A230: i in Seg (the_rank_of F5()) and
A231: j in Seg (the_rank_of F5()) and
A232: i <> j ; ::_thesis: (Segm (A9,(Seg (card rngf)),rngf)) * (i,j) = 0. F1()
A233: i in (dom f) \ {j} by A223, A230, A232, ZFMISC_1:56;
A234: (idseq (the_rank_of F5())) . i = i by A230, FINSEQ_2:49;
[i,j] in Indices (Segm (A9,(Seg (card rngf)),rngf)) by A229, A230, A231, ZFMISC_1:87;
then (Segm (A9,(Seg (card rngf)),rngf)) * (i,j) = A9 * (((Sgm (Seg (the_rank_of F5()))) . i),(f . j)) by A224, A227, MATRIX13:def_1
.= A9 * (i,(f . j)) by A234, FINSEQ_3:48
.= A9 * (i,(f /. j)) by A223, A231, PARTFUN1:def_6 ;
hence (Segm (A9,(Seg (card rngf)),rngf)) * (i,j) = 0. F1() by A221, A223, A231, A233; ::_thesis: verum
end;
hence Segm (A9,(Seg (card rngf)),rngf) is diagonal by A224, A228, MATRIX_7:def_2; ::_thesis: ( ( for i being Nat st i in Seg (card rngf) holds
A9 * (i,((Sgm rngf) /. i)) <> 0. F1() ) & ( for i being Nat st i in dom A9 & i > card rngf holds
Line (A9,i) = F3() |-> (0. F1()) ) & ( for i, j being Nat st i in Seg (card rngf) & j in Seg (width A9) & j < (Sgm rngf) . i holds
A9 * (i,j) = 0. F1() ) )
thus for i being Nat st i in Seg (card rngf) holds
A9 * (i,((Sgm rngf) /. i)) <> 0. F1() by A221, A224, A227, A223; ::_thesis: ( ( for i being Nat st i in dom A9 & i > card rngf holds
Line (A9,i) = F3() |-> (0. F1()) ) & ( for i, j being Nat st i in Seg (card rngf) & j in Seg (width A9) & j < (Sgm rngf) . i holds
A9 * (i,j) = 0. F1() ) )
thus for i being Nat st i in dom A9 & i > card rngf holds
Line (A9,i) = F3() |-> (0. F1()) by A216, A224; ::_thesis: for i, j being Nat st i in Seg (card rngf) & j in Seg (width A9) & j < (Sgm rngf) . i holds
A9 * (i,j) = 0. F1()
let i, j be Nat; ::_thesis: ( i in Seg (card rngf) & j in Seg (width A9) & j < (Sgm rngf) . i implies A9 * (i,j) = 0. F1() )
assume that
A235: i in Seg (card rngf) and
A236: j in Seg (width A9) and
A237: j < (Sgm rngf) . i ; ::_thesis: A9 * (i,j) = 0. F1()
j < f /. i by A224, A227, A223, A235, A237, PARTFUN1:def_6;
hence A9 * (i,j) = 0. F1() by A221, A224, A223, A235, A236; ::_thesis: verum
end;
scheme :: MATRIX15:sch 2
GAUSS2{ F1() -> Field, F2() -> Nat, F3() -> Nat, F4() -> Nat, F5() -> Matrix of F2(),F3(),F1(), F6() -> Matrix of F2(),F4(),F1(), F7( Matrix of F2(),F4(),F1(), Nat, Nat, Element of F1()) -> Matrix of F2(),F4(),F1(), P1[ set , set ] } :
ex A9 being Matrix of F2(),F3(),F1() ex B9 being Matrix of F2(),F4(),F1() ex N being finite without_zero Subset of NAT st
( N c= Seg F3() & the_rank_of F5() = the_rank_of A9 & the_rank_of F5() = card N & P1[A9,B9] & Segm (A9,(Seg (card N)),N) = 1. (F1(),(card N)) & ( for i being Nat st i in dom A9 & i > card N holds
Line (A9,i) = F3() |-> (0. F1()) ) & ( for i, j being Nat st i in Seg (card N) & j in Seg (width A9) & j < (Sgm N) . i holds
A9 * (i,j) = 0. F1() ) )
provided
A1: P1[F5(),F6()] and
A2: for A9 being Matrix of F2(),F3(),F1()
for B9 being Matrix of F2(),F4(),F1() st P1[A9,B9] holds
for a being Element of F1()
for i, j being Nat st j in dom A9 & ( i = j implies a <> - (1_ F1()) ) holds
P1[ RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j))))),F7(B9,i,j,a)]
proof
set r = the_rank_of F5();
A3: for A9 being Matrix of F2(),F3(),F1()
for B9 being Matrix of F2(),F4(),F1() st P1[A9,B9] holds
for i, j being Nat st i <> j & j in dom A9 holds
for a being Element of F1() holds P1[ RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j))))),F7(B9,i,j,a)] by A2;
consider A9 being Matrix of F2(),F3(),F1(), B9 being Matrix of F2(),F4(),F1(), N being finite without_zero Subset of NAT such that
A4: N c= Seg F3() and
A5: ( the_rank_of F5() = the_rank_of A9 & the_rank_of F5() = card N ) and
A6: ( P1[A9,B9] & Segm (A9,(Seg (card N)),N) is diagonal & ( for i being Nat st i in Seg (card N) holds
A9 * (i,((Sgm N) /. i)) <> 0. F1() ) & ( for i being Nat st i in dom A9 & i > card N holds
Line (A9,i) = F3() |-> (0. F1()) ) & ( for i, j being Nat st i in Seg (card N) & j in Seg (width A9) & j < (Sgm N) . i holds
A9 * (i,j) = 0. F1() ) ) from MATRIX15:sch_1(A1, A3);
set ONE = 1. (F1(),(card N));
A7: Indices (1. (F1(),(card N))) = [:(Seg (the_rank_of F5())),(Seg (the_rank_of F5())):] by A5, MATRIX_1:24;
defpred S1[ Nat] means ( $1 <= card N implies ex A9 being Matrix of F2(),F3(),F1() ex B9 being Matrix of F2(),F4(),F1() st
( the_rank_of F5() = the_rank_of A9 & ( for i being Nat st i in Seg (card N) & i <= $1 holds
A9 * (i,((Sgm N) /. i)) = 1_ F1() ) & P1[A9,B9] & Segm (A9,(Seg (card N)),N) is diagonal & ( for i being Nat st i in Seg (card N) holds
A9 * (i,((Sgm N) /. i)) <> 0. F1() ) & ( for i being Nat st i in dom A9 & i > card N holds
Line (A9,i) = F3() |-> (0. F1()) ) & ( for i, j being Nat st i in Seg (card N) & j in Seg (width A9) & j < (Sgm N) . i holds
A9 * (i,j) = 0. F1() ) ) );
A8: for n being Nat st S1[n] holds
S1[n + 1]
proof
set f = Sgm N;
let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A9: S1[n] ; ::_thesis: S1[n + 1]
set n1 = n + 1;
assume A10: n + 1 <= card N ; ::_thesis: ex A9 being Matrix of F2(),F3(),F1() ex B9 being Matrix of F2(),F4(),F1() st
( the_rank_of F5() = the_rank_of A9 & ( for i being Nat st i in Seg (card N) & i <= n + 1 holds
A9 * (i,((Sgm N) /. i)) = 1_ F1() ) & P1[A9,B9] & Segm (A9,(Seg (card N)),N) is diagonal & ( for i being Nat st i in Seg (card N) holds
A9 * (i,((Sgm N) /. i)) <> 0. F1() ) & ( for i being Nat st i in dom A9 & i > card N holds
Line (A9,i) = F3() |-> (0. F1()) ) & ( for i, j being Nat st i in Seg (card N) & j in Seg (width A9) & j < (Sgm N) . i holds
A9 * (i,j) = 0. F1() ) )
then consider A1 being Matrix of F2(),F3(),F1(), A25 being Matrix of F2(),F4(),F1() such that
A11: the_rank_of F5() = the_rank_of A1 and
A12: for i being Nat st i in Seg (card N) & i <= n holds
A1 * (i,((Sgm N) /. i)) = 1_ F1() and
A13: P1[A1,A25] and
A14: Segm (A1,(Seg (card N)),N) is diagonal and
A15: for i being Nat st i in Seg (card N) holds
A1 * (i,((Sgm N) /. i)) <> 0. F1() and
A16: for i being Nat st i in dom A1 & i > card N holds
Line (A1,i) = F3() |-> (0. F1()) and
A17: for i, j being Nat st i in Seg (card N) & j in Seg (width A1) & j < (Sgm N) . i holds
A1 * (i,j) = 0. F1() by A9, NAT_1:13;
set L = Line (A1,(n + 1));
set LL = (Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))));
set R = RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))));
take RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1)))))) ; ::_thesis: ex B9 being Matrix of F2(),F4(),F1() st
( the_rank_of F5() = the_rank_of (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) & ( for i being Nat st i in Seg (card N) & i <= n + 1 holds
(RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,((Sgm N) /. i)) = 1_ F1() ) & P1[ RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1)))))),B9] & Segm ((RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))),(Seg (card N)),N) is diagonal & ( for i being Nat st i in Seg (card N) holds
(RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,((Sgm N) /. i)) <> 0. F1() ) & ( for i being Nat st i in dom (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) & i > card N holds
Line ((RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))),i) = F3() |-> (0. F1()) ) & ( for i, j being Nat st i in Seg (card N) & j in Seg (width (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1)))))))) & j < (Sgm N) . i holds
(RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,j) = 0. F1() ) )
take FB = F7(A25,(n + 1),(n + 1),(((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1()))); ::_thesis: ( the_rank_of F5() = the_rank_of (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) & ( for i being Nat st i in Seg (card N) & i <= n + 1 holds
(RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,((Sgm N) /. i)) = 1_ F1() ) & P1[ RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1)))))),FB] & Segm ((RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))),(Seg (card N)),N) is diagonal & ( for i being Nat st i in Seg (card N) holds
(RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,((Sgm N) /. i)) <> 0. F1() ) & ( for i being Nat st i in dom (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) & i > card N holds
Line ((RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))),i) = F3() |-> (0. F1()) ) & ( for i, j being Nat st i in Seg (card N) & j in Seg (width (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1)))))))) & j < (Sgm N) . i holds
(RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,j) = 0. F1() ) )
A18: len A1 = F2() by MATRIX_1:def_2;
set SA = Segm (A1,(Seg (card N)),N);
set S = Segm ((RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))),(Seg (card N)),N);
A19: len ((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))) = width A1 by CARD_1:def_7;
( the_rank_of F5() <= len A9 & len A9 = F2() ) by A5, MATRIX13:74, MATRIX_1:def_2;
then A20: Seg (the_rank_of F5()) c= Seg F2() by FINSEQ_1:5;
1 <= 1 + n by NAT_1:11;
then A21: n + 1 in Seg (card N) by A10;
then A22: F2() <> 0 by A5, A20;
then A23: width (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) = F3() by MATRIX_1:23;
A24: ((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1()) <> - (1. F1())
proof
assume A25: ((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1()) = - (1. F1()) ; ::_thesis: contradiction
A26: 0. F1() = (1_ F1()) + (- (1_ F1())) by VECTSP_1:19
.= ((1_ F1()) + (- (1_ F1()))) + ((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") by A25, RLVECT_1:def_3
.= (0. F1()) + ((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") by VECTSP_1:19
.= (A1 * ((n + 1),((Sgm N) /. (n + 1)))) " by RLVECT_1:def_4 ;
A1 * ((n + 1),((Sgm N) /. (n + 1))) <> 0. F1() by A15, A21;
hence contradiction by A26, VECTSP_1:25; ::_thesis: verum
end;
hence the_rank_of (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) = the_rank_of F5() by A5, A11, A21, A20, A18, MATRIX13:92; ::_thesis: ( ( for i being Nat st i in Seg (card N) & i <= n + 1 holds
(RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,((Sgm N) /. i)) = 1_ F1() ) & P1[ RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1)))))),FB] & Segm ((RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))),(Seg (card N)),N) is diagonal & ( for i being Nat st i in Seg (card N) holds
(RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,((Sgm N) /. i)) <> 0. F1() ) & ( for i being Nat st i in dom (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) & i > card N holds
Line ((RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))),i) = F3() |-> (0. F1()) ) & ( for i, j being Nat st i in Seg (card N) & j in Seg (width (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1)))))))) & j < (Sgm N) . i holds
(RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,j) = 0. F1() ) )
A27: width A1 = F3() by A22, MATRIX_1:23;
A28: (Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1)))) = ((1_ F1()) * (Line (A1,(n + 1)))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1)))) by FVSUM_1:57
.= ((1_ F1()) + ((- (1_ F1())) + ((A1 * ((n + 1),((Sgm N) /. (n + 1)))) "))) * (Line (A1,(n + 1))) by FVSUM_1:55
.= (((1_ F1()) + (- (1_ F1()))) + ((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ")) * (Line (A1,(n + 1))) by RLVECT_1:def_3
.= ((0. F1()) + ((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ")) * (Line (A1,(n + 1))) by VECTSP_1:19
.= ((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") * (Line (A1,(n + 1))) by RLVECT_1:def_4 ;
A29: dom A1 = Seg (len A1) by FINSEQ_1:def_3;
A30: rng (Sgm N) = N by A4, FINSEQ_1:def_13;
A31: dom (Sgm N) = Seg (card N) by A4, FINSEQ_3:40;
thus A32: for i being Nat st i in Seg (card N) & i <= n + 1 holds
(RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,((Sgm N) /. i)) = 1_ F1() ::_thesis: ( P1[ RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1)))))),FB] & Segm ((RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))),(Seg (card N)),N) is diagonal & ( for i being Nat st i in Seg (card N) holds
(RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,((Sgm N) /. i)) <> 0. F1() ) & ( for i being Nat st i in dom (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) & i > card N holds
Line ((RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))),i) = F3() |-> (0. F1()) ) & ( for i, j being Nat st i in Seg (card N) & j in Seg (width (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1)))))))) & j < (Sgm N) . i holds
(RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,j) = 0. F1() ) )
proof
let i be Nat; ::_thesis: ( i in Seg (card N) & i <= n + 1 implies (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,((Sgm N) /. i)) = 1_ F1() )
assume that
A33: i in Seg (card N) and
A34: i <= n + 1 ; ::_thesis: (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,((Sgm N) /. i)) = 1_ F1()
A35: ( (Sgm N) . i = (Sgm N) /. i & (Sgm N) . i in rng (Sgm N) ) by A31, A33, FUNCT_1:def_3, PARTFUN1:def_6;
then A36: [i,((Sgm N) /. i)] in Indices A1 by A4, A5, A30, A20, A29, A18, A27, A33, ZFMISC_1:87;
percases ( i <= n or i = n + 1 ) by A34, NAT_1:8;
supposeA37: i <= n ; ::_thesis: (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,((Sgm N) /. i)) = 1_ F1()
then i < n + 1 by NAT_1:13;
hence (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,((Sgm N) /. i)) = A1 * (i,((Sgm N) /. i)) by A19, A36, MATRIX11:def_3
.= 1_ F1() by A12, A33, A37 ;
::_thesis: verum
end;
supposeA38: i = n + 1 ; ::_thesis: (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,((Sgm N) /. i)) = 1_ F1()
A39: A1 * (i,((Sgm N) /. i)) <> 0. F1() by A15, A33;
( (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,((Sgm N) /. i)) = (((A1 * (i,((Sgm N) /. i))) ") * (Line (A1,(n + 1)))) . ((Sgm N) /. i) & (Line (A1,(n + 1))) . ((Sgm N) /. i) = A1 * (i,((Sgm N) /. i)) ) by A4, A30, A19, A28, A27, A35, A36, A38, MATRIX11:def_3, MATRIX_1:def_7;
hence (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,((Sgm N) /. i)) = ((A1 * (i,((Sgm N) /. i))) ") * (A1 * (i,((Sgm N) /. i))) by A4, A30, A27, A35, FVSUM_1:51
.= 1_ F1() by A39, VECTSP_1:def_10 ;
::_thesis: verum
end;
end;
end;
thus P1[ RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1)))))),FB] by A2, A5, A13, A21, A20, A29, A18, A24; ::_thesis: ( Segm ((RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))),(Seg (card N)),N) is diagonal & ( for i being Nat st i in Seg (card N) holds
(RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,((Sgm N) /. i)) <> 0. F1() ) & ( for i being Nat st i in dom (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) & i > card N holds
Line ((RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))),i) = F3() |-> (0. F1()) ) & ( for i, j being Nat st i in Seg (card N) & j in Seg (width (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1)))))))) & j < (Sgm N) . i holds
(RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,j) = 0. F1() ) )
thus Segm ((RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))),(Seg (card N)),N) is diagonal ::_thesis: ( ( for i being Nat st i in Seg (card N) holds
(RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,((Sgm N) /. i)) <> 0. F1() ) & ( for i being Nat st i in dom (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) & i > card N holds
Line ((RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))),i) = F3() |-> (0. F1()) ) & ( for i, j being Nat st i in Seg (card N) & j in Seg (width (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1)))))))) & j < (Sgm N) . i holds
(RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,j) = 0. F1() ) )
proof
A40: Indices A1 = Indices (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) by MATRIX_1:26;
let i be Nat; :: according to MATRIX_1:def_14 ::_thesis: for b1 being set holds
( not [i,b1] in Indices (Segm ((RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))),(Seg (card N)),N)) or (Segm ((RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))),(Seg (card N)),N)) * (i,b1) = 0. F1() or i = b1 )
let j be Nat; ::_thesis: ( not [i,j] in Indices (Segm ((RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))),(Seg (card N)),N)) or (Segm ((RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))),(Seg (card N)),N)) * (i,j) = 0. F1() or i = j )
assume that
A41: [i,j] in Indices (Segm ((RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))),(Seg (card N)),N)) and
A42: (Segm ((RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))),(Seg (card N)),N)) * (i,j) <> 0. F1() ; ::_thesis: i = j
reconsider I = i, J = j as Element of NAT by ORDINAL1:def_12;
Indices (Segm ((RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))),(Seg (card N)),N)) = [:(Seg (card (Seg (card N)))),(Seg (width (Segm ((RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))),(Seg (card N)),N)))):] by MATRIX_1:25;
then i in Seg (card (Seg (card N))) by A41, ZFMISC_1:87;
then i in Seg (card (card N)) by FINSEQ_1:55;
then A43: ( Sgm (Seg (card N)) = idseq (card N) & (idseq (card N)) . I = I ) by FINSEQ_2:49, FINSEQ_3:48;
then A44: (Segm ((RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))),(Seg (card N)),N)) * (i,j) = (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (I,((Sgm N) . J)) by A41, MATRIX13:def_1;
( rng (Sgm (Seg (card N))) = Seg (card N) & [:(Seg (card N)),N:] c= Indices A1 ) by A4, A5, A20, A29, A18, A27, FINSEQ_1:def_13, ZFMISC_1:96;
then A45: [I,((Sgm N) . J)] in Indices A1 by A30, A41, A40, A43, MATRIX13:17;
then A46: (Sgm N) . J in Seg (width A1) by ZFMISC_1:87;
then reconsider SgmNJ = (Sgm N) . j as Element of NAT ;
A47: Indices (Segm ((RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))),(Seg (card N)),N)) = Indices (Segm (A1,(Seg (card N)),N)) by MATRIX_1:26;
percases ( I = n + 1 or I <> n + 1 ) ;
supposeA48: I = n + 1 ; ::_thesis: i = j
thus i = j ::_thesis: verum
proof
assume A49: i <> j ; ::_thesis: contradiction
( (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (I,((Sgm N) . J)) = (((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") * (Line (A1,(n + 1)))) . SgmNJ & (Line (A1,(n + 1))) . SgmNJ = A1 * (I,SgmNJ) ) by A19, A28, A45, A46, A48, MATRIX11:def_3, MATRIX_1:def_7;
then (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (I,((Sgm N) . J)) = ((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") * (A1 * (I,SgmNJ)) by A46, FVSUM_1:51
.= ((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") * ((Segm (A1,(Seg (card N)),N)) * (i,j)) by A41, A47, A43, MATRIX13:def_1
.= ((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") * (0. F1()) by A14, A41, A47, A49, MATRIX_1:def_14
.= 0. F1() by VECTSP_1:6 ;
hence contradiction by A41, A42, A43, MATRIX13:def_1; ::_thesis: verum
end;
end;
suppose I <> n + 1 ; ::_thesis: i = j
then (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (I,((Sgm N) . J)) = A1 * (((Sgm (Seg (card N))) . I),((Sgm N) . J)) by A19, A43, A45, MATRIX11:def_3
.= (Segm (A1,(Seg (card N)),N)) * (i,j) by A41, A47, MATRIX13:def_1 ;
hence i = j by A14, A41, A42, A47, A44, MATRIX_1:def_14; ::_thesis: verum
end;
end;
end;
thus for i being Nat st i in Seg (card N) holds
(RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,((Sgm N) /. i)) <> 0. F1() ::_thesis: ( ( for i being Nat st i in dom (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) & i > card N holds
Line ((RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))),i) = F3() |-> (0. F1()) ) & ( for i, j being Nat st i in Seg (card N) & j in Seg (width (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1)))))))) & j < (Sgm N) . i holds
(RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,j) = 0. F1() ) )
proof
let i be Nat; ::_thesis: ( i in Seg (card N) implies (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,((Sgm N) /. i)) <> 0. F1() )
assume A50: i in Seg (card N) ; ::_thesis: (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,((Sgm N) /. i)) <> 0. F1()
( (Sgm N) . i = (Sgm N) /. i & (Sgm N) . i in rng (Sgm N) ) by A31, A50, FUNCT_1:def_3, PARTFUN1:def_6;
then A51: [i,((Sgm N) /. i)] in Indices A1 by A4, A5, A30, A20, A29, A18, A27, A50, ZFMISC_1:87;
percases ( i = n + 1 or i <> n + 1 ) ;
suppose i = n + 1 ; ::_thesis: (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,((Sgm N) /. i)) <> 0. F1()
hence (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,((Sgm N) /. i)) <> 0. F1() by A32, A50; ::_thesis: verum
end;
suppose i <> n + 1 ; ::_thesis: (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,((Sgm N) /. i)) <> 0. F1()
then (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,((Sgm N) /. i)) = A1 * (i,((Sgm N) /. i)) by A19, A51, MATRIX11:def_3;
hence (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,((Sgm N) /. i)) <> 0. F1() by A15, A50; ::_thesis: verum
end;
end;
end;
thus for i being Nat st i in dom (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) & i > card N holds
Line ((RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))),i) = F3() |-> (0. F1()) ::_thesis: for i, j being Nat st i in Seg (card N) & j in Seg (width (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1)))))))) & j < (Sgm N) . i holds
(RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,j) = 0. F1()
proof
A52: ( dom (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) = Seg (len (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1)))))))) & len (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) = F2() ) by FINSEQ_1:def_3, MATRIX_1:def_2;
let i be Nat; ::_thesis: ( i in dom (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) & i > card N implies Line ((RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))),i) = F3() |-> (0. F1()) )
assume A53: ( i in dom (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) & i > card N ) ; ::_thesis: Line ((RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))),i) = F3() |-> (0. F1())
thus Line ((RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))),i) = Line (A1,i) by A10, A53, A52, MATRIX11:28
.= F3() |-> (0. F1()) by A16, A29, A18, A53, A52 ; ::_thesis: verum
end;
let i, j be Nat; ::_thesis: ( i in Seg (card N) & j in Seg (width (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1)))))))) & j < (Sgm N) . i implies (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,j) = 0. F1() )
assume that
A54: i in Seg (card N) and
A55: j in Seg (width (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1)))))))) and
A56: j < (Sgm N) . i ; ::_thesis: (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,j) = 0. F1()
A57: [i,j] in Indices A1 by A5, A20, A29, A18, A27, A23, A54, A55, ZFMISC_1:87;
percases ( i = n + 1 or i <> n + 1 ) ;
suppose i = n + 1 ; ::_thesis: (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,j) = 0. F1()
then ( (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,j) = (((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") * (Line (A1,(n + 1)))) . j & (Line (A1,(n + 1))) . j = A1 * (i,j) ) by A19, A28, A27, A23, A55, A57, MATRIX11:def_3, MATRIX_1:def_7;
hence (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,j) = ((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") * (A1 * (i,j)) by A27, A23, A55, FVSUM_1:51
.= ((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") * (0. F1()) by A17, A27, A23, A54, A55, A56
.= 0. F1() by VECTSP_1:6 ;
::_thesis: verum
end;
suppose i <> n + 1 ; ::_thesis: (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,j) = 0. F1()
hence (RLine (A1,(n + 1),((Line (A1,(n + 1))) + ((((A1 * ((n + 1),((Sgm N) /. (n + 1)))) ") - (1_ F1())) * (Line (A1,(n + 1))))))) * (i,j) = A1 * (i,j) by A19, A57, MATRIX11:def_3
.= 0. F1() by A17, A27, A23, A54, A55, A56 ;
::_thesis: verum
end;
end;
end;
for i being Nat st i in Seg (card N) & i <= 0 holds
A9 * (i,((Sgm N) /. i)) = 1_ F1() ;
then A58: S1[ 0 ] by A5, A6;
for n being Nat holds S1[n] from NAT_1:sch_2(A58, A8);
then consider A being Matrix of F2(),F3(),F1(), B being Matrix of F2(),F4(),F1() such that
A59: the_rank_of F5() = the_rank_of A and
A60: for i being Nat st i in Seg (card N) & i <= card N holds
A * (i,((Sgm N) /. i)) = 1_ F1() and
A61: P1[A,B] and
A62: Segm (A,(Seg (card N)),N) is diagonal and
for i being Nat st i in Seg (card N) holds
A * (i,((Sgm N) /. i)) <> 0. F1() and
A63: ( ( for i being Nat st i in dom A & i > card N holds
Line (A,i) = F3() |-> (0. F1()) ) & ( for i, j being Nat st i in Seg (card N) & j in Seg (width A) & j < (Sgm N) . i holds
A * (i,j) = 0. F1() ) ) ;
take A ; ::_thesis: ex B9 being Matrix of F2(),F4(),F1() ex N being finite without_zero Subset of NAT st
( N c= Seg F3() & the_rank_of F5() = the_rank_of A & the_rank_of F5() = card N & P1[A,B9] & Segm (A,(Seg (card N)),N) = 1. (F1(),(card N)) & ( for i being Nat st i in dom A & i > card N holds
Line (A,i) = F3() |-> (0. F1()) ) & ( for i, j being Nat st i in Seg (card N) & j in Seg (width A) & j < (Sgm N) . i holds
A * (i,j) = 0. F1() ) )
take B ; ::_thesis: ex N being finite without_zero Subset of NAT st
( N c= Seg F3() & the_rank_of F5() = the_rank_of A & the_rank_of F5() = card N & P1[A,B] & Segm (A,(Seg (card N)),N) = 1. (F1(),(card N)) & ( for i being Nat st i in dom A & i > card N holds
Line (A,i) = F3() |-> (0. F1()) ) & ( for i, j being Nat st i in Seg (card N) & j in Seg (width A) & j < (Sgm N) . i holds
A * (i,j) = 0. F1() ) )
take N ; ::_thesis: ( N c= Seg F3() & the_rank_of F5() = the_rank_of A & the_rank_of F5() = card N & P1[A,B] & Segm (A,(Seg (card N)),N) = 1. (F1(),(card N)) & ( for i being Nat st i in dom A & i > card N holds
Line (A,i) = F3() |-> (0. F1()) ) & ( for i, j being Nat st i in Seg (card N) & j in Seg (width A) & j < (Sgm N) . i holds
A * (i,j) = 0. F1() ) )
thus ( N c= Seg F3() & the_rank_of F5() = the_rank_of A & the_rank_of F5() = card N & P1[A,B] ) by A4, A5, A59, A61; ::_thesis: ( Segm (A,(Seg (card N)),N) = 1. (F1(),(card N)) & ( for i being Nat st i in dom A & i > card N holds
Line (A,i) = F3() |-> (0. F1()) ) & ( for i, j being Nat st i in Seg (card N) & j in Seg (width A) & j < (Sgm N) . i holds
A * (i,j) = 0. F1() ) )
set S = Segm (A,(Seg (card N)),N);
A64: card (Seg (card N)) = card N by FINSEQ_1:57;
then A65: Indices (1. (F1(),(card N))) = Indices (Segm (A,(Seg (card N)),N)) by MATRIX_1:26;
now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_(1._(F1(),(card_N)))_holds_
(Segm_(A,(Seg_(card_N)),N))_*_(i,j)_=_(1._(F1(),(card_N)))_*_(i,j)
A66: dom (Sgm N) = Seg (the_rank_of F5()) by A4, A5, FINSEQ_3:40;
let i, j be Nat; ::_thesis: ( [i,j] in Indices (1. (F1(),(card N))) implies (Segm (A,(Seg (card N)),N)) * (i,j) = (1. (F1(),(card N))) * (i,j) )
assume A67: [i,j] in Indices (1. (F1(),(card N))) ; ::_thesis: (Segm (A,(Seg (card N)),N)) * (i,j) = (1. (F1(),(card N))) * (i,j)
A68: j in Seg (the_rank_of F5()) by A7, A67, ZFMISC_1:87;
reconsider i9 = i, j9 = j as Element of NAT by ORDINAL1:def_12;
A69: i in Seg (the_rank_of F5()) by A7, A67, ZFMISC_1:87;
then A70: (idseq (the_rank_of F5())) . i9 = i9 by FINSEQ_2:49;
A71: i <= the_rank_of F5() by A69, FINSEQ_1:1;
A72: (Segm (A,(Seg (card N)),N)) * (i,j) = A * (((Sgm (Seg (the_rank_of F5()))) . i9),((Sgm N) . j9)) by A5, A65, A67, MATRIX13:def_1
.= A * (i9,((Sgm N) . j9)) by A70, FINSEQ_3:48
.= A * (i9,((Sgm N) /. j9)) by A68, A66, PARTFUN1:def_6 ;
now__::_thesis:_(Segm_(A,(Seg_(card_N)),N))_*_(i,j)_=_(1._(F1(),(card_N)))_*_(i,j)
percases ( i9 = j9 or i9 <> j9 ) ;
supposeA73: i9 = j9 ; ::_thesis: (Segm (A,(Seg (card N)),N)) * (i,j) = (1. (F1(),(card N))) * (i,j)
hence (Segm (A,(Seg (card N)),N)) * (i,j) = 1_ F1() by A5, A60, A69, A71, A72
.= (1. (F1(),(card N))) * (i,j) by A67, A73, MATRIX_1:def_11 ;
::_thesis: verum
end;
supposeA74: i9 <> j9 ; ::_thesis: (Segm (A,(Seg (card N)),N)) * (i,j) = (1. (F1(),(card N))) * (i,j)
hence (Segm (A,(Seg (card N)),N)) * (i,j) = 0. F1() by A62, A65, A67, MATRIX_1:def_14
.= (1. (F1(),(card N))) * (i,j) by A67, A74, MATRIX_1:def_11 ;
::_thesis: verum
end;
end;
end;
hence (Segm (A,(Seg (card N)),N)) * (i,j) = (1. (F1(),(card N))) * (i,j) ; ::_thesis: verum
end;
hence ( Segm (A,(Seg (card N)),N) = 1. (F1(),(card N)) & ( for i being Nat st i in dom A & i > card N holds
Line (A,i) = F3() |-> (0. F1()) ) & ( for i, j being Nat st i in Seg (card N) & j in Seg (width A) & j < (Sgm N) . i holds
A * (i,j) = 0. F1() ) ) by A63, A64, MATRIX_1:27; ::_thesis: verum
end;
begin
theorem Th57: :: MATRIX15:57
for K being Field
for A, B being Matrix of K st len A = len B & ( width A = 0 implies width B = 0 ) holds
( the_rank_of A = the_rank_of (A ^^ B) iff not Solutions_of (A,B) is empty )
proof
let K be Field; ::_thesis: for A, B being Matrix of K st len A = len B & ( width A = 0 implies width B = 0 ) holds
( the_rank_of A = the_rank_of (A ^^ B) iff not Solutions_of (A,B) is empty )
let A, B be Matrix of K; ::_thesis: ( len A = len B & ( width A = 0 implies width B = 0 ) implies ( the_rank_of A = the_rank_of (A ^^ B) iff not Solutions_of (A,B) is empty ) )
assume that
A1: len A = len B and
A2: ( width A = 0 implies width B = 0 ) ; ::_thesis: ( the_rank_of A = the_rank_of (A ^^ B) iff not Solutions_of (A,B) is empty )
set wB = width B;
set L = len A;
reconsider B9 = B as Matrix of len A, width B,K by A1, MATRIX_2:7;
set wA = width A;
reconsider A9 = A as Matrix of len A, width A,K by MATRIX_2:7;
deffunc H1( Matrix of len A, width B,K, Nat, Nat, Element of K) -> Matrix of len A, width B, the carrier of K = RLine ($1,$2,((Line ($1,$2)) + ($4 * (Line ($1,$3)))));
defpred S1[ set , set ] means for A1 being Matrix of len A, width A,K
for B1 being Matrix of len A, width B,K st A1 = $1 & B1 = $2 holds
( the_rank_of (A9 ^^ B9) = the_rank_of (A1 ^^ B1) & Solutions_of (A9,B9) = Solutions_of (A1,B1) );
A3: for A1 being Matrix of len A, width A,K
for B1 being Matrix of len A, width B,K st S1[A1,B1] holds
for a being Element of K
for i, j being Nat st j in dom A1 & ( i = j implies a <> - (1_ K) ) holds
S1[ RLine (A1,i,((Line (A1,i)) + (a * (Line (A1,j))))),H1(B1,i,j,a)]
proof
let A1 be Matrix of len A, width A,K; ::_thesis: for B1 being Matrix of len A, width B,K st S1[A1,B1] holds
for a being Element of K
for i, j being Nat st j in dom A1 & ( i = j implies a <> - (1_ K) ) holds
S1[ RLine (A1,i,((Line (A1,i)) + (a * (Line (A1,j))))),H1(B1,i,j,a)]
let B1 be Matrix of len A, width B,K; ::_thesis: ( S1[A1,B1] implies for a being Element of K
for i, j being Nat st j in dom A1 & ( i = j implies a <> - (1_ K) ) holds
S1[ RLine (A1,i,((Line (A1,i)) + (a * (Line (A1,j))))),H1(B1,i,j,a)] )
assume A4: S1[A1,B1] ; ::_thesis: for a being Element of K
for i, j being Nat st j in dom A1 & ( i = j implies a <> - (1_ K) ) holds
S1[ RLine (A1,i,((Line (A1,i)) + (a * (Line (A1,j))))),H1(B1,i,j,a)]
let a be Element of K; ::_thesis: for i, j being Nat st j in dom A1 & ( i = j implies a <> - (1_ K) ) holds
S1[ RLine (A1,i,((Line (A1,i)) + (a * (Line (A1,j))))),H1(B1,i,j,a)]
let i, j be Nat; ::_thesis: ( j in dom A1 & ( i = j implies a <> - (1_ K) ) implies S1[ RLine (A1,i,((Line (A1,i)) + (a * (Line (A1,j))))),H1(B1,i,j,a)] )
assume that
A5: j in dom A1 and
A6: ( i = j implies a <> - (1_ K) ) ; ::_thesis: S1[ RLine (A1,i,((Line (A1,i)) + (a * (Line (A1,j))))),H1(B1,i,j,a)]
set LAj = Line (A1,j);
set LAi = Line (A1,i);
set RA = RLine (A1,i,((Line (A1,i)) + (a * (Line (A1,j)))));
A7: dom A1 = Seg (len A1) by FINSEQ_1:def_3
.= Seg (len A) by MATRIX_1:def_2 ;
then A8: Solutions_of (A1,B1) = Solutions_of ((RLine (A1,i,((Line (A1,i)) + (a * (Line (A1,j)))))),H1(B1,i,j,a)) by A5, A6, Th40;
set RB = H1(B1,i,j,a);
set LBj = Line (B1,j);
set LBi = Line (B1,i);
A9: ( len A1 = len A & len B1 = len A ) by MATRIX_1:def_2;
percases ( not i in Seg (len A) or i in Seg (len A) ) ;
suppose not i in Seg (len A) ; ::_thesis: S1[ RLine (A1,i,((Line (A1,i)) + (a * (Line (A1,j))))),H1(B1,i,j,a)]
then ( RLine (A1,i,((Line (A1,i)) + (a * (Line (A1,j))))) = A1 & H1(B1,i,j,a) = B1 ) by A9, MATRIX13:40;
hence S1[ RLine (A1,i,((Line (A1,i)) + (a * (Line (A1,j))))),H1(B1,i,j,a)] by A4; ::_thesis: verum
end;
supposeA10: i in Seg (len A) ; ::_thesis: S1[ RLine (A1,i,((Line (A1,i)) + (a * (Line (A1,j))))),H1(B1,i,j,a)]
A11: len (A1 ^^ B1) = len A by MATRIX_1:def_2;
A12: ( len (a * (Line (A1,j))) = width A1 & len (a * (Line (B1,j))) = width B1 ) by CARD_1:def_7;
A13: ( len (Line (A1,i)) = width A1 & len (Line (B1,i)) = width B1 ) by CARD_1:def_7;
( len ((Line (A1,i)) + (a * (Line (A1,j)))) = width A1 & len ((Line (B1,i)) + (a * (Line (B1,j)))) = width B1 ) by CARD_1:def_7;
then (RLine (A1,i,((Line (A1,i)) + (a * (Line (A1,j)))))) ^^ H1(B1,i,j,a) = RLine ((A1 ^^ B1),i,(((Line (A1,i)) + (a * (Line (A1,j)))) ^ ((Line (B1,i)) + (a * (Line (B1,j)))))) by Th18
.= RLine ((A1 ^^ B1),i,(((Line (A1,i)) ^ (Line (B1,i))) + ((a * (Line (A1,j))) ^ (a * (Line (B1,j)))))) by A13, A12, Th3
.= RLine ((A1 ^^ B1),i,(((Line (A1,i)) ^ (Line (B1,i))) + (a * ((Line (A1,j)) ^ (Line (B1,j)))))) by Th4
.= RLine ((A1 ^^ B1),i,((Line ((A1 ^^ B1),i)) + (a * ((Line (A1,j)) ^ (Line (B1,j)))))) by A10, Th15
.= RLine ((A1 ^^ B1),i,((Line ((A1 ^^ B1),i)) + (a * (Line ((A1 ^^ B1),j))))) by A5, A7, Th15 ;
then the_rank_of ((RLine (A1,i,((Line (A1,i)) + (a * (Line (A1,j)))))) ^^ H1(B1,i,j,a)) = the_rank_of (A1 ^^ B1) by A5, A6, A7, A11, MATRIX13:92;
hence S1[ RLine (A1,i,((Line (A1,i)) + (a * (Line (A1,j))))),H1(B1,i,j,a)] by A4, A8; ::_thesis: verum
end;
end;
end;
A14: S1[A9,B9] ;
consider A1 being Matrix of len A, width A,K, B1 being Matrix of len A, width B,K, N being finite without_zero Subset of NAT such that
A15: N c= Seg (width A) and
A16: ( the_rank_of A9 = the_rank_of A1 & the_rank_of A9 = card N ) and
A17: ( S1[A1,B1] & Segm (A1,(Seg (card N)),N) = 1. (K,(card N)) ) and
A18: for i being Nat st i in dom A1 & i > card N holds
Line (A1,i) = (width A) |-> (0. K) and
for i, j being Nat st i in Seg (card N) & j in Seg (width A1) & j < (Sgm N) . i holds
A1 * (i,j) = 0. K from MATRIX15:sch_2(A14, A3);
percases ( len A = 0 or len A > 0 ) ;
supposeA19: len A = 0 ; ::_thesis: ( the_rank_of A = the_rank_of (A ^^ B) iff not Solutions_of (A,B) is empty )
then ( len (A9 ^^ B9) = 0 & the_rank_of A = 0 ) by MATRIX13:74, MATRIX_1:def_2;
hence ( the_rank_of A = the_rank_of (A ^^ B) iff not Solutions_of (A,B) is empty ) by A19, Th51, MATRIX13:74; ::_thesis: verum
end;
supposeA20: len A > 0 ; ::_thesis: ( the_rank_of A = the_rank_of (A ^^ B) iff not Solutions_of (A,B) is empty )
percases ( N <> {} or N = {} ) ;
supposeA21: N <> {} ; ::_thesis: ( the_rank_of A = the_rank_of (A ^^ B) iff not Solutions_of (A,B) is empty )
set SN = Seg (card N);
set SS = (Seg (len A)) \ (Seg (card N));
A22: card (Seg (card N)) = card N by FINSEQ_1:57;
reconsider P2 = (Sgm (Seg (card N))) " (Seg (card N)), Q2 = (Sgm (Seg (width A))) " N as finite without_zero Subset of NAT by MATRIX13:53;
( dom (Sgm (Seg (card N))) = Seg (card (Seg (card N))) & rng (Sgm (Seg (card N))) = Seg (card N) ) by FINSEQ_1:def_13, FINSEQ_3:40;
then A23: P2 = Seg (card N) by A22, RELAT_1:134;
rng (Sgm (Seg (width A))) = Seg (width A) by FINSEQ_1:def_13;
then A24: (Sgm (Seg (width A))) .: Q2 = N by A15, FUNCT_1:77;
( Q2 c= dom (Sgm (Seg (width A))) & Sgm (Seg (width A)) is one-to-one ) by FINSEQ_3:92, RELAT_1:132;
then N,Q2 are_equipotent by A24, CARD_1:33;
then A25: ( dom (Sgm (Seg (width A))) = Seg (card (Seg (width A))) & card N = card Q2 ) by CARD_1:5, FINSEQ_3:40;
A26: ( Seg (len A1) = dom A1 & len A1 = len A ) by A20, FINSEQ_1:def_3, MATRIX_1:23;
A27: width A1 = width A by A20, MATRIX_1:23;
A28: now__::_thesis:_for_i_being_Nat_st_i_in_(Seg_(len_A))_\_(Seg_(card_N))_holds_
Line_(A1,i)_=_(width_A1)_|->_(0._K)
let i be Nat; ::_thesis: ( i in (Seg (len A)) \ (Seg (card N)) implies Line (A1,i) = (width A1) |-> (0. K) )
assume A29: i in (Seg (len A)) \ (Seg (card N)) ; ::_thesis: Line (A1,i) = (width A1) |-> (0. K)
not i in Seg (card N) by A29, XBOOLE_0:def_5;
then A30: ( i < 1 or i > card N ) by A29;
i in Seg (len A) by A29, XBOOLE_0:def_5;
hence Line (A1,i) = (width A1) |-> (0. K) by A18, A26, A27, A30, FINSEQ_1:1; ::_thesis: verum
end;
card N <= len A by A16, MATRIX13:74;
then A31: Seg (card N) c= Seg (len A) by FINSEQ_1:5;
A32: len B1 = len A by A20, MATRIX_1:23;
A33: ( Seg (len B1) = dom B1 & width B1 = width B ) by A20, FINSEQ_1:def_3, MATRIX_1:23;
thus ( the_rank_of A = the_rank_of (A ^^ B) implies not Solutions_of (A,B) is empty ) ::_thesis: ( not Solutions_of (A,B) is empty implies the_rank_of A = the_rank_of (A ^^ B) )
proof
assume the_rank_of A = the_rank_of (A ^^ B) ; ::_thesis: not Solutions_of (A,B) is empty
then the_rank_of A1 = the_rank_of (A1 ^^ B1) by A16, A17;
then for i being Nat st i in (dom A1) \ (Seg (card N)) holds
( Line (A1,i) = (width A1) |-> (0. K) & Line (B1,i) = (width B1) |-> (0. K) ) by A26, A32, A28, Th24, XBOOLE_1:36;
then A34: Solutions_of (A1,B1) = Solutions_of ((Segm (A1,(Seg (card N)),(Seg (width A)))),(Segm (B1,(Seg (card N)),(Seg (width B))))) by A21, A31, A26, A32, A27, A33, Th45;
Segm ((Segm (A1,(Seg (card N)),(Seg (width A)))),P2,Q2) = 1. (K,(card N)) by A15, A17, MATRIX13:56;
then ex X being Matrix of card (Seg (width A)), card (Seg (width B)),K st
( Segm (X,((Seg (card (Seg (width A)))) \ Q2),(Seg (card (Seg (width B))))) = 0. (K,((card (Seg (width A))) -' (card (Seg (card N)))),(card (Seg (width B)))) & Segm (X,Q2,(Seg (card (Seg (width B))))) = Segm (B1,(Seg (card N)),(Seg (width B))) & X in Solutions_of ((Segm (A1,(Seg (card N)),(Seg (width A)))),(Segm (B1,(Seg (card N)),(Seg (width B))))) ) by A21, A22, A23, A25, Th50, RELAT_1:132;
hence not Solutions_of (A,B) is empty by A17, A34; ::_thesis: verum
end;
A35: (Seg (len A)) \ (Seg (card N)) c= Seg (len A) by XBOOLE_1:36;
thus ( not Solutions_of (A,B) is empty implies the_rank_of A = the_rank_of (A ^^ B) ) ::_thesis: verum
proof
assume not Solutions_of (A,B) is empty ; ::_thesis: the_rank_of A = the_rank_of (A ^^ B)
then not Solutions_of (A1,B1) is empty by A17;
then consider x being set such that
A36: x in Solutions_of (A1,B1) by XBOOLE_0:def_1;
set AB = A1 ^^ B1;
A37: len (Segm ((A1 ^^ B1),(Seg (card N)),(Seg (width (A1 ^^ B1))))) = card (Seg (card N)) by MATRIX_1:def_2;
A38: ( dom (A1 ^^ B1) = Seg (len (A1 ^^ B1)) & len (A1 ^^ B1) = len A ) by A20, FINSEQ_1:def_3, MATRIX_1:23;
reconsider x = x as Matrix of width A, width B,K by A20, A36, Th53;
A39: the_rank_of (Segm ((A1 ^^ B1),(Seg (len A)),(Seg (width A1)))) = card N by A16, Th19;
A40: width (A1 ^^ B1) = (width A1) + (width B1) by A20, MATRIX_1:23;
now__::_thesis:_for_i_being_Nat_st_i_in_(Seg_(len_A))_\_(Seg_(card_N))_holds_
Line_((A1_^^_B1),i)_=_(width_(A1_^^_B1))_|->_(0._K)
let i be Nat; ::_thesis: ( i in (Seg (len A)) \ (Seg (card N)) implies Line ((A1 ^^ B1),i) = (width (A1 ^^ B1)) |-> (0. K) )
assume A41: i in (Seg (len A)) \ (Seg (card N)) ; ::_thesis: Line ((A1 ^^ B1),i) = (width (A1 ^^ B1)) |-> (0. K)
A42: Line (A1,i) = (width A1) |-> (0. K) by A28, A41;
A43: ( x in Solutions_of (A1,B1) & i in dom A1 & Line (A1,i) = (width A1) |-> (0. K) implies Line (B1,i) = (width B1) |-> (0. K) ) by Th41;
thus Line ((A1 ^^ B1),i) = (Line (A1,i)) ^ (Line (B1,i)) by A35, A41, Th15
.= (width (A1 ^^ B1)) |-> (0. K) by A26, A35, A36, A40, A41, A42, A43, FINSEQ_2:123 ; ::_thesis: verum
end;
then the_rank_of (Segm ((A1 ^^ B1),(Seg (card N)),(Seg (width (A1 ^^ B1))))) = the_rank_of (A1 ^^ B1) by A31, A38, Th11;
then the_rank_of (A1 ^^ B1) <= card (Seg (card N)) by A37, MATRIX13:74;
then A44: the_rank_of (A1 ^^ B1) <= card N by FINSEQ_1:57;
width A1 <= width (A1 ^^ B1) by A40, NAT_1:11;
then Seg (width A1) c= Seg (width (A1 ^^ B1)) by FINSEQ_1:5;
then [:(Seg (len A)),(Seg (width A1)):] c= Indices (A1 ^^ B1) by A38, ZFMISC_1:95;
then card N <= the_rank_of (A1 ^^ B1) by A39, MATRIX13:79;
then the_rank_of (A1 ^^ B1) = card N by A44, XXREAL_0:1;
hence the_rank_of A = the_rank_of (A ^^ B) by A16, A17; ::_thesis: verum
end;
end;
supposeA45: N = {} ; ::_thesis: ( the_rank_of A = the_rank_of (A ^^ B) iff not Solutions_of (A,B) is empty )
set ZERO = 0. (K,(len A),(width A));
A46: now__::_thesis:_for_i_being_Nat_st_1_<=_i_&_i_<=_len_A_holds_
(0._(K,(len_A),(width_A)))_._i_=_A1_._i
let i be Nat; ::_thesis: ( 1 <= i & i <= len A implies (0. (K,(len A),(width A))) . i = A1 . i )
assume A47: ( 1 <= i & i <= len A ) ; ::_thesis: (0. (K,(len A),(width A))) . i = A1 . i
A48: ( dom A1 = Seg (len A1) & len A1 = len A ) by FINSEQ_1:def_3, MATRIX_1:def_2;
i in NAT by ORDINAL1:def_12;
then A49: i in Seg (len A) by A47;
hence (0. (K,(len A),(width A))) . i = (width A) |-> (0. K) by FINSEQ_2:57
.= Line (A1,i) by A18, A45, A49, A48
.= A1 . i by A49, MATRIX_2:8 ;
::_thesis: verum
end;
A50: len (0. (K,(len A),(width A))) = len A by A20, MATRIX_1:23;
A51: width A1 = width A by A20, MATRIX_1:23;
len A1 = len A by A20, MATRIX_1:23;
then 0. (K,(len A),(width A)) = A1 by A50, A46, FINSEQ_1:14;
then A52: the_rank_of A = 0 by A16, A50, A51, MATRIX13:95;
then A53: 0. (K,(len A),(width A)) = A by MATRIX13:95;
A54: Indices (A9 ^^ B9) = [:(Seg (len A)),(Seg ((width A) + (width B))):] by A20, MATRIX_1:23;
thus ( the_rank_of A = the_rank_of (A ^^ B) implies not Solutions_of (A,B) is empty ) ::_thesis: ( not Solutions_of (A,B) is empty implies the_rank_of A = the_rank_of (A ^^ B) )
proof
set x = the Matrix of width A, width B,K;
assume A55: the_rank_of A = the_rank_of (A ^^ B) ; ::_thesis: not Solutions_of (A,B) is empty
(Seg ((width A) + (width B))) \ (Seg (width A)) c= Seg ((width A) + (width B)) by XBOOLE_1:36;
then A56: [:(Seg (len A)),((Seg ((width A) + (width B))) \ (Seg (width A))):] c= Indices (A ^^ B) by A54, ZFMISC_1:95;
Segm ((A9 ^^ B9),(Seg (len A)),((Seg ((width A) + (width B))) \ (Seg (width A)))) = B by Th19;
then 0 = the_rank_of B by A52, A55, A56, MATRIX13:79;
then A57: B = 0. (K,(len A),(width B)) by A1, MATRIX13:95;
then ( ( width A = 0 & width B = 0 ) or Solutions_of (A9,B9) = { X where X is Matrix of width A, width B,K : verum } ) by A2, A20, A53, Th54;
then ( Solutions_of (A9,B9) = {{}} or the Matrix of width A, width B,K in Solutions_of (A9,B9) ) by A53, A57, Th56;
hence not Solutions_of (A,B) is empty ; ::_thesis: verum
end;
A58: Indices B9 = [:(Seg (len A)),(Seg (width B)):] by A20, MATRIX_1:23;
A59: (width A) + (width B) = width (A9 ^^ B9) by A20, MATRIX_1:23;
A60: Indices (0. (K,(len A),(width A))) = [:(Seg (len A)),(Seg (width A)):] by A20, MATRIX_1:23;
thus ( not Solutions_of (A,B) is empty implies the_rank_of A = the_rank_of (A ^^ B) ) ::_thesis: verum
proof
assume A61: not Solutions_of (A,B) is empty ; ::_thesis: the_rank_of A = the_rank_of (A ^^ B)
assume the_rank_of A <> the_rank_of (A ^^ B) ; ::_thesis: contradiction
then consider i, j being Nat such that
A62: [i,j] in Indices (A9 ^^ B9) and
A63: (A9 ^^ B9) * (i,j) <> 0. K by A52, MATRIX13:94;
A64: j in Seg ((width A) + (width B)) by A59, A62, ZFMISC_1:87;
A65: dom (Line ((A9 ^^ B9),i)) = Seg ((width A) + (width B)) by A59, FINSEQ_2:124;
A66: len (Line (A9,i)) = width A by CARD_1:def_7;
A67: dom (Line (B9,i)) = Seg (width B) by FINSEQ_2:124;
A68: dom (Line (A9,i)) = Seg (width A) by FINSEQ_2:124;
A69: i in Seg (len A) by A54, A62, ZFMISC_1:87;
then A70: Line ((A9 ^^ B9),i) = (Line (A9,i)) ^ (Line (B9,i)) by Th15;
percases ( j in dom (Line (A9,i)) or ex n being Nat st
( n in dom (Line (B9,i)) & j = (width A) + n ) ) by A64, A66, A65, A70, FINSEQ_1:25;
supposeA71: j in dom (Line (A9,i)) ; ::_thesis: contradiction
then A72: [i,j] in Indices (0. (K,(len A),(width A))) by A60, A69, A68, ZFMISC_1:87;
(A9 ^^ B9) * (i,j) = (Line ((A9 ^^ B9),i)) . j by A59, A64, MATRIX_1:def_7
.= (Line (A9,i)) . j by A70, A71, FINSEQ_1:def_7
.= A9 * (i,j) by A68, A71, MATRIX_1:def_7
.= 0. K by A53, A72, MATRIX_3:1 ;
hence contradiction by A63; ::_thesis: verum
end;
suppose ex n being Nat st
( n in dom (Line (B9,i)) & j = (width A) + n ) ; ::_thesis: contradiction
then consider n being Nat such that
A73: n in dom (Line (B9,i)) and
A74: j = (width A) + n ;
A75: [i,n] in Indices B by A58, A69, A67, A73, ZFMISC_1:87;
A76: B = 0. (K,(len A),(width B)) by A53, A61, Th52;
(A9 ^^ B9) * (i,j) = (Line ((A9 ^^ B9),i)) . j by A59, A64, MATRIX_1:def_7
.= (Line (B9,i)) . n by A66, A70, A73, A74, FINSEQ_1:def_7
.= B9 * (i,n) by A67, A73, MATRIX_1:def_7
.= 0. K by A75, A76, MATRIX_3:1 ;
hence contradiction by A63; ::_thesis: verum
end;
end;
end;
end;
end;
end;
end;
end;
begin
definition
let K be Field;
let A be Matrix of K;
let b be FinSequence of K;
func Solutions_of (A,b) -> set equals :: MATRIX15:def 4
{ f where f is FinSequence of K : ColVec2Mx f in Solutions_of (A,(ColVec2Mx b)) } ;
coherence
{ f where f is FinSequence of K : ColVec2Mx f in Solutions_of (A,(ColVec2Mx b)) } is set ;
end;
:: deftheorem defines Solutions_of MATRIX15:def_4_:_
for K being Field
for A being Matrix of K
for b being FinSequence of K holds Solutions_of (A,b) = { f where f is FinSequence of K : ColVec2Mx f in Solutions_of (A,(ColVec2Mx b)) } ;
theorem Th58: :: MATRIX15:58
for K being Field
for A being Matrix of K
for b being FinSequence of K
for x being set st x in Solutions_of (A,(ColVec2Mx b)) holds
ex f being FinSequence of K st
( x = ColVec2Mx f & len f = width A )
proof
let K be Field; ::_thesis: for A being Matrix of K
for b being FinSequence of K
for x being set st x in Solutions_of (A,(ColVec2Mx b)) holds
ex f being FinSequence of K st
( x = ColVec2Mx f & len f = width A )
let A be Matrix of K; ::_thesis: for b being FinSequence of K
for x being set st x in Solutions_of (A,(ColVec2Mx b)) holds
ex f being FinSequence of K st
( x = ColVec2Mx f & len f = width A )
let b be FinSequence of K; ::_thesis: for x being set st x in Solutions_of (A,(ColVec2Mx b)) holds
ex f being FinSequence of K st
( x = ColVec2Mx f & len f = width A )
let x be set ; ::_thesis: ( x in Solutions_of (A,(ColVec2Mx b)) implies ex f being FinSequence of K st
( x = ColVec2Mx f & len f = width A ) )
assume A1: x in Solutions_of (A,(ColVec2Mx b)) ; ::_thesis: ex f being FinSequence of K st
( x = ColVec2Mx f & len f = width A )
consider X being Matrix of K such that
A2: X = x and
A3: len X = width A and
A4: width X = width (ColVec2Mx b) and
A * X = ColVec2Mx b by A1;
percases ( len X = 0 or len X > 0 ) ;
supposeA5: len X = 0 ; ::_thesis: ex f being FinSequence of K st
( x = ColVec2Mx f & len f = width A )
take f = 0 |-> (0. K); ::_thesis: ( x = ColVec2Mx f & len f = width A )
len (ColVec2Mx f) = 0 by MATRIX_1:def_2;
hence ( x = ColVec2Mx f & len f = width A ) by A2, A3, A5, CARD_2:64; ::_thesis: verum
end;
supposeA6: len X > 0 ; ::_thesis: ex f being FinSequence of K st
( x = ColVec2Mx f & len f = width A )
take Col (X,1) ; ::_thesis: ( x = ColVec2Mx (Col (X,1)) & len (Col (X,1)) = width A )
A7: len A = len (ColVec2Mx b) by A1, Th33;
len A <> 0 by A3, A6, MATRIX_1:def_3;
then len b > 0 by A7, MATRIX_1:def_2;
then width X = 1 by A4, MATRIX_1:23;
hence ( x = ColVec2Mx (Col (X,1)) & len (Col (X,1)) = width A ) by A2, A3, A6, Th26, MATRIX_1:def_8; ::_thesis: verum
end;
end;
end;
theorem Th59: :: MATRIX15:59
for K being Field
for A being Matrix of K
for b, f being FinSequence of K st ColVec2Mx f in Solutions_of (A,(ColVec2Mx b)) holds
len f = width A
proof
let K be Field; ::_thesis: for A being Matrix of K
for b, f being FinSequence of K st ColVec2Mx f in Solutions_of (A,(ColVec2Mx b)) holds
len f = width A
let A be Matrix of K; ::_thesis: for b, f being FinSequence of K st ColVec2Mx f in Solutions_of (A,(ColVec2Mx b)) holds
len f = width A
let b, f be FinSequence of K; ::_thesis: ( ColVec2Mx f in Solutions_of (A,(ColVec2Mx b)) implies len f = width A )
assume ColVec2Mx f in Solutions_of (A,(ColVec2Mx b)) ; ::_thesis: len f = width A
then A1: ex g being FinSequence of K st
( ColVec2Mx f = ColVec2Mx g & len g = width A ) by Th58;
len (ColVec2Mx f) = len f by MATRIX_1:def_2;
hence len f = width A by A1, MATRIX_1:def_2; ::_thesis: verum
end;
definition
let K be Field;
let A be Matrix of K;
let b be FinSequence of K;
:: original: Solutions_of
redefine func Solutions_of (A,b) -> Subset of ((width A) -VectSp_over K);
coherence
Solutions_of (A,b) is Subset of ((width A) -VectSp_over K)
proof
Solutions_of (A,b) c= the carrier of ((width A) -VectSp_over K)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Solutions_of (A,b) or x in the carrier of ((width A) -VectSp_over K) )
assume x in Solutions_of (A,b) ; ::_thesis: x in the carrier of ((width A) -VectSp_over K)
then consider f being FinSequence of K such that
A1: x = f and
A2: ColVec2Mx f in Solutions_of (A,(ColVec2Mx b)) ;
len f = width A by A2, Th59;
then ( (width A) -tuples_on the carrier of K = the carrier of ((width A) -VectSp_over K) & f is Element of (width A) -tuples_on the carrier of K ) by FINSEQ_2:92, MATRIX13:102;
hence x in the carrier of ((width A) -VectSp_over K) by A1; ::_thesis: verum
end;
hence Solutions_of (A,b) is Subset of ((width A) -VectSp_over K) ; ::_thesis: verum
end;
end;
registration
let K be Field;
let A be Matrix of K;
let k be Element of NAT ;
cluster Solutions_of (A,(k |-> (0. K))) -> linearly-closed for Subset of ((width A) -VectSp_over K);
coherence
for b1 being Subset of ((width A) -VectSp_over K) st b1 = Solutions_of (A,(k |-> (0. K))) holds
b1 is linearly-closed
proof
set V = (width A) -VectSp_over K;
set k0 = k |-> (0. K);
set S = Solutions_of (A,(k |-> (0. K)));
A1: ColVec2Mx (k |-> (0. K)) = 0. (K,k,1) by Th32;
A2: now__::_thesis:_for_a_being_Element_of_K
for_v_being_Element_of_((width_A)_-VectSp_over_K)_st_v_in_Solutions_of_(A,(k_|->_(0._K)))_holds_
a_*_v_in_Solutions_of_(A,(k_|->_(0._K)))
let a be Element of K; ::_thesis: for v being Element of ((width A) -VectSp_over K) st v in Solutions_of (A,(k |-> (0. K))) holds
a * v in Solutions_of (A,(k |-> (0. K)))
let v be Element of ((width A) -VectSp_over K); ::_thesis: ( v in Solutions_of (A,(k |-> (0. K))) implies a * v in Solutions_of (A,(k |-> (0. K))) )
assume v in Solutions_of (A,(k |-> (0. K))) ; ::_thesis: a * v in Solutions_of (A,(k |-> (0. K)))
then consider f being FinSequence of K such that
A3: v = f and
A4: ColVec2Mx f in Solutions_of (A,(ColVec2Mx (k |-> (0. K)))) ;
reconsider f = f as Element of (width A) -tuples_on the carrier of K by A3, MATRIX13:102;
now__::_thesis:_0._(K,k,1)_=_a_*_(0._(K,k,1))
percases ( k = 0 or k > 0 ) ;
suppose k = 0 ; ::_thesis: 0. (K,k,1) = a * (0. (K,k,1))
then ( len (a * (0. (K,k,1))) = len (0. (K,k,1)) & 0. (K,k,1) = {} ) by MATRIX_3:def_5;
hence 0. (K,k,1) = a * (0. (K,k,1)) ; ::_thesis: verum
end;
supposeA5: k > 0 ; ::_thesis: a * (0. (K,k,1)) = 0. (K,k,1)
then A6: ( len (0. (K,k,1)) = k & width (0. (K,k,1)) = 1 ) by MATRIX_1:23;
hence a * (0. (K,k,1)) = a * ((0. K) * (0. (K,k,1))) by A5, MATRIX_5:24
.= (a * (0. K)) * (0. (K,k,1)) by MATRIX_5:11
.= (0. K) * (0. (K,k,1)) by VECTSP_1:6
.= 0. (K,k,1) by A5, A6, MATRIX_5:24 ;
::_thesis: verum
end;
end;
end;
then a * (ColVec2Mx f) in Solutions_of (A,(0. (K,k,1))) by A1, A4, Th35;
then ColVec2Mx (a * f) in Solutions_of (A,(0. (K,k,1))) by Th30;
then a * f in Solutions_of (A,(k |-> (0. K))) by A1;
hence a * v in Solutions_of (A,(k |-> (0. K))) by A3, MATRIX13:102; ::_thesis: verum
end;
now__::_thesis:_for_v,_u_being_Element_of_((width_A)_-VectSp_over_K)_st_v_in_Solutions_of_(A,(k_|->_(0._K)))_&_u_in_Solutions_of_(A,(k_|->_(0._K)))_holds_
v_+_u_in_Solutions_of_(A,(k_|->_(0._K)))
let v, u be Element of ((width A) -VectSp_over K); ::_thesis: ( v in Solutions_of (A,(k |-> (0. K))) & u in Solutions_of (A,(k |-> (0. K))) implies v + u in Solutions_of (A,(k |-> (0. K))) )
assume that
A7: v in Solutions_of (A,(k |-> (0. K))) and
A8: u in Solutions_of (A,(k |-> (0. K))) ; ::_thesis: v + u in Solutions_of (A,(k |-> (0. K)))
consider f being FinSequence of K such that
A9: v = f and
A10: ColVec2Mx f in Solutions_of (A,(ColVec2Mx (k |-> (0. K)))) by A7;
consider g being FinSequence of K such that
A11: u = g and
A12: ColVec2Mx g in Solutions_of (A,(ColVec2Mx (k |-> (0. K)))) by A8;
A13: len g = width A by A12, Th59;
reconsider f = f, g = g as Element of (width A) -tuples_on the carrier of K by A9, A11, MATRIX13:102;
(ColVec2Mx f) + (ColVec2Mx g) in Solutions_of (A,((0. (K,k,1)) + (0. (K,k,1)))) by A1, A10, A12, Th37;
then (ColVec2Mx f) + (ColVec2Mx g) in Solutions_of (A,(0. (K,k,1))) by MATRIX_3:4;
then ColVec2Mx (f + g) in Solutions_of (A,(0. (K,k,1))) by A10, A13, Th28, Th59;
then f + g in Solutions_of (A,(k |-> (0. K))) by A1;
hence v + u in Solutions_of (A,(k |-> (0. K))) by A9, A11, MATRIX13:102; ::_thesis: verum
end;
hence for b1 being Subset of ((width A) -VectSp_over K) st b1 = Solutions_of (A,(k |-> (0. K))) holds
b1 is linearly-closed by A2, VECTSP_4:def_1; ::_thesis: verum
end;
end;
theorem Th60: :: MATRIX15:60
for K being Field
for A being Matrix of K
for b being FinSequence of K st not Solutions_of (A,b) is empty & width A = 0 holds
len A = 0
proof
let K be Field; ::_thesis: for A being Matrix of K
for b being FinSequence of K st not Solutions_of (A,b) is empty & width A = 0 holds
len A = 0
let A be Matrix of K; ::_thesis: for b being FinSequence of K st not Solutions_of (A,b) is empty & width A = 0 holds
len A = 0
let b be FinSequence of K; ::_thesis: ( not Solutions_of (A,b) is empty & width A = 0 implies len A = 0 )
set S = Solutions_of (A,b);
assume that
A1: not Solutions_of (A,b) is empty and
A2: width A = 0 ; ::_thesis: len A = 0
consider x being set such that
A3: x in Solutions_of (A,b) by A1, XBOOLE_0:def_1;
consider f being FinSequence of K such that
x = f and
A4: ColVec2Mx f in Solutions_of (A,(ColVec2Mx b)) by A3;
consider X being Matrix of K such that
ColVec2Mx f = X and
A5: len X = width A and
width X = width (ColVec2Mx b) and
A6: A * X = ColVec2Mx b by A4;
width (A * X) = width X by A5, MATRIX_3:def_4
.= 0 by A2, A5, MATRIX_1:def_3 ;
hence 0 = len b by A6, MATRIX_1:23
.= len (ColVec2Mx b) by MATRIX_1:def_2
.= len A by A4, Th33 ;
::_thesis: verum
end;
theorem Th61: :: MATRIX15:61
for K being Field
for A being Matrix of K st ( width A <> 0 or len A = 0 ) holds
not Solutions_of (A,((len A) |-> (0. K))) is empty
proof
let K be Field; ::_thesis: for A being Matrix of K st ( width A <> 0 or len A = 0 ) holds
not Solutions_of (A,((len A) |-> (0. K))) is empty
let A be Matrix of K; ::_thesis: ( ( width A <> 0 or len A = 0 ) implies not Solutions_of (A,((len A) |-> (0. K))) is empty )
set L = (len A) |-> (0. K);
A1: len ((len A) |-> (0. K)) = len A by CARD_1:def_7;
reconsider A9 = A as Matrix of len A, width A,K by MATRIX_2:7;
assume A2: ( width A <> 0 or len A = 0 ) ; ::_thesis: not Solutions_of (A,((len A) |-> (0. K))) is empty
percases ( len A = 0 or width A > 0 ) by A2;
suppose len A = 0 ; ::_thesis: not Solutions_of (A,((len A) |-> (0. K))) is empty
then Solutions_of (A9,(ColVec2Mx ((len A) |-> (0. K)))) = {{}} by A1, Th51;
then A3: {} in Solutions_of (A9,(ColVec2Mx ((len A) |-> (0. K)))) by TARSKI:def_1;
then consider f being FinSequence of K such that
A4: {} = ColVec2Mx f and
len f = width A by Th58;
f in Solutions_of (A,((len A) |-> (0. K))) by A3, A4;
hence not Solutions_of (A,((len A) |-> (0. K))) is empty ; ::_thesis: verum
end;
supposeA5: width A > 0 ; ::_thesis: not Solutions_of (A,((len A) |-> (0. K))) is empty
ColVec2Mx ((len A) |-> (0. K)) = 0. (K,(len A),1) by Th32;
then ( len (ColVec2Mx ((len A) |-> (0. K))) = len ((len A) |-> (0. K)) & the_rank_of A = the_rank_of (A ^^ (ColVec2Mx ((len A) |-> (0. K)))) ) by Th23, MATRIX_1:def_2;
then not Solutions_of (A,(ColVec2Mx ((len A) |-> (0. K)))) is empty by A1, A5, Th57;
then consider x being set such that
A6: x in Solutions_of (A,(ColVec2Mx ((len A) |-> (0. K)))) by XBOOLE_0:def_1;
consider f being FinSequence of K such that
A7: x = ColVec2Mx f and
len f = width A by A6, Th58;
f in Solutions_of (A,((len A) |-> (0. K))) by A6, A7;
hence not Solutions_of (A,((len A) |-> (0. K))) is empty ; ::_thesis: verum
end;
end;
end;
definition
let K be Field;
let A be Matrix of K;
assume A1: ( width A = 0 implies len A = 0 ) ;
func Space_of_Solutions_of A -> strict Subspace of (width A) -VectSp_over K means :Def5: :: MATRIX15:def 5
the carrier of it = Solutions_of (A,((len A) |-> (0. K)));
existence
ex b1 being strict Subspace of (width A) -VectSp_over K st the carrier of b1 = Solutions_of (A,((len A) |-> (0. K)))
proof
not Solutions_of (A,((len A) |-> (0. K))) is empty by A1, Th61;
hence ex b1 being strict Subspace of (width A) -VectSp_over K st the carrier of b1 = Solutions_of (A,((len A) |-> (0. K))) by VECTSP_4:34; ::_thesis: verum
end;
uniqueness
for b1, b2 being strict Subspace of (width A) -VectSp_over K st the carrier of b1 = Solutions_of (A,((len A) |-> (0. K))) & the carrier of b2 = Solutions_of (A,((len A) |-> (0. K))) holds
b1 = b2 by VECTSP_4:29;
end;
:: deftheorem Def5 defines Space_of_Solutions_of MATRIX15:def_5_:_
for K being Field
for A being Matrix of K st ( width A = 0 implies len A = 0 ) holds
for b3 being strict Subspace of (width A) -VectSp_over K holds
( b3 = Space_of_Solutions_of A iff the carrier of b3 = Solutions_of (A,((len A) |-> (0. K))) );
theorem :: MATRIX15:62
for K being Field
for A being Matrix of K
for b being FinSequence of K st not Solutions_of (A,b) is empty holds
Solutions_of (A,b) is Coset of Space_of_Solutions_of A
proof
let K be Field; ::_thesis: for A being Matrix of K
for b being FinSequence of K st not Solutions_of (A,b) is empty holds
Solutions_of (A,b) is Coset of Space_of_Solutions_of A
let A be Matrix of K; ::_thesis: for b being FinSequence of K st not Solutions_of (A,b) is empty holds
Solutions_of (A,b) is Coset of Space_of_Solutions_of A
let b be FinSequence of K; ::_thesis: ( not Solutions_of (A,b) is empty implies Solutions_of (A,b) is Coset of Space_of_Solutions_of A )
set V = (width A) -VectSp_over K;
reconsider B = b as Element of (len b) -tuples_on the carrier of K by FINSEQ_2:92;
set CB = ColVec2Mx B;
assume not Solutions_of (A,b) is empty ; ::_thesis: Solutions_of (A,b) is Coset of Space_of_Solutions_of A
then consider x being set such that
A1: x in Solutions_of (A,B) by XBOOLE_0:def_1;
consider f being FinSequence of K such that
x = f and
A2: ColVec2Mx f in Solutions_of (A,(ColVec2Mx B)) by A1;
set Cf = ColVec2Mx f;
A3: len f = width A by A2, Th59;
then reconsider f = f as Element of (width A) -tuples_on the carrier of K by FINSEQ_2:92;
reconsider F = f as Element of ((width A) -VectSp_over K) by MATRIX13:102;
A4: ( len (ColVec2Mx B) = len B & len A = len (ColVec2Mx B) ) by A2, Th33, MATRIX_1:def_2;
( width A = 0 implies len A = 0 ) by A1, Th60;
then A5: the carrier of (Space_of_Solutions_of A) = Solutions_of (A,((len A) |-> (0. K))) by Def5;
A6: Solutions_of (A,b) c= F + (Space_of_Solutions_of A)
proof
( len B = len ((- (1_ K)) * B) & (- (1_ K)) * (ColVec2Mx B) = ColVec2Mx ((- (1_ K)) * B) ) by Th30, CARD_1:def_7;
then A7: (ColVec2Mx B) + ((- (1_ K)) * (ColVec2Mx B)) = ColVec2Mx (B + ((- (1_ K)) * B)) by Th28
.= ColVec2Mx (B + (- B)) by FVSUM_1:59
.= ColVec2Mx ((len A) |-> (0. K)) by A4, FVSUM_1:26 ;
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in Solutions_of (A,b) or y in F + (Space_of_Solutions_of A) )
assume y in Solutions_of (A,b) ; ::_thesis: y in F + (Space_of_Solutions_of A)
then consider g being FinSequence of K such that
A8: y = g and
A9: ColVec2Mx g in Solutions_of (A,(ColVec2Mx B)) ;
len g = width A by A9, Th59;
then reconsider g = g as Element of (width A) -tuples_on the carrier of K by FINSEQ_2:92;
set Cg = ColVec2Mx g;
reconsider GF = g + ((- (1_ K)) * f) as Element of ((width A) -VectSp_over K) by MATRIX13:102;
A10: len f = len ((- (1_ K)) * f) by A3, CARD_1:def_7;
( width (ColVec2Mx B) = width ((- (1_ K)) * (ColVec2Mx B)) & (- (1_ K)) * (ColVec2Mx f) in Solutions_of (A,((- (1_ K)) * (ColVec2Mx B))) ) by A2, Th35, MATRIX_3:def_5;
then A11: (ColVec2Mx g) + ((- (1_ K)) * (ColVec2Mx f)) in Solutions_of (A,(ColVec2Mx ((len A) |-> (0. K)))) by A9, A7, Th37;
(ColVec2Mx g) + ((- (1_ K)) * (ColVec2Mx f)) = (ColVec2Mx g) + (ColVec2Mx ((- (1_ K)) * f)) by Th30
.= ColVec2Mx (g + ((- (1_ K)) * f)) by A3, A9, A10, Th28, Th59 ;
then g + ((- (1_ K)) * f) in Solutions_of (A,((len A) |-> (0. K))) by A11;
then A12: GF in Space_of_Solutions_of A by A5, STRUCT_0:def_5;
f + (g + ((- (1_ K)) * f)) = f + (((- (1_ K)) * f) + g) by FINSEQOP:33
.= f + ((- f) + g) by FVSUM_1:59
.= (f + (- f)) + g by FINSEQOP:28
.= ((width A) |-> (0. K)) + g by FVSUM_1:26
.= g by FVSUM_1:21 ;
then F + GF = g by MATRIX13:102;
hence y in F + (Space_of_Solutions_of A) by A8, A12; ::_thesis: verum
end;
F + (Space_of_Solutions_of A) c= Solutions_of (A,b)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in F + (Space_of_Solutions_of A) or y in Solutions_of (A,b) )
A13: ( len A = 0 or len A <> 0 ) ;
len ((len A) |-> (0. K)) = len A by CARD_1:def_7;
then A14: ( ( width (ColVec2Mx B) = 1 & width (ColVec2Mx ((len A) |-> (0. K))) = 1 ) or ( width (ColVec2Mx B) = 0 & width (ColVec2Mx ((len A) |-> (0. K))) = 0 ) ) by A4, A13, Th26, MATRIX_1:22;
ColVec2Mx ((len A) |-> (0. K)) = 0. (K,(len A),1) by Th32;
then A15: (ColVec2Mx B) + (ColVec2Mx ((len A) |-> (0. K))) = ColVec2Mx B by A4, MATRIX_3:4;
assume y in F + (Space_of_Solutions_of A) ; ::_thesis: y in Solutions_of (A,b)
then consider U being Element of ((width A) -VectSp_over K) such that
A16: U in Space_of_Solutions_of A and
A17: y = F + U by VECTSP_4:42;
reconsider u = U as Element of (width A) -tuples_on the carrier of K by MATRIX13:102;
u in Solutions_of (A,((len A) |-> (0. K))) by A5, A16, STRUCT_0:def_5;
then consider g being FinSequence of K such that
A18: u = g and
A19: ColVec2Mx g in Solutions_of (A,(ColVec2Mx ((len A) |-> (0. K)))) ;
width A = len g by A19, Th59;
then ColVec2Mx (f + g) = (ColVec2Mx f) + (ColVec2Mx g) by A2, Th28, Th59;
then ColVec2Mx (f + g) in Solutions_of (A,(ColVec2Mx B)) by A2, A19, A14, A15, Th37;
then f + g in Solutions_of (A,b) ;
hence y in Solutions_of (A,b) by A17, A18, MATRIX13:102; ::_thesis: verum
end;
then F + (Space_of_Solutions_of A) = Solutions_of (A,b) by A6, XBOOLE_0:def_10;
hence Solutions_of (A,b) is Coset of Space_of_Solutions_of A by VECTSP_4:def_6; ::_thesis: verum
end;
theorem Th63: :: MATRIX15:63
for K being Field
for A being Matrix of K st ( width A = 0 implies len A = 0 ) & the_rank_of A = 0 holds
Space_of_Solutions_of A = (width A) -VectSp_over K
proof
let K be Field; ::_thesis: for A being Matrix of K st ( width A = 0 implies len A = 0 ) & the_rank_of A = 0 holds
Space_of_Solutions_of A = (width A) -VectSp_over K
let A be Matrix of K; ::_thesis: ( ( width A = 0 implies len A = 0 ) & the_rank_of A = 0 implies Space_of_Solutions_of A = (width A) -VectSp_over K )
assume that
A1: ( width A = 0 implies len A = 0 ) and
A2: the_rank_of A = 0 ; ::_thesis: Space_of_Solutions_of A = (width A) -VectSp_over K
set L = (len A) |-> (0. K);
the carrier of ((width A) -VectSp_over K) c= Solutions_of (A,((len A) |-> (0. K)))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of ((width A) -VectSp_over K) or x in Solutions_of (A,((len A) |-> (0. K))) )
assume A3: x in the carrier of ((width A) -VectSp_over K) ; ::_thesis: x in Solutions_of (A,((len A) |-> (0. K)))
reconsider x9 = x as Element of (width A) -tuples_on the carrier of K by A3, MATRIX13:102;
A4: ( A = 0. (K,(len A),(width A)) & ColVec2Mx ((len A) |-> (0. K)) = 0. (K,(len A),1) ) by A2, Th32, MATRIX13:95;
percases ( len A = 0 or len A > 0 ) ;
supposeA5: len A = 0 ; ::_thesis: x in Solutions_of (A,((len A) |-> (0. K)))
then Solutions_of (A,(ColVec2Mx ((len A) |-> (0. K)))) = {{}} by A4, Th51;
then A6: {} in Solutions_of (A,(ColVec2Mx ((len A) |-> (0. K)))) by TARSKI:def_1;
then consider f being FinSequence of K such that
A7: {} = ColVec2Mx f and
A8: len f = width A by Th58;
width A = 0 by A5, MATRIX_1:def_3;
then the carrier of ((width A) -VectSp_over K) = 0 -tuples_on the carrier of K by MATRIX13:102
.= {(<*> the carrier of K)} by FINSEQ_2:94 ;
then x = <*> the carrier of K by A3, TARSKI:def_1;
then f = x by A5, A8, MATRIX_1:def_3;
hence x in Solutions_of (A,((len A) |-> (0. K))) by A6, A7; ::_thesis: verum
end;
supposeA9: len A > 0 ; ::_thesis: x in Solutions_of (A,((len A) |-> (0. K)))
A10: len x9 = width A by CARD_1:def_7;
Solutions_of (A,(ColVec2Mx ((len A) |-> (0. K)))) = { X where X is Matrix of width A,1,K : verum } by A1, A4, A9, Th54;
then ColVec2Mx x9 in Solutions_of (A,(ColVec2Mx ((len A) |-> (0. K)))) by A10;
hence x in Solutions_of (A,((len A) |-> (0. K))) ; ::_thesis: verum
end;
end;
end;
then the carrier of ((width A) -VectSp_over K) = Solutions_of (A,((len A) |-> (0. K))) by XBOOLE_0:def_10
.= the carrier of (Space_of_Solutions_of A) by A1, Def5 ;
hence Space_of_Solutions_of A = (width A) -VectSp_over K by VECTSP_4:31; ::_thesis: verum
end;
theorem :: MATRIX15:64
for K being Field
for A being Matrix of K st Space_of_Solutions_of A = (width A) -VectSp_over K holds
the_rank_of A = 0
proof
let K be Field; ::_thesis: for A being Matrix of K st Space_of_Solutions_of A = (width A) -VectSp_over K holds
the_rank_of A = 0
let A be Matrix of K; ::_thesis: ( Space_of_Solutions_of A = (width A) -VectSp_over K implies the_rank_of A = 0 )
assume A1: Space_of_Solutions_of A = (width A) -VectSp_over K ; ::_thesis: the_rank_of A = 0
assume the_rank_of A <> 0 ; ::_thesis: contradiction
then consider i, j being Nat such that
A2: [i,j] in Indices A and
A3: A * (i,j) <> 0. K by MATRIX13:94;
A4: j in Seg (width A) by A2, ZFMISC_1:87;
then A5: width A <> 0 ;
set L = Line ((1. (K,(width A))),j);
A6: width (1. (K,(width A))) = width A by MATRIX_1:24;
then A7: dom (Line ((1. (K,(width A))),j)) = Seg (width A) by FINSEQ_2:124;
A8: Indices (1. (K,(width A))) = [:(Seg (width A)),(Seg (width A)):] by MATRIX_1:24;
A9: now__::_thesis:_for_k_being_Nat_st_k_in_dom_(Line_((1._(K,(width_A))),j))_&_k_<>_j_holds_
0._K_=_(Line_((1._(K,(width_A))),j))_._k
let k be Nat; ::_thesis: ( k in dom (Line ((1. (K,(width A))),j)) & k <> j implies 0. K = (Line ((1. (K,(width A))),j)) . k )
assume that
A10: k in dom (Line ((1. (K,(width A))),j)) and
A11: k <> j ; ::_thesis: 0. K = (Line ((1. (K,(width A))),j)) . k
[j,k] in Indices (1. (K,(width A))) by A4, A7, A8, A10, ZFMISC_1:87;
hence 0. K = (1. (K,(width A))) * (j,k) by A11, MATRIX_1:def_11
.= (Line ((1. (K,(width A))),j)) . k by A6, A7, A10, MATRIX_1:def_7 ;
::_thesis: verum
end;
A12: dom (Line (A,i)) = Seg (width A) by FINSEQ_2:124;
[j,j] in Indices (1. (K,(width A))) by A4, A8, ZFMISC_1:87;
then 1_ K = (1. (K,(width A))) * (j,j) by MATRIX_1:def_11
.= (Line ((1. (K,(width A))),j)) . j by A4, A6, MATRIX_1:def_7 ;
then A13: Sum (mlt ((Line ((1. (K,(width A))),j)),(Line (A,i)))) = (Line (A,i)) . j by A4, A7, A12, A9, MATRIX_3:17
.= A * (i,j) by A4, MATRIX_1:def_7 ;
A14: ColVec2Mx ((len A) |-> (0. K)) = 0. (K,(len A),1) by Th32;
A15: i in dom A by A2, ZFMISC_1:87;
Line ((1. (K,(width A))),j) in (width A) -tuples_on the carrier of K by A6;
then Line ((1. (K,(width A))),j) in the carrier of (Space_of_Solutions_of A) by A1, MATRIX13:102;
then Line ((1. (K,(width A))),j) in Solutions_of (A,((len A) |-> (0. K))) by Def5, A5;
then consider f being FinSequence of K such that
A16: f = Line ((1. (K,(width A))),j) and
A17: ColVec2Mx f in Solutions_of (A,(ColVec2Mx ((len A) |-> (0. K)))) ;
consider X being Matrix of K such that
A18: X = ColVec2Mx f and
A19: len X = width A and
width X = width (ColVec2Mx ((len A) |-> (0. K))) and
A20: A * X = ColVec2Mx ((len A) |-> (0. K)) by A17;
A21: 1 in Seg 1 ;
A22: dom A = Seg (len A) by FINSEQ_1:def_3;
then len A <> 0 by A2, ZFMISC_1:87;
then Indices (ColVec2Mx ((len A) |-> (0. K))) = [:(Seg (len A)),(Seg 1):] by A14, MATRIX_1:23;
then A23: [i,1] in Indices (ColVec2Mx ((len A) |-> (0. K))) by A15, A22, A21, ZFMISC_1:87;
then (Line (A,i)) "*" (Col (X,1)) = (0. (K,(len A),1)) * (i,1) by A19, A20, A14, MATRIX_3:def_4
.= 0. K by A14, A23, MATRIX_3:1 ;
then 0. K = (Col (X,1)) "*" (Line (A,i)) by FVSUM_1:90
.= Sum (mlt (f,(Line (A,i)))) by A18, A19, Th26, A5 ;
hence contradiction by A3, A16, A13; ::_thesis: verum
end;
theorem Th65: :: MATRIX15:65
for m, n being Nat
for K being Field
for a being Element of K
for A9 being Matrix of m,n,K
for i, j being Nat st j in Seg m & n > 0 & ( i = j implies a <> - (1_ K) ) holds
Space_of_Solutions_of A9 = Space_of_Solutions_of (RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j))))))
proof
let m, n be Nat; ::_thesis: for K being Field
for a being Element of K
for A9 being Matrix of m,n,K
for i, j being Nat st j in Seg m & n > 0 & ( i = j implies a <> - (1_ K) ) holds
Space_of_Solutions_of A9 = Space_of_Solutions_of (RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j))))))
let K be Field; ::_thesis: for a being Element of K
for A9 being Matrix of m,n,K
for i, j being Nat st j in Seg m & n > 0 & ( i = j implies a <> - (1_ K) ) holds
Space_of_Solutions_of A9 = Space_of_Solutions_of (RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j))))))
let a be Element of K; ::_thesis: for A9 being Matrix of m,n,K
for i, j being Nat st j in Seg m & n > 0 & ( i = j implies a <> - (1_ K) ) holds
Space_of_Solutions_of A9 = Space_of_Solutions_of (RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j))))))
let A9 be Matrix of m,n,K; ::_thesis: for i, j being Nat st j in Seg m & n > 0 & ( i = j implies a <> - (1_ K) ) holds
Space_of_Solutions_of A9 = Space_of_Solutions_of (RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j))))))
let i, j be Nat; ::_thesis: ( j in Seg m & n > 0 & ( i = j implies a <> - (1_ K) ) implies Space_of_Solutions_of A9 = Space_of_Solutions_of (RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))) )
assume that
A1: j in Seg m and
A2: n > 0 and
A3: ( i = j implies a <> - (1_ K) ) ; ::_thesis: Space_of_Solutions_of A9 = Space_of_Solutions_of (RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j))))))
set L = (len A9) |-> (0. K);
set R = RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))));
A4: m <> 0 by A1;
then A5: width (RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))) = n by MATRIX_1:23;
len ((len A9) |-> (0. K)) = len A9 by CARD_1:def_7;
then reconsider C = ColVec2Mx ((len A9) |-> (0. K)) as Matrix of m,1,K by MATRIX_1:def_2;
set RC = RLine (C,i,((Line (C,i)) + (a * (Line (C,j)))));
A6: C = 0. (K,(len A9),1) by Th32;
now__::_thesis:_for_i9,_j9_being_Nat_st_[i9,j9]_in_Indices_C_holds_
C_*_(i9,j9)_=_(RLine_(C,i,((Line_(C,i))_+_(a_*_(Line_(C,j))))))_*_(i9,j9)
let i9, j9 be Nat; ::_thesis: ( [i9,j9] in Indices C implies C * (i9,j9) = (RLine (C,i,((Line (C,i)) + (a * (Line (C,j)))))) * (i9,j9) )
assume A7: [i9,j9] in Indices C ; ::_thesis: C * (i9,j9) = (RLine (C,i,((Line (C,i)) + (a * (Line (C,j)))))) * (i9,j9)
reconsider I = i9, J = j9 as Element of NAT by ORDINAL1:def_12;
A8: len ((Line (C,i)) + (a * (Line (C,j)))) = width C by CARD_1:def_7;
now__::_thesis:_C_*_(I,J)_=_(RLine_(C,i,((Line_(C,i))_+_(a_*_(Line_(C,j))))))_*_(I,J)
percases ( i9 = i or i <> i9 ) ;
supposeA9: i9 = i ; ::_thesis: C * (I,J) = (RLine (C,i,((Line (C,i)) + (a * (Line (C,j)))))) * (I,J)
A10: 1 = width C by A4, MATRIX_1:23;
then A11: j9 in Seg 1 by A7, ZFMISC_1:87;
then (Line (C,j)) . j9 = C * (j,j9) by A10, MATRIX_1:def_7;
then A12: (a * (Line (C,j))) . j9 = a * (C * (j,j9)) by A10, A11, FVSUM_1:51;
Indices C = [:(Seg m),(Seg 1):] by A4, MATRIX_1:23;
then A13: [j,j9] in Indices C by A1, A11, ZFMISC_1:87;
(Line (C,i)) . j9 = C * (i,j9) by A10, A11, MATRIX_1:def_7;
then ((Line (C,i)) + (a * (Line (C,j)))) . j9 = (C * (i,j9)) + (a * (C * (j,j9))) by A10, A11, A12, FVSUM_1:18
.= (0. K) + (a * (C * (j,j9))) by A6, A7, A9, MATRIX_3:1
.= (0. K) + (a * (0. K)) by A6, A13, MATRIX_3:1
.= (0. K) + (0. K) by VECTSP_1:6
.= 0. K by RLVECT_1:def_4
.= C * (i9,j9) by A6, A7, MATRIX_3:1 ;
hence C * (I,J) = (RLine (C,i,((Line (C,i)) + (a * (Line (C,j)))))) * (I,J) by A7, A8, A9, MATRIX11:def_3; ::_thesis: verum
end;
suppose i <> i9 ; ::_thesis: C * (I,J) = (RLine (C,i,((Line (C,i)) + (a * (Line (C,j)))))) * (I,J)
hence C * (I,J) = (RLine (C,i,((Line (C,i)) + (a * (Line (C,j)))))) * (I,J) by A7, A8, MATRIX11:def_3; ::_thesis: verum
end;
end;
end;
hence C * (i9,j9) = (RLine (C,i,((Line (C,i)) + (a * (Line (C,j)))))) * (i9,j9) ; ::_thesis: verum
end;
then RLine (C,i,((Line (C,i)) + (a * (Line (C,j))))) = C by MATRIX_1:27;
then A14: Solutions_of (A9,C) = Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),C) by A1, A3, Th40;
set SR = Space_of_Solutions_of (RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j))))));
( len A9 = m & len (RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))) = m ) by A4, MATRIX_1:23;
then A15: the carrier of (Space_of_Solutions_of (RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j))))))) = Solutions_of ((RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))),((len A9) |-> (0. K))) by A2, A5, Def5;
set SA = Space_of_Solutions_of A9;
A16: width A9 = n by A4, MATRIX_1:23;
then the carrier of (Space_of_Solutions_of A9) = Solutions_of (A9,((len A9) |-> (0. K))) by A2, Def5;
hence Space_of_Solutions_of A9 = Space_of_Solutions_of (RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))) by A16, A5, A14, A15, VECTSP_4:29; ::_thesis: verum
end;
theorem Th66: :: MATRIX15:66
for K being Field
for A being Matrix of K
for N being finite without_zero Subset of NAT st N c= dom A & not N is empty & width A > 0 & ( for i being Nat st i in (dom A) \ N holds
Line (A,i) = (width A) |-> (0. K) ) holds
Space_of_Solutions_of A = Space_of_Solutions_of (Segm (A,N,(Seg (width A))))
proof
let K be Field; ::_thesis: for A being Matrix of K
for N being finite without_zero Subset of NAT st N c= dom A & not N is empty & width A > 0 & ( for i being Nat st i in (dom A) \ N holds
Line (A,i) = (width A) |-> (0. K) ) holds
Space_of_Solutions_of A = Space_of_Solutions_of (Segm (A,N,(Seg (width A))))
let A be Matrix of K; ::_thesis: for N being finite without_zero Subset of NAT st N c= dom A & not N is empty & width A > 0 & ( for i being Nat st i in (dom A) \ N holds
Line (A,i) = (width A) |-> (0. K) ) holds
Space_of_Solutions_of A = Space_of_Solutions_of (Segm (A,N,(Seg (width A))))
let N be finite without_zero Subset of NAT; ::_thesis: ( N c= dom A & not N is empty & width A > 0 & ( for i being Nat st i in (dom A) \ N holds
Line (A,i) = (width A) |-> (0. K) ) implies Space_of_Solutions_of A = Space_of_Solutions_of (Segm (A,N,(Seg (width A)))) )
assume that
A1: N c= dom A and
A2: not N is empty and
A3: width A > 0 and
A4: for i being Nat st i in (dom A) \ N holds
Line (A,i) = (width A) |-> (0. K) ; ::_thesis: Space_of_Solutions_of A = Space_of_Solutions_of (Segm (A,N,(Seg (width A))))
set L = (len A) |-> (0. K);
set C = ColVec2Mx ((len A) |-> (0. K));
A5: len ((len A) |-> (0. K)) = len A by CARD_1:def_7;
set S = Segm (A,N,(Seg (width A)));
A6: width (Segm (A,N,(Seg (width A)))) = card (Seg (width A)) by A2, MATRIX_1:23;
then A7: width A = width (Segm (A,N,(Seg (width A)))) by FINSEQ_1:57;
set SS = Space_of_Solutions_of (Segm (A,N,(Seg (width A))));
len (Segm (A,N,(Seg (width A)))) = card N by MATRIX_1:def_2;
then A8: the carrier of (Space_of_Solutions_of (Segm (A,N,(Seg (width A))))) = Solutions_of ((Segm (A,N,(Seg (width A)))),((card N) |-> (0. K))) by A3, A6, Def5;
set SA = Space_of_Solutions_of A;
A9: the carrier of (Space_of_Solutions_of A) = Solutions_of (A,((len A) |-> (0. K))) by A3, Def5;
A10: ColVec2Mx ((len A) |-> (0. K)) = 0. (K,(len A),1) by Th32;
len (ColVec2Mx ((len A) |-> (0. K))) = len ((len A) |-> (0. K)) by MATRIX_1:def_2;
then A11: dom (ColVec2Mx ((len A) |-> (0. K))) = dom A by A5, FINSEQ_3:29;
A12: dom A = Seg (len A) by FINSEQ_1:def_3;
then A13: Seg (len A) <> {} by A1, A2, XBOOLE_1:3;
then A14: width (ColVec2Mx ((len A) |-> (0. K))) = 1 by Th26;
then A15: card (Seg (width (ColVec2Mx ((len A) |-> (0. K))))) = 1 by FINSEQ_1:57;
now__::_thesis:_for_k,_l_being_Nat_st_[k,l]_in_Indices_(Segm_((ColVec2Mx_((len_A)_|->_(0._K))),N,(Seg_(width_(ColVec2Mx_((len_A)_|->_(0._K)))))))_holds_
(Segm_((ColVec2Mx_((len_A)_|->_(0._K))),N,(Seg_(width_(ColVec2Mx_((len_A)_|->_(0._K)))))))_*_(k,l)_=_(0._(K,(card_N),1))_*_(k,l)
A16: rng (Sgm (Seg 1)) = Seg 1 by FINSEQ_1:def_13;
A17: Indices (Segm ((ColVec2Mx ((len A) |-> (0. K))),N,(Seg (width (ColVec2Mx ((len A) |-> (0. K))))))) = Indices (0. (K,(card N),1)) by A15, MATRIX_1:26;
let k, l be Nat; ::_thesis: ( [k,l] in Indices (Segm ((ColVec2Mx ((len A) |-> (0. K))),N,(Seg (width (ColVec2Mx ((len A) |-> (0. K))))))) implies (Segm ((ColVec2Mx ((len A) |-> (0. K))),N,(Seg (width (ColVec2Mx ((len A) |-> (0. K))))))) * (k,l) = (0. (K,(card N),1)) * (k,l) )
assume A18: [k,l] in Indices (Segm ((ColVec2Mx ((len A) |-> (0. K))),N,(Seg (width (ColVec2Mx ((len A) |-> (0. K))))))) ; ::_thesis: (Segm ((ColVec2Mx ((len A) |-> (0. K))),N,(Seg (width (ColVec2Mx ((len A) |-> (0. K))))))) * (k,l) = (0. (K,(card N),1)) * (k,l)
reconsider kk = k, ll = l as Element of NAT by ORDINAL1:def_12;
( [:N,(Seg (width (ColVec2Mx ((len A) |-> (0. K))))):] c= Indices (ColVec2Mx ((len A) |-> (0. K))) & rng (Sgm N) = N ) by A1, A12, A11, FINSEQ_1:def_13, ZFMISC_1:95;
then A19: [((Sgm N) . kk),((Sgm (Seg (width (ColVec2Mx ((len A) |-> (0. K)))))) . ll)] in Indices (ColVec2Mx ((len A) |-> (0. K))) by A14, A18, A16, MATRIX13:17;
thus (Segm ((ColVec2Mx ((len A) |-> (0. K))),N,(Seg (width (ColVec2Mx ((len A) |-> (0. K))))))) * (k,l) = (ColVec2Mx ((len A) |-> (0. K))) * (((Sgm N) . kk),((Sgm (Seg (width (ColVec2Mx ((len A) |-> (0. K)))))) . ll)) by A18, MATRIX13:def_1
.= 0. K by A10, A19, MATRIX_3:1
.= (0. (K,(card N),1)) * (k,l) by A18, A17, MATRIX_3:1 ; ::_thesis: verum
end;
then A20: Segm ((ColVec2Mx ((len A) |-> (0. K))),N,(Seg (width (ColVec2Mx ((len A) |-> (0. K)))))) = 0. (K,(card N),1) by A15, MATRIX_1:27
.= ColVec2Mx ((card N) |-> (0. K)) by Th32 ;
now__::_thesis:_for_i_being_Nat_st_i_in_(dom_A)_\_N_holds_
(_Line_(A,i)_=_(width_A)_|->_(0._K)_&_Line_((ColVec2Mx_((len_A)_|->_(0._K))),i)_=_(width_(ColVec2Mx_((len_A)_|->_(0._K))))_|->_(0._K)_)
let i be Nat; ::_thesis: ( i in (dom A) \ N implies ( Line (A,i) = (width A) |-> (0. K) & Line ((ColVec2Mx ((len A) |-> (0. K))),i) = (width (ColVec2Mx ((len A) |-> (0. K)))) |-> (0. K) ) )
assume A21: i in (dom A) \ N ; ::_thesis: ( Line (A,i) = (width A) |-> (0. K) & Line ((ColVec2Mx ((len A) |-> (0. K))),i) = (width (ColVec2Mx ((len A) |-> (0. K)))) |-> (0. K) )
A22: i in dom A by A21, XBOOLE_0:def_5;
then Line ((ColVec2Mx ((len A) |-> (0. K))),i) = (ColVec2Mx ((len A) |-> (0. K))) . i by A5, A12, MATRIX_2:8
.= ((len A) |-> ((width (ColVec2Mx ((len A) |-> (0. K)))) |-> (0. K))) . i by A10, A13, Th26
.= (width (ColVec2Mx ((len A) |-> (0. K)))) |-> (0. K) by A12, A22, FINSEQ_2:57 ;
hence ( Line (A,i) = (width A) |-> (0. K) & Line ((ColVec2Mx ((len A) |-> (0. K))),i) = (width (ColVec2Mx ((len A) |-> (0. K)))) |-> (0. K) ) by A4, A21; ::_thesis: verum
end;
then Solutions_of (A,(ColVec2Mx ((len A) |-> (0. K)))) = Solutions_of ((Segm (A,N,(Seg (width A)))),(Segm ((ColVec2Mx ((len A) |-> (0. K))),N,(Seg (width (ColVec2Mx ((len A) |-> (0. K)))))))) by A1, A2, A11, Th45;
hence Space_of_Solutions_of A = Space_of_Solutions_of (Segm (A,N,(Seg (width A)))) by A20, A7, A9, A8, VECTSP_4:29; ::_thesis: verum
end;
Lm7: for K being Field
for g being FinSequence of K
for A being set st A c= dom g holds
ex ga, gb being FinSequence of K st
( ga = g * (Sgm A) & gb = g * (Sgm ((dom g) \ A)) & Sum g = (Sum ga) + (Sum gb) )
proof
let K be Field; ::_thesis: for g being FinSequence of K
for A being set st A c= dom g holds
ex ga, gb being FinSequence of K st
( ga = g * (Sgm A) & gb = g * (Sgm ((dom g) \ A)) & Sum g = (Sum ga) + (Sum gb) )
let g be FinSequence of K; ::_thesis: for A being set st A c= dom g holds
ex ga, gb being FinSequence of K st
( ga = g * (Sgm A) & gb = g * (Sgm ((dom g) \ A)) & Sum g = (Sum ga) + (Sum gb) )
A1: rng g c= the carrier of K by FINSEQ_1:def_4;
set Ad = the addF of K;
A2: dom g = Seg (len g) by FINSEQ_1:def_3;
then A3: ( dom g = rng (idseq (len g)) & dom g = dom (idseq (len g)) ) by RELAT_1:45;
let A be set ; ::_thesis: ( A c= dom g implies ex ga, gb being FinSequence of K st
( ga = g * (Sgm A) & gb = g * (Sgm ((dom g) \ A)) & Sum g = (Sum ga) + (Sum gb) ) )
assume A4: A c= dom g ; ::_thesis: ex ga, gb being FinSequence of K st
( ga = g * (Sgm A) & gb = g * (Sgm ((dom g) \ A)) & Sum g = (Sum ga) + (Sum gb) )
A5: rng (Sgm A) = A by A4, A2, FINSEQ_1:def_13;
A6: (idseq (len g)) " A = A by A4, A2, FUNCT_2:94;
A7: (dom g) \ A c= dom g by XBOOLE_1:36;
then A8: rng (Sgm ((dom g) \ A)) = (dom g) \ A by A2, FINSEQ_1:def_13;
then reconsider ga = g * (Sgm A), gb = g * (Sgm ((dom g) \ A)) as FinSequence by A4, A5, FINSEQ_1:16, XBOOLE_1:36;
(idseq (len g)) " ((dom g) \ A) = (dom g) \ A by A2, A7, FUNCT_2:94;
then A9: (Sgm A) ^ (Sgm ((dom g) \ A)) is Permutation of (dom g) by A6, A3, FINSEQ_3:114;
then reconsider gS = g * ((Sgm A) ^ (Sgm ((dom g) \ A))) as FinSequence of K by A2, FINSEQ_2:46;
rng ga c= rng g by RELAT_1:26;
then A10: rng ga c= the carrier of K by A1, XBOOLE_1:1;
rng gb c= rng g by RELAT_1:26;
then rng gb c= the carrier of K by A1, XBOOLE_1:1;
then reconsider ga = ga, gb = gb as FinSequence of K by A10, FINSEQ_1:def_4;
take ga ; ::_thesis: ex gb being FinSequence of K st
( ga = g * (Sgm A) & gb = g * (Sgm ((dom g) \ A)) & Sum g = (Sum ga) + (Sum gb) )
take gb ; ::_thesis: ( ga = g * (Sgm A) & gb = g * (Sgm ((dom g) \ A)) & Sum g = (Sum ga) + (Sum gb) )
the addF of K $$ g = the addF of K "**" gS by A9, FINSOP_1:7, FVSUM_1:8
.= the addF of K "**" (ga ^ gb) by A4, A7, A5, A8, Th5
.= (Sum ga) + (Sum gb) by FINSOP_1:5, FVSUM_1:8 ;
hence ( ga = g * (Sgm A) & gb = g * (Sgm ((dom g) \ A)) & Sum g = (Sum ga) + (Sum gb) ) ; ::_thesis: verum
end;
theorem Th67: :: MATRIX15:67
for n, m being Nat
for K being Field
for A being Matrix of n,m,K
for N being finite without_zero Subset of NAT st card N = n & N c= Seg m & Segm (A,(Seg n),N) = 1. (K,n) & n > 0 & m -' n > 0 holds
ex MVectors being Matrix of m -' n,m,K st
( Segm (MVectors,(Seg (m -' n)),((Seg m) \ N)) = 1. (K,(m -' n)) & Segm (MVectors,(Seg (m -' n)),N) = - ((Segm (A,(Seg n),((Seg m) \ N))) @) & Lin (lines MVectors) = Space_of_Solutions_of A )
proof
let n, m be Nat; ::_thesis: for K being Field
for A being Matrix of n,m,K
for N being finite without_zero Subset of NAT st card N = n & N c= Seg m & Segm (A,(Seg n),N) = 1. (K,n) & n > 0 & m -' n > 0 holds
ex MVectors being Matrix of m -' n,m,K st
( Segm (MVectors,(Seg (m -' n)),((Seg m) \ N)) = 1. (K,(m -' n)) & Segm (MVectors,(Seg (m -' n)),N) = - ((Segm (A,(Seg n),((Seg m) \ N))) @) & Lin (lines MVectors) = Space_of_Solutions_of A )
let K be Field; ::_thesis: for A being Matrix of n,m,K
for N being finite without_zero Subset of NAT st card N = n & N c= Seg m & Segm (A,(Seg n),N) = 1. (K,n) & n > 0 & m -' n > 0 holds
ex MVectors being Matrix of m -' n,m,K st
( Segm (MVectors,(Seg (m -' n)),((Seg m) \ N)) = 1. (K,(m -' n)) & Segm (MVectors,(Seg (m -' n)),N) = - ((Segm (A,(Seg n),((Seg m) \ N))) @) & Lin (lines MVectors) = Space_of_Solutions_of A )
let A be Matrix of n,m,K; ::_thesis: for N being finite without_zero Subset of NAT st card N = n & N c= Seg m & Segm (A,(Seg n),N) = 1. (K,n) & n > 0 & m -' n > 0 holds
ex MVectors being Matrix of m -' n,m,K st
( Segm (MVectors,(Seg (m -' n)),((Seg m) \ N)) = 1. (K,(m -' n)) & Segm (MVectors,(Seg (m -' n)),N) = - ((Segm (A,(Seg n),((Seg m) \ N))) @) & Lin (lines MVectors) = Space_of_Solutions_of A )
let N be finite without_zero Subset of NAT; ::_thesis: ( card N = n & N c= Seg m & Segm (A,(Seg n),N) = 1. (K,n) & n > 0 & m -' n > 0 implies ex MVectors being Matrix of m -' n,m,K st
( Segm (MVectors,(Seg (m -' n)),((Seg m) \ N)) = 1. (K,(m -' n)) & Segm (MVectors,(Seg (m -' n)),N) = - ((Segm (A,(Seg n),((Seg m) \ N))) @) & Lin (lines MVectors) = Space_of_Solutions_of A ) )
assume that
A1: card N = n and
A2: N c= Seg m and
A3: Segm (A,(Seg n),N) = 1. (K,n) and
A4: n > 0 and
A5: m -' n > 0 ; ::_thesis: ex MVectors being Matrix of m -' n,m,K st
( Segm (MVectors,(Seg (m -' n)),((Seg m) \ N)) = 1. (K,(m -' n)) & Segm (MVectors,(Seg (m -' n)),N) = - ((Segm (A,(Seg n),((Seg m) \ N))) @) & Lin (lines MVectors) = Space_of_Solutions_of A )
Seg m <> {} by A1, A2, A4, CARD_1:27, XBOOLE_1:3;
then A6: m <> 0 ;
consider MV being Matrix of m -' n,m,K such that
A7: Segm (MV,(Seg (m -' n)),((Seg m) \ N)) = 1. (K,(m -' n)) and
A8: Segm (MV,(Seg (m -' n)),N) = - ((Segm (A,(Seg n),((Seg m) \ N))) @) and
A9: for l being Nat
for M being Matrix of m,l,K st ( for i being Nat holds
( not i in Seg l or ex j being Nat st
( j in Seg (m -' n) & Col (M,i) = Line (MV,j) ) or Col (M,i) = m |-> (0. K) ) ) holds
M in Solutions_of (A,(0. (K,n,l))) by A1, A2, A3, A4, Th49;
A10: width MV = m by A5, MATRIX_1:23;
A11: ( len MV = m -' n & len MV >= the_rank_of MV ) by MATRIX13:74, MATRIX_1:def_2;
A12: Indices MV = [:(Seg (m -' n)),(Seg m):] by A5, MATRIX_1:23;
(Seg m) \ N c= Seg m by XBOOLE_1:36;
then A13: [:(Seg (m -' n)),((Seg m) \ N):] c= Indices MV by A12, ZFMISC_1:95;
A14: width A = m by A4, MATRIX_1:23;
A15: len A = n by A4, MATRIX_1:23;
lines MV c= Solutions_of (A,((len A) |-> (0. K)))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in lines MV or x in Solutions_of (A,((len A) |-> (0. K))) )
assume x in lines MV ; ::_thesis: x in Solutions_of (A,((len A) |-> (0. K)))
then consider k being Nat such that
A16: k in Seg (m -' n) and
A17: x = Line (MV,k) by MATRIX13:103;
set C = ColVec2Mx (Line (MV,k));
A18: m = width MV by A5, MATRIX_1:23
.= len (Line (MV,k)) by CARD_1:def_7 ;
now__::_thesis:_for_i_being_Nat_holds_
(_not_i_in_Seg_1_or_ex_j_being_Nat_st_
(_j_in_Seg_(m_-'_n)_&_Col_((ColVec2Mx_(Line_(MV,k))),i)_=_Line_(MV,j)_)_or_Col_((ColVec2Mx_(Line_(MV,k))),i)_=_m_|->_(0._K)_)
let i be Nat; ::_thesis: ( not i in Seg 1 or ex j being Nat st
( j in Seg (m -' n) & Col ((ColVec2Mx (Line (MV,k))),i) = Line (MV,j) ) or Col ((ColVec2Mx (Line (MV,k))),i) = m |-> (0. K) )
assume i in Seg 1 ; ::_thesis: ( ex j being Nat st
( j in Seg (m -' n) & Col ((ColVec2Mx (Line (MV,k))),i) = Line (MV,j) ) or Col ((ColVec2Mx (Line (MV,k))),i) = m |-> (0. K) )
then A19: i = 1 by FINSEQ_1:2, TARSKI:def_1;
Col ((ColVec2Mx (Line (MV,k))),1) = Line (MV,k) by A6, A18, Th26;
hence ( ex j being Nat st
( j in Seg (m -' n) & Col ((ColVec2Mx (Line (MV,k))),i) = Line (MV,j) ) or Col ((ColVec2Mx (Line (MV,k))),i) = m |-> (0. K) ) by A16, A19; ::_thesis: verum
end;
then ColVec2Mx (Line (MV,k)) in Solutions_of (A,(0. (K,n,1))) by A9, A18;
then ColVec2Mx (Line (MV,k)) in Solutions_of (A,(ColVec2Mx ((len A) |-> (0. K)))) by A15, Th32;
hence x in Solutions_of (A,((len A) |-> (0. K))) by A17; ::_thesis: verum
end;
then Lin (lines MV) is Subspace of Lin (Solutions_of (A,((len A) |-> (0. K)))) by A14, VECTSP_7:13;
then A20: the carrier of (Lin (lines MV)) c= the carrier of (Lin (Solutions_of (A,((len A) |-> (0. K))))) by VECTSP_4:def_2;
(m -' n) + 0 > 0 by A5;
then m -' n >= 1 by NAT_1:19;
then A21: Det (1. (K,(m -' n))) = 1_ K by MATRIX_7:16;
A22: 0. K <> 1_ K ;
A23: card (Seg m) = m by FINSEQ_1:57;
then A24: m -' n = m - n by A1, A2, NAT_1:43, XREAL_1:233;
A25: card ((Seg m) \ N) = m - n by A1, A2, A23, CARD_2:44;
then A26: card (Seg (m -' n)) = card ((Seg m) \ N) by A24, FINSEQ_1:57;
EqSegm (MV,(Seg (m -' n)),((Seg m) \ N)) = 1. (K,(m -' n)) by A7, A24, A25, FINSEQ_1:57, MATRIX13:def_3;
then m -' n <= the_rank_of MV by A24, A25, A26, A13, A21, A22, MATRIX13:def_4;
then A27: the_rank_of MV = m -' n by A11, XXREAL_0:1;
A28: the carrier of (Space_of_Solutions_of A) c= the carrier of (Lin (lines MV))
proof
set SN = (Seg m) \ N;
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in the carrier of (Space_of_Solutions_of A) or y in the carrier of (Lin (lines MV)) )
assume y in the carrier of (Space_of_Solutions_of A) ; ::_thesis: y in the carrier of (Lin (lines MV))
then y in Solutions_of (A,((len A) |-> (0. K))) by A6, A14, Def5;
then consider f being FinSequence of K such that
A29: f = y and
A30: ColVec2Mx f in Solutions_of (A,(ColVec2Mx ((len A) |-> (0. K)))) ;
A31: len f = m by A14, A30, Th59;
deffunc H1( Nat) -> Element of (width MV) -tuples_on the carrier of K = (f /. ((Sgm ((Seg m) \ N)) . $1)) * (Line (MV,$1));
A32: dom f = Seg (len f) by FINSEQ_1:def_3;
consider M being FinSequence of (width MV) -tuples_on the carrier of K such that
A33: len M = m -' n and
A34: for j being Nat st j in dom M holds
M . j = H1(j) from FINSEQ_2:sch_1();
A35: dom M = Seg (m -' n) by A33, FINSEQ_1:def_3;
reconsider M = M as FinSequence of the carrier of (m -VectSp_over K) by A10, MATRIX13:102;
reconsider M1 = FinS2MX M as Matrix of m -' n,m,K by A33;
now__::_thesis:_for_i_being_Nat_st_i_in_Seg_(m_-'_n)_holds_
ex_a_being_Element_of_the_carrier_of_K_st_Line_(M1,i)_=_a_*_(Line_(MV,i))
let i be Nat; ::_thesis: ( i in Seg (m -' n) implies ex a being Element of the carrier of K st Line (M1,i) = a * (Line (MV,i)) )
assume A36: i in Seg (m -' n) ; ::_thesis: ex a being Element of the carrier of K st Line (M1,i) = a * (Line (MV,i))
take a = f /. ((Sgm ((Seg m) \ N)) . i); ::_thesis: Line (M1,i) = a * (Line (MV,i))
thus Line (M1,i) = M1 . i by A36, MATRIX_2:8
.= a * (Line (MV,i)) by A34, A35, A36 ; ::_thesis: verum
end;
then consider L being Linear_Combination of lines MV such that
A37: L (#) (MX2FinS MV) = M1 by A27, MATRIX13:105, MATRIX13:108;
reconsider SumL = Sum L, f = f as Element of m -tuples_on the carrier of K by A31, FINSEQ_2:92, MATRIX13:102;
now__::_thesis:_for_i_being_Nat_st_i_in_Seg_m_holds_
SumL_._i_=_f_._i
let i be Nat; ::_thesis: ( i in Seg m implies SumL . i = f . i )
assume A38: i in Seg m ; ::_thesis: SumL . i = f . i
A39: (Seg m) \ N c= Seg m by XBOOLE_1:36;
A40: now__::_thesis:_Sum_(Col_(M1,i))_=_f_._i
percases ( i in N or not i in N ) ;
supposeA41: i in N ; ::_thesis: Sum (Col (M1,i)) = f . i
consider X being Matrix of K such that
A42: X = ColVec2Mx f and
A43: len X = width A and
width X = width (ColVec2Mx ((len A) |-> (0. K))) and
A44: A * X = ColVec2Mx ((len A) |-> (0. K)) by A30;
A45: f = Col (X,1) by A6, A14, A42, A43, Th26;
reconsider F = f as Element of (width A) -tuples_on the carrier of K by A4, MATRIX_1:23;
A46: rng (Sgm (Seg (m -' n))) = Seg (m -' n) by FINSEQ_1:def_13;
A47: rng (Sgm N) = N by A2, FINSEQ_1:def_13;
then consider x being set such that
A48: x in dom (Sgm N) and
A49: (Sgm N) . x = i by A41, FUNCT_1:def_3;
reconsider x = x as Element of NAT by A48;
set L = Line (A,x);
A50: dom (mlt ((Line (A,x)),F)) = Seg m by A14, FINSEQ_2:124;
then consider mN, mSN being FinSequence of K such that
A51: mN = (mlt ((Line (A,x)),F)) * (Sgm N) and
A52: mSN = (mlt ((Line (A,x)),F)) * (Sgm ((Seg m) \ N)) and
A53: Sum (mlt ((Line (A,x)),F)) = (Sum mN) + (Sum mSN) by A2, Lm7;
A54: dom (Sgm N) = Seg (card N) by A2, FINSEQ_3:40;
then A55: dom mN = Seg n by A1, A2, A47, A50, A51, RELAT_1:27;
Indices (1. (K,n)) = [:(Seg n),(Seg n):] by MATRIX_1:24;
then A56: [x,x] in Indices (1. (K,n)) by A1, A54, A48, ZFMISC_1:87;
A57: (Sgm N) . x in N by A47, A48, FUNCT_1:def_3;
then A58: ( F . ((Sgm N) . x) = F /. ((Sgm N) . x) & (Line (A,x)) . ((Sgm N) . x) = A * (x,((Sgm N) . x)) ) by A2, A14, A31, A32, MATRIX_1:def_7, PARTFUN1:def_6;
A59: x = (idseq n) . x by A1, A54, A48, FINSEQ_2:49
.= (Sgm (Seg n)) . x by FINSEQ_3:48 ;
now__::_thesis:_for_j_being_Nat_st_j_in_dom_mN_&_j_<>_x_holds_
mN_._j_=_0._K
let j be Nat; ::_thesis: ( j in dom mN & j <> x implies mN . j = 0. K )
assume that
A60: j in dom mN and
A61: j <> x ; ::_thesis: mN . j = 0. K
Indices (1. (K,n)) = [:(Seg n),(Seg n):] by MATRIX_1:24;
then A62: [x,j] in Indices (1. (K,n)) by A1, A54, A48, A55, A60, ZFMISC_1:87;
A63: (Sgm N) . j in N by A1, A47, A54, A55, A60, FUNCT_1:def_3;
then A64: ( F . ((Sgm N) . j) = F /. ((Sgm N) . j) & (Line (A,x)) . ((Sgm N) . j) = A * (x,((Sgm N) . j)) ) by A2, A14, A31, A32, MATRIX_1:def_7, PARTFUN1:def_6;
thus mN . j = (mlt ((Line (A,x)),F)) . ((Sgm N) . j) by A51, A60, FUNCT_1:12
.= (F /. ((Sgm N) . j)) * (A * (((Sgm (Seg n)) . x),((Sgm N) . j))) by A2, A14, A59, A63, A64, FVSUM_1:61
.= (F /. ((Sgm N) . j)) * ((1. (K,n)) * (x,j)) by A3, A62, MATRIX13:def_1
.= (F /. ((Sgm N) . j)) * (0. K) by A61, A62, MATRIX_1:def_11
.= 0. K by VECTSP_1:6 ; ::_thesis: verum
end;
then Sum mN = mN . x by A1, A54, A48, A55, MATRIX_3:12;
then A65: Sum mN = (mlt ((Line (A,x)),F)) . ((Sgm N) . x) by A1, A54, A48, A51, A55, FUNCT_1:12
.= (F /. ((Sgm N) . x)) * (A * (((Sgm (Seg n)) . x),((Sgm N) . x))) by A2, A14, A59, A57, A58, FVSUM_1:61
.= (F /. ((Sgm N) . x)) * ((1. (K,n)) * (x,x)) by A3, A56, MATRIX13:def_1
.= (F /. ((Sgm N) . x)) * (1_ K) by A56, MATRIX_1:def_11
.= F /. i by A49, VECTSP_1:def_4
.= f . i by A31, A32, A38, PARTFUN1:def_6 ;
A66: dom (Sgm ((Seg m) \ N)) = Seg (card ((Seg m) \ N)) by FINSEQ_3:40, XBOOLE_1:36;
A67: rng (Sgm ((Seg m) \ N)) = (Seg m) \ N by A39, FINSEQ_1:def_13;
then dom mSN = Seg (m -' n) by A24, A25, A50, A52, A66, RELAT_1:27, XBOOLE_1:36;
then A68: len mSN = m -' n by FINSEQ_1:def_3;
A69: ColVec2Mx ((len A) |-> (0. K)) = 0. (K,(len A),1) by Th32;
( Indices (0. (K,(len A),1)) = [:(Seg (len A)),(Seg 1):] & 1 in Seg 1 ) by A4, A15, MATRIX_1:23;
then A70: [x,1] in Indices (0. (K,(len A),1)) by A1, A15, A54, A48, ZFMISC_1:87;
then A71: 0. K = (ColVec2Mx ((len A) |-> (0. K))) * (x,1) by A69, MATRIX_3:1
.= (Line (A,x)) "*" (Col (X,1)) by A43, A44, A69, A70, MATRIX_3:def_4
.= Sum (mlt ((Line (A,x)),(Col (X,1)))) ;
reconsider mSN = mSN as Element of (m -' n) -tuples_on the carrier of K by A68, FINSEQ_2:92;
A72: width M1 = m by A5, MATRIX_1:23;
now__::_thesis:_for_j_being_Nat_st_j_in_Seg_(m_-'_n)_holds_
(Col_(M1,i))_._j_=_(-_mSN)_._j
let j be Nat; ::_thesis: ( j in Seg (m -' n) implies (Col (M1,i)) . j = (- mSN) . j )
assume A73: j in Seg (m -' n) ; ::_thesis: (Col (M1,i)) . j = (- mSN) . j
A74: j = (idseq (m -' n)) . j by A73, FINSEQ_2:49
.= (Sgm (Seg (m -' n))) . j by FINSEQ_3:48 ;
A75: Line (M1,j) = M1 . j by A73, MATRIX_2:8
.= (f /. ((Sgm ((Seg m) \ N)) . j)) * (Line (MV,j)) by A34, A35, A73 ;
A76: (Sgm ((Seg m) \ N)) . j in (Seg m) \ N by A24, A25, A67, A66, A73, FUNCT_1:def_3;
then ( f /. ((Sgm ((Seg m) \ N)) . j) = F . ((Sgm ((Seg m) \ N)) . j) & A * (x,((Sgm ((Seg m) \ N)) . j)) = (Line (A,x)) . ((Sgm ((Seg m) \ N)) . j) ) by A14, A31, A32, A39, MATRIX_1:def_7, PARTFUN1:def_6;
then A77: (f /. ((Sgm ((Seg m) \ N)) . j)) * (A * (x,((Sgm ((Seg m) \ N)) . j))) = (mlt ((Line (A,x)),F)) . ((Sgm ((Seg m) \ N)) . j) by A14, A39, A76, FVSUM_1:61
.= mSN . j by A24, A25, A52, A66, A73, FUNCT_1:13 ;
A78: x = (idseq n) . x by A1, A54, A48, FINSEQ_2:49
.= (Sgm (Seg n)) . x by FINSEQ_3:48 ;
A79: [:(Seg (m -' n)),N:] c= Indices MV by A2, A12, ZFMISC_1:95;
[j,i] in Indices MV by A2, A12, A41, A73, ZFMISC_1:87;
then A80: [j,x] in Indices (Segm (MV,(Seg (m -' n)),N)) by A47, A49, A46, A74, A79, MATRIX13:17;
then A81: [j,x] in Indices ((Segm (A,(Seg n),((Seg m) \ N))) @) by A8, Lm1;
then A82: [x,j] in Indices (Segm (A,(Seg n),((Seg m) \ N))) by MATRIX_1:def_6;
A83: (Line (MV,j)) . i = MV * (((Sgm (Seg (m -' n))) . j),((Sgm N) . x)) by A10, A38, A49, A74, MATRIX_1:def_7
.= (- ((Segm (A,(Seg n),((Seg m) \ N))) @)) * (j,x) by A8, A80, MATRIX13:def_1
.= - (((Segm (A,(Seg n),((Seg m) \ N))) @) * (j,x)) by A81, MATRIX_3:def_2
.= - ((Segm (A,(Seg n),((Seg m) \ N))) * (x,j)) by A82, MATRIX_1:def_6
.= - (A * (x,((Sgm ((Seg m) \ N)) . j))) by A82, A78, MATRIX13:def_1 ;
dom M1 = Seg (m -' n) by A33, FINSEQ_1:def_3;
hence (Col (M1,i)) . j = M1 * (j,i) by A73, MATRIX_1:def_8
.= (Line (M1,j)) . i by A38, A72, MATRIX_1:def_7
.= (f /. ((Sgm ((Seg m) \ N)) . j)) * (- (A * (x,((Sgm ((Seg m) \ N)) . j)))) by A2, A10, A41, A75, A83, FVSUM_1:51
.= - ((f /. ((Sgm ((Seg m) \ N)) . j)) * (A * (x,((Sgm ((Seg m) \ N)) . j)))) by VECTSP_1:8
.= (- mSN) . j by A73, A77, FVSUM_1:23 ;
::_thesis: verum
end;
then Col (M1,i) = - mSN by A33, FINSEQ_2:119;
hence Sum (Col (M1,i)) = - (Sum mSN) by FVSUM_1:75
.= (- (Sum mSN)) + ((Sum mSN) + (Sum mN)) by A71, A45, A53, RLVECT_1:def_4
.= ((- (Sum mSN)) + (Sum mSN)) + (Sum mN) by RLVECT_1:def_3
.= (0. K) + (Sum mN) by VECTSP_1:19
.= f . i by A65, RLVECT_1:def_4 ;
::_thesis: verum
end;
supposeA84: not i in N ; ::_thesis: Sum (Col (M1,i)) = f . i
A85: rng (Sgm ((Seg m) \ N)) = (Seg m) \ N by A39, FINSEQ_1:def_13;
i in (Seg m) \ N by A38, A84, XBOOLE_0:def_5;
then consider x being set such that
A86: x in dom (Sgm ((Seg m) \ N)) and
A87: (Sgm ((Seg m) \ N)) . x = i by A85, FUNCT_1:def_3;
reconsider x = x as Element of NAT by A86;
A88: dom (Sgm ((Seg m) \ N)) = Seg (card ((Seg m) \ N)) by FINSEQ_3:40, XBOOLE_1:36;
then A89: Line (M1,x) = M1 . x by A24, A25, A86, MATRIX_2:8
.= (f /. ((Sgm ((Seg m) \ N)) . x)) * (Line (MV,x)) by A24, A25, A34, A35, A88, A86 ;
[x,x] in [:(Seg (m -' n)),(Seg (m -' n)):] by A24, A25, A88, A86, ZFMISC_1:87;
then A90: [x,x] in Indices (1. (K,(m -' n))) by MATRIX_1:24;
x = (idseq (m -' n)) . x by A24, A25, A88, A86, FINSEQ_2:49
.= (Sgm (Seg (m -' n))) . x by FINSEQ_3:48 ;
then A91: (Line (MV,x)) . i = MV * (((Sgm (Seg (m -' n))) . x),((Sgm ((Seg m) \ N)) . x)) by A10, A38, A87, MATRIX_1:def_7
.= (1. (K,(m -' n))) * (x,x) by A7, A90, MATRIX13:def_1
.= 1_ K by A90, MATRIX_1:def_11 ;
A92: dom (Col (M1,i)) = Seg (len M1) by FINSEQ_2:124;
A93: dom M1 = Seg (len M1) by FINSEQ_1:def_3;
A94: len M1 = m - n by A5, A24, MATRIX_1:23;
A95: width M1 = m by A5, MATRIX_1:23;
A96: now__::_thesis:_for_j_being_Nat_st_j_in_dom_(Col_(M1,i))_&_x_<>_j_holds_
(Col_(M1,i))_._j_=_0._K
let j be Nat; ::_thesis: ( j in dom (Col (M1,i)) & x <> j implies (Col (M1,i)) . j = 0. K )
assume that
A97: j in dom (Col (M1,i)) and
A98: x <> j ; ::_thesis: (Col (M1,i)) . j = 0. K
A99: Line (M1,j) = M1 . j by A92, A97, MATRIX_2:8
.= (f /. ((Sgm ((Seg m) \ N)) . j)) * (Line (MV,j)) by A34, A92, A93, A97 ;
[j,x] in [:(Seg (m -' n)),(Seg (m -' n)):] by A24, A25, A88, A86, A92, A94, A97, ZFMISC_1:87;
then A100: [j,x] in Indices (1. (K,(m -' n))) by MATRIX_1:24;
j = (idseq (m -' n)) . j by A24, A92, A94, A97, FINSEQ_2:49
.= (Sgm (Seg (m -' n))) . j by FINSEQ_3:48 ;
then A101: (Line (MV,j)) . i = MV * (((Sgm (Seg (m -' n))) . j),((Sgm ((Seg m) \ N)) . x)) by A10, A38, A87, MATRIX_1:def_7
.= (1. (K,(m -' n))) * (j,x) by A7, A100, MATRIX13:def_1
.= 0. K by A98, A100, MATRIX_1:def_11 ;
thus (Col (M1,i)) . j = M1 * (j,i) by A92, A93, A97, MATRIX_1:def_8
.= ((f /. ((Sgm ((Seg m) \ N)) . j)) * (Line (MV,j))) . i by A38, A95, A99, MATRIX_1:def_7
.= (f /. ((Sgm ((Seg m) \ N)) . j)) * (0. K) by A10, A38, A101, FVSUM_1:51
.= 0. K by VECTSP_1:6 ; ::_thesis: verum
end;
(Col (M1,i)) . x = M1 * (x,i) by A25, A88, A86, A94, A93, MATRIX_1:def_8
.= (Line (M1,x)) . i by A38, A95, MATRIX_1:def_7
.= (f /. ((Sgm ((Seg m) \ N)) . x)) * (1_ K) by A10, A38, A91, A89, FVSUM_1:51
.= f /. i by A87, VECTSP_1:def_4
.= f . i by A31, A32, A38, PARTFUN1:def_6 ;
hence Sum (Col (M1,i)) = f . i by A25, A88, A86, A92, A94, A96, MATRIX_3:12; ::_thesis: verum
end;
end;
end;
Carrier L c= lines MV by VECTSP_6:def_4;
hence SumL . i = f . i by A27, A37, A38, A40, MATRIX13:105, MATRIX13:107; ::_thesis: verum
end;
then A102: SumL = f by FINSEQ_2:119;
the carrier of (Lin (lines MV)) = { (Sum l) where l is Linear_Combination of lines MV : verum } by VECTSP_7:def_2;
hence y in the carrier of (Lin (lines MV)) by A29, A102; ::_thesis: verum
end;
take MV ; ::_thesis: ( Segm (MV,(Seg (m -' n)),((Seg m) \ N)) = 1. (K,(m -' n)) & Segm (MV,(Seg (m -' n)),N) = - ((Segm (A,(Seg n),((Seg m) \ N))) @) & Lin (lines MV) = Space_of_Solutions_of A )
Solutions_of (A,((len A) |-> (0. K))) = the carrier of (Space_of_Solutions_of A) by A6, A14, Def5;
then the carrier of (Lin (lines MV)) c= the carrier of (Space_of_Solutions_of A) by A20, VECTSP_7:11;
then the carrier of (Lin (lines MV)) = the carrier of (Space_of_Solutions_of A) by A28, XBOOLE_0:def_10;
hence ( Segm (MV,(Seg (m -' n)),((Seg m) \ N)) = 1. (K,(m -' n)) & Segm (MV,(Seg (m -' n)),N) = - ((Segm (A,(Seg n),((Seg m) \ N))) @) & Lin (lines MV) = Space_of_Solutions_of A ) by A14, A7, A8, VECTSP_4:29; ::_thesis: verum
end;
Lm8: for n being Nat
for K being Field holds dim (Space_of_Solutions_of (1. (K,n))) = 0
proof
let n be Nat; ::_thesis: for K being Field holds dim (Space_of_Solutions_of (1. (K,n))) = 0
let K be Field; ::_thesis: dim (Space_of_Solutions_of (1. (K,n))) = 0
set ONE = 1. (K,n);
set SS = Space_of_Solutions_of (1. (K,n));
A1: the carrier of (Space_of_Solutions_of (1. (K,n))) c= the carrier of ((0). (Space_of_Solutions_of (1. (K,n))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of (Space_of_Solutions_of (1. (K,n))) or x in the carrier of ((0). (Space_of_Solutions_of (1. (K,n)))) )
assume A2: x in the carrier of (Space_of_Solutions_of (1. (K,n))) ; ::_thesis: x in the carrier of ((0). (Space_of_Solutions_of (1. (K,n))))
A3: len (1. (K,n)) = n by MATRIX_1:24;
A4: width (1. (K,n)) = n by MATRIX_1:24;
then ( width (1. (K,n)) = 0 implies len (1. (K,n)) = 0 ) by MATRIX_1:24;
then x in Solutions_of ((1. (K,n)),(n |-> (0. K))) by A2, A3, Def5;
then consider f being FinSequence of K such that
A5: f = x and
A6: ColVec2Mx f in Solutions_of ((1. (K,n)),(ColVec2Mx (n |-> (0. K)))) ;
consider X being Matrix of K such that
A7: X = ColVec2Mx f and
A8: len X = width (1. (K,n)) and
width X = width (ColVec2Mx (n |-> (0. K))) and
A9: (1. (K,n)) * X = ColVec2Mx (n |-> (0. K)) by A6;
A10: now__::_thesis:_n_|->_(0._K)_=_f
percases ( n > 0 or n = 0 ) ;
supposeA11: n > 0 ; ::_thesis: n |-> (0. K) = f
(1. (K,n)) * X = X by A4, A8, MATRIXR2:68;
hence n |-> (0. K) = Col (X,1) by A9, A11, Th26
.= f by A4, A7, A8, A11, Th26 ;
::_thesis: verum
end;
supposeA12: n = 0 ; ::_thesis: f = n |-> (0. K)
then f = {} by A4, A7, A8, MATRIX_1:def_2;
hence f = n |-> (0. K) by A12; ::_thesis: verum
end;
end;
end;
0. (Space_of_Solutions_of (1. (K,n))) = 0. ((width (1. (K,n))) -VectSp_over K) by VECTSP_4:11
.= n |-> (0. K) by A4, MATRIX13:102 ;
then f in {(0. (Space_of_Solutions_of (1. (K,n))))} by A10, TARSKI:def_1;
hence x in the carrier of ((0). (Space_of_Solutions_of (1. (K,n)))) by A5, VECTSP_4:def_3; ::_thesis: verum
end;
the carrier of ((0). (Space_of_Solutions_of (1. (K,n)))) c= the carrier of (Space_of_Solutions_of (1. (K,n))) by VECTSP_4:def_2;
then the carrier of (Space_of_Solutions_of (1. (K,n))) = the carrier of ((0). (Space_of_Solutions_of (1. (K,n)))) by A1, XBOOLE_0:def_10;
then (0). (Space_of_Solutions_of (1. (K,n))) = (Omega). (Space_of_Solutions_of (1. (K,n))) by VECTSP_4:29;
hence dim (Space_of_Solutions_of (1. (K,n))) = 0 by VECTSP_9:29; ::_thesis: verum
end;
theorem Th68: :: MATRIX15:68
for K being Field
for A being Matrix of K st ( width A = 0 implies len A = 0 ) holds
dim (Space_of_Solutions_of A) = (width A) - (the_rank_of A)
proof
let K be Field; ::_thesis: for A being Matrix of K st ( width A = 0 implies len A = 0 ) holds
dim (Space_of_Solutions_of A) = (width A) - (the_rank_of A)
let A be Matrix of K; ::_thesis: ( ( width A = 0 implies len A = 0 ) implies dim (Space_of_Solutions_of A) = (width A) - (the_rank_of A) )
assume A1: ( width A = 0 implies len A = 0 ) ; ::_thesis: dim (Space_of_Solutions_of A) = (width A) - (the_rank_of A)
set W = width A;
set L = len A;
reconsider A9 = A as Matrix of len A, width A,K by MATRIX_2:7;
percases ( the_rank_of A = 0 or the_rank_of A > 0 ) ;
supposeA2: the_rank_of A = 0 ; ::_thesis: dim (Space_of_Solutions_of A) = (width A) - (the_rank_of A)
dim ((width A) -VectSp_over K) = width A by MATRIX13:112;
hence dim (Space_of_Solutions_of A) = (width A) - (the_rank_of A) by A1, A2, Th63; ::_thesis: verum
end;
supposeA3: the_rank_of A > 0 ; ::_thesis: dim (Space_of_Solutions_of A) = (width A) - (the_rank_of A)
defpred S1[ set , set ] means for A1 being Matrix of len A, width A,K st A1 = $1 holds
Space_of_Solutions_of A9 = Space_of_Solutions_of A1;
deffunc H1( Matrix of len A, width A,K, Nat, Nat, Element of K) -> Matrix of len A, width A,K = $1;
A4: width A > 0 by A3, MATRIX13:74;
A5: for A1, B1 being Matrix of len A, width A,K st S1[A1,B1] holds
for a being Element of K
for i, j being Nat st j in dom A1 & ( i = j implies a <> - (1_ K) ) holds
S1[ RLine (A1,i,((Line (A1,i)) + (a * (Line (A1,j))))),H1(B1,i,j,a)]
proof
let A1, B1 be Matrix of len A, width A,K; ::_thesis: ( S1[A1,B1] implies for a being Element of K
for i, j being Nat st j in dom A1 & ( i = j implies a <> - (1_ K) ) holds
S1[ RLine (A1,i,((Line (A1,i)) + (a * (Line (A1,j))))),H1(B1,i,j,a)] )
assume A6: S1[A1,B1] ; ::_thesis: for a being Element of K
for i, j being Nat st j in dom A1 & ( i = j implies a <> - (1_ K) ) holds
S1[ RLine (A1,i,((Line (A1,i)) + (a * (Line (A1,j))))),H1(B1,i,j,a)]
let a be Element of K; ::_thesis: for i, j being Nat st j in dom A1 & ( i = j implies a <> - (1_ K) ) holds
S1[ RLine (A1,i,((Line (A1,i)) + (a * (Line (A1,j))))),H1(B1,i,j,a)]
A7: dom A1 = Seg (len A1) by FINSEQ_1:def_3
.= Seg (len A) by MATRIX_1:def_2 ;
let i, j be Nat; ::_thesis: ( j in dom A1 & ( i = j implies a <> - (1_ K) ) implies S1[ RLine (A1,i,((Line (A1,i)) + (a * (Line (A1,j))))),H1(B1,i,j,a)] )
assume ( j in dom A1 & ( i = j implies a <> - (1_ K) ) ) ; ::_thesis: S1[ RLine (A1,i,((Line (A1,i)) + (a * (Line (A1,j))))),H1(B1,i,j,a)]
then Space_of_Solutions_of (RLine (A1,i,((Line (A1,i)) + (a * (Line (A1,j)))))) = Space_of_Solutions_of A1 by A4, A7, Th65
.= Space_of_Solutions_of A9 by A6 ;
hence S1[ RLine (A1,i,((Line (A1,i)) + (a * (Line (A1,j))))),H1(B1,i,j,a)] ; ::_thesis: verum
end;
A8: S1[A9,A9] ;
consider A1, B1 being Matrix of len A, width A,K, N being finite without_zero Subset of NAT such that
A9: N c= Seg (width A) and
A10: ( the_rank_of A9 = the_rank_of A1 & the_rank_of A9 = card N ) and
A11: ( S1[A1,B1] & Segm (A1,(Seg (card N)),N) = 1. (K,(card N)) ) and
A12: for i being Nat st i in dom A1 & i > card N holds
Line (A1,i) = (width A) |-> (0. K) and
for i, j being Nat st i in Seg (card N) & j in Seg (width A1) & j < (Sgm N) . i holds
A1 * (i,j) = 0. K from MATRIX15:sch_2(A8, A5);
A13: 0 < len A by A3, MATRIX13:74;
then A14: width A1 = width A by MATRIX_1:23;
then A15: card (Seg (width A1)) = width A by FINSEQ_1:57;
then A16: card ((Seg (width A)) \ N) = (width A) - (card N) by A9, A14, CARD_2:44;
set SN = Segm (A1,(Seg (card N)),(Seg (width A1)));
A17: Seg (card N) c= Seg (card N) ;
A18: (width A) -' (card N) = (width A) - (card N) by A9, A14, A15, NAT_1:43, XREAL_1:233;
Sgm (Seg (card N)) = idseq (card N) by FINSEQ_3:48
.= id (Seg (card N)) ;
then A19: Seg (card N) = (Sgm (Seg (card N))) " (Seg (card N)) by A17, FUNCT_2:94;
Sgm (Seg (width A1)) = idseq (width A1) by FINSEQ_3:48
.= id (Seg (width A1)) ;
then N = (Sgm (Seg (width A1))) " N by A9, A14, FUNCT_2:94;
then A20: Segm ((Segm (A1,(Seg (card N)),(Seg (width A1)))),(Seg (card N)),N) = 1. (K,(card N)) by A9, A11, A14, A19, MATRIX13:56;
A21: card (Seg (card N)) = card N by FINSEQ_1:57;
A22: Seg (len A1) = dom A1 by FINSEQ_1:def_3;
A23: now__::_thesis:_for_i_being_Nat_st_i_in_(dom_A1)_\_(Seg_(card_N))_holds_
Line_(A1,i)_=_(width_A1)_|->_(0._K)
let i be Nat; ::_thesis: ( i in (dom A1) \ (Seg (card N)) implies Line (A1,i) = (width A1) |-> (0. K) )
assume A24: i in (dom A1) \ (Seg (card N)) ; ::_thesis: Line (A1,i) = (width A1) |-> (0. K)
not i in Seg (card N) by A24, XBOOLE_0:def_5;
then A25: ( i < 1 or i > card N ) by A24;
i in dom A1 by A24, XBOOLE_0:def_5;
hence Line (A1,i) = (width A1) |-> (0. K) by A12, A22, A14, A25, FINSEQ_1:1; ::_thesis: verum
end;
card N <= len A1 by A10, MATRIX13:74;
then A26: Seg (card N) c= Seg (len A1) by FINSEQ_1:5;
width A1 > 0 by A4, A13, MATRIX_1:23;
then A27: Space_of_Solutions_of (Segm (A1,(Seg (card N)),(Seg (width A1)))) = Space_of_Solutions_of A1 by A3, A10, A26, A22, A23, Th66
.= Space_of_Solutions_of A9 by A11 ;
percases ( (width A) -' (card N) = 0 or (width A) -' (card N) > 0 ) ;
supposeA28: (width A) -' (card N) = 0 ; ::_thesis: dim (Space_of_Solutions_of A) = (width A) - (the_rank_of A)
then Segm (A1,(Seg (card N)),(Seg (width A1))) = 1. (K,(card N)) by A9, A11, A14, A21, A18, CARD_FIN:1;
hence dim (Space_of_Solutions_of A) = (width A) - (the_rank_of A) by A10, A27, A18, A28, Lm8; ::_thesis: verum
end;
supposeA29: (width A) -' (card N) > 0 ; ::_thesis: dim (Space_of_Solutions_of A) = (width A) - (the_rank_of A)
then ((width A) -' (card N)) + 0 > 0 ;
then (width A) -' (card N) >= 1 by NAT_1:19;
then A30: Det (1. (K,((width A) -' (card N)))) = 1_ K by MATRIX_7:16;
A31: ( card (Seg ((width A) -' (card N))) = (width A) -' (card N) & 0. K <> 1_ K ) by FINSEQ_1:57;
consider MVectors being Matrix of (width A) -' (card N), width A,K such that
A32: Segm (MVectors,(Seg ((width A) -' (card N))),((Seg (width A)) \ N)) = 1. (K,((width A) -' (card N))) and
Segm (MVectors,(Seg ((width A) -' (card N))),N) = - ((Segm ((Segm (A1,(Seg (card N)),(Seg (width A1)))),(Seg (card N)),((Seg (width A)) \ N))) @) and
A33: Lin (lines MVectors) = Space_of_Solutions_of A9 by A3, A9, A10, A27, A21, A15, A20, A29, Th67;
len MVectors = (width A) -' (card N) by A29, MATRIX_1:23;
then A34: (width A) -' (card N) >= the_rank_of MVectors by MATRIX13:74;
A35: (Seg (width A)) \ N c= Seg (width A) by XBOOLE_1:36;
Indices MVectors = [:(Seg ((width A) -' (card N))),(Seg (width A)):] by A29, MATRIX_1:23;
then A36: [:(Seg ((width A) -' (card N))),((Seg (width A)) \ N):] c= Indices MVectors by A35, ZFMISC_1:95;
reconsider B = lines MVectors as Subset of ((width A) -VectSp_over K) ;
A37: ( dom MVectors = Seg (len MVectors) & len MVectors = (width A) -' (card N) ) by FINSEQ_1:def_3, MATRIX_1:def_2;
EqSegm (MVectors,(Seg ((width A) -' (card N))),((Seg (width A)) \ N)) = 1. (K,((width A) -' (card N))) by A16, A18, A32, FINSEQ_1:57, MATRIX13:def_3;
then A38: (width A) -' (card N) <= the_rank_of MVectors by A16, A18, A36, A30, A31, MATRIX13:def_4;
then MVectors is without_repeated_line by A34, MATRIX13:105, XXREAL_0:1;
then Seg ((width A) -' (card N)),B are_equipotent by A37, WELLORD2:def_4;
then A39: card B = card (Seg ((width A) -' (card N))) by CARD_1:5
.= (width A) -' (card N) by FINSEQ_1:57 ;
the_rank_of MVectors = (width A) -' (card N) by A38, A34, XXREAL_0:1;
then lines MVectors is linearly-independent by MATRIX13:121;
hence dim (Space_of_Solutions_of A) = (width A) - (the_rank_of A) by A10, A18, A33, A39, VECTSP_9:26; ::_thesis: verum
end;
end;
end;
end;
end;
theorem Th69: :: MATRIX15:69
for n, m being Nat
for K being Field
for M being Matrix of n,m,K
for i, j being Nat
for a being Element of K st M is without_repeated_line & j in dom M & ( i = j implies a <> - (1_ K) ) holds
Lin (lines M) = Lin (lines (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))))
proof
let n, m be Nat; ::_thesis: for K being Field
for M being Matrix of n,m,K
for i, j being Nat
for a being Element of K st M is without_repeated_line & j in dom M & ( i = j implies a <> - (1_ K) ) holds
Lin (lines M) = Lin (lines (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))))
let K be Field; ::_thesis: for M being Matrix of n,m,K
for i, j being Nat
for a being Element of K st M is without_repeated_line & j in dom M & ( i = j implies a <> - (1_ K) ) holds
Lin (lines M) = Lin (lines (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))))
let M be Matrix of n,m,K; ::_thesis: for i, j being Nat
for a being Element of K st M is without_repeated_line & j in dom M & ( i = j implies a <> - (1_ K) ) holds
Lin (lines M) = Lin (lines (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))))
let i, j be Nat; ::_thesis: for a being Element of K st M is without_repeated_line & j in dom M & ( i = j implies a <> - (1_ K) ) holds
Lin (lines M) = Lin (lines (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))))
let a be Element of K; ::_thesis: ( M is without_repeated_line & j in dom M & ( i = j implies a <> - (1_ K) ) implies Lin (lines M) = Lin (lines (RLine (M,i,((Line (M,i)) + (a * (Line (M,j))))))) )
assume that
A1: M is without_repeated_line and
A2: j in dom M and
A3: ( i = j implies a <> - (1_ K) ) ; ::_thesis: Lin (lines M) = Lin (lines (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))))
A4: len M = n by MATRIX_1:def_2;
set L = (Line (M,i)) + (a * (Line (M,j)));
A5: dom M = Seg (len M) by FINSEQ_1:def_3;
set R = RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))));
percases ( not i in dom M or i in dom M ) ;
suppose not i in dom M ; ::_thesis: Lin (lines M) = Lin (lines (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))))
hence Lin (lines M) = Lin (lines (RLine (M,i,((Line (M,i)) + (a * (Line (M,j))))))) by A5, MATRIX13:40; ::_thesis: verum
end;
supposeA6: i in dom M ; ::_thesis: Lin (lines M) = Lin (lines (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))))
then n <> 0 by A5, A4;
then A7: width M = m by MATRIX_1:23;
then reconsider Li = Line (M,i), Lj = Line (M,j) as Vector of (m -VectSp_over K) by MATRIX13:102;
a * Lj = a * (Line (M,j)) by A7, MATRIX13:102;
then A8: (Line (M,i)) + (a * (Line (M,j))) = Li + (a * Lj) by A7, MATRIX13:102;
A9: ( not Li = Lj or a <> - (1_ K) or Li = 0. (m -VectSp_over K) )
proof
assume A10: Li = Lj ; ::_thesis: ( a <> - (1_ K) or Li = 0. (m -VectSp_over K) )
( Li = M . i & Lj = M . j ) by A2, A5, A4, A6, MATRIX_2:8;
hence ( a <> - (1_ K) or Li = 0. (m -VectSp_over K) ) by A1, A2, A3, A6, A10, FUNCT_1:def_4; ::_thesis: verum
end;
reconsider L9 = (Line (M,i)) + (a * (Line (M,j))) as Element of the carrier of K * by FINSEQ_1:def_11;
reconsider LL = L9 as set ;
set iL = {i} --> L9;
len ((Line (M,i)) + (a * (Line (M,j)))) = width M by CARD_1:def_7;
then A11: RLine (M,i,((Line (M,i)) + (a * (Line (M,j))))) = M +* (i,LL) by MATRIX11:29
.= M +* (i .--> LL) by A6, FUNCT_7:def_3
.= M +* ({i} --> L9) by FUNCOP_1:def_9 ;
M .: ((dom M) \ (dom ({i} --> L9))) = (M .: (dom M)) \ (M .: (dom ({i} --> L9))) by A1, FUNCT_1:64
.= (rng M) \ (M .: (dom ({i} --> L9))) by RELAT_1:113
.= (rng M) \ (Im (M,i)) by FUNCOP_1:13
.= (rng M) \ {(M . i)} by A6, FUNCT_1:59
.= (rng M) \ {(Line (M,i))} by A5, A4, A6, MATRIX_2:8 ;
then A12: lines (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))) = ((lines M) \ {(Line (M,i))}) \/ (rng ({i} --> L9)) by A11, FRECHET:12
.= ((lines M) \ {(Line (M,i))}) \/ {((Line (M,i)) + (a * (Line (M,j))))} by FUNCOP_1:8 ;
A13: Lj in lines M by A2, A5, A4, MATRIX13:103;
Li in lines M by A5, A4, A6, MATRIX13:103;
hence Lin (lines M) = Lin (lines (RLine (M,i,((Line (M,i)) + (a * (Line (M,j))))))) by A8, A12, A13, A9, Th14; ::_thesis: verum
end;
end;
end;
theorem Th70: :: MATRIX15:70
for m being Nat
for K being Field
for W being Subspace of m -VectSp_over K ex A being Matrix of dim W,m,K ex N being finite without_zero Subset of NAT st
( N c= Seg m & dim W = card N & Segm (A,(Seg (dim W)),N) = 1. (K,(dim W)) & the_rank_of A = dim W & lines A is Basis of W )
proof
let m be Nat; ::_thesis: for K being Field
for W being Subspace of m -VectSp_over K ex A being Matrix of dim W,m,K ex N being finite without_zero Subset of NAT st
( N c= Seg m & dim W = card N & Segm (A,(Seg (dim W)),N) = 1. (K,(dim W)) & the_rank_of A = dim W & lines A is Basis of W )
let K be Field; ::_thesis: for W being Subspace of m -VectSp_over K ex A being Matrix of dim W,m,K ex N being finite without_zero Subset of NAT st
( N c= Seg m & dim W = card N & Segm (A,(Seg (dim W)),N) = 1. (K,(dim W)) & the_rank_of A = dim W & lines A is Basis of W )
let W be Subspace of m -VectSp_over K; ::_thesis: ex A being Matrix of dim W,m,K ex N being finite without_zero Subset of NAT st
( N c= Seg m & dim W = card N & Segm (A,(Seg (dim W)),N) = 1. (K,(dim W)) & the_rank_of A = dim W & lines A is Basis of W )
consider I being finite Subset of W such that
A1: I is Basis of W by MATRLIN:def_1;
I is linearly-independent by A1, VECTSP_7:def_3;
then reconsider U = I as linearly-independent Subset of (m -VectSp_over K) by VECTSP_9:11;
defpred S1[ set , set ] means for A, B being Matrix of card I,m,K st $1 = A holds
( A is without_repeated_line & lines A is linearly-independent & Lin (lines A) = (Omega). W );
deffunc H1( Matrix of card I,m,K, Nat, Nat, Element of K) -> Matrix of card I,m,K = $1;
consider M being Matrix of card I,m,K such that
A2: ( M is without_repeated_line & lines M = U ) by MATRIX13:104;
A3: for A9, B9 being Matrix of card I,m,K st S1[A9,B9] holds
for a being Element of K
for i, j being Nat st j in dom A9 & ( i = j implies a <> - (1_ K) ) holds
S1[ RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j))))),H1(B9,i,j,a)]
proof
let A9, B9 be Matrix of card I,m,K; ::_thesis: ( S1[A9,B9] implies for a being Element of K
for i, j being Nat st j in dom A9 & ( i = j implies a <> - (1_ K) ) holds
S1[ RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j))))),H1(B9,i,j,a)] )
assume A4: S1[A9,B9] ; ::_thesis: for a being Element of K
for i, j being Nat st j in dom A9 & ( i = j implies a <> - (1_ K) ) holds
S1[ RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j))))),H1(B9,i,j,a)]
A5: dom A9 = Seg (len A9) by FINSEQ_1:def_3;
let a be Element of K; ::_thesis: for i, j being Nat st j in dom A9 & ( i = j implies a <> - (1_ K) ) holds
S1[ RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j))))),H1(B9,i,j,a)]
let i, j be Nat; ::_thesis: ( j in dom A9 & ( i = j implies a <> - (1_ K) ) implies S1[ RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j))))),H1(B9,i,j,a)] )
assume A6: ( j in dom A9 & ( i = j implies a <> - (1_ K) ) ) ; ::_thesis: S1[ RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j))))),H1(B9,i,j,a)]
set R = RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))));
A7: A9 is without_repeated_line by A4;
then A8: Lin (lines A9) = Lin (lines (RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j))))))) by A6, Th69;
lines A9 is linearly-independent by A4;
then card I = the_rank_of A9 by A7, MATRIX13:121
.= the_rank_of (RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j)))))) by A6, A5, MATRIX13:92 ;
hence S1[ RLine (A9,i,((Line (A9,i)) + (a * (Line (A9,j))))),H1(B9,i,j,a)] by A4, A8, MATRIX13:121; ::_thesis: verum
end;
Lin I = VectSpStr(# the carrier of W, the addF of W, the ZeroF of W, the lmult of W #) by A1, VECTSP_7:def_3;
then A9: S1[M,M] by A2, VECTSP_9:17;
consider A9, B9 being Matrix of card I,m,K, N being finite without_zero Subset of NAT such that
A10: N c= Seg m and
A11: ( the_rank_of M = the_rank_of A9 & the_rank_of M = card N & S1[A9,B9] ) and
A12: Segm (A9,(Seg (card N)),N) = 1. (K,(card N)) and
for i being Nat st i in dom A9 & i > card N holds
Line (A9,i) = m |-> (0. K) and
for i, j being Nat st i in Seg (card N) & j in Seg (width A9) & j < (Sgm N) . i holds
A9 * (i,j) = 0. K from MATRIX15:sch_2(A9, A3);
reconsider A9 = A9 as Matrix of dim W,m,K by A1, VECTSP_9:def_1;
lines A9 c= the carrier of (Lin (lines A9))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in lines A9 or x in the carrier of (Lin (lines A9)) )
assume x in lines A9 ; ::_thesis: x in the carrier of (Lin (lines A9))
then x in Lin (lines A9) by VECTSP_7:8;
hence x in the carrier of (Lin (lines A9)) by STRUCT_0:def_5; ::_thesis: verum
end;
then reconsider lA = lines A9 as linearly-independent Subset of W by A11, VECTSP_9:12;
take A9 ; ::_thesis: ex N being finite without_zero Subset of NAT st
( N c= Seg m & dim W = card N & Segm (A9,(Seg (dim W)),N) = 1. (K,(dim W)) & the_rank_of A9 = dim W & lines A9 is Basis of W )
take N ; ::_thesis: ( N c= Seg m & dim W = card N & Segm (A9,(Seg (dim W)),N) = 1. (K,(dim W)) & the_rank_of A9 = dim W & lines A9 is Basis of W )
A13: Lin lA = Lin (lines A9) by VECTSP_9:17;
A14: the_rank_of M = card I by A2, MATRIX13:121;
A15: card I = dim W by A1, VECTSP_9:def_1;
Lin (lines A9) = VectSpStr(# the carrier of W, the addF of W, the ZeroF of W, the lmult of W #) by A11;
hence ( N c= Seg m & dim W = card N & Segm (A9,(Seg (dim W)),N) = 1. (K,(dim W)) & the_rank_of A9 = dim W & lines A9 is Basis of W ) by A15, A10, A11, A12, A14, A13, VECTSP_7:def_3; ::_thesis: verum
end;
theorem :: MATRIX15:71
for m being Nat
for K being Field
for W being strict Subspace of m -VectSp_over K st dim W < m holds
ex A being Matrix of m -' (dim W),m,K ex N being finite without_zero Subset of NAT st
( card N = m -' (dim W) & N c= Seg m & Segm (A,(Seg (m -' (dim W))),N) = 1. (K,(m -' (dim W))) & W = Space_of_Solutions_of A )
proof
let m be Nat; ::_thesis: for K being Field
for W being strict Subspace of m -VectSp_over K st dim W < m holds
ex A being Matrix of m -' (dim W),m,K ex N being finite without_zero Subset of NAT st
( card N = m -' (dim W) & N c= Seg m & Segm (A,(Seg (m -' (dim W))),N) = 1. (K,(m -' (dim W))) & W = Space_of_Solutions_of A )
let K be Field; ::_thesis: for W being strict Subspace of m -VectSp_over K st dim W < m holds
ex A being Matrix of m -' (dim W),m,K ex N being finite without_zero Subset of NAT st
( card N = m -' (dim W) & N c= Seg m & Segm (A,(Seg (m -' (dim W))),N) = 1. (K,(m -' (dim W))) & W = Space_of_Solutions_of A )
let W be strict Subspace of m -VectSp_over K; ::_thesis: ( dim W < m implies ex A being Matrix of m -' (dim W),m,K ex N being finite without_zero Subset of NAT st
( card N = m -' (dim W) & N c= Seg m & Segm (A,(Seg (m -' (dim W))),N) = 1. (K,(m -' (dim W))) & W = Space_of_Solutions_of A ) )
assume A1: dim W < m ; ::_thesis: ex A being Matrix of m -' (dim W),m,K ex N being finite without_zero Subset of NAT st
( card N = m -' (dim W) & N c= Seg m & Segm (A,(Seg (m -' (dim W))),N) = 1. (K,(m -' (dim W))) & W = Space_of_Solutions_of A )
percases ( dim W = 0 or dim W > 0 ) ;
supposeA2: dim W = 0 ; ::_thesis: ex A being Matrix of m -' (dim W),m,K ex N being finite without_zero Subset of NAT st
( card N = m -' (dim W) & N c= Seg m & Segm (A,(Seg (m -' (dim W))),N) = 1. (K,(m -' (dim W))) & W = Space_of_Solutions_of A )
then reconsider ONE = 1. (K,m) as Matrix of m -' (dim W),m,K by NAT_D:40;
take ONE ; ::_thesis: ex N being finite without_zero Subset of NAT st
( card N = m -' (dim W) & N c= Seg m & Segm (ONE,(Seg (m -' (dim W))),N) = 1. (K,(m -' (dim W))) & W = Space_of_Solutions_of ONE )
take Seg m ; ::_thesis: ( card (Seg m) = m -' (dim W) & Seg m c= Seg m & Segm (ONE,(Seg (m -' (dim W))),(Seg m)) = 1. (K,(m -' (dim W))) & W = Space_of_Solutions_of ONE )
A3: len (1. (K,m)) = m by MATRIX_1:24;
A4: dim (Space_of_Solutions_of ONE) = 0 by Lm8;
A5: m -' (dim W) = m by A2, NAT_D:40;
A6: width (1. (K,m)) = m by MATRIX_1:24;
Space_of_Solutions_of ONE = (Omega). (Space_of_Solutions_of ONE)
.= (0). (Space_of_Solutions_of ONE) by A4, VECTSP_9:29
.= (0). W by A6, VECTSP_4:37
.= (Omega). W by A2, VECTSP_9:29
.= W ;
hence ( card (Seg m) = m -' (dim W) & Seg m c= Seg m & Segm (ONE,(Seg (m -' (dim W))),(Seg m)) = 1. (K,(m -' (dim W))) & W = Space_of_Solutions_of ONE ) by A5, A3, A6, FINSEQ_1:57, MATRIX13:46; ::_thesis: verum
end;
supposeA7: dim W > 0 ; ::_thesis: ex A being Matrix of m -' (dim W),m,K ex N being finite without_zero Subset of NAT st
( card N = m -' (dim W) & N c= Seg m & Segm (A,(Seg (m -' (dim W))),N) = 1. (K,(m -' (dim W))) & W = Space_of_Solutions_of A )
set ZERO = 0. (K,(m -' (dim W)),m);
A8: m - (dim W) > (dim W) - (dim W) by A1, XREAL_1:9;
A9: m -' (dim W) = m - (dim W) by A1, XREAL_1:233;
then A10: ( len (0. (K,(m -' (dim W)),m)) = m -' (dim W) & width (0. (K,(m -' (dim W)),m)) = m ) by A8, MATRIX_1:23;
A11: card (Seg m) = m by FINSEQ_1:57;
consider A being Matrix of dim W,m,K, N being finite without_zero Subset of NAT such that
A12: N c= Seg m and
A13: dim W = card N and
A14: Segm (A,(Seg (dim W)),N) = 1. (K,(dim W)) and
the_rank_of A = dim W and
A15: lines A is Basis of W by Th70;
set SA = Segm (A,(Seg (dim W)),((Seg m) \ N));
A16: card ((Seg m) \ N) = (card (Seg m)) - (card N) by A12, CARD_2:44;
then A17: width (Segm (A,(Seg (dim W)),((Seg m) \ N))) = m - (card N) by A7, A11, MATRIX_1:23;
A18: card (Seg (dim W)) = dim W by FINSEQ_1:57;
then len (Segm (A,(Seg (dim W)),((Seg m) \ N))) = dim W by A7, MATRIX_1:23;
then width ((Segm (A,(Seg (dim W)),((Seg m) \ N))) @) = dim W by A13, A8, A17, MATRIX_2:10;
then A19: width (- ((Segm (A,(Seg (dim W)),((Seg m) \ N))) @)) = dim W by MATRIX_3:def_2;
A20: card (Seg (m -' (dim W))) = m -' (dim W) by FINSEQ_1:57;
then reconsider CC = 1. (K,(m -' (dim W))) as Matrix of card (Seg (m -' (dim W))), card ((Seg m) \ N),K by A1, A13, A16, A11, XREAL_1:233;
A21: ( (Seg m) \ ((Seg m) \ N) = (Seg m) /\ N & m -' (m -' (dim W)) = m - (m -' (dim W)) ) by NAT_D:35, XBOOLE_1:48, XREAL_1:233;
A22: Indices (0. (K,(m -' (dim W)),m)) = [:(Seg (m -' (dim W))),(Seg m):] by A9, A8, MATRIX_1:23;
then A23: [:(Seg (m -' (dim W))),N:] c= Indices (0. (K,(m -' (dim W)),m)) by A12, ZFMISC_1:95;
len ((Segm (A,(Seg (dim W)),((Seg m) \ N))) @) = m - (dim W) by A13, A8, A17, MATRIX_2:10;
then len (- ((Segm (A,(Seg (dim W)),((Seg m) \ N))) @)) = m -' (dim W) by A9, MATRIX_3:def_2;
then reconsider BB = - ((Segm (A,(Seg (dim W)),((Seg m) \ N))) @) as Matrix of card (Seg (m -' (dim W))), card N,K by A13, A20, A19, MATRIX_2:7;
A24: N misses (Seg m) \ N by XBOOLE_1:79;
A25: (Seg m) \ N c= Seg m by XBOOLE_1:36;
then A26: [:(Seg (m -' (dim W))),((Seg m) \ N):] c= Indices (0. (K,(m -' (dim W)),m)) by A22, ZFMISC_1:95;
[:(Seg (m -' (dim W))),N:] /\ [:(Seg (m -' (dim W))),((Seg m) \ N):] = [:(Seg (m -' (dim W))),(N /\ ((Seg m) \ N)):] by ZFMISC_1:99
.= [:(Seg (m -' (dim W))),{}:] by A24, XBOOLE_0:def_7
.= {} by ZFMISC_1:90 ;
then for i, j, bi, bj, ci, cj being Nat st [i,j] in [:(Seg (m -' (dim W))),N:] /\ [:(Seg (m -' (dim W))),((Seg m) \ N):] & bi = ((Sgm (Seg (m -' (dim W)))) ") . i & bj = ((Sgm N) ") . j & ci = ((Sgm (Seg (m -' (dim W)))) ") . i & cj = ((Sgm ((Seg m) \ N)) ") . j holds
BB * (bi,bj) = CC * (ci,cj) ;
then consider M being Matrix of m -' (dim W),m,K such that
A27: Segm (M,(Seg (m -' (dim W))),N) = BB and
A28: Segm (M,(Seg (m -' (dim W))),((Seg m) \ N)) = CC and
for i, j being Nat st [i,j] in (Indices M) \ ([:(Seg (m -' (dim W))),N:] \/ [:(Seg (m -' (dim W))),((Seg m) \ N):]) holds
M * (i,j) = (0. (K,(m -' (dim W)),m)) * (i,j) by A10, A23, A26, Th9;
(Seg m) /\ N = N by A12, XBOOLE_1:28;
then consider MV being Matrix of dim W,m,K such that
A29: Segm (MV,(Seg (dim W)),N) = 1. (K,(dim W)) and
A30: Segm (MV,(Seg (dim W)),((Seg m) \ N)) = - ((Segm (M,(Seg (m -' (dim W))),N)) @) and
A31: Lin (lines MV) = Space_of_Solutions_of M by A7, A13, A9, A8, A16, A11, A28, A21, Th67, XBOOLE_1:36;
A32: now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_A_holds_
A_*_(i,j)_=_MV_*_(i,j)
A33: Indices A = [:(Seg (dim W)),(Seg m):] by A7, MATRIX_1:23;
let i, j be Nat; ::_thesis: ( [i,j] in Indices A implies A * (i,j) = MV * (i,j) )
assume A34: [i,j] in Indices A ; ::_thesis: A * (i,j) = MV * (i,j)
A35: i in Seg (dim W) by A34, A33, ZFMISC_1:87;
A36: Indices A = Indices MV by MATRIX_1:26;
A37: rng (Sgm (Seg (dim W))) = Seg (dim W) by FINSEQ_1:def_13;
dom (Sgm (Seg (dim W))) = Seg (dim W) by A18, FINSEQ_3:40;
then consider x being set such that
A38: x in Seg (dim W) and
A39: (Sgm (Seg (dim W))) . x = i by A35, A37, FUNCT_1:def_3;
reconsider x = x as Element of NAT by A38;
A40: j in Seg m by A34, A33, ZFMISC_1:87;
now__::_thesis:_A_*_(i,j)_=_MV_*_(i,j)
percases ( j in N or not j in N ) ;
supposeA41: j in N ; ::_thesis: A * (i,j) = MV * (i,j)
then A42: [i,j] in [:(Seg (dim W)),N:] by A35, ZFMISC_1:87;
A43: rng (Sgm N) = N by A12, FINSEQ_1:def_13;
dom (Sgm N) = Seg (dim W) by A12, A13, FINSEQ_3:40;
then consider y being set such that
A44: y in Seg (dim W) and
A45: (Sgm N) . y = j by A41, A43, FUNCT_1:def_3;
reconsider y = y as Element of NAT by A44;
A46: [:(Seg (dim W)),N:] c= Indices A by A12, A33, ZFMISC_1:95;
then A47: [x,y] in Indices (Segm (MV,(Seg (dim W)),N)) by A36, A37, A39, A43, A45, A42, MATRIX13:17;
[x,y] in Indices (Segm (A,(Seg (dim W)),N)) by A37, A39, A43, A45, A46, A42, MATRIX13:17;
hence A * (i,j) = (Segm (MV,(Seg (dim W)),N)) * (x,y) by A14, A29, A39, A45, MATRIX13:def_1
.= MV * (i,j) by A39, A45, A47, MATRIX13:def_1 ;
::_thesis: verum
end;
suppose not j in N ; ::_thesis: MV * (i,j) = A * (i,j)
then A48: j in (Seg m) \ N by A40, XBOOLE_0:def_5;
then A49: [i,j] in [:(Seg (dim W)),((Seg m) \ N):] by A35, ZFMISC_1:87;
A50: rng (Sgm ((Seg m) \ N)) = (Seg m) \ N by A25, FINSEQ_1:def_13;
dom (Sgm ((Seg m) \ N)) = Seg (m -' (dim W)) by A13, A9, A16, A11, FINSEQ_3:40, XBOOLE_1:36;
then consider y being set such that
A51: y in Seg (m -' (dim W)) and
A52: (Sgm ((Seg m) \ N)) . y = j by A48, A50, FUNCT_1:def_3;
reconsider y = y as Element of NAT by A51;
A53: [:(Seg (dim W)),((Seg m) \ N):] c= Indices A by A25, A33, ZFMISC_1:95;
then A54: [x,y] in Indices (Segm (A,(Seg (dim W)),((Seg m) \ N))) by A37, A39, A50, A52, A49, MATRIX13:17;
A55: [x,y] in Indices (Segm (MV,(Seg (dim W)),((Seg m) \ N))) by A36, A37, A39, A50, A52, A53, A49, MATRIX13:17;
then A56: [x,y] in Indices ((Segm (M,(Seg (m -' (dim W))),N)) @) by A30, Lm1;
then A57: [y,x] in Indices (Segm (M,(Seg (m -' (dim W))),N)) by MATRIX_1:def_6;
then A58: [y,x] in Indices ((Segm (A,(Seg (dim W)),((Seg m) \ N))) @) by A27, Lm1;
thus MV * (i,j) = (- ((Segm (M,(Seg (m -' (dim W))),N)) @)) * (x,y) by A30, A39, A52, A55, MATRIX13:def_1
.= - (((Segm (M,(Seg (m -' (dim W))),N)) @) * (x,y)) by A56, MATRIX_3:def_2
.= - ((- ((Segm (A,(Seg (dim W)),((Seg m) \ N))) @)) * (y,x)) by A27, A57, MATRIX_1:def_6
.= - (- (((Segm (A,(Seg (dim W)),((Seg m) \ N))) @) * (y,x))) by A58, MATRIX_3:def_2
.= ((Segm (A,(Seg (dim W)),((Seg m) \ N))) @) * (y,x) by RLVECT_1:17
.= (Segm (A,(Seg (dim W)),((Seg m) \ N))) * (x,y) by A54, MATRIX_1:def_6
.= A * (i,j) by A39, A52, A54, MATRIX13:def_1 ; ::_thesis: verum
end;
end;
end;
hence A * (i,j) = MV * (i,j) ; ::_thesis: verum
end;
then reconsider lA = lines MV as Subset of W by A15, MATRIX_1:27;
take M ; ::_thesis: ex N being finite without_zero Subset of NAT st
( card N = m -' (dim W) & N c= Seg m & Segm (M,(Seg (m -' (dim W))),N) = 1. (K,(m -' (dim W))) & W = Space_of_Solutions_of M )
take (Seg m) \ N ; ::_thesis: ( card ((Seg m) \ N) = m -' (dim W) & (Seg m) \ N c= Seg m & Segm (M,(Seg (m -' (dim W))),((Seg m) \ N)) = 1. (K,(m -' (dim W))) & W = Space_of_Solutions_of M )
MV = A by A32, MATRIX_1:27;
then Lin lA = VectSpStr(# the carrier of W, the addF of W, the ZeroF of W, the lmult of W #) by A15, VECTSP_7:def_3;
hence ( card ((Seg m) \ N) = m -' (dim W) & (Seg m) \ N c= Seg m & Segm (M,(Seg (m -' (dim W))),((Seg m) \ N)) = 1. (K,(m -' (dim W))) & W = Space_of_Solutions_of M ) by A1, A13, A16, A11, A28, A31, VECTSP_9:17, XBOOLE_1:36, XREAL_1:233; ::_thesis: verum
end;
end;
end;
theorem Th72: :: MATRIX15:72
for K being Field
for A, B being Matrix of K st width A = len B & ( width A = 0 implies len A = 0 ) & ( width B = 0 implies len B = 0 ) holds
Space_of_Solutions_of B is Subspace of Space_of_Solutions_of (A * B)
proof
let K be Field; ::_thesis: for A, B being Matrix of K st width A = len B & ( width A = 0 implies len A = 0 ) & ( width B = 0 implies len B = 0 ) holds
Space_of_Solutions_of B is Subspace of Space_of_Solutions_of (A * B)
let A, B be Matrix of K; ::_thesis: ( width A = len B & ( width A = 0 implies len A = 0 ) & ( width B = 0 implies len B = 0 ) implies Space_of_Solutions_of B is Subspace of Space_of_Solutions_of (A * B) )
assume that
A1: width A = len B and
A2: ( width A = 0 implies len A = 0 ) and
A3: ( width B = 0 implies len B = 0 ) ; ::_thesis: Space_of_Solutions_of B is Subspace of Space_of_Solutions_of (A * B)
set AB = A * B;
A4: len (A * B) = len A by A1, MATRIX_3:def_4;
A5: width (A * B) = width B by A1, MATRIX_3:def_4;
then reconsider AB = A * B as Matrix of len A, width B,K by A4, MATRIX_2:7;
the carrier of (Space_of_Solutions_of B) c= the carrier of (Space_of_Solutions_of AB)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of (Space_of_Solutions_of B) or x in the carrier of (Space_of_Solutions_of AB) )
assume x in the carrier of (Space_of_Solutions_of B) ; ::_thesis: x in the carrier of (Space_of_Solutions_of AB)
then x in Solutions_of (B,((len B) |-> (0. K))) by A3, Def5;
then consider f being FinSequence of K such that
A6: f = x and
A7: ColVec2Mx f in Solutions_of (B,(ColVec2Mx ((len B) |-> (0. K)))) ;
consider X being Matrix of K such that
A8: X = ColVec2Mx f and
A9: len X = width B and
A10: width X = width (ColVec2Mx ((len B) |-> (0. K))) and
A11: B * X = ColVec2Mx ((len B) |-> (0. K)) by A7;
A12: ColVec2Mx ((len AB) |-> (0. K)) = 0. (K,(len A),1) by A4, Th32;
A13: ColVec2Mx ((len B) |-> (0. K)) = 0. (K,(len B),1) by Th32;
now__::_thesis:_X_in_Solutions_of_(AB,(ColVec2Mx_((len_AB)_|->_(0._K))))
percases ( len A = 0 or len A <> 0 ) ;
suppose len A = 0 ; ::_thesis: X in Solutions_of (AB,(ColVec2Mx ((len AB) |-> (0. K))))
then ( Solutions_of (AB,(ColVec2Mx ((len AB) |-> (0. K)))) = {{}} & X = {} ) by A4, A5, A9, A12, Th51, MATRIX_1:def_3;
hence X in Solutions_of (AB,(ColVec2Mx ((len AB) |-> (0. K)))) by TARSKI:def_1; ::_thesis: verum
end;
supposeA14: len A <> 0 ; ::_thesis: X in Solutions_of (AB,(ColVec2Mx ((len AB) |-> (0. K))))
then A15: width (ColVec2Mx ((len AB) |-> (0. K))) = 1 by A4, Th26
.= width (ColVec2Mx ((len B) |-> (0. K))) by A1, A2, A14, Th26 ;
ColVec2Mx ((len AB) |-> (0. K)) = A * (B * X) by A1, A2, A11, A13, A12, A14, MATRIXR2:18
.= AB * X by A1, A9, MATRIX_3:33 ;
hence X in Solutions_of (AB,(ColVec2Mx ((len AB) |-> (0. K)))) by A5, A9, A10, A15; ::_thesis: verum
end;
end;
end;
then f in Solutions_of (AB,((len AB) |-> (0. K))) by A8;
hence x in the carrier of (Space_of_Solutions_of AB) by A1, A2, A3, A4, A5, A6, Def5; ::_thesis: verum
end;
hence Space_of_Solutions_of B is Subspace of Space_of_Solutions_of (A * B) by A5, VECTSP_4:27; ::_thesis: verum
end;
theorem :: MATRIX15:73
for K being Field
for A, B being Matrix of K st width A = len B holds
( the_rank_of (A * B) <= the_rank_of A & the_rank_of (A * B) <= the_rank_of B )
proof
let K be Field; ::_thesis: for A, B being Matrix of K st width A = len B holds
( the_rank_of (A * B) <= the_rank_of A & the_rank_of (A * B) <= the_rank_of B )
let A, B be Matrix of K; ::_thesis: ( width A = len B implies ( the_rank_of (A * B) <= the_rank_of A & the_rank_of (A * B) <= the_rank_of B ) )
assume A1: width A = len B ; ::_thesis: ( the_rank_of (A * B) <= the_rank_of A & the_rank_of (A * B) <= the_rank_of B )
set AB = A * B;
A2: width (A * B) = width B by A1, MATRIX_3:def_4;
percases ( the_rank_of (A * B) = 0 or the_rank_of (A * B) > 0 ) ;
suppose the_rank_of (A * B) = 0 ; ::_thesis: ( the_rank_of (A * B) <= the_rank_of A & the_rank_of (A * B) <= the_rank_of B )
hence ( the_rank_of (A * B) <= the_rank_of A & the_rank_of (A * B) <= the_rank_of B ) ; ::_thesis: verum
end;
supposeA3: the_rank_of (A * B) > 0 ; ::_thesis: ( the_rank_of (A * B) <= the_rank_of A & the_rank_of (A * B) <= the_rank_of B )
set AT = A @ ;
A4: width (A * B) > 0 by A3, MATRIX13:74;
then A5: width A > 0 by A1, A2, MATRIX_1:def_3;
then A6: len (A @) = width A by MATRIX_2:10;
set BT = B @ ;
set BA = (B @) * (A @);
width (A @) = len A by A5, MATRIX_2:10;
then A7: ( width (A @) = 0 implies len (A @) = 0 ) by A5, MATRIX_1:def_3;
then A8: dim (Space_of_Solutions_of (A @)) = (width (A @)) - (the_rank_of (A @)) by Th68;
A9: width (B @) = len B by A2, A4, MATRIX_2:10;
then ( width (B @) = 0 implies len (B @) = 0 ) by A2, A4, MATRIX_1:def_3;
then A10: Space_of_Solutions_of (A @) is Subspace of Space_of_Solutions_of ((B @) * (A @)) by A1, A6, A9, A7, Th72;
A11: width ((B @) * (A @)) = width (A @) by A1, A6, A9, MATRIX_3:def_4;
then dim (Space_of_Solutions_of ((B @) * (A @))) = (width ((B @) * (A @))) - (the_rank_of ((B @) * (A @))) by A5, A7, Th68, MATRIX_2:10;
then (width (A @)) - (the_rank_of (A @)) <= (width (A @)) - (the_rank_of ((B @) * (A @))) by A11, A10, A8, VECTSP_9:25;
then the_rank_of (A @) >= the_rank_of ((B @) * (A @)) by XREAL_1:10;
then A12: the_rank_of A >= the_rank_of ((B @) * (A @)) by MATRIX13:84;
( width A = 0 implies len A = 0 ) by A1, A2, A4, MATRIX_1:def_3;
then A13: Space_of_Solutions_of B is Subspace of Space_of_Solutions_of (A * B) by A1, A2, A4, Th72;
( dim (Space_of_Solutions_of B) = (width B) - (the_rank_of B) & dim (Space_of_Solutions_of (A * B)) = (width (A * B)) - (the_rank_of (A * B)) ) by A2, A4, Th68;
then A14: (width B) - (the_rank_of B) <= (width B) - (the_rank_of (A * B)) by A2, A13, VECTSP_9:25;
(B @) * (A @) = (A * B) @ by A1, A2, A4, MATRIX_3:22;
hence ( the_rank_of (A * B) <= the_rank_of A & the_rank_of (A * B) <= the_rank_of B ) by A14, A12, MATRIX13:84, XREAL_1:10; ::_thesis: verum
end;
end;
end;
theorem Th74: :: MATRIX15:74
for n being Nat
for K being Field
for A being Matrix of n,n,K
for B being Matrix of K st Det A <> 0. K & width A = len B & ( width B = 0 implies len B = 0 ) holds
Space_of_Solutions_of B = Space_of_Solutions_of (A * B)
proof
let n be Nat; ::_thesis: for K being Field
for A being Matrix of n,n,K
for B being Matrix of K st Det A <> 0. K & width A = len B & ( width B = 0 implies len B = 0 ) holds
Space_of_Solutions_of B = Space_of_Solutions_of (A * B)
let K be Field; ::_thesis: for A being Matrix of n,n,K
for B being Matrix of K st Det A <> 0. K & width A = len B & ( width B = 0 implies len B = 0 ) holds
Space_of_Solutions_of B = Space_of_Solutions_of (A * B)
let A be Matrix of n,n,K; ::_thesis: for B being Matrix of K st Det A <> 0. K & width A = len B & ( width B = 0 implies len B = 0 ) holds
Space_of_Solutions_of B = Space_of_Solutions_of (A * B)
let B be Matrix of K; ::_thesis: ( Det A <> 0. K & width A = len B & ( width B = 0 implies len B = 0 ) implies Space_of_Solutions_of B = Space_of_Solutions_of (A * B) )
assume that
A1: Det A <> 0. K and
A2: width A = len B and
A3: ( width B = 0 implies len B = 0 ) ; ::_thesis: Space_of_Solutions_of B = Space_of_Solutions_of (A * B)
set AB = A * B;
A4: len (A * B) = len A by A2, MATRIX_3:def_4;
A5: width (A * B) = width B by A2, MATRIX_3:def_4;
A6: len A = n by MATRIX_1:24;
reconsider AB = A * B as Matrix of n, width B,K by A4, A5, MATRIX_1:24, MATRIX_2:7;
A7: width A = n by MATRIX_1:24;
A8: the carrier of (Space_of_Solutions_of AB) c= the carrier of (Space_of_Solutions_of B)
proof
A is invertible by A1, LAPLACE:34;
then A is_reverse_of A ~ by MATRIX_6:def_4;
then A9: 1. (K,n) = (A ~) * A by MATRIX_6:def_2;
A10: len (A ~) = n by MATRIX_1:24;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of (Space_of_Solutions_of AB) or x in the carrier of (Space_of_Solutions_of B) )
assume x in the carrier of (Space_of_Solutions_of AB) ; ::_thesis: x in the carrier of (Space_of_Solutions_of B)
then x in Solutions_of (AB,((len AB) |-> (0. K))) by A2, A3, A6, A7, A4, A5, Def5;
then consider f being FinSequence of K such that
A11: f = x and
A12: ColVec2Mx f in Solutions_of (AB,(ColVec2Mx ((len AB) |-> (0. K)))) ;
consider X being Matrix of K such that
A13: X = ColVec2Mx f and
A14: len X = width AB and
A15: width X = width (ColVec2Mx ((len AB) |-> (0. K))) and
A16: AB * X = ColVec2Mx ((len AB) |-> (0. K)) by A12;
A17: width (A ~) = n by MATRIX_1:24;
set BX = B * X;
A18: len (B * X) = len B by A5, A14, MATRIX_3:def_4;
then A19: B * X = (1. (K,n)) * (B * X) by A2, A7, MATRIXR2:68
.= (A ~) * (A * (B * X)) by A2, A6, A9, A17, A18, MATRIX_3:33
.= (A ~) * (ColVec2Mx ((len AB) |-> (0. K))) by A2, A5, A14, A16, MATRIX_3:33
.= (A ~) * (0. (K,(len AB),1)) by Th32 ;
now__::_thesis:_B_*_X_=_0._(K,(len_AB),1)
percases ( n = 0 or n > 0 ) ;
supposeA20: n = 0 ; ::_thesis: B * X = 0. (K,(len AB),1)
then 0. (K,(len AB),1) = {} by A6, A4;
hence B * X = 0. (K,(len AB),1) by A2, A18, A20, MATRIX_1:24; ::_thesis: verum
end;
suppose n > 0 ; ::_thesis: B * X = 0. (K,(len AB),1)
hence B * X = 0. (K,(len AB),1) by A6, A4, A10, A17, A19, MATRIXR2:18; ::_thesis: verum
end;
end;
end;
then B * X = ColVec2Mx ((len AB) |-> (0. K)) by Th32;
then ColVec2Mx f in Solutions_of (B,(ColVec2Mx ((len B) |-> (0. K)))) by A2, A6, A7, A4, A5, A13, A14, A15;
then f in Solutions_of (B,((len B) |-> (0. K))) ;
hence x in the carrier of (Space_of_Solutions_of B) by A3, A11, Def5; ::_thesis: verum
end;
( width A = 0 implies len A = 0 ) by A6, MATRIX_1:24;
then Space_of_Solutions_of B is Subspace of Space_of_Solutions_of (A * B) by A2, A3, Th72;
then the carrier of (Space_of_Solutions_of B) c= the carrier of (Space_of_Solutions_of (A * B)) by VECTSP_4:def_2;
then the carrier of (Space_of_Solutions_of B) = the carrier of (Space_of_Solutions_of (A * B)) by A8, XBOOLE_0:def_10;
hence Space_of_Solutions_of B = Space_of_Solutions_of (A * B) by A5, VECTSP_4:29; ::_thesis: verum
end;
theorem Th75: :: MATRIX15:75
for n being Nat
for K being Field
for A being Matrix of n,n,K
for B being Matrix of K st width A = len B & Det A <> 0. K holds
the_rank_of (A * B) = the_rank_of B
proof
let n be Nat; ::_thesis: for K being Field
for A being Matrix of n,n,K
for B being Matrix of K st width A = len B & Det A <> 0. K holds
the_rank_of (A * B) = the_rank_of B
let K be Field; ::_thesis: for A being Matrix of n,n,K
for B being Matrix of K st width A = len B & Det A <> 0. K holds
the_rank_of (A * B) = the_rank_of B
let A be Matrix of n,n,K; ::_thesis: for B being Matrix of K st width A = len B & Det A <> 0. K holds
the_rank_of (A * B) = the_rank_of B
let B be Matrix of K; ::_thesis: ( width A = len B & Det A <> 0. K implies the_rank_of (A * B) = the_rank_of B )
assume that
A1: width A = len B and
A2: Det A <> 0. K ; ::_thesis: the_rank_of (A * B) = the_rank_of B
set AB = A * B;
A3: len (A * B) = len A by A1, MATRIX_3:def_4;
A4: ( len A = n & width A = n ) by MATRIX_1:24;
A5: width (A * B) = width B by A1, MATRIX_3:def_4;
percases ( width (A * B) = 0 or width (A * B) > 0 ) ;
suppose width (A * B) = 0 ; ::_thesis: the_rank_of (A * B) = the_rank_of B
hence the_rank_of (A * B) = the_rank_of B by A1, A3, A5, A4, Lm3; ::_thesis: verum
end;
supposeA6: width (A * B) > 0 ; ::_thesis: the_rank_of (A * B) = the_rank_of B
then ( Space_of_Solutions_of B = Space_of_Solutions_of (A * B) & dim (Space_of_Solutions_of B) = (width B) - (the_rank_of B) ) by A1, A2, A5, Th68, Th74;
then (width B) - (the_rank_of B) = (width B) - (the_rank_of (A * B)) by A5, A6, Th68;
hence the_rank_of (A * B) = the_rank_of B ; ::_thesis: verum
end;
end;
end;
theorem :: MATRIX15:76
for n being Nat
for K being Field
for A being Matrix of n,n,K
for B being Matrix of K st len A = width B & Det A <> 0. K holds
the_rank_of (B * A) = the_rank_of B
proof
let n be Nat; ::_thesis: for K being Field
for A being Matrix of n,n,K
for B being Matrix of K st len A = width B & Det A <> 0. K holds
the_rank_of (B * A) = the_rank_of B
let K be Field; ::_thesis: for A being Matrix of n,n,K
for B being Matrix of K st len A = width B & Det A <> 0. K holds
the_rank_of (B * A) = the_rank_of B
let A be Matrix of n,n,K; ::_thesis: for B being Matrix of K st len A = width B & Det A <> 0. K holds
the_rank_of (B * A) = the_rank_of B
let B be Matrix of K; ::_thesis: ( len A = width B & Det A <> 0. K implies the_rank_of (B * A) = the_rank_of B )
assume that
A1: width B = len A and
A2: Det A <> 0. K ; ::_thesis: the_rank_of (B * A) = the_rank_of B
set BA = B * A;
A3: len (B * A) = len B by A1, MATRIX_3:def_4;
A4: width (B * A) = width A by A1, MATRIX_3:def_4;
A5: ( len A = n & width A = n ) by MATRIX_1:24;
percases ( width (B * A) = 0 or width (B * A) > 0 ) ;
suppose width (B * A) = 0 ; ::_thesis: the_rank_of (B * A) = the_rank_of B
hence the_rank_of (B * A) = the_rank_of B by A1, A3, A4, A5, Lm3; ::_thesis: verum
end;
supposeA6: width (B * A) > 0 ; ::_thesis: the_rank_of (B * A) = the_rank_of B
then A7: ( width (A @) = len A & len (B @) = width B ) by A1, A4, A5, MATRIX_2:10;
A8: Det (A @) <> 0. K by A2, MATRIXR2:43;
thus the_rank_of (B * A) = the_rank_of ((B * A) @) by MATRIX13:84
.= the_rank_of ((A @) * (B @)) by A1, A4, A6, MATRIX_3:22
.= the_rank_of (B @) by A1, A8, A7, Th75
.= the_rank_of B by MATRIX13:84 ; ::_thesis: verum
end;
end;
end;