:: POLYNOM2 semantic presentation
begin
Lm1: for X being set
for A being Subset of X
for O being Order of X holds
( O is_reflexive_in A & O is_antisymmetric_in A & O is_transitive_in A )
proof
let X be set ; ::_thesis: for A being Subset of X
for O being Order of X holds
( O is_reflexive_in A & O is_antisymmetric_in A & O is_transitive_in A )
let A be Subset of X; ::_thesis: for O being Order of X holds
( O is_reflexive_in A & O is_antisymmetric_in A & O is_transitive_in A )
let O be Order of X; ::_thesis: ( O is_reflexive_in A & O is_antisymmetric_in A & O is_transitive_in A )
A1: field O = X by ORDERS_1:12;
then O is_antisymmetric_in X by RELAT_2:def_12;
then A2: for x, y being set st x in A & y in A & [x,y] in O & [y,x] in O holds
x = y by RELAT_2:def_4;
O is_transitive_in X by A1, RELAT_2:def_16;
then A3: for x, y, z being set st x in A & y in A & z in A & [x,y] in O & [y,z] in O holds
[x,z] in O by RELAT_2:def_8;
O is_reflexive_in X by A1, RELAT_2:def_9;
then for x being set st x in A holds
[x,x] in O by RELAT_2:def_1;
hence ( O is_reflexive_in A & O is_antisymmetric_in A & O is_transitive_in A ) by A2, A3, RELAT_2:def_1, RELAT_2:def_4, RELAT_2:def_8; ::_thesis: verum
end;
Lm2: for X being set
for A being Subset of X
for O being Order of X st O is_connected_in X holds
O is_connected_in A
proof
let X be set ; ::_thesis: for A being Subset of X
for O being Order of X st O is_connected_in X holds
O is_connected_in A
let A be Subset of X; ::_thesis: for O being Order of X st O is_connected_in X holds
O is_connected_in A
let O be Order of X; ::_thesis: ( O is_connected_in X implies O is_connected_in A )
assume O is_connected_in X ; ::_thesis: O is_connected_in A
then for x, y being set st x in A & y in A & x <> y & not [x,y] in O holds
[y,x] in O by RELAT_2:def_6;
hence O is_connected_in A by RELAT_2:def_6; ::_thesis: verum
end;
Lm3: for X being set
for S being Subset of X
for R being Order of X st R is being_linear-order holds
R linearly_orders S
proof
let X be set ; ::_thesis: for S being Subset of X
for R being Order of X st R is being_linear-order holds
R linearly_orders S
let S be Subset of X; ::_thesis: for R being Order of X st R is being_linear-order holds
R linearly_orders S
let R be Order of X; ::_thesis: ( R is being_linear-order implies R linearly_orders S )
A1: field R = X by ORDERS_1:12;
then A2: R is_reflexive_in X by RELAT_2:def_9;
R is_transitive_in X by A1, RELAT_2:def_16;
then for x, y, z being set st x in S & y in S & z in S & [x,y] in R & [y,z] in R holds
[x,z] in R by RELAT_2:def_8;
then A3: R is_transitive_in S by RELAT_2:def_8;
R is_antisymmetric_in X by A1, RELAT_2:def_12;
then for x, y being set st x in S & y in S & [x,y] in R & [y,x] in R holds
x = y by RELAT_2:def_4;
then A4: R is_antisymmetric_in S by RELAT_2:def_4;
assume R is being_linear-order ; ::_thesis: R linearly_orders S
then R is connected by ORDERS_1:def_5;
then A5: R is_connected_in field R by RELAT_2:def_14;
for x, y being set st x in S & y in S & x <> y & not [x,y] in R holds
[y,x] in R
proof
let x, y be set ; ::_thesis: ( x in S & y in S & x <> y & not [x,y] in R implies [y,x] in R )
assume that
A6: x in S and
A7: y in S and
A8: x <> y ; ::_thesis: ( [x,y] in R or [y,x] in R )
A9: field R = (dom R) \/ (rng R) by RELAT_1:def_6;
[y,y] in R by A2, A7, RELAT_2:def_1;
then y in dom R by XTUPLE_0:def_12;
then A10: y in field R by A9, XBOOLE_0:def_3;
[x,x] in R by A2, A6, RELAT_2:def_1;
then x in dom R by XTUPLE_0:def_12;
then x in field R by A9, XBOOLE_0:def_3;
hence ( [x,y] in R or [y,x] in R ) by A5, A8, A10, RELAT_2:def_6; ::_thesis: verum
end;
then A11: R is_connected_in S by RELAT_2:def_6;
for x being set st x in S holds
[x,x] in R by A2, RELAT_2:def_1;
then R is_reflexive_in S by RELAT_2:def_1;
hence R linearly_orders S by A4, A3, A11, ORDERS_1:def_8; ::_thesis: verum
end;
theorem Th1: :: POLYNOM2:1
for L being non empty unital associative multMagma
for a being Element of L
for n, m being Element of NAT holds (power L) . (a,(n + m)) = ((power L) . (a,n)) * ((power L) . (a,m))
proof
let L be non empty unital associative multMagma ; ::_thesis: for a being Element of L
for n, m being Element of NAT holds (power L) . (a,(n + m)) = ((power L) . (a,n)) * ((power L) . (a,m))
let a be Element of L; ::_thesis: for n, m being Element of NAT holds (power L) . (a,(n + m)) = ((power L) . (a,n)) * ((power L) . (a,m))
let n, m be Element of NAT ; ::_thesis: (power L) . (a,(n + m)) = ((power L) . (a,n)) * ((power L) . (a,m))
defpred S1[ Element of NAT ] means (power L) . (a,(n + $1)) = ((power L) . (a,n)) * ((power L) . (a,$1));
A1: now__::_thesis:_for_m_being_Element_of_NAT_st_S1[m]_holds_
S1[m_+_1]
let m be Element of NAT ; ::_thesis: ( S1[m] implies S1[m + 1] )
assume A2: S1[m] ; ::_thesis: S1[m + 1]
(power L) . (a,(n + (m + 1))) = (power L) . (a,((n + m) + 1))
.= (((power L) . (a,n)) * ((power L) . (a,m))) * a by A2, GROUP_1:def_7
.= ((power L) . (a,n)) * (((power L) . (a,m)) * a) by GROUP_1:def_3
.= ((power L) . (a,n)) * ((power L) . (a,(m + 1))) by GROUP_1:def_7 ;
hence S1[m + 1] ; ::_thesis: verum
end;
(power L) . (a,(n + 0)) = ((power L) . (a,n)) * (1_ L) by GROUP_1:def_4
.= ((power L) . (a,n)) * ((power L) . (a,0)) by GROUP_1:def_7 ;
then A3: S1[ 0 ] ;
for m being Element of NAT holds S1[m] from NAT_1:sch_1(A3, A1);
hence (power L) . (a,(n + m)) = ((power L) . (a,n)) * ((power L) . (a,m)) ; ::_thesis: verum
end;
registration
cluster non empty non trivial right_complementable associative commutative Abelian add-associative right_zeroed well-unital distributive for doubleLoopStr ;
existence
ex b1 being non trivial doubleLoopStr st
( b1 is Abelian & b1 is right_zeroed & b1 is add-associative & b1 is right_complementable & b1 is well-unital & b1 is distributive & b1 is commutative & b1 is associative )
proof
take F_Real ; ::_thesis: ( F_Real is Abelian & F_Real is right_zeroed & F_Real is add-associative & F_Real is right_complementable & F_Real is well-unital & F_Real is distributive & F_Real is commutative & F_Real is associative )
thus ( F_Real is Abelian & F_Real is right_zeroed & F_Real is add-associative & F_Real is right_complementable & F_Real is well-unital & F_Real is distributive & F_Real is commutative & F_Real is associative ) ; ::_thesis: verum
end;
end;
begin
theorem Th2: :: POLYNOM2:2
for p being FinSequence
for k being Element of NAT st k in dom p holds
for i being Element of NAT st 1 <= i & i <= k holds
i in dom p
proof
let p be FinSequence; ::_thesis: for k being Element of NAT st k in dom p holds
for i being Element of NAT st 1 <= i & i <= k holds
i in dom p
let k be Element of NAT ; ::_thesis: ( k in dom p implies for i being Element of NAT st 1 <= i & i <= k holds
i in dom p )
assume A1: k in dom p ; ::_thesis: for i being Element of NAT st 1 <= i & i <= k holds
i in dom p
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= k implies i in dom p )
assume that
A2: 1 <= i and
A3: i <= k ; ::_thesis: i in dom p
consider l being Nat such that
A4: dom p = Seg l by FINSEQ_1:def_2;
k <= l by A1, A4, FINSEQ_1:1;
then i <= l by A3, XXREAL_0:2;
hence i in dom p by A2, A4, FINSEQ_1:1; ::_thesis: verum
end;
Lm4: for X being set
for A being finite Subset of X
for a being Element of X
for R being Order of X st R linearly_orders {a} \/ A holds
R linearly_orders A
proof
let X be set ; ::_thesis: for A being finite Subset of X
for a being Element of X
for R being Order of X st R linearly_orders {a} \/ A holds
R linearly_orders A
let A be finite Subset of X; ::_thesis: for a being Element of X
for R being Order of X st R linearly_orders {a} \/ A holds
R linearly_orders A
let a be Element of X; ::_thesis: for R being Order of X st R linearly_orders {a} \/ A holds
R linearly_orders A
let R be Order of X; ::_thesis: ( R linearly_orders {a} \/ A implies R linearly_orders A )
for x being set st x in A holds
x in {a} \/ A by XBOOLE_0:def_3;
then A1: A c= {a} \/ A by TARSKI:def_3;
assume R linearly_orders {a} \/ A ; ::_thesis: R linearly_orders A
hence R linearly_orders A by A1, ORDERS_1:38; ::_thesis: verum
end;
theorem Th3: :: POLYNOM2:3
for L being non empty right_zeroed left_zeroed addLoopStr
for p being FinSequence of the carrier of L
for i being Element of NAT st i in dom p & ( for i9 being Element of NAT st i9 in dom p & i9 <> i holds
p /. i9 = 0. L ) holds
Sum p = p /. i
proof
let L be non empty right_zeroed left_zeroed addLoopStr ; ::_thesis: for p being FinSequence of the carrier of L
for i being Element of NAT st i in dom p & ( for i9 being Element of NAT st i9 in dom p & i9 <> i holds
p /. i9 = 0. L ) holds
Sum p = p /. i
let p be FinSequence of the carrier of L; ::_thesis: for i being Element of NAT st i in dom p & ( for i9 being Element of NAT st i9 in dom p & i9 <> i holds
p /. i9 = 0. L ) holds
Sum p = p /. i
let i be Element of NAT ; ::_thesis: ( i in dom p & ( for i9 being Element of NAT st i9 in dom p & i9 <> i holds
p /. i9 = 0. L ) implies Sum p = p /. i )
assume that
A1: i in dom p and
A2: for i9 being Element of NAT st i9 in dom p & i9 <> i holds
p /. i9 = 0. L ; ::_thesis: Sum p = p /. i
consider fp being Function of NAT, the carrier of L such that
A3: Sum p = fp . (len p) and
A4: fp . 0 = 0. L and
A5: for j being Element of NAT
for v being Element of L st j < len p & v = p . (j + 1) holds
fp . (j + 1) = (fp . j) + v by RLVECT_1:def_12;
defpred S1[ Element of NAT ] means ( ( $1 < i & fp . $1 = 0. L ) or ( i <= $1 & fp . $1 = p /. i ) );
consider l being Nat such that
A6: dom p = Seg l by FINSEQ_1:def_2;
reconsider l = l as Element of NAT by ORDINAL1:def_12;
A7: len p = l by A6, FINSEQ_1:def_3;
i in { z where z is Element of NAT : ( 1 <= z & z <= l ) } by A1, A6, FINSEQ_1:def_1;
then A8: ex i9 being Element of NAT st
( i9 = i & 1 <= i9 & i9 <= l ) ;
A9: for j being Element of NAT st 0 <= j & j < len p & S1[j] holds
S1[j + 1]
proof
let j be Element of NAT ; ::_thesis: ( 0 <= j & j < len p & S1[j] implies S1[j + 1] )
assume that
0 <= j and
A10: j < len p ; ::_thesis: ( not S1[j] or S1[j + 1] )
assume A11: S1[j] ; ::_thesis: S1[j + 1]
percases ( j < i or i <= j ) ;
supposeA12: j < i ; ::_thesis: S1[j + 1]
now__::_thesis:_(_(_j_+_1_=_i_&_S1[j_+_1]_)_or_(_j_+_1_<>_i_&_S1[j_+_1]_)_)
percases ( j + 1 = i or j + 1 <> i ) ;
caseA13: j + 1 = i ; ::_thesis: S1[j + 1]
then p . (j + 1) = p /. (j + 1) by A1, PARTFUN1:def_6;
then fp . (j + 1) = (0. L) + (p /. (j + 1)) by A5, A10, A11, A12
.= p /. (j + 1) by ALGSTR_1:def_2 ;
hence S1[j + 1] by A13; ::_thesis: verum
end;
caseA14: j + 1 <> i ; ::_thesis: S1[j + 1]
A15: j + 1 < i
proof
assume i <= j + 1 ; ::_thesis: contradiction
then i < j + 1 by A14, XXREAL_0:1;
hence contradiction by A12, NAT_1:13; ::_thesis: verum
end;
j + 1 <= i by A12, NAT_1:13;
then A16: j + 1 <= l by A8, XXREAL_0:2;
0 + 1 <= j + 1 by XREAL_1:6;
then A17: j + 1 in Seg l by A16, FINSEQ_1:1;
then p . (j + 1) = p /. (j + 1) by A6, PARTFUN1:def_6;
then fp . (j + 1) = (0. L) + (p /. (j + 1)) by A5, A10, A11, A12
.= p /. (j + 1) by ALGSTR_1:def_2
.= 0. L by A2, A6, A14, A17 ;
hence S1[j + 1] by A15; ::_thesis: verum
end;
end;
end;
hence S1[j + 1] ; ::_thesis: verum
end;
supposeA18: i <= j ; ::_thesis: S1[j + 1]
j < l by A6, A10, FINSEQ_1:def_3;
then A19: j + 1 <= l by NAT_1:13;
A20: i < j + 1 by A18, NAT_1:13;
0 + 1 <= j + 1 by XREAL_1:6;
then A21: j + 1 in dom p by A6, A19, FINSEQ_1:1;
then p . (j + 1) = p /. (j + 1) by PARTFUN1:def_6;
then fp . (j + 1) = (p /. i) + (p /. (j + 1)) by A5, A10, A11, A18
.= (p /. i) + (0. L) by A2, A20, A21
.= p /. i by RLVECT_1:def_4 ;
hence S1[j + 1] by A18, NAT_1:13; ::_thesis: verum
end;
end;
end;
A22: S1[ 0 ] by A4, A8;
for j being Element of NAT st 0 <= j & j <= len p holds
S1[j] from INT_1:sch_7(A22, A9);
hence Sum p = p /. i by A3, A8, A7; ::_thesis: verum
end;
theorem :: POLYNOM2:4
for L being non empty right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr
for p being FinSequence of the carrier of L st ex i being Element of NAT st
( i in dom p & p /. i = 0. L ) holds
Product p = 0. L
proof
let L be non empty right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr ; ::_thesis: for p being FinSequence of the carrier of L st ex i being Element of NAT st
( i in dom p & p /. i = 0. L ) holds
Product p = 0. L
let p be FinSequence of the carrier of L; ::_thesis: ( ex i being Element of NAT st
( i in dom p & p /. i = 0. L ) implies Product p = 0. L )
given i being Element of NAT such that A1: i in dom p and
A2: p /. i = 0. L ; ::_thesis: Product p = 0. L
defpred S1[ Element of NAT ] means for p being FinSequence of the carrier of L st len p = $1 & ex i being Element of NAT st
( i in dom p & p /. i = 0. L ) holds
Product p = 0. L;
A3: for j being Element of NAT st S1[j] holds
S1[j + 1]
proof
let j be Element of NAT ; ::_thesis: ( S1[j] implies S1[j + 1] )
assume A4: S1[j] ; ::_thesis: S1[j + 1]
for p being FinSequence of the carrier of L st len p = j + 1 & ex i being Element of NAT st
( i in dom p & p /. i = 0. L ) holds
Product p = 0. L
proof
let p be FinSequence of the carrier of L; ::_thesis: ( len p = j + 1 & ex i being Element of NAT st
( i in dom p & p /. i = 0. L ) implies Product p = 0. L )
assume that
A5: len p = j + 1 and
A6: ex i being Element of NAT st
( i in dom p & p /. i = 0. L ) ; ::_thesis: Product p = 0. L
A7: ex fp being Function of NAT, the carrier of L st
( fp . 1 = p . 1 & ( for i being Element of NAT st 0 <> i & i < len p holds
fp . (i + 1) = the multF of L . ((fp . i),(p . (i + 1))) ) & the multF of L "**" p = fp . (len p) ) by A5, FINSOP_1:1, NAT_1:14;
A8: len p >= 1 by A5, NAT_1:14;
then A9: 1 in dom p by FINSEQ_3:25;
A10: dom p = Seg (len p) by FINSEQ_1:def_3;
then A11: j + 1 in dom p by A5, A8, FINSEQ_1:1;
percases ( j = 0 or j <> 0 ) ;
supposeA12: j = 0 ; ::_thesis: Product p = 0. L
then A13: dom p = {1} by A5, FINSEQ_1:2, FINSEQ_1:def_3;
Product p = p . 1 by A5, A7, A12, GROUP_4:def_2
.= p /. 1 by A9, PARTFUN1:def_6 ;
hence Product p = 0. L by A6, A13, TARSKI:def_1; ::_thesis: verum
end;
suppose j <> 0 ; ::_thesis: Product p = 0. L
then A14: 1 <= j by NAT_1:14;
reconsider p9 = p | (Seg j) as FinSequence of the carrier of L by FINSEQ_1:18;
A15: j <= j + 1 by XREAL_1:29;
then A16: dom p9 = Seg j by A5, FINSEQ_1:17;
p = p9 ^ <*(p . (len p))*> by A5, FINSEQ_3:55;
then A17: p = p9 ^ <*(p /. (len p))*> by A5, A11, PARTFUN1:def_6;
A18: len p9 = j by A5, A15, FINSEQ_1:17;
now__::_thesis:_(_(_p_/._(len_p)_=_0._L_&_Product_p_=_0._L_)_or_(_p_/._(len_p)_<>_0._L_&_Product_p_=_0._L_)_)
percases ( p /. (len p) = 0. L or p /. (len p) <> 0. L ) ;
case p /. (len p) = 0. L ; ::_thesis: Product p = 0. L
hence Product p = (Product p9) * (0. L) by A17, GROUP_4:6
.= 0. L by VECTSP_1:6 ;
::_thesis: verum
end;
caseA19: p /. (len p) <> 0. L ; ::_thesis: Product p = 0. L
consider i being Element of NAT such that
A20: i in dom p and
A21: p /. i = 0. L by A6;
i <= len p by A10, A20, FINSEQ_1:1;
then i < len p by A19, A21, XXREAL_0:1;
then A22: i <= j by A5, NAT_1:13;
A23: 1 <= i by A10, A20, FINSEQ_1:1;
then A24: i in dom p9 by A16, A22, FINSEQ_1:1;
A25: j in dom p by A5, A10, A14, A15, FINSEQ_1:1;
i in Seg j by A23, A22, FINSEQ_1:1;
then (p | j) . i = p . i by A25, RFINSEQ:6;
then p9 . i = p . i by FINSEQ_1:def_15;
then p9 /. i = p . i by A24, PARTFUN1:def_6;
then A26: p9 /. i = 0. L by A20, A21, PARTFUN1:def_6;
thus Product p = (Product p9) * (p /. (len p)) by A17, GROUP_4:6
.= (0. L) * (p /. (len p)) by A4, A18, A24, A26
.= 0. L by VECTSP_1:7 ; ::_thesis: verum
end;
end;
end;
hence Product p = 0. L ; ::_thesis: verum
end;
end;
end;
hence S1[j + 1] ; ::_thesis: verum
end;
A27: ex l being Element of NAT st l = len p ;
A28: S1[ 0 ]
proof
let p be FinSequence of L; ::_thesis: ( len p = 0 & ex i being Element of NAT st
( i in dom p & p /. i = 0. L ) implies Product p = 0. L )
assume len p = 0 ; ::_thesis: ( for i being Element of NAT holds
( not i in dom p or not p /. i = 0. L ) or Product p = 0. L )
then p = {} ;
hence ( for i being Element of NAT holds
( not i in dom p or not p /. i = 0. L ) or Product p = 0. L ) ; ::_thesis: verum
end;
for j being Element of NAT holds S1[j] from NAT_1:sch_1(A28, A3);
hence Product p = 0. L by A1, A2, A27; ::_thesis: verum
end;
theorem Th5: :: POLYNOM2:5
for L being non empty Abelian add-associative addLoopStr
for a being Element of L
for p, q being FinSequence of the carrier of L st len p = len q & ex i being Element of NAT st
( i in dom p & q /. i = a + (p /. i) & ( for i9 being Element of NAT st i9 in dom p & i9 <> i holds
q /. i9 = p /. i9 ) ) holds
Sum q = a + (Sum p)
proof
let L be non empty Abelian add-associative addLoopStr ; ::_thesis: for a being Element of L
for p, q being FinSequence of the carrier of L st len p = len q & ex i being Element of NAT st
( i in dom p & q /. i = a + (p /. i) & ( for i9 being Element of NAT st i9 in dom p & i9 <> i holds
q /. i9 = p /. i9 ) ) holds
Sum q = a + (Sum p)
let a be Element of L; ::_thesis: for p, q being FinSequence of the carrier of L st len p = len q & ex i being Element of NAT st
( i in dom p & q /. i = a + (p /. i) & ( for i9 being Element of NAT st i9 in dom p & i9 <> i holds
q /. i9 = p /. i9 ) ) holds
Sum q = a + (Sum p)
let p, q be FinSequence of the carrier of L; ::_thesis: ( len p = len q & ex i being Element of NAT st
( i in dom p & q /. i = a + (p /. i) & ( for i9 being Element of NAT st i9 in dom p & i9 <> i holds
q /. i9 = p /. i9 ) ) implies Sum q = a + (Sum p) )
assume that
A1: len p = len q and
A2: ex i being Element of NAT st
( i in dom p & q /. i = a + (p /. i) & ( for i9 being Element of NAT st i9 in dom p & i9 <> i holds
q /. i9 = p /. i9 ) ) ; ::_thesis: Sum q = a + (Sum p)
consider i being Element of NAT such that
A3: i in dom p and
A4: q /. i = a + (p /. i) and
A5: for i9 being Element of NAT st i9 in dom p & i9 <> i holds
q /. i9 = p /. i9 by A2;
consider fq being Function of NAT, the carrier of L such that
A6: Sum q = fq . (len q) and
A7: fq . 0 = 0. L and
A8: for j being Element of NAT
for v being Element of L st j < len q & v = q . (j + 1) holds
fq . (j + 1) = (fq . j) + v by RLVECT_1:def_12;
consider l being Nat such that
A9: dom p = Seg l by FINSEQ_1:def_2;
i in { z where z is Element of NAT : ( 1 <= z & z <= l ) } by A3, A9, FINSEQ_1:def_1;
then A10: ex i9 being Element of NAT st
( i9 = i & 1 <= i9 & i9 <= l ) ;
consider l9 being Nat such that
A11: dom q = Seg l9 by FINSEQ_1:def_2;
reconsider l = l, l9 = l9 as Element of NAT by ORDINAL1:def_12;
consider fp being Function of NAT, the carrier of L such that
A12: Sum p = fp . (len p) and
A13: fp . 0 = 0. L and
A14: for j being Element of NAT
for v being Element of L st j < len p & v = p . (j + 1) holds
fp . (j + 1) = (fp . j) + v by RLVECT_1:def_12;
A15: len p = l by A9, FINSEQ_1:def_3;
defpred S1[ Element of NAT ] means ( ( $1 < i & fp . $1 = fq . $1 ) or ( i <= $1 & fq . $1 = a + (fp . $1) ) );
A16: l = len p by A9, FINSEQ_1:def_3
.= l9 by A1, A11, FINSEQ_1:def_3 ;
A17: for j being Element of NAT st 0 <= j & j < len p & S1[j] holds
S1[j + 1]
proof
let j be Element of NAT ; ::_thesis: ( 0 <= j & j < len p & S1[j] implies S1[j + 1] )
assume that
0 <= j and
A18: j < len p ; ::_thesis: ( not S1[j] or S1[j + 1] )
assume A19: S1[j] ; ::_thesis: S1[j + 1]
percases ( j < i or i <= j ) ;
supposeA20: j < i ; ::_thesis: S1[j + 1]
now__::_thesis:_(_(_j_+_1_=_i_&_S1[j_+_1]_)_or_(_j_+_1_<>_i_&_S1[j_+_1]_)_)
percases ( j + 1 = i or j + 1 <> i ) ;
caseA21: j + 1 = i ; ::_thesis: S1[j + 1]
then A22: p . (j + 1) = p /. (j + 1) by A3, PARTFUN1:def_6;
q . (j + 1) = q /. (j + 1) by A3, A9, A11, A16, A21, PARTFUN1:def_6;
then fq . (j + 1) = (fq . j) + (a + (p /. (j + 1))) by A1, A4, A8, A18, A21
.= a + ((fq . j) + (p /. (j + 1))) by RLVECT_1:def_3
.= a + (fp . (j + 1)) by A14, A18, A19, A20, A22 ;
hence S1[j + 1] by A21; ::_thesis: verum
end;
caseA23: j + 1 <> i ; ::_thesis: S1[j + 1]
A24: j + 1 < i
proof
assume i <= j + 1 ; ::_thesis: contradiction
then i < j + 1 by A23, XXREAL_0:1;
hence contradiction by A20, NAT_1:13; ::_thesis: verum
end;
j + 1 <= i by A20, NAT_1:13;
then A25: j + 1 <= l by A10, XXREAL_0:2;
0 + 1 <= j + 1 by XREAL_1:6;
then A26: j + 1 in Seg l by A25, FINSEQ_1:1;
then A27: p . (j + 1) = p /. (j + 1) by A9, PARTFUN1:def_6;
q . (j + 1) = q /. (j + 1) by A11, A16, A26, PARTFUN1:def_6;
then fq . (j + 1) = (fq . j) + (q /. (j + 1)) by A1, A8, A18
.= fp . (j + 1) by A5, A14, A9, A18, A19, A20, A23, A26, A27 ;
hence S1[j + 1] by A24; ::_thesis: verum
end;
end;
end;
hence S1[j + 1] ; ::_thesis: verum
end;
supposeA28: i <= j ; ::_thesis: S1[j + 1]
j < l by A9, A18, FINSEQ_1:def_3;
then A29: j + 1 <= l by NAT_1:13;
0 + 1 <= j + 1 by XREAL_1:6;
then A30: j + 1 in dom p by A9, A29, FINSEQ_1:1;
then A31: p . (j + 1) = p /. (j + 1) by PARTFUN1:def_6;
A32: i < j + 1 by A28, NAT_1:13;
q . (j + 1) = q /. (j + 1) by A9, A11, A16, A30, PARTFUN1:def_6;
then fq . (j + 1) = (fq . j) + (q /. (j + 1)) by A1, A8, A18
.= (a + (fp . j)) + (p /. (j + 1)) by A5, A19, A28, A32, A30
.= a + ((fp . j) + (p /. (j + 1))) by RLVECT_1:def_3
.= a + (fp . (j + 1)) by A14, A18, A31 ;
hence S1[j + 1] by A28, NAT_1:13; ::_thesis: verum
end;
end;
end;
A33: S1[ 0 ] by A13, A7, A10;
for j being Element of NAT st 0 <= j & j <= len p holds
S1[j] from INT_1:sch_7(A33, A17);
hence Sum q = a + (Sum p) by A1, A12, A6, A10, A15; ::_thesis: verum
end;
theorem Th6: :: POLYNOM2:6
for L being non empty associative commutative doubleLoopStr
for a being Element of L
for p, q being FinSequence of the carrier of L st len p = len q & ex i being Element of NAT st
( i in dom p & q /. i = a * (p /. i) & ( for i9 being Element of NAT st i9 in dom p & i9 <> i holds
q /. i9 = p /. i9 ) ) holds
Product q = a * (Product p)
proof
let L be non empty associative commutative doubleLoopStr ; ::_thesis: for a being Element of L
for p, q being FinSequence of the carrier of L st len p = len q & ex i being Element of NAT st
( i in dom p & q /. i = a * (p /. i) & ( for i9 being Element of NAT st i9 in dom p & i9 <> i holds
q /. i9 = p /. i9 ) ) holds
Product q = a * (Product p)
let a be Element of L; ::_thesis: for p, q being FinSequence of the carrier of L st len p = len q & ex i being Element of NAT st
( i in dom p & q /. i = a * (p /. i) & ( for i9 being Element of NAT st i9 in dom p & i9 <> i holds
q /. i9 = p /. i9 ) ) holds
Product q = a * (Product p)
let p, q be FinSequence of the carrier of L; ::_thesis: ( len p = len q & ex i being Element of NAT st
( i in dom p & q /. i = a * (p /. i) & ( for i9 being Element of NAT st i9 in dom p & i9 <> i holds
q /. i9 = p /. i9 ) ) implies Product q = a * (Product p) )
assume that
A1: len p = len q and
A2: ex i being Element of NAT st
( i in dom p & q /. i = a * (p /. i) & ( for i9 being Element of NAT st i9 in dom p & i9 <> i holds
q /. i9 = p /. i9 ) ) ; ::_thesis: Product q = a * (Product p)
consider i being Element of NAT such that
A3: i in dom p and
A4: q /. i = a * (p /. i) and
A5: for i9 being Element of NAT st i9 in dom p & i9 <> i holds
q /. i9 = p /. i9 by A2;
A6: Product p = the multF of L "**" p by GROUP_4:def_2;
A7: Product q = the multF of L "**" q by GROUP_4:def_2;
percases ( len p = 0 or len p <> 0 ) ;
suppose len p = 0 ; ::_thesis: Product q = a * (Product p)
then p = {} ;
hence Product q = a * (Product p) by A3; ::_thesis: verum
end;
supposeA8: len p <> 0 ; ::_thesis: Product q = a * (Product p)
then A9: len p >= 1 by NAT_1:14;
consider l9 being Nat such that
A10: dom q = Seg l9 by FINSEQ_1:def_2;
consider fp being Function of NAT, the carrier of L such that
A11: fp . 1 = p . 1 and
A12: for i being Element of NAT st 0 <> i & i < len p holds
fp . (i + 1) = the multF of L . ((fp . i),(p . (i + 1))) and
A13: Product p = fp . (len p) by A6, A8, FINSOP_1:1, NAT_1:14;
consider fq being Function of NAT, the carrier of L such that
A14: fq . 1 = q . 1 and
A15: for i being Element of NAT st 0 <> i & i < len q holds
fq . (i + 1) = the multF of L . ((fq . i),(q . (i + 1))) and
A16: Product q = fq . (len p) by A1, A7, A8, FINSOP_1:1, NAT_1:14;
defpred S1[ Element of NAT ] means ( ( $1 < i & fp . $1 = fq . $1 ) or ( i <= $1 & fq . $1 = a * (fp . $1) ) );
consider l being Nat such that
A17: dom p = Seg l by FINSEQ_1:def_2;
i in { z where z is Element of NAT : ( 1 <= z & z <= l ) } by A3, A17, FINSEQ_1:def_1;
then A18: ex i9 being Element of NAT st
( i9 = i & 1 <= i9 & i9 <= l ) ;
reconsider l = l, l9 = l9 as Element of NAT by ORDINAL1:def_12;
A19: len p = l by A17, FINSEQ_1:def_3;
A20: 1 <= l by A18, XXREAL_0:2;
then A21: 1 in dom p by A17, FINSEQ_1:1;
A22: l = len p by A17, FINSEQ_1:def_3
.= l9 by A1, A10, FINSEQ_1:def_3 ;
A23: for j being Element of NAT st 1 <= j & j < len p & S1[j] holds
S1[j + 1]
proof
let j be Element of NAT ; ::_thesis: ( 1 <= j & j < len p & S1[j] implies S1[j + 1] )
assume that
A24: 1 <= j and
A25: j < len p ; ::_thesis: ( not S1[j] or S1[j + 1] )
assume A26: S1[j] ; ::_thesis: S1[j + 1]
percases ( j < i or i <= j ) ;
supposeA27: j < i ; ::_thesis: S1[j + 1]
now__::_thesis:_(_(_j_+_1_=_i_&_S1[j_+_1]_)_or_(_j_+_1_<>_i_&_S1[j_+_1]_)_)
percases ( j + 1 = i or j + 1 <> i ) ;
caseA28: j + 1 = i ; ::_thesis: S1[j + 1]
then A29: p . (j + 1) = p /. (j + 1) by A3, PARTFUN1:def_6;
q . (j + 1) = q /. (j + 1) by A3, A17, A10, A22, A28, PARTFUN1:def_6;
then fq . (j + 1) = (fp . j) * (a * (p /. (j + 1))) by A1, A4, A15, A24, A25, A26, A27, A28
.= ((fp . j) * (p /. (j + 1))) * a by GROUP_1:def_3
.= (fp . (j + 1)) * a by A12, A24, A25, A29 ;
hence S1[j + 1] by A28; ::_thesis: verum
end;
caseA30: j + 1 <> i ; ::_thesis: S1[j + 1]
A31: j + 1 < i
proof
assume i <= j + 1 ; ::_thesis: contradiction
then i < j + 1 by A30, XXREAL_0:1;
hence contradiction by A27, NAT_1:13; ::_thesis: verum
end;
j + 1 <= i by A27, NAT_1:13;
then A32: j + 1 <= l by A18, XXREAL_0:2;
0 + 1 <= j + 1 by XREAL_1:6;
then A33: j + 1 in Seg l by A32, FINSEQ_1:1;
then A34: p . (j + 1) = p /. (j + 1) by A17, PARTFUN1:def_6;
q . (j + 1) = q /. (j + 1) by A10, A22, A33, PARTFUN1:def_6;
then fq . (j + 1) = (fq . j) * (q /. (j + 1)) by A1, A15, A24, A25
.= the multF of L . ((fq . j),(p . (j + 1))) by A5, A17, A30, A33, A34
.= fp . (j + 1) by A12, A24, A25, A26, A27 ;
hence S1[j + 1] by A31; ::_thesis: verum
end;
end;
end;
hence S1[j + 1] ; ::_thesis: verum
end;
supposeA35: i <= j ; ::_thesis: S1[j + 1]
j < l by A17, A25, FINSEQ_1:def_3;
then A36: j + 1 <= l by NAT_1:13;
0 + 1 <= j + 1 by XREAL_1:6;
then A37: j + 1 in dom p by A17, A36, FINSEQ_1:1;
then A38: p . (j + 1) = p /. (j + 1) by PARTFUN1:def_6;
A39: i < j + 1 by A35, NAT_1:13;
q . (j + 1) = q /. (j + 1) by A17, A10, A22, A37, PARTFUN1:def_6;
then fq . (j + 1) = (fq . j) * (q /. (j + 1)) by A1, A15, A24, A25
.= (a * (fp . j)) * (p /. (j + 1)) by A5, A26, A35, A39, A37
.= a * ((fp . j) * (p /. (j + 1))) by GROUP_1:def_3
.= a * (fp . (j + 1)) by A12, A24, A25, A38 ;
hence S1[j + 1] by A35, NAT_1:13; ::_thesis: verum
end;
end;
end;
A40: 1 in dom q by A10, A22, A20, FINSEQ_1:1;
now__::_thesis:_(_(_1_<_i_&_fp_._1_=_fq_._1_)_or_(_i_<=_1_&_fq_._1_=_a_*_(fp_._1)_)_)
percases ( 1 < i or i <= 1 ) ;
caseA41: 1 < i ; ::_thesis: fp . 1 = fq . 1
thus fp . 1 = p /. 1 by A11, A21, PARTFUN1:def_6
.= q /. 1 by A5, A21, A41
.= fq . 1 by A14, A40, PARTFUN1:def_6 ; ::_thesis: verum
end;
case i <= 1 ; ::_thesis: fq . 1 = a * (fp . 1)
then i = 1 by A18, XXREAL_0:1;
hence fq . 1 = a * (p /. 1) by A3, A4, A14, A17, A10, A22, PARTFUN1:def_6
.= a * (fp . 1) by A11, A21, PARTFUN1:def_6 ;
::_thesis: verum
end;
end;
end;
then A42: S1[1] ;
for j being Element of NAT st 1 <= j & j <= len p holds
S1[j] from INT_1:sch_7(A42, A23);
hence Product q = a * (Product p) by A9, A13, A16, A18, A19; ::_thesis: verum
end;
end;
end;
theorem Th7: :: POLYNOM2:7
for X being set
for A being empty Subset of X
for R being Order of X st R linearly_orders A holds
SgmX (R,A) = {}
proof
let X be set ; ::_thesis: for A being empty Subset of X
for R being Order of X st R linearly_orders A holds
SgmX (R,A) = {}
let A be empty Subset of X; ::_thesis: for R being Order of X st R linearly_orders A holds
SgmX (R,A) = {}
let R be Order of X; ::_thesis: ( R linearly_orders A implies SgmX (R,A) = {} )
assume R linearly_orders A ; ::_thesis: SgmX (R,A) = {}
then rng (SgmX (R,A)) = {} by PRE_POLY:def_2;
hence SgmX (R,A) = {} ; ::_thesis: verum
end;
theorem Th8: :: POLYNOM2:8
for X being set
for A being finite Subset of X
for R being Order of X st R linearly_orders A holds
for i, j being Element of NAT st i in dom (SgmX (R,A)) & j in dom (SgmX (R,A)) & (SgmX (R,A)) /. i = (SgmX (R,A)) /. j holds
i = j
proof
let X be set ; ::_thesis: for A being finite Subset of X
for R being Order of X st R linearly_orders A holds
for i, j being Element of NAT st i in dom (SgmX (R,A)) & j in dom (SgmX (R,A)) & (SgmX (R,A)) /. i = (SgmX (R,A)) /. j holds
i = j
let A be finite Subset of X; ::_thesis: for R being Order of X st R linearly_orders A holds
for i, j being Element of NAT st i in dom (SgmX (R,A)) & j in dom (SgmX (R,A)) & (SgmX (R,A)) /. i = (SgmX (R,A)) /. j holds
i = j
let R be Order of X; ::_thesis: ( R linearly_orders A implies for i, j being Element of NAT st i in dom (SgmX (R,A)) & j in dom (SgmX (R,A)) & (SgmX (R,A)) /. i = (SgmX (R,A)) /. j holds
i = j )
assume A1: R linearly_orders A ; ::_thesis: for i, j being Element of NAT st i in dom (SgmX (R,A)) & j in dom (SgmX (R,A)) & (SgmX (R,A)) /. i = (SgmX (R,A)) /. j holds
i = j
let i, j be Element of NAT ; ::_thesis: ( i in dom (SgmX (R,A)) & j in dom (SgmX (R,A)) & (SgmX (R,A)) /. i = (SgmX (R,A)) /. j implies i = j )
assume that
A2: i in dom (SgmX (R,A)) and
A3: j in dom (SgmX (R,A)) and
A4: (SgmX (R,A)) /. i = (SgmX (R,A)) /. j ; ::_thesis: i = j
assume i <> j ; ::_thesis: contradiction
then ( i < j or j < i ) by XXREAL_0:1;
hence contradiction by A1, A2, A3, A4, PRE_POLY:def_2; ::_thesis: verum
end;
Lm5: for D being set
for p being FinSequence of D st dom p <> {} holds
1 in dom p
proof
let D be set ; ::_thesis: for p being FinSequence of D st dom p <> {} holds
1 in dom p
let p be FinSequence of D; ::_thesis: ( dom p <> {} implies 1 in dom p )
assume dom p <> {} ; ::_thesis: 1 in dom p
then p <> {} ;
hence 1 in dom p by FINSEQ_5:6; ::_thesis: verum
end;
theorem Th9: :: POLYNOM2:9
for X being set
for A being finite Subset of X
for a being Element of X st not a in A holds
for B being finite Subset of X st B = {a} \/ A holds
for R being Order of X st R linearly_orders B holds
for k being Element of NAT st k in dom (SgmX (R,B)) & (SgmX (R,B)) /. k = a holds
for i being Element of NAT st 1 <= i & i <= k - 1 holds
(SgmX (R,B)) /. i = (SgmX (R,A)) /. i
proof
let X be set ; ::_thesis: for A being finite Subset of X
for a being Element of X st not a in A holds
for B being finite Subset of X st B = {a} \/ A holds
for R being Order of X st R linearly_orders B holds
for k being Element of NAT st k in dom (SgmX (R,B)) & (SgmX (R,B)) /. k = a holds
for i being Element of NAT st 1 <= i & i <= k - 1 holds
(SgmX (R,B)) /. i = (SgmX (R,A)) /. i
let A be finite Subset of X; ::_thesis: for a being Element of X st not a in A holds
for B being finite Subset of X st B = {a} \/ A holds
for R being Order of X st R linearly_orders B holds
for k being Element of NAT st k in dom (SgmX (R,B)) & (SgmX (R,B)) /. k = a holds
for i being Element of NAT st 1 <= i & i <= k - 1 holds
(SgmX (R,B)) /. i = (SgmX (R,A)) /. i
let a be Element of X; ::_thesis: ( not a in A implies for B being finite Subset of X st B = {a} \/ A holds
for R being Order of X st R linearly_orders B holds
for k being Element of NAT st k in dom (SgmX (R,B)) & (SgmX (R,B)) /. k = a holds
for i being Element of NAT st 1 <= i & i <= k - 1 holds
(SgmX (R,B)) /. i = (SgmX (R,A)) /. i )
assume A1: not a in A ; ::_thesis: for B being finite Subset of X st B = {a} \/ A holds
for R being Order of X st R linearly_orders B holds
for k being Element of NAT st k in dom (SgmX (R,B)) & (SgmX (R,B)) /. k = a holds
for i being Element of NAT st 1 <= i & i <= k - 1 holds
(SgmX (R,B)) /. i = (SgmX (R,A)) /. i
let B be finite Subset of X; ::_thesis: ( B = {a} \/ A implies for R being Order of X st R linearly_orders B holds
for k being Element of NAT st k in dom (SgmX (R,B)) & (SgmX (R,B)) /. k = a holds
for i being Element of NAT st 1 <= i & i <= k - 1 holds
(SgmX (R,B)) /. i = (SgmX (R,A)) /. i )
assume A2: B = {a} \/ A ; ::_thesis: for R being Order of X st R linearly_orders B holds
for k being Element of NAT st k in dom (SgmX (R,B)) & (SgmX (R,B)) /. k = a holds
for i being Element of NAT st 1 <= i & i <= k - 1 holds
(SgmX (R,B)) /. i = (SgmX (R,A)) /. i
let R be Order of X; ::_thesis: ( R linearly_orders B implies for k being Element of NAT st k in dom (SgmX (R,B)) & (SgmX (R,B)) /. k = a holds
for i being Element of NAT st 1 <= i & i <= k - 1 holds
(SgmX (R,B)) /. i = (SgmX (R,A)) /. i )
assume A3: R linearly_orders B ; ::_thesis: for k being Element of NAT st k in dom (SgmX (R,B)) & (SgmX (R,B)) /. k = a holds
for i being Element of NAT st 1 <= i & i <= k - 1 holds
(SgmX (R,B)) /. i = (SgmX (R,A)) /. i
then A4: R linearly_orders A by A2, Lm4;
field R = X by ORDERS_1:12;
then A5: R is_antisymmetric_in X by RELAT_2:def_12;
set sgb = SgmX (R,B);
set sga = SgmX (R,A);
consider lensga being Nat such that
A6: dom (SgmX (R,A)) = Seg lensga by FINSEQ_1:def_2;
consider lensgb being Nat such that
A7: dom (SgmX (R,B)) = Seg lensgb by FINSEQ_1:def_2;
reconsider lensga = lensga, lensgb = lensgb as Element of NAT by ORDINAL1:def_12;
lensgb = len (SgmX (R,B)) by A7, FINSEQ_1:def_3
.= card B by A3, PRE_POLY:11
.= (card A) + 1 by A1, A2, CARD_2:41
.= (len (SgmX (R,A))) + 1 by A2, A3, Lm4, PRE_POLY:11
.= lensga + 1 by A6, FINSEQ_1:def_3 ;
then A8: lensga <= lensgb by NAT_1:11;
defpred S1[ Element of NAT ] means (SgmX (R,B)) /. $1 = (SgmX (R,A)) /. $1;
let k be Element of NAT ; ::_thesis: ( k in dom (SgmX (R,B)) & (SgmX (R,B)) /. k = a implies for i being Element of NAT st 1 <= i & i <= k - 1 holds
(SgmX (R,B)) /. i = (SgmX (R,A)) /. i )
assume that
A9: k in dom (SgmX (R,B)) and
A10: (SgmX (R,B)) /. k = a ; ::_thesis: for i being Element of NAT st 1 <= i & i <= k - 1 holds
(SgmX (R,B)) /. i = (SgmX (R,A)) /. i
k in Seg (len (SgmX (R,B))) by A9, FINSEQ_1:def_3;
then A11: 1 <= k by FINSEQ_1:1;
then 1 - 1 <= k - 1 by XREAL_1:9;
then reconsider k9 = k - 1 as Element of NAT by INT_1:3;
A12: k9 + 1 = k + 0 ;
A13: for j being Element of NAT st 1 <= j & j < k9 & ( for j9 being Element of NAT st 1 <= j9 & j9 <= j holds
S1[j9] ) holds
S1[j + 1]
proof
let i9 be Element of NAT ; ::_thesis: ( 1 <= i9 & i9 < k9 & ( for j9 being Element of NAT st 1 <= j9 & j9 <= i9 holds
S1[j9] ) implies S1[i9 + 1] )
assume that
A14: 1 <= i9 and
A15: i9 < k9 ; ::_thesis: ( ex j9 being Element of NAT st
( 1 <= j9 & j9 <= i9 & not S1[j9] ) or S1[i9 + 1] )
A16: 1 <= i9 + 1 by A14, XREAL_1:29;
A17: i9 + 1 < k by A12, A15, XREAL_1:6;
then A18: i9 + 1 in dom (SgmX (R,B)) by A9, A16, Th2;
(SgmX (R,B)) /. (i9 + 1) = (SgmX (R,B)) . (i9 + 1) by A9, A17, A16, Th2, PARTFUN1:def_6;
then (SgmX (R,B)) /. (i9 + 1) in rng (SgmX (R,B)) by A18, FUNCT_1:def_3;
then A19: (SgmX (R,B)) /. (i9 + 1) in B by A3, PRE_POLY:def_2;
(SgmX (R,B)) /. (i9 + 1) <> a by A3, A9, A10, A17, A18, PRE_POLY:def_2;
then not (SgmX (R,B)) /. (i9 + 1) in {a} by TARSKI:def_1;
then (SgmX (R,B)) /. (i9 + 1) in A by A2, A19, XBOOLE_0:def_3;
then (SgmX (R,B)) /. (i9 + 1) in rng (SgmX (R,A)) by A4, PRE_POLY:def_2;
then consider l being set such that
A20: l in dom (SgmX (R,A)) and
A21: (SgmX (R,A)) . l = (SgmX (R,B)) /. (i9 + 1) by FUNCT_1:def_3;
reconsider l = l as Element of NAT by A20;
A22: 1 <= l by A6, A20, FINSEQ_1:1;
l <= lensga by A6, A20, FINSEQ_1:1;
then l <= lensgb by A8, XXREAL_0:2;
then A23: l in dom (SgmX (R,B)) by A7, A22, FINSEQ_1:1;
assume A24: for j9 being Element of NAT st 1 <= j9 & j9 <= i9 holds
S1[j9] ; ::_thesis: S1[i9 + 1]
assume A25: (SgmX (R,B)) /. (i9 + 1) <> (SgmX (R,A)) /. (i9 + 1) ; ::_thesis: contradiction
then A26: l <> i9 + 1 by A20, A21, PARTFUN1:def_6;
percases ( l < i9 + 1 or i9 + 1 <= l ) ;
suppose l < i9 + 1 ; ::_thesis: contradiction
then l <= i9 by NAT_1:13;
then (SgmX (R,B)) /. l = (SgmX (R,A)) /. l by A24, A22
.= (SgmX (R,B)) /. (i9 + 1) by A20, A21, PARTFUN1:def_6 ;
hence contradiction by A3, A18, A26, A23, Th8; ::_thesis: verum
end;
supposeA27: i9 + 1 <= l ; ::_thesis: contradiction
then A28: i9 + 1 in dom (SgmX (R,A)) by A16, A20, Th2;
(SgmX (R,A)) /. (i9 + 1) = (SgmX (R,A)) . (i9 + 1) by A16, A20, A27, Th2, PARTFUN1:def_6;
then (SgmX (R,A)) /. (i9 + 1) in rng (SgmX (R,A)) by A28, FUNCT_1:def_3;
then (SgmX (R,A)) /. (i9 + 1) in A by A4, PRE_POLY:def_2;
then (SgmX (R,A)) /. (i9 + 1) in B by A2, XBOOLE_0:def_3;
then (SgmX (R,A)) /. (i9 + 1) in rng (SgmX (R,B)) by A3, PRE_POLY:def_2;
then consider l9 being set such that
A29: l9 in dom (SgmX (R,B)) and
A30: (SgmX (R,B)) . l9 = (SgmX (R,A)) /. (i9 + 1) by FUNCT_1:def_3;
reconsider l9 = l9 as Element of NAT by A29;
i9 + 1 < l by A26, A27, XXREAL_0:1;
then [((SgmX (R,A)) /. (i9 + 1)),((SgmX (R,A)) /. l)] in R by A4, A20, A28, PRE_POLY:def_2;
then [((SgmX (R,B)) /. l9),((SgmX (R,A)) /. l)] in R by A29, A30, PARTFUN1:def_6;
then A31: [((SgmX (R,B)) /. l9),((SgmX (R,B)) /. (i9 + 1))] in R by A20, A21, PARTFUN1:def_6;
(SgmX (R,B)) /. l9 = (SgmX (R,B)) . l9 by A29, PARTFUN1:def_6;
then (SgmX (R,B)) /. l9 in rng (SgmX (R,B)) by A29, FUNCT_1:def_3;
then A32: (SgmX (R,B)) /. l9 in B by A3, PRE_POLY:def_2;
A33: 1 <= l9 by A7, A29, FINSEQ_1:1;
A34: i9 + 1 < l9
proof
assume A35: l9 <= i9 + 1 ; ::_thesis: contradiction
now__::_thesis:_(_(_l9_=_i9_+_1_&_contradiction_)_or_(_l9_<_i9_+_1_&_contradiction_)_)
percases ( l9 = i9 + 1 or l9 < i9 + 1 ) by A35, XXREAL_0:1;
case l9 = i9 + 1 ; ::_thesis: contradiction
hence contradiction by A25, A29, A30, PARTFUN1:def_6; ::_thesis: verum
end;
caseA36: l9 < i9 + 1 ; ::_thesis: contradiction
then l9 <= i9 by NAT_1:13;
then A37: (SgmX (R,A)) /. l9 = (SgmX (R,B)) /. l9 by A24, A33
.= (SgmX (R,A)) /. (i9 + 1) by A29, A30, PARTFUN1:def_6 ;
l9 in dom (SgmX (R,A)) by A28, A33, A36, Th2;
hence contradiction by A2, A3, A28, A36, A37, Lm4, Th8; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
then [((SgmX (R,B)) /. (i9 + 1)),((SgmX (R,B)) /. l9)] in R by A3, A18, A29, PRE_POLY:def_2;
then (SgmX (R,B)) /. l9 = (SgmX (R,B)) /. (i9 + 1) by A5, A31, A32, RELAT_2:def_4;
hence contradiction by A3, A18, A29, A34, Th8; ::_thesis: verum
end;
end;
end;
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= k - 1 implies (SgmX (R,B)) /. i = (SgmX (R,A)) /. i )
assume that
A38: 1 <= i and
A39: i <= k - 1 ; ::_thesis: (SgmX (R,B)) /. i = (SgmX (R,A)) /. i
A40: 1 in dom (SgmX (R,B)) by A9, Lm5;
A41: S1[1]
proof
(SgmX (R,B)) /. 1 = (SgmX (R,B)) . 1 by A9, Lm5, PARTFUN1:def_6;
then (SgmX (R,B)) /. 1 in rng (SgmX (R,B)) by A40, FUNCT_1:def_3;
then A42: (SgmX (R,B)) /. 1 in B by A3, PRE_POLY:def_2;
k <> 1 by A38, A39, XXREAL_0:2;
then 1 < k by A11, XXREAL_0:1;
then (SgmX (R,B)) /. 1 <> a by A3, A9, A10, A40, PRE_POLY:def_2;
then not (SgmX (R,B)) /. 1 in {a} by TARSKI:def_1;
then (SgmX (R,B)) /. 1 in A by A2, A42, XBOOLE_0:def_3;
then (SgmX (R,B)) /. 1 in rng (SgmX (R,A)) by A4, PRE_POLY:def_2;
then consider l being set such that
A43: l in dom (SgmX (R,A)) and
A44: (SgmX (R,A)) . l = (SgmX (R,B)) /. 1 by FUNCT_1:def_3;
A45: (SgmX (R,A)) /. 1 = (SgmX (R,A)) . 1 by A43, Lm5, PARTFUN1:def_6;
assume A46: (SgmX (R,B)) /. 1 <> (SgmX (R,A)) /. 1 ; ::_thesis: contradiction
reconsider l = l as Element of NAT by A43;
A47: 1 in dom (SgmX (R,A)) by A43, Lm5;
1 <= l by A6, A43, FINSEQ_1:1;
then 1 < l by A46, A44, A45, XXREAL_0:1;
then [((SgmX (R,A)) /. 1),((SgmX (R,A)) /. l)] in R by A4, A43, A47, PRE_POLY:def_2;
then A48: [((SgmX (R,A)) /. 1),((SgmX (R,B)) /. 1)] in R by A43, A44, PARTFUN1:def_6;
not (SgmX (R,A)) /. 1 in B
proof
A49: (SgmX (R,B)) /. 1 = (SgmX (R,B)) . 1 by A9, Lm5, PARTFUN1:def_6;
assume (SgmX (R,A)) /. 1 in B ; ::_thesis: contradiction
then (SgmX (R,A)) /. 1 in rng (SgmX (R,B)) by A3, PRE_POLY:def_2;
then consider l9 being set such that
A50: l9 in dom (SgmX (R,B)) and
A51: (SgmX (R,B)) . l9 = (SgmX (R,A)) /. 1 by FUNCT_1:def_3;
reconsider l9 = l9 as Element of NAT by A50;
1 <= l9 by A7, A50, FINSEQ_1:1;
then 1 < l9 by A46, A51, A49, XXREAL_0:1;
then [((SgmX (R,B)) /. 1),((SgmX (R,B)) /. l9)] in R by A3, A40, A50, PRE_POLY:def_2;
then [((SgmX (R,B)) /. 1),((SgmX (R,A)) /. 1)] in R by A50, A51, PARTFUN1:def_6;
hence contradiction by A5, A46, A42, A48, RELAT_2:def_4; ::_thesis: verum
end;
then A52: not (SgmX (R,A)) /. 1 in A by A2, XBOOLE_0:def_3;
(SgmX (R,A)) /. 1 in rng (SgmX (R,A)) by A47, A45, FUNCT_1:def_3;
hence contradiction by A4, A52, PRE_POLY:def_2; ::_thesis: verum
end;
for j being Element of NAT st 1 <= j & j <= k9 holds
S1[j] from INT_1:sch_8(A41, A13);
hence (SgmX (R,B)) /. i = (SgmX (R,A)) /. i by A38, A39; ::_thesis: verum
end;
theorem Th10: :: POLYNOM2:10
for X being set
for A being finite Subset of X
for a being Element of X st not a in A holds
for B being finite Subset of X st B = {a} \/ A holds
for R being Order of X st R linearly_orders B holds
for k being Element of NAT st k in dom (SgmX (R,B)) & (SgmX (R,B)) /. k = a holds
for i being Element of NAT st k <= i & i <= len (SgmX (R,A)) holds
(SgmX (R,B)) /. (i + 1) = (SgmX (R,A)) /. i
proof
let X be set ; ::_thesis: for A being finite Subset of X
for a being Element of X st not a in A holds
for B being finite Subset of X st B = {a} \/ A holds
for R being Order of X st R linearly_orders B holds
for k being Element of NAT st k in dom (SgmX (R,B)) & (SgmX (R,B)) /. k = a holds
for i being Element of NAT st k <= i & i <= len (SgmX (R,A)) holds
(SgmX (R,B)) /. (i + 1) = (SgmX (R,A)) /. i
let A be finite Subset of X; ::_thesis: for a being Element of X st not a in A holds
for B being finite Subset of X st B = {a} \/ A holds
for R being Order of X st R linearly_orders B holds
for k being Element of NAT st k in dom (SgmX (R,B)) & (SgmX (R,B)) /. k = a holds
for i being Element of NAT st k <= i & i <= len (SgmX (R,A)) holds
(SgmX (R,B)) /. (i + 1) = (SgmX (R,A)) /. i
let a be Element of X; ::_thesis: ( not a in A implies for B being finite Subset of X st B = {a} \/ A holds
for R being Order of X st R linearly_orders B holds
for k being Element of NAT st k in dom (SgmX (R,B)) & (SgmX (R,B)) /. k = a holds
for i being Element of NAT st k <= i & i <= len (SgmX (R,A)) holds
(SgmX (R,B)) /. (i + 1) = (SgmX (R,A)) /. i )
assume A1: not a in A ; ::_thesis: for B being finite Subset of X st B = {a} \/ A holds
for R being Order of X st R linearly_orders B holds
for k being Element of NAT st k in dom (SgmX (R,B)) & (SgmX (R,B)) /. k = a holds
for i being Element of NAT st k <= i & i <= len (SgmX (R,A)) holds
(SgmX (R,B)) /. (i + 1) = (SgmX (R,A)) /. i
let B be finite Subset of X; ::_thesis: ( B = {a} \/ A implies for R being Order of X st R linearly_orders B holds
for k being Element of NAT st k in dom (SgmX (R,B)) & (SgmX (R,B)) /. k = a holds
for i being Element of NAT st k <= i & i <= len (SgmX (R,A)) holds
(SgmX (R,B)) /. (i + 1) = (SgmX (R,A)) /. i )
assume A2: B = {a} \/ A ; ::_thesis: for R being Order of X st R linearly_orders B holds
for k being Element of NAT st k in dom (SgmX (R,B)) & (SgmX (R,B)) /. k = a holds
for i being Element of NAT st k <= i & i <= len (SgmX (R,A)) holds
(SgmX (R,B)) /. (i + 1) = (SgmX (R,A)) /. i
let R be Order of X; ::_thesis: ( R linearly_orders B implies for k being Element of NAT st k in dom (SgmX (R,B)) & (SgmX (R,B)) /. k = a holds
for i being Element of NAT st k <= i & i <= len (SgmX (R,A)) holds
(SgmX (R,B)) /. (i + 1) = (SgmX (R,A)) /. i )
assume A3: R linearly_orders B ; ::_thesis: for k being Element of NAT st k in dom (SgmX (R,B)) & (SgmX (R,B)) /. k = a holds
for i being Element of NAT st k <= i & i <= len (SgmX (R,A)) holds
(SgmX (R,B)) /. (i + 1) = (SgmX (R,A)) /. i
then A4: R linearly_orders A by A2, Lm4;
field R = X by ORDERS_1:12;
then A5: R is_antisymmetric_in X by RELAT_2:def_12;
set sgb = SgmX (R,B);
set sga = SgmX (R,A);
consider lensga being Nat such that
A6: dom (SgmX (R,A)) = Seg lensga by FINSEQ_1:def_2;
defpred S1[ Element of NAT ] means (SgmX (R,B)) /. ($1 + 1) = (SgmX (R,A)) /. $1;
consider lensgb being Nat such that
A7: dom (SgmX (R,B)) = Seg lensgb by FINSEQ_1:def_2;
let k be Element of NAT ; ::_thesis: ( k in dom (SgmX (R,B)) & (SgmX (R,B)) /. k = a implies for i being Element of NAT st k <= i & i <= len (SgmX (R,A)) holds
(SgmX (R,B)) /. (i + 1) = (SgmX (R,A)) /. i )
assume that
A8: k in dom (SgmX (R,B)) and
A9: (SgmX (R,B)) /. k = a ; ::_thesis: for i being Element of NAT st k <= i & i <= len (SgmX (R,A)) holds
(SgmX (R,B)) /. (i + 1) = (SgmX (R,A)) /. i
k in Seg (len (SgmX (R,B))) by A8, FINSEQ_1:def_3;
then A10: 1 <= k by FINSEQ_1:1;
then 1 - 1 <= k - 1 by XREAL_1:9;
then reconsider k9 = k - 1 as Element of NAT by INT_1:3;
reconsider lensga = lensga, lensgb = lensgb as Element of NAT by ORDINAL1:def_12;
A11: k9 + 1 = k + 0 ;
A12: lensgb = len (SgmX (R,B)) by A7, FINSEQ_1:def_3
.= card B by A3, PRE_POLY:11
.= (card A) + 1 by A1, A2, CARD_2:41
.= (len (SgmX (R,A))) + 1 by A2, A3, Lm4, PRE_POLY:11
.= lensga + 1 by A6, FINSEQ_1:def_3 ;
A13: for j being Element of NAT st k <= j & j < len (SgmX (R,A)) & ( for j9 being Element of NAT st k <= j9 & j9 <= j holds
S1[j9] ) holds
S1[j + 1]
proof
let j be Element of NAT ; ::_thesis: ( k <= j & j < len (SgmX (R,A)) & ( for j9 being Element of NAT st k <= j9 & j9 <= j holds
S1[j9] ) implies S1[j + 1] )
assume that
A14: k <= j and
A15: j < len (SgmX (R,A)) ; ::_thesis: ( ex j9 being Element of NAT st
( k <= j9 & j9 <= j & not S1[j9] ) or S1[j + 1] )
A16: (j + 1) + 1 = j + (1 + 1) ;
A17: 1 <= j + 2 by NAT_1:12;
len (SgmX (R,B)) = card B by A3, PRE_POLY:11
.= (card A) + 1 by A1, A2, CARD_2:41
.= (len (SgmX (R,A))) + 1 by A2, A3, Lm4, PRE_POLY:11 ;
then j + 1 < len (SgmX (R,B)) by A15, XREAL_1:6;
then j + 2 <= len (SgmX (R,B)) by A16, NAT_1:13;
then j + 2 <= lensgb by A7, FINSEQ_1:def_3;
then A18: j + 2 in dom (SgmX (R,B)) by A7, A17, FINSEQ_1:1;
now__::_thesis:_not_(SgmX_(R,B))_/._(j_+_2)_=_a
assume (SgmX (R,B)) /. (j + 2) = a ; ::_thesis: contradiction
then j + 2 = k by A3, A8, A9, A18, Th8;
hence contradiction by A14, NAT_1:19; ::_thesis: verum
end;
then A19: not (SgmX (R,B)) /. (j + 2) in {a} by TARSKI:def_1;
(SgmX (R,B)) /. (j + 2) = (SgmX (R,B)) . (j + 2) by A18, PARTFUN1:def_6;
then (SgmX (R,B)) /. (j + 2) in rng (SgmX (R,B)) by A18, FUNCT_1:def_3;
then (SgmX (R,B)) /. (j + 2) in B by A3, PRE_POLY:def_2;
then (SgmX (R,B)) /. (j + 2) in A by A2, A19, XBOOLE_0:def_3;
then (SgmX (R,B)) /. (j + 2) in rng (SgmX (R,A)) by A4, PRE_POLY:def_2;
then consider l being set such that
A20: l in dom (SgmX (R,A)) and
A21: (SgmX (R,A)) . l = (SgmX (R,B)) /. (j + 2) by FUNCT_1:def_3;
reconsider l = l as Element of NAT by A20;
A22: (SgmX (R,A)) /. l = (SgmX (R,A)) . l by A20, PARTFUN1:def_6;
A23: 1 <= l by A6, A20, FINSEQ_1:1;
j + 1 <= len (SgmX (R,A)) by A15, NAT_1:13;
then A24: j + 1 <= lensga by A6, FINSEQ_1:def_3;
1 <= j + 1 by NAT_1:12;
then A25: j + 1 in dom (SgmX (R,A)) by A6, A24, FINSEQ_1:1;
then A26: (SgmX (R,A)) /. (j + 1) = (SgmX (R,A)) . (j + 1) by PARTFUN1:def_6;
assume A27: for j9 being Element of NAT st k <= j9 & j9 <= j holds
S1[j9] ; ::_thesis: S1[j + 1]
l <= lensga by A6, A20, FINSEQ_1:1;
then A28: l + 1 <= lensgb by A12, XREAL_1:6;
1 <= l + 1 by NAT_1:12;
then A29: l + 1 in dom (SgmX (R,B)) by A7, A28, FINSEQ_1:1;
l <= l + 1 by XREAL_1:29;
then A30: l in dom (SgmX (R,B)) by A23, A29, Th2;
assume A31: (SgmX (R,B)) /. ((j + 1) + 1) <> (SgmX (R,A)) /. (j + 1) ; ::_thesis: contradiction
then A32: l <> j + 1 by A20, A21, PARTFUN1:def_6;
percases ( l <= j + 1 or l > j + 1 ) ;
supposeA33: l <= j + 1 ; ::_thesis: contradiction
then l < j + 1 by A32, XXREAL_0:1;
then A34: l <= j by NAT_1:13;
now__::_thesis:_(_(_k_<=_l_&_contradiction_)_or_(_l_<_k_&_contradiction_)_)
percases ( k <= l or l < k ) ;
case k <= l ; ::_thesis: contradiction
then (SgmX (R,B)) /. (l + 1) = (SgmX (R,A)) /. l by A27, A34;
then j + 2 = l + 1 by A3, A18, A20, A21, A29, Th8, PARTFUN1:def_6;
hence contradiction by A31, A20, A21, PARTFUN1:def_6; ::_thesis: verum
end;
case l < k ; ::_thesis: contradiction
then l <= k9 by A11, NAT_1:13;
then A35: (SgmX (R,B)) /. l = (SgmX (R,A)) /. l by A1, A2, A3, A8, A9, A23, Th9;
j + 1 < (j + 1) + 1 by XREAL_1:29;
hence contradiction by A3, A18, A20, A21, A30, A33, A35, Th8, PARTFUN1:def_6; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
supposeA36: l > j + 1 ; ::_thesis: contradiction
A37: for i9 being Element of NAT st 1 <= i9 & i9 <= j + 2 holds
(SgmX (R,A)) /. (j + 1) <> (SgmX (R,B)) /. i9
proof
let i9 be Element of NAT ; ::_thesis: ( 1 <= i9 & i9 <= j + 2 implies (SgmX (R,A)) /. (j + 1) <> (SgmX (R,B)) /. i9 )
assume that
A38: 1 <= i9 and
A39: i9 <= j + 2 ; ::_thesis: (SgmX (R,A)) /. (j + 1) <> (SgmX (R,B)) /. i9
assume A40: (SgmX (R,A)) /. (j + 1) = (SgmX (R,B)) /. i9 ; ::_thesis: contradiction
percases ( i9 = j + 2 or i9 <> j + 2 ) ;
suppose i9 = j + 2 ; ::_thesis: contradiction
hence contradiction by A31, A40; ::_thesis: verum
end;
supposeA41: i9 <> j + 2 ; ::_thesis: contradiction
then i9 < j + 2 by A39, XXREAL_0:1;
then A42: i9 <= j + 1 by A16, NAT_1:13;
then i9 <= lensga by A24, XXREAL_0:2;
then A43: i9 in dom (SgmX (R,A)) by A6, A38, FINSEQ_1:1;
now__::_thesis:_(_(_i9_<=_k_&_contradiction_)_or_(_k_<_i9_&_contradiction_)_)
percases ( i9 <= k or k < i9 ) ;
caseA44: i9 <= k ; ::_thesis: contradiction
now__::_thesis:_(_(_i9_=_k_&_contradiction_)_or_(_i9_<>_k_&_contradiction_)_)
percases ( i9 = k or i9 <> k ) ;
case i9 = k ; ::_thesis: contradiction
then (SgmX (R,A)) . (j + 1) = a by A9, A25, A40, PARTFUN1:def_6;
then a in rng (SgmX (R,A)) by A25, FUNCT_1:def_3;
hence contradiction by A1, A4, PRE_POLY:def_2; ::_thesis: verum
end;
case i9 <> k ; ::_thesis: contradiction
then i9 < k by A44, XXREAL_0:1;
then i9 <= k9 by A11, NAT_1:13;
then (SgmX (R,B)) /. i9 = (SgmX (R,A)) /. i9 by A1, A2, A3, A8, A9, A38, Th9;
then A45: i9 = j + 1 by A2, A3, A25, A40, A43, Lm4, Th8;
i9 <= j by A14, A44, XXREAL_0:2;
hence contradiction by A45, XREAL_1:29; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
caseA46: k < i9 ; ::_thesis: contradiction
A47: i9 - 1 <= (j + 1) - 1 by A42, XREAL_1:9;
A48: i9 - 1 <= i9 by XREAL_1:146;
1 <= i9 by A10, A46, XXREAL_0:2;
then 1 - 1 <= i9 - 1 by XREAL_1:9;
then A49: i9 - 1 is Element of NAT by INT_1:3;
A50: (i9 - 1) + 1 = i9 + 0 ;
then k <= i9 - 1 by A46, A49, NAT_1:13;
then 1 <= i9 - 1 by A10, XXREAL_0:2;
then A51: i9 - 1 in dom (SgmX (R,A)) by A43, A49, A48, Th2;
k <= i9 - 1 by A46, A49, A50, NAT_1:13;
then (SgmX (R,A)) /. (i9 - 1) = (SgmX (R,A)) /. (j + 1) by A27, A40, A49, A50, A47;
hence contradiction by A2, A3, A16, A25, A41, A50, A51, Lm4, Th8; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
end;
end;
(SgmX (R,A)) /. (j + 1) in rng (SgmX (R,A)) by A25, A26, FUNCT_1:def_3;
then A52: (SgmX (R,A)) /. (j + 1) in A by A4, PRE_POLY:def_2;
then (SgmX (R,A)) /. (j + 1) in B by A2, XBOOLE_0:def_3;
then (SgmX (R,A)) /. (j + 1) in rng (SgmX (R,B)) by A3, PRE_POLY:def_2;
then consider l9 being set such that
A53: l9 in dom (SgmX (R,B)) and
A54: (SgmX (R,B)) . l9 = (SgmX (R,A)) /. (j + 1) by FUNCT_1:def_3;
reconsider l9 = l9 as Element of NAT by A53;
A55: (SgmX (R,B)) /. l9 = (SgmX (R,B)) . l9 by A53, PARTFUN1:def_6;
A56: 1 <= j + 1 by NAT_1:12;
j + 1 <= len (SgmX (R,A)) by A15, NAT_1:13;
then j + 1 in Seg (len (SgmX (R,A))) by A56, FINSEQ_1:1;
then j + 1 in dom (SgmX (R,A)) by FINSEQ_1:def_3;
then A57: [((SgmX (R,A)) /. (j + 1)),((SgmX (R,A)) /. l)] in R by A4, A20, A36, PRE_POLY:def_2;
1 <= l9 by A7, A53, FINSEQ_1:1;
then l9 > j + 2 by A37, A54, A55;
then [((SgmX (R,A)) /. l),((SgmX (R,A)) /. (j + 1))] in R by A3, A18, A21, A22, A53, A54, A55, PRE_POLY:def_2;
then (SgmX (R,A)) /. l = (SgmX (R,A)) /. (j + 1) by A5, A57, A52, RELAT_2:def_4;
hence contradiction by A2, A3, A25, A20, A36, Lm4, Th8; ::_thesis: verum
end;
end;
end;
let i be Element of NAT ; ::_thesis: ( k <= i & i <= len (SgmX (R,A)) implies (SgmX (R,B)) /. (i + 1) = (SgmX (R,A)) /. i )
assume that
A58: k <= i and
A59: i <= len (SgmX (R,A)) ; ::_thesis: (SgmX (R,B)) /. (i + 1) = (SgmX (R,A)) /. i
k <= len (SgmX (R,A)) by A58, A59, XXREAL_0:2;
then A60: k <= lensga by A6, FINSEQ_1:def_3;
then A61: k in dom (SgmX (R,A)) by A10, A6, FINSEQ_1:1;
A62: lensga <= lensgb by A12, NAT_1:11;
A63: S1[k]
proof
A64: (SgmX (R,A)) /. k = (SgmX (R,A)) . k by A61, PARTFUN1:def_6;
then (SgmX (R,A)) /. k in rng (SgmX (R,A)) by A61, FUNCT_1:def_3;
then (SgmX (R,A)) /. k in A by A4, PRE_POLY:def_2;
then (SgmX (R,A)) /. k in B by A2, XBOOLE_0:def_3;
then (SgmX (R,A)) /. k in rng (SgmX (R,B)) by A3, PRE_POLY:def_2;
then consider l being set such that
A65: l in dom (SgmX (R,B)) and
A66: (SgmX (R,B)) . l = (SgmX (R,A)) /. k by FUNCT_1:def_3;
reconsider l = l as Element of NAT by A65;
A67: (SgmX (R,B)) /. l = (SgmX (R,B)) . l by A65, PARTFUN1:def_6;
A68: 1 <= l by A7, A65, FINSEQ_1:1;
assume A69: not S1[k] ; ::_thesis: contradiction
then A70: l <> k + 1 by A65, A66, PARTFUN1:def_6;
percases ( l = k or l < k or k < l ) by XXREAL_0:1;
suppose l = k ; ::_thesis: contradiction
then (SgmX (R,A)) . k = a by A8, A9, A64, A66, PARTFUN1:def_6;
then a in rng (SgmX (R,A)) by A61, FUNCT_1:def_3;
hence contradiction by A1, A4, PRE_POLY:def_2; ::_thesis: verum
end;
supposeA71: l < k ; ::_thesis: contradiction
then l <= lensga by A60, XXREAL_0:2;
then A72: l in dom (SgmX (R,A)) by A6, A68, FINSEQ_1:1;
l <= k9 by A11, A71, NAT_1:13;
then (SgmX (R,A)) /. l = (SgmX (R,A)) /. k by A1, A2, A3, A8, A9, A66, A68, A67, Th9;
hence contradiction by A2, A3, A61, A71, A72, Lm4, Th8; ::_thesis: verum
end;
suppose k < l ; ::_thesis: contradiction
then A73: k + 1 <= l by NAT_1:13;
A74: 1 <= k + 1 by NAT_1:12;
then A75: k + 1 in dom (SgmX (R,B)) by A65, A73, Th2;
now__::_thesis:_not_(SgmX_(R,B))_/._(k_+_1)_=_a
assume (SgmX (R,B)) /. (k + 1) = a ; ::_thesis: contradiction
then k + 1 = k by A3, A8, A9, A75, Th8;
hence contradiction ; ::_thesis: verum
end;
then A76: not (SgmX (R,B)) /. (k + 1) in {a} by TARSKI:def_1;
k + 1 < l by A70, A73, XXREAL_0:1;
then A77: [((SgmX (R,B)) /. (k + 1)),((SgmX (R,B)) /. l)] in R by A3, A65, A75, PRE_POLY:def_2;
(SgmX (R,B)) /. l in rng (SgmX (R,B)) by A65, A67, FUNCT_1:def_3;
then A78: (SgmX (R,B)) /. l in B by A3, PRE_POLY:def_2;
(SgmX (R,B)) /. (k + 1) = (SgmX (R,B)) . (k + 1) by A65, A73, A74, Th2, PARTFUN1:def_6;
then (SgmX (R,B)) /. (k + 1) in rng (SgmX (R,B)) by A75, FUNCT_1:def_3;
then (SgmX (R,B)) /. (k + 1) in B by A3, PRE_POLY:def_2;
then (SgmX (R,B)) /. (k + 1) in A by A2, A76, XBOOLE_0:def_3;
then (SgmX (R,B)) /. (k + 1) in rng (SgmX (R,A)) by A4, PRE_POLY:def_2;
then consider l9 being set such that
A79: l9 in dom (SgmX (R,A)) and
A80: (SgmX (R,A)) . l9 = (SgmX (R,B)) /. (k + 1) by FUNCT_1:def_3;
reconsider l9 = l9 as Element of NAT by A79;
A81: (SgmX (R,A)) /. l9 = (SgmX (R,A)) . l9 by A79, PARTFUN1:def_6;
A82: 1 <= l9 by A6, A79, FINSEQ_1:1;
l9 <= lensga by A6, A79, FINSEQ_1:1;
then l9 <= lensgb by A62, XXREAL_0:2;
then A83: l9 in dom (SgmX (R,B)) by A7, A82, FINSEQ_1:1;
now__::_thesis:_not_l9_<_k
assume A84: l9 < k ; ::_thesis: contradiction
then l9 <= k9 by A11, NAT_1:13;
then (SgmX (R,B)) /. l9 = (SgmX (R,A)) /. l9 by A1, A2, A3, A8, A9, A82, Th9;
then l9 = k + 1 by A3, A75, A79, A80, A83, Th8, PARTFUN1:def_6;
hence contradiction by A84, XREAL_1:29; ::_thesis: verum
end;
then l9 > k by A69, A80, A81, XXREAL_0:1;
then [((SgmX (R,B)) /. l),((SgmX (R,B)) /. (k + 1))] in R by A4, A61, A66, A67, A79, A80, A81, PRE_POLY:def_2;
then (SgmX (R,B)) /. l = (SgmX (R,B)) /. (k + 1) by A5, A77, A78, RELAT_2:def_4;
hence contradiction by A69, A65, A66, PARTFUN1:def_6; ::_thesis: verum
end;
end;
end;
for j being Element of NAT st k <= j & j <= len (SgmX (R,A)) holds
S1[j] from INT_1:sch_8(A63, A13);
hence (SgmX (R,B)) /. (i + 1) = (SgmX (R,A)) /. i by A58, A59; ::_thesis: verum
end;
theorem Th11: :: POLYNOM2:11
for X being non empty set
for A being finite Subset of X
for a being Element of X st not a in A holds
for B being finite Subset of X st B = {a} \/ A holds
for R being Order of X st R linearly_orders B holds
for k being Element of NAT st k + 1 in dom (SgmX (R,B)) & (SgmX (R,B)) /. (k + 1) = a holds
SgmX (R,B) = Ins ((SgmX (R,A)),k,a)
proof
let X be non empty set ; ::_thesis: for A being finite Subset of X
for a being Element of X st not a in A holds
for B being finite Subset of X st B = {a} \/ A holds
for R being Order of X st R linearly_orders B holds
for k being Element of NAT st k + 1 in dom (SgmX (R,B)) & (SgmX (R,B)) /. (k + 1) = a holds
SgmX (R,B) = Ins ((SgmX (R,A)),k,a)
let A be finite Subset of X; ::_thesis: for a being Element of X st not a in A holds
for B being finite Subset of X st B = {a} \/ A holds
for R being Order of X st R linearly_orders B holds
for k being Element of NAT st k + 1 in dom (SgmX (R,B)) & (SgmX (R,B)) /. (k + 1) = a holds
SgmX (R,B) = Ins ((SgmX (R,A)),k,a)
let a be Element of X; ::_thesis: ( not a in A implies for B being finite Subset of X st B = {a} \/ A holds
for R being Order of X st R linearly_orders B holds
for k being Element of NAT st k + 1 in dom (SgmX (R,B)) & (SgmX (R,B)) /. (k + 1) = a holds
SgmX (R,B) = Ins ((SgmX (R,A)),k,a) )
assume A1: not a in A ; ::_thesis: for B being finite Subset of X st B = {a} \/ A holds
for R being Order of X st R linearly_orders B holds
for k being Element of NAT st k + 1 in dom (SgmX (R,B)) & (SgmX (R,B)) /. (k + 1) = a holds
SgmX (R,B) = Ins ((SgmX (R,A)),k,a)
let B be finite Subset of X; ::_thesis: ( B = {a} \/ A implies for R being Order of X st R linearly_orders B holds
for k being Element of NAT st k + 1 in dom (SgmX (R,B)) & (SgmX (R,B)) /. (k + 1) = a holds
SgmX (R,B) = Ins ((SgmX (R,A)),k,a) )
assume A2: B = {a} \/ A ; ::_thesis: for R being Order of X st R linearly_orders B holds
for k being Element of NAT st k + 1 in dom (SgmX (R,B)) & (SgmX (R,B)) /. (k + 1) = a holds
SgmX (R,B) = Ins ((SgmX (R,A)),k,a)
let R be Order of X; ::_thesis: ( R linearly_orders B implies for k being Element of NAT st k + 1 in dom (SgmX (R,B)) & (SgmX (R,B)) /. (k + 1) = a holds
SgmX (R,B) = Ins ((SgmX (R,A)),k,a) )
assume A3: R linearly_orders B ; ::_thesis: for k being Element of NAT st k + 1 in dom (SgmX (R,B)) & (SgmX (R,B)) /. (k + 1) = a holds
SgmX (R,B) = Ins ((SgmX (R,A)),k,a)
let k be Element of NAT ; ::_thesis: ( k + 1 in dom (SgmX (R,B)) & (SgmX (R,B)) /. (k + 1) = a implies SgmX (R,B) = Ins ((SgmX (R,A)),k,a) )
assume that
A4: k + 1 in dom (SgmX (R,B)) and
A5: (SgmX (R,B)) /. (k + 1) = a ; ::_thesis: SgmX (R,B) = Ins ((SgmX (R,A)),k,a)
set sgb = SgmX (R,B);
set sga = Ins ((SgmX (R,A)),k,a);
A6: dom (SgmX (R,B)) = Seg (len (SgmX (R,B))) by FINSEQ_1:def_3;
then k + 1 <= len (SgmX (R,B)) by A4, FINSEQ_1:1;
then A7: (k + 1) - 1 <= (len (SgmX (R,B))) - 1 by XREAL_1:9;
A8: (k + 1) - 1 = k + 0 ;
A9: len (SgmX (R,B)) = card B by A3, PRE_POLY:11
.= (card A) + 1 by A1, A2, CARD_2:41
.= (len (SgmX (R,A))) + 1 by A2, A3, Lm4, PRE_POLY:11 ;
then A10: dom (SgmX (R,B)) = Seg (len (Ins ((SgmX (R,A)),k,a))) by A6, FINSEQ_5:69
.= dom (Ins ((SgmX (R,A)),k,a)) by FINSEQ_1:def_3 ;
A11: for i being Nat st 1 <= i & i <= len (SgmX (R,B)) holds
(SgmX (R,B)) . i = (Ins ((SgmX (R,A)),k,a)) . i
proof
let i be Nat; ::_thesis: ( 1 <= i & i <= len (SgmX (R,B)) implies (SgmX (R,B)) . i = (Ins ((SgmX (R,A)),k,a)) . i )
assume that
A12: 1 <= i and
A13: i <= len (SgmX (R,B)) ; ::_thesis: (SgmX (R,B)) . i = (Ins ((SgmX (R,A)),k,a)) . i
A14: i in Seg (len (SgmX (R,B))) by A12, A13, FINSEQ_1:1;
then A15: i in dom (SgmX (R,B)) by FINSEQ_1:def_3;
A16: i in dom (Ins ((SgmX (R,A)),k,a)) by A10, A14, FINSEQ_1:def_3;
percases ( i = k + 1 or i <> k + 1 ) ;
supposeA17: i = k + 1 ; ::_thesis: (SgmX (R,B)) . i = (Ins ((SgmX (R,A)),k,a)) . i
thus (SgmX (R,B)) . i = (SgmX (R,B)) /. i by A15, PARTFUN1:def_6
.= (Ins ((SgmX (R,A)),k,a)) /. (k + 1) by A5, A9, A7, A17, FINSEQ_5:73
.= (Ins ((SgmX (R,A)),k,a)) . i by A16, A17, PARTFUN1:def_6 ; ::_thesis: verum
end;
supposeA18: i <> k + 1 ; ::_thesis: (SgmX (R,B)) . i = (Ins ((SgmX (R,A)),k,a)) . i
now__::_thesis:_(_(_i_<_k_+_1_&_(SgmX_(R,B))_._i_=_(Ins_((SgmX_(R,A)),k,a))_._i_)_or_(_k_+_1_<=_i_&_(SgmX_(R,B))_._i_=_(Ins_((SgmX_(R,A)),k,a))_._i_)_)
percases ( i < k + 1 or k + 1 <= i ) ;
case i < k + 1 ; ::_thesis: (SgmX (R,B)) . i = (Ins ((SgmX (R,A)),k,a)) . i
then A19: i <= k by NAT_1:13;
(SgmX (R,A)) | (Seg k) is FinSequence by FINSEQ_1:15;
then dom ((SgmX (R,A)) | (Seg k)) = Seg k by A9, A7, FINSEQ_1:17;
then i in dom ((SgmX (R,A)) | (Seg k)) by A12, A19, FINSEQ_1:1;
then A20: i in dom ((SgmX (R,A)) | k) by FINSEQ_1:def_15;
thus (SgmX (R,B)) . i = (SgmX (R,B)) /. i by A15, PARTFUN1:def_6
.= (SgmX (R,A)) /. i by A1, A2, A3, A4, A5, A8, A12, A14, A19, Th9
.= (Ins ((SgmX (R,A)),k,a)) /. i by A20, FINSEQ_5:72
.= (Ins ((SgmX (R,A)),k,a)) . i by A16, PARTFUN1:def_6 ; ::_thesis: verum
end;
caseA21: k + 1 <= i ; ::_thesis: (SgmX (R,B)) . i = (Ins ((SgmX (R,A)),k,a)) . i
1 - 1 <= i - 1 by A12, XREAL_1:9;
then reconsider i9 = i - 1 as Element of NAT by INT_1:3;
A22: i9 + 1 = i + 0 ;
k + 1 < i by A18, A21, XXREAL_0:1;
then A23: k + 1 <= i9 by A22, NAT_1:13;
A24: i9 <= (len (SgmX (R,B))) - 1 by A13, XREAL_1:9;
thus (SgmX (R,B)) . i = (SgmX (R,B)) /. (i9 + 1) by A15, PARTFUN1:def_6
.= (SgmX (R,A)) /. i9 by A1, A2, A3, A4, A5, A9, A24, A23, Th10
.= (Ins ((SgmX (R,A)),k,a)) /. (i9 + 1) by A9, A24, A23, FINSEQ_5:74
.= (Ins ((SgmX (R,A)),k,a)) . i by A16, PARTFUN1:def_6 ; ::_thesis: verum
end;
end;
end;
hence (SgmX (R,B)) . i = (Ins ((SgmX (R,A)),k,a)) . i ; ::_thesis: verum
end;
end;
end;
len (SgmX (R,B)) = len (Ins ((SgmX (R,A)),k,a)) by A9, FINSEQ_5:69;
hence SgmX (R,B) = Ins ((SgmX (R,A)),k,a) by A11, FINSEQ_1:14; ::_thesis: verum
end;
begin
theorem Th12: :: POLYNOM2:12
for X being set
for b being bag of X st support b = {} holds
b = EmptyBag X
proof
let X be set ; ::_thesis: for b being bag of X st support b = {} holds
b = EmptyBag X
let b be bag of X; ::_thesis: ( support b = {} implies b = EmptyBag X )
A1: X = dom (EmptyBag X) by PARTFUN1:def_2;
assume A2: support b = {} ; ::_thesis: b = EmptyBag X
A3: for u being set st u in X holds
b . u = (EmptyBag X) . u
proof
let u be set ; ::_thesis: ( u in X implies b . u = (EmptyBag X) . u )
assume u in X ; ::_thesis: b . u = (EmptyBag X) . u
b . u = 0 by A2, PRE_POLY:def_7;
hence b . u = (EmptyBag X) . u by PRE_POLY:52; ::_thesis: verum
end;
X = dom b by PARTFUN1:def_2;
hence b = EmptyBag X by A1, A3, FUNCT_1:2; ::_thesis: verum
end;
Lm6: for X being set
for b being bag of X holds b is PartFunc of X,NAT
proof
let X be set ; ::_thesis: for b being bag of X holds b is PartFunc of X,NAT
let b be bag of X; ::_thesis: b is PartFunc of X,NAT
for u being set st u in b holds
u in [:X,NAT:]
proof
let u be set ; ::_thesis: ( u in b implies u in [:X,NAT:] )
A1: rng b c= NAT by VALUED_0:def_6;
assume A2: u in b ; ::_thesis: u in [:X,NAT:]
then consider u1, u2 being set such that
A3: u = [u1,u2] by RELAT_1:def_1;
u1 in dom b by A2, A3, XTUPLE_0:def_12;
then A4: u1 in X ;
u2 in rng b by A2, A3, XTUPLE_0:def_13;
hence u in [:X,NAT:] by A3, A4, A1, ZFMISC_1:def_2; ::_thesis: verum
end;
hence b is PartFunc of X,NAT by TARSKI:def_3; ::_thesis: verum
end;
definition
let X be set ;
let b be bag of X;
attrb is empty means :Def1: :: POLYNOM2:def 1
b = EmptyBag X;
end;
:: deftheorem Def1 defines empty POLYNOM2:def_1_:_
for X being set
for b being bag of X holds
( b is empty iff b = EmptyBag X );
registration
let X be non empty set ;
cluster Relation-like X -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support non empty for set ;
existence
not for b1 being bag of X holds b1 is empty
proof
set x = the Element of X;
set b = (EmptyBag X) +* ( the Element of X,1);
take (EmptyBag X) +* ( the Element of X,1) ; ::_thesis: not (EmptyBag X) +* ( the Element of X,1) is empty
dom (EmptyBag X) = X by PARTFUN1:def_2;
then A1: ((EmptyBag X) +* ( the Element of X,1)) . the Element of X = ((EmptyBag X) +* ( the Element of X .--> 1)) . the Element of X by FUNCT_7:def_3;
dom ( the Element of X .--> 1) = { the Element of X} by FUNCOP_1:13;
then the Element of X in dom ( the Element of X .--> 1) by TARSKI:def_1;
then ((EmptyBag X) +* ( the Element of X,1)) . the Element of X = ( the Element of X .--> 1) . the Element of X by A1, FUNCT_4:13
.= 1 by FUNCOP_1:72 ;
then ((EmptyBag X) +* ( the Element of X,1)) . the Element of X <> (EmptyBag X) . the Element of X by PRE_POLY:52;
hence not (EmptyBag X) +* ( the Element of X,1) is empty by Def1; ::_thesis: verum
end;
end;
definition
let X be set ;
let b be bag of X;
:: original: support
redefine func support b -> finite Subset of X;
coherence
support b is finite Subset of X
proof
A1: support b c= dom b by PRE_POLY:37;
for x being set st x in support b holds
x in X
proof
let x be set ; ::_thesis: ( x in support b implies x in X )
assume x in support b ; ::_thesis: x in X
then x in dom b by A1;
hence x in X ; ::_thesis: verum
end;
hence support b is finite Subset of X by TARSKI:def_3; ::_thesis: verum
end;
end;
theorem Th13: :: POLYNOM2:13
for n being Ordinal
for b being bag of n holds RelIncl n linearly_orders support b
proof
let n be Ordinal; ::_thesis: for b being bag of n holds RelIncl n linearly_orders support b
let b be bag of n; ::_thesis: RelIncl n linearly_orders support b
set R = RelIncl n;
set s = support b;
for x, y being set st x in support b & y in support b & x <> y & not [x,y] in RelIncl n holds
[y,x] in RelIncl n
proof
let x, y be set ; ::_thesis: ( x in support b & y in support b & x <> y & not [x,y] in RelIncl n implies [y,x] in RelIncl n )
assume that
A1: x in support b and
A2: y in support b and
x <> y ; ::_thesis: ( [x,y] in RelIncl n or [y,x] in RelIncl n )
assume A3: not [x,y] in RelIncl n ; ::_thesis: [y,x] in RelIncl n
reconsider x = x, y = y as Ordinal by A1, A2;
y c= x by A1, A2, A3, WELLORD2:def_1;
hence [y,x] in RelIncl n by A1, A2, WELLORD2:def_1; ::_thesis: verum
end;
then A4: RelIncl n is_connected_in support b by RELAT_2:def_6;
A5: RelIncl n is_antisymmetric_in support b by Lm1;
A6: RelIncl n is_transitive_in support b by Lm1;
RelIncl n is_reflexive_in support b by Lm1;
hence RelIncl n linearly_orders support b by A4, A5, A6, ORDERS_1:def_8; ::_thesis: verum
end;
definition
let X be set ;
let x be FinSequence of X;
let b be bag of X;
:: original: *
redefine funcb * x -> PartFunc of NAT,NAT;
coherence
x * b is PartFunc of NAT,NAT
proof
reconsider b = b as PartFunc of X,NAT by Lm6;
b * x c= [:NAT,NAT:] ;
hence x * b is PartFunc of NAT,NAT ; ::_thesis: verum
end;
end;
definition
let n be Ordinal;
let b be bag of n;
let L be non trivial well-unital doubleLoopStr ;
let x be Function of n,L;
func eval (b,x) -> Element of L means :Def2: :: POLYNOM2:def 2
ex y being FinSequence of the carrier of L st
( len y = len (SgmX ((RelIncl n),(support b))) & it = Product y & ( for i being Element of NAT st 1 <= i & i <= len y holds
y /. i = (power L) . (((x * (SgmX ((RelIncl n),(support b)))) /. i),((b * (SgmX ((RelIncl n),(support b)))) /. i)) ) );
existence
ex b1 being Element of L ex y being FinSequence of the carrier of L st
( len y = len (SgmX ((RelIncl n),(support b))) & b1 = Product y & ( for i being Element of NAT st 1 <= i & i <= len y holds
y /. i = (power L) . (((x * (SgmX ((RelIncl n),(support b)))) /. i),((b * (SgmX ((RelIncl n),(support b)))) /. i)) ) )
proof
set S = SgmX ((RelIncl n),(support b));
set l = len (SgmX ((RelIncl n),(support b)));
defpred S1[ Element of NAT , Element of L] means $2 = (power L) . (((x * (SgmX ((RelIncl n),(support b)))) /. $1),((b * (SgmX ((RelIncl n),(support b)))) /. $1));
A1: for k being Element of NAT st k in Seg (len (SgmX ((RelIncl n),(support b)))) holds
ex x being Element of L st S1[k,x]
proof
let k be Element of NAT ; ::_thesis: ( k in Seg (len (SgmX ((RelIncl n),(support b)))) implies ex x being Element of L st S1[k,x] )
assume k in Seg (len (SgmX ((RelIncl n),(support b)))) ; ::_thesis: ex x being Element of L st S1[k,x]
then A2: k in dom (SgmX ((RelIncl n),(support b))) by FINSEQ_1:def_3;
take (power L) . (((x * (SgmX ((RelIncl n),(support b)))) /. k),((b * (SgmX ((RelIncl n),(support b)))) /. k)) ; ::_thesis: ( (power L) . (((x * (SgmX ((RelIncl n),(support b)))) /. k),((b * (SgmX ((RelIncl n),(support b)))) /. k)) is Element of L & S1[k,(power L) . (((x * (SgmX ((RelIncl n),(support b)))) /. k),((b * (SgmX ((RelIncl n),(support b)))) /. k))] )
dom b = n by PARTFUN1:def_2;
then rng (SgmX ((RelIncl n),(support b))) c= dom b by FINSEQ_1:def_4;
then dom (b * (SgmX ((RelIncl n),(support b)))) = dom (SgmX ((RelIncl n),(support b))) by RELAT_1:27;
then (b * (SgmX ((RelIncl n),(support b)))) /. k = (b * (SgmX ((RelIncl n),(support b)))) . k by A2, PARTFUN1:def_6;
hence ( (power L) . (((x * (SgmX ((RelIncl n),(support b)))) /. k),((b * (SgmX ((RelIncl n),(support b)))) /. k)) is Element of L & S1[k,(power L) . (((x * (SgmX ((RelIncl n),(support b)))) /. k),((b * (SgmX ((RelIncl n),(support b)))) /. k))] ) by BINOP_1:17; ::_thesis: verum
end;
consider p being FinSequence of the carrier of L such that
A3: ( dom p = Seg (len (SgmX ((RelIncl n),(support b)))) & ( for k being Element of NAT st k in Seg (len (SgmX ((RelIncl n),(support b)))) holds
S1[k,p /. k] ) ) from RECDEF_1:sch_17(A1);
take Product p ; ::_thesis: ex y being FinSequence of the carrier of L st
( len y = len (SgmX ((RelIncl n),(support b))) & Product p = Product y & ( for i being Element of NAT st 1 <= i & i <= len y holds
y /. i = (power L) . (((x * (SgmX ((RelIncl n),(support b)))) /. i),((b * (SgmX ((RelIncl n),(support b)))) /. i)) ) )
A4: len p = len (SgmX ((RelIncl n),(support b))) by A3, FINSEQ_1:def_3;
now__::_thesis:_for_m_being_Element_of_NAT_st_1_<=_m_&_m_<=_len_p_holds_
p_/._m_=_(power_L)_._(((x_*_(SgmX_((RelIncl_n),(support_b))))_/._m),((b_*_(SgmX_((RelIncl_n),(support_b))))_/._m))
let m be Element of NAT ; ::_thesis: ( 1 <= m & m <= len p implies p /. m = (power L) . (((x * (SgmX ((RelIncl n),(support b)))) /. m),((b * (SgmX ((RelIncl n),(support b)))) /. m)) )
assume ( 1 <= m & m <= len p ) ; ::_thesis: p /. m = (power L) . (((x * (SgmX ((RelIncl n),(support b)))) /. m),((b * (SgmX ((RelIncl n),(support b)))) /. m))
then m in Seg (len (SgmX ((RelIncl n),(support b)))) by A4, FINSEQ_1:1;
hence p /. m = (power L) . (((x * (SgmX ((RelIncl n),(support b)))) /. m),((b * (SgmX ((RelIncl n),(support b)))) /. m)) by A3; ::_thesis: verum
end;
hence ex y being FinSequence of the carrier of L st
( len y = len (SgmX ((RelIncl n),(support b))) & Product p = Product y & ( for i being Element of NAT st 1 <= i & i <= len y holds
y /. i = (power L) . (((x * (SgmX ((RelIncl n),(support b)))) /. i),((b * (SgmX ((RelIncl n),(support b)))) /. i)) ) ) by A4; ::_thesis: verum
end;
uniqueness
for b1, b2 being Element of L st ex y being FinSequence of the carrier of L st
( len y = len (SgmX ((RelIncl n),(support b))) & b1 = Product y & ( for i being Element of NAT st 1 <= i & i <= len y holds
y /. i = (power L) . (((x * (SgmX ((RelIncl n),(support b)))) /. i),((b * (SgmX ((RelIncl n),(support b)))) /. i)) ) ) & ex y being FinSequence of the carrier of L st
( len y = len (SgmX ((RelIncl n),(support b))) & b2 = Product y & ( for i being Element of NAT st 1 <= i & i <= len y holds
y /. i = (power L) . (((x * (SgmX ((RelIncl n),(support b)))) /. i),((b * (SgmX ((RelIncl n),(support b)))) /. i)) ) ) holds
b1 = b2
proof
set S = SgmX ((RelIncl n),(support b));
let a, c be Element of L; ::_thesis: ( ex y being FinSequence of the carrier of L st
( len y = len (SgmX ((RelIncl n),(support b))) & a = Product y & ( for i being Element of NAT st 1 <= i & i <= len y holds
y /. i = (power L) . (((x * (SgmX ((RelIncl n),(support b)))) /. i),((b * (SgmX ((RelIncl n),(support b)))) /. i)) ) ) & ex y being FinSequence of the carrier of L st
( len y = len (SgmX ((RelIncl n),(support b))) & c = Product y & ( for i being Element of NAT st 1 <= i & i <= len y holds
y /. i = (power L) . (((x * (SgmX ((RelIncl n),(support b)))) /. i),((b * (SgmX ((RelIncl n),(support b)))) /. i)) ) ) implies a = c )
assume that
A5: ex y1 being FinSequence of the carrier of L st
( len y1 = len (SgmX ((RelIncl n),(support b))) & a = Product y1 & ( for i being Element of NAT st 1 <= i & i <= len y1 holds
y1 /. i = (power L) . (((x * (SgmX ((RelIncl n),(support b)))) /. i),((b * (SgmX ((RelIncl n),(support b)))) /. i)) ) ) and
A6: ex y2 being FinSequence of the carrier of L st
( len y2 = len (SgmX ((RelIncl n),(support b))) & c = Product y2 & ( for i being Element of NAT st 1 <= i & i <= len y2 holds
y2 /. i = (power L) . (((x * (SgmX ((RelIncl n),(support b)))) /. i),((b * (SgmX ((RelIncl n),(support b)))) /. i)) ) ) ; ::_thesis: a = c
consider y1 being FinSequence of the carrier of L such that
A7: len y1 = len (SgmX ((RelIncl n),(support b))) and
A8: Product y1 = a and
A9: for i being Element of NAT st 1 <= i & i <= len y1 holds
y1 /. i = (power L) . (((x * (SgmX ((RelIncl n),(support b)))) /. i),((b * (SgmX ((RelIncl n),(support b)))) /. i)) by A5;
consider y2 being FinSequence of the carrier of L such that
A10: len y2 = len (SgmX ((RelIncl n),(support b))) and
A11: Product y2 = c and
A12: for i being Element of NAT st 1 <= i & i <= len y2 holds
y2 /. i = (power L) . (((x * (SgmX ((RelIncl n),(support b)))) /. i),((b * (SgmX ((RelIncl n),(support b)))) /. i)) by A6;
now__::_thesis:_for_i_being_Nat_st_1_<=_i_&_i_<=_len_y1_holds_
y1_._i_=_y2_._i
let i be Nat; ::_thesis: ( 1 <= i & i <= len y1 implies y1 . i = y2 . i )
assume that
A13: 1 <= i and
A14: i <= len y1 ; ::_thesis: y1 . i = y2 . i
i in Seg (len y2) by A7, A10, A13, A14, FINSEQ_1:1;
then A15: i in dom y2 by FINSEQ_1:def_3;
A16: i in Seg (len y1) by A13, A14, FINSEQ_1:1;
then i in dom y1 by FINSEQ_1:def_3;
hence y1 . i = y1 /. i by PARTFUN1:def_6
.= (power L) . (((x * (SgmX ((RelIncl n),(support b)))) /. i),((b * (SgmX ((RelIncl n),(support b)))) /. i)) by A9, A13, A14, A16
.= y2 /. i by A7, A10, A12, A13, A14, A16
.= y2 . i by A15, PARTFUN1:def_6 ;
::_thesis: verum
end;
hence a = c by A7, A8, A10, A11, FINSEQ_1:14; ::_thesis: verum
end;
end;
:: deftheorem Def2 defines eval POLYNOM2:def_2_:_
for n being Ordinal
for b being bag of n
for L being non trivial well-unital doubleLoopStr
for x being Function of n,L
for b5 being Element of L holds
( b5 = eval (b,x) iff ex y being FinSequence of the carrier of L st
( len y = len (SgmX ((RelIncl n),(support b))) & b5 = Product y & ( for i being Element of NAT st 1 <= i & i <= len y holds
y /. i = (power L) . (((x * (SgmX ((RelIncl n),(support b)))) /. i),((b * (SgmX ((RelIncl n),(support b)))) /. i)) ) ) );
Lm7: for X being set holds support (EmptyBag X) = {}
proof
let X be set ; ::_thesis: support (EmptyBag X) = {}
assume support (EmptyBag X) <> {} ; ::_thesis: contradiction
then reconsider S = support (EmptyBag X) as non empty set ;
set u = the Element of S;
(EmptyBag X) . the Element of S <> 0 by PRE_POLY:def_7;
hence contradiction by PRE_POLY:52; ::_thesis: verum
end;
theorem Th14: :: POLYNOM2:14
for n being Ordinal
for L being non trivial well-unital doubleLoopStr
for x being Function of n,L holds eval ((EmptyBag n),x) = 1. L
proof
let n be Ordinal; ::_thesis: for L being non trivial well-unital doubleLoopStr
for x being Function of n,L holds eval ((EmptyBag n),x) = 1. L
let L be non trivial well-unital doubleLoopStr ; ::_thesis: for x being Function of n,L holds eval ((EmptyBag n),x) = 1. L
let x be Function of n,L; ::_thesis: eval ((EmptyBag n),x) = 1. L
set b = EmptyBag n;
reconsider s = support (EmptyBag n) as empty Subset of n by Lm7;
consider y being FinSequence of the carrier of L such that
A1: len y = len (SgmX ((RelIncl n),(support (EmptyBag n)))) and
A2: eval ((EmptyBag n),x) = Product y and
for i being Element of NAT st 1 <= i & i <= len y holds
y /. i = (power L) . (((x * (SgmX ((RelIncl n),(support (EmptyBag n))))) /. i),(((EmptyBag n) * (SgmX ((RelIncl n),(support (EmptyBag n))))) /. i)) by Def2;
SgmX ((RelIncl n),s) = {} by Th7, Th13;
then y = <*> the carrier of L by A1;
then eval ((EmptyBag n),x) = 1_ L by A2, GROUP_4:8;
hence eval ((EmptyBag n),x) = 1. L ; ::_thesis: verum
end;
theorem Th15: :: POLYNOM2:15
for n being Ordinal
for L being non trivial well-unital doubleLoopStr
for u being set
for b being bag of n st support b = {u} holds
for x being Function of n,L holds eval (b,x) = (power L) . ((x . u),(b . u))
proof
let n be Ordinal; ::_thesis: for L being non trivial well-unital doubleLoopStr
for u being set
for b being bag of n st support b = {u} holds
for x being Function of n,L holds eval (b,x) = (power L) . ((x . u),(b . u))
let L be non trivial well-unital doubleLoopStr ; ::_thesis: for u being set
for b being bag of n st support b = {u} holds
for x being Function of n,L holds eval (b,x) = (power L) . ((x . u),(b . u))
let u be set ; ::_thesis: for b being bag of n st support b = {u} holds
for x being Function of n,L holds eval (b,x) = (power L) . ((x . u),(b . u))
let b be bag of n; ::_thesis: ( support b = {u} implies for x being Function of n,L holds eval (b,x) = (power L) . ((x . u),(b . u)) )
reconsider sb = support b as finite Subset of n ;
set sg = SgmX ((RelIncl n),sb);
assume A1: support b = {u} ; ::_thesis: for x being Function of n,L holds eval (b,x) = (power L) . ((x . u),(b . u))
then A2: u in support b by TARSKI:def_1;
let x be Function of n,L; ::_thesis: eval (b,x) = (power L) . ((x . u),(b . u))
A3: rng x c= the carrier of L by RELAT_1:def_19;
A4: n = dom x by FUNCT_2:def_1;
then x . u in rng x by A2, FUNCT_1:def_3;
then reconsider xu = x . u as Element of L by A3;
A5: RelIncl n linearly_orders sb by Th13;
then A6: rng (SgmX ((RelIncl n),sb)) = {u} by A1, PRE_POLY:def_2;
then A7: u in rng (SgmX ((RelIncl n),sb)) by TARSKI:def_1;
then A8: 1 in dom (SgmX ((RelIncl n),sb)) by FINSEQ_3:31;
then A9: (SgmX ((RelIncl n),sb)) . 1 in rng (SgmX ((RelIncl n),sb)) by FUNCT_1:def_3;
then A10: (SgmX ((RelIncl n),sb)) . 1 = u by A6, TARSKI:def_1;
then 1 in dom (x * (SgmX ((RelIncl n),sb))) by A4, A8, A2, FUNCT_1:11;
then A11: (x * (SgmX ((RelIncl n),sb))) /. 1 = (x * (SgmX ((RelIncl n),sb))) . 1 by PARTFUN1:def_6
.= x . ((SgmX ((RelIncl n),sb)) . 1) by A8, FUNCT_1:13
.= x . u by A6, A9, TARSKI:def_1 ;
dom b = n by PARTFUN1:def_2;
then 1 in dom (b * (SgmX ((RelIncl n),sb))) by A8, A10, A2, FUNCT_1:11;
then A12: (b * (SgmX ((RelIncl n),sb))) /. 1 = (b * (SgmX ((RelIncl n),sb))) . 1 by PARTFUN1:def_6
.= b . ((SgmX ((RelIncl n),sb)) . 1) by A8, FUNCT_1:13
.= b . u by A6, A9, TARSKI:def_1 ;
A13: (power L) . (xu,(b . u)) = (power L) . [xu,(b . u)] ;
A14: for v being set st v in dom (SgmX ((RelIncl n),sb)) holds
v in {1}
proof
let v be set ; ::_thesis: ( v in dom (SgmX ((RelIncl n),sb)) implies v in {1} )
assume A15: v in dom (SgmX ((RelIncl n),sb)) ; ::_thesis: v in {1}
assume A16: not v in {1} ; ::_thesis: contradiction
reconsider v = v as Element of NAT by A15;
(SgmX ((RelIncl n),sb)) /. v = (SgmX ((RelIncl n),sb)) . v by A15, PARTFUN1:def_6;
then A17: (SgmX ((RelIncl n),sb)) /. v in rng (SgmX ((RelIncl n),sb)) by A15, FUNCT_1:def_3;
A18: v <> 1 by A16, TARSKI:def_1;
A19: 1 < v
proof
consider k being Nat such that
A20: dom (SgmX ((RelIncl n),sb)) = Seg k by FINSEQ_1:def_2;
Seg k = { l where l is Element of NAT : ( 1 <= l & l <= k ) } by FINSEQ_1:def_1;
then ex m9 being Element of NAT st
( m9 = v & 1 <= m9 & m9 <= k ) by A15, A20;
hence 1 < v by A18, XXREAL_0:1; ::_thesis: verum
end;
(SgmX ((RelIncl n),sb)) /. 1 = (SgmX ((RelIncl n),sb)) . 1 by A7, A15, FINSEQ_3:31, PARTFUN1:def_6;
then (SgmX ((RelIncl n),sb)) /. 1 in rng (SgmX ((RelIncl n),sb)) by A8, FUNCT_1:def_3;
then (SgmX ((RelIncl n),sb)) /. 1 = u by A6, TARSKI:def_1
.= (SgmX ((RelIncl n),sb)) /. v by A6, A17, TARSKI:def_1 ;
hence contradiction by A5, A8, A15, A19, PRE_POLY:def_2; ::_thesis: verum
end;
consider y being FinSequence of the carrier of L such that
A21: len y = len (SgmX ((RelIncl n),(support b))) and
A22: eval (b,x) = Product y and
A23: for i being Element of NAT st 1 <= i & i <= len y holds
y /. i = (power L) . (((x * (SgmX ((RelIncl n),(support b)))) /. i),((b * (SgmX ((RelIncl n),(support b)))) /. i)) by Def2;
for v being set st v in {1} holds
v in dom (SgmX ((RelIncl n),sb)) by A8, TARSKI:def_1;
then dom (SgmX ((RelIncl n),sb)) = Seg 1 by A14, FINSEQ_1:2, TARSKI:1;
then A24: len (SgmX ((RelIncl n),sb)) = 1 by FINSEQ_1:def_3;
then y . 1 = y /. 1 by A21, FINSEQ_4:15
.= (power L) . (((x * (SgmX ((RelIncl n),sb))) /. 1),((b * (SgmX ((RelIncl n),sb))) /. 1)) by A21, A23, A24 ;
then y = <*((power L) . ((x . u),(b . u)))*> by A21, A24, A12, A11, FINSEQ_1:40;
hence eval (b,x) = (power L) . ((x . u),(b . u)) by A22, A13, GROUP_4:9; ::_thesis: verum
end;
Lm8: for n being Ordinal
for L being non empty non trivial right_complementable associative commutative add-associative right_zeroed well-unital distributive doubleLoopStr
for b1, b2 being bag of n
for u being set st not u in support b1 & support b2 = (support b1) \/ {u} & ( for u9 being set st u9 <> u holds
b2 . u9 = b1 . u9 ) holds
for x being Function of n,L
for a being Element of L st a = (power L) . ((x . u),(b2 . u)) holds
eval (b2,x) = a * (eval (b1,x))
proof
let n be Ordinal; ::_thesis: for L being non empty non trivial right_complementable associative commutative add-associative right_zeroed well-unital distributive doubleLoopStr
for b1, b2 being bag of n
for u being set st not u in support b1 & support b2 = (support b1) \/ {u} & ( for u9 being set st u9 <> u holds
b2 . u9 = b1 . u9 ) holds
for x being Function of n,L
for a being Element of L st a = (power L) . ((x . u),(b2 . u)) holds
eval (b2,x) = a * (eval (b1,x))
let L be non trivial right_complementable associative commutative add-associative right_zeroed well-unital distributive doubleLoopStr ; ::_thesis: for b1, b2 being bag of n
for u being set st not u in support b1 & support b2 = (support b1) \/ {u} & ( for u9 being set st u9 <> u holds
b2 . u9 = b1 . u9 ) holds
for x being Function of n,L
for a being Element of L st a = (power L) . ((x . u),(b2 . u)) holds
eval (b2,x) = a * (eval (b1,x))
let b1, b2 be bag of n; ::_thesis: for u being set st not u in support b1 & support b2 = (support b1) \/ {u} & ( for u9 being set st u9 <> u holds
b2 . u9 = b1 . u9 ) holds
for x being Function of n,L
for a being Element of L st a = (power L) . ((x . u),(b2 . u)) holds
eval (b2,x) = a * (eval (b1,x))
let u be set ; ::_thesis: ( not u in support b1 & support b2 = (support b1) \/ {u} & ( for u9 being set st u9 <> u holds
b2 . u9 = b1 . u9 ) implies for x being Function of n,L
for a being Element of L st a = (power L) . ((x . u),(b2 . u)) holds
eval (b2,x) = a * (eval (b1,x)) )
assume that
A1: not u in support b1 and
A2: support b2 = (support b1) \/ {u} and
A3: for u9 being set st u9 <> u holds
b2 . u9 = b1 . u9 ; ::_thesis: for x being Function of n,L
for a being Element of L st a = (power L) . ((x . u),(b2 . u)) holds
eval (b2,x) = a * (eval (b1,x))
u in {u} by TARSKI:def_1;
then A4: u in support b2 by A2, XBOOLE_0:def_3;
set sb2 = SgmX ((RelIncl n),(support b2));
set sb1 = SgmX ((RelIncl n),(support b1));
let x be Function of n,L; ::_thesis: for a being Element of L st a = (power L) . ((x . u),(b2 . u)) holds
eval (b2,x) = a * (eval (b1,x))
A5: n = dom x by FUNCT_2:def_1;
let a be Element of L; ::_thesis: ( a = (power L) . ((x . u),(b2 . u)) implies eval (b2,x) = a * (eval (b1,x)) )
assume A6: a = (power L) . ((x . u),(b2 . u)) ; ::_thesis: eval (b2,x) = a * (eval (b1,x))
RelIncl n linearly_orders support b2 by Th13;
then u in rng (SgmX ((RelIncl n),(support b2))) by A4, PRE_POLY:def_2;
then consider k being Nat such that
A7: k in dom (SgmX ((RelIncl n),(support b2))) and
A8: (SgmX ((RelIncl n),(support b2))) . k = u by FINSEQ_2:10;
A9: (SgmX ((RelIncl n),(support b2))) /. k = u by A7, A8, PARTFUN1:def_6;
reconsider u = u as Element of n by A4;
A10: dom (SgmX ((RelIncl n),(support b2))) = Seg (len (SgmX ((RelIncl n),(support b2)))) by FINSEQ_1:def_3;
then A11: k <= len (SgmX ((RelIncl n),(support b2))) by A7, FINSEQ_1:1;
A12: 1 <= k by A7, A10, FINSEQ_1:1;
then 1 - 1 <= k - 1 by XREAL_1:9;
then reconsider k9 = k - 1 as Element of NAT by INT_1:3;
A13: k9 + 1 = k + 0 ;
A14: RelIncl n linearly_orders support b1 by Th13;
percases ( n = {} or n <> {} ) ;
supposeA15: n = {} ; ::_thesis: eval (b2,x) = a * (eval (b1,x))
A16: b2 in Bags n by PRE_POLY:def_12;
u in {u} by TARSKI:def_1;
then u in support b2 by A2, XBOOLE_0:def_3;
then A17: b2 . u <> 0 by PRE_POLY:def_7;
Bags n = {(EmptyBag n)} by A15, PRE_POLY:51;
then b2 = EmptyBag n by A16, TARSKI:def_1;
hence eval (b2,x) = a * (eval (b1,x)) by A17, PRE_POLY:52; ::_thesis: verum
end;
suppose n <> {} ; ::_thesis: eval (b2,x) = a * (eval (b1,x))
then reconsider n9 = n as non empty Ordinal ;
reconsider x9 = x as Function of n9,L ;
reconsider b1 = b1, b2 = b2 as bag of n9 ;
reconsider sb2 = SgmX ((RelIncl n),(support b2)), sb1 = SgmX ((RelIncl n),(support b1)) as FinSequence of n9 ;
reconsider u = u as Element of n9 ;
consider yb2 being FinSequence of the carrier of L such that
A18: len yb2 = len sb2 and
A19: eval (b2,x9) = Product yb2 and
A20: for i being Element of NAT st 1 <= i & i <= len yb2 holds
yb2 /. i = (power L) . (((x * sb2) /. i),((b2 * sb2) /. i)) by Def2;
consider yb1 being FinSequence of the carrier of L such that
A21: len yb1 = len sb1 and
A22: eval (b1,x9) = Product yb1 and
A23: for i being Element of NAT st 1 <= i & i <= len yb1 holds
yb1 /. i = (power L) . (((x * sb1) /. i),((b1 * sb1) /. i)) by Def2;
set ytmp = Ins (yb1,k9,a);
A24: (len sb1) + 1 = (card (support b1)) + 1 by Th13, PRE_POLY:11
.= card (support b2) by A1, A2, CARD_2:41
.= len sb2 by Th13, PRE_POLY:11 ;
then A25: len yb2 = len (Ins (yb1,k9,a)) by A18, A21, FINSEQ_5:69;
A26: sb2 = Ins (sb1,k9,u) by A1, A2, A7, A9, A13, Th11, Th13;
A27: for i being Nat st 1 <= i & i <= len yb2 holds
yb2 . i = (Ins (yb1,k9,a)) . i
proof
let i be Nat; ::_thesis: ( 1 <= i & i <= len yb2 implies yb2 . i = (Ins (yb1,k9,a)) . i )
assume that
A28: 1 <= i and
A29: i <= len yb2 ; ::_thesis: yb2 . i = (Ins (yb1,k9,a)) . i
A30: i in Seg (len yb2) by A28, A29, FINSEQ_1:1;
then A31: yb2 /. i = (power L) . (((x * (Ins (sb1,k9,u))) /. i),((b2 * (Ins (sb1,k9,u))) /. i)) by A26, A20, A28, A29;
A32: i in dom yb2 by A30, FINSEQ_1:def_3;
i in Seg (len (Ins (yb1,k9,a))) by A25, A28, A29, FINSEQ_1:1;
then A33: i in dom (Ins (yb1,k9,a)) by FINSEQ_1:def_3;
A34: 1 - 1 <= i - 1 by A28, XREAL_1:9;
then A35: i - 1 is Element of NAT by INT_1:3;
now__::_thesis:_(_(_i_=_k_&_(Ins_(yb1,k9,a))_._i_=_yb2_._i_)_or_(_i_<>_k_&_yb2_._i_=_(Ins_(yb1,k9,a))_._i_)_)
percases ( i = k or i <> k ) ;
caseA36: i = k ; ::_thesis: (Ins (yb1,k9,a)) . i = yb2 . i
A37: sb2 . k in {u} by A8, TARSKI:def_1;
then A38: k in dom (x * sb2) by A5, A7, A8, FUNCT_1:11;
then A39: (x * sb2) /. k = (x * sb2) . k by PARTFUN1:def_6
.= x . u by A8, A38, FUNCT_1:12 ;
A40: support b2 c= dom b2 by PRE_POLY:37;
sb2 . k in support b2 by A2, A37, XBOOLE_0:def_3;
then A41: k in dom (b2 * sb2) by A7, A40, FUNCT_1:11;
then (b2 * sb2) /. k = (b2 * sb2) . k by PARTFUN1:def_6
.= b2 . u by A8, A41, FUNCT_1:12 ;
then A42: yb2 /. i = (power L) . ((x . u),(b2 . u)) by A20, A28, A29, A30, A36, A39;
A43: k9 <= len yb1 by A13, A18, A21, A24, A29, A36, XREAL_1:6;
thus (Ins (yb1,k9,a)) . i = (Ins (yb1,k9,a)) /. i by A33, PARTFUN1:def_6
.= yb2 /. i by A6, A13, A36, A43, A42, FINSEQ_5:73
.= yb2 . i by A32, PARTFUN1:def_6 ; ::_thesis: verum
end;
caseA44: i <> k ; ::_thesis: yb2 . i = (Ins (yb1,k9,a)) . i
len (Ins (sb1,k9,u)) = (len sb1) + 1 by FINSEQ_5:69;
then A45: dom (Ins (sb1,k9,u)) = Seg ((len sb1) + 1) by FINSEQ_1:def_3;
now__::_thesis:_(_(_i_<_k_&_yb2_._i_=_(Ins_(yb1,k9,a))_._i_)_or_(_i_>_k_&_yb2_._i_=_(Ins_(yb1,k9,a))_._i_)_)
percases ( i < k or i > k ) by A44, XXREAL_0:1;
caseA46: i < k ; ::_thesis: yb2 . i = (Ins (yb1,k9,a)) . i
then A47: i <= k9 by A13, NAT_1:13;
then A48: i in Seg k9 by A28, FINSEQ_1:1;
A49: yb1 | (Seg k9) is FinSequence by FINSEQ_1:15;
k9 <= len yb1 by A11, A13, A21, A24, XREAL_1:6;
then i in dom (yb1 | (Seg k9)) by A48, A49, FINSEQ_1:17;
then A50: i in dom (yb1 | k9) by FINSEQ_1:def_15;
A51: sb1 | (Seg k9) is FinSequence by FINSEQ_1:15;
A52: rng sb1 c= n by FINSEQ_1:def_4;
A53: i < len yb2 by A11, A18, A46, XXREAL_0:2;
then A54: i <= len yb1 by A18, A21, A24, NAT_1:13;
i <= len sb1 by A18, A24, A53, NAT_1:13;
then i in Seg (len sb1) by A28, FINSEQ_1:1;
then A55: i in dom sb1 by FINSEQ_1:def_3;
then A56: sb1 . i in rng sb1 by FUNCT_1:def_3;
then A57: i in dom (x * sb1) by A5, A55, A52, FUNCT_1:11;
A58: now__::_thesis:_not_sb1_/._i_=_u
assume sb1 /. i = u ; ::_thesis: contradiction
then sb1 . i = u by A55, PARTFUN1:def_6;
then u in rng sb1 by A55, FUNCT_1:def_3;
hence contradiction by A1, A14, PRE_POLY:def_2; ::_thesis: verum
end;
A59: k - 1 <= ((len sb1) + 1) - 1 by A11, A24, XREAL_1:9;
A60: rng (Ins (sb1,k9,u)) c= n by FINSEQ_1:def_4;
sb1 . i in n by A56, A52;
then sb1 . i in dom b1 by PARTFUN1:def_2;
then A61: i in dom (b1 * sb1) by A55, FUNCT_1:11;
i in Seg k9 by A28, A47, FINSEQ_1:1;
then i in dom (sb1 | (Seg k9)) by A59, A51, FINSEQ_1:17;
then A62: i in dom (sb1 | k9) by FINSEQ_1:def_15;
i <= (len sb1) + 1 by A11, A24, A46, XXREAL_0:2;
then A63: i in dom (Ins (sb1,k9,u)) by A28, A45, FINSEQ_1:1;
then A64: (Ins (sb1,k9,u)) . i in rng (Ins (sb1,k9,u)) by FUNCT_1:def_3;
then A65: i in dom (x * (Ins (sb1,k9,u))) by A5, A63, A60, FUNCT_1:11;
then A66: (x * (Ins (sb1,k9,u))) /. i = (x * (Ins (sb1,k9,u))) . i by PARTFUN1:def_6
.= x . ((Ins (sb1,k9,u)) . i) by A65, FUNCT_1:12
.= x . ((Ins (sb1,k9,u)) /. i) by A63, PARTFUN1:def_6
.= x . (sb1 /. i) by A62, FINSEQ_5:72
.= x . (sb1 . i) by A55, PARTFUN1:def_6
.= (x * sb1) . i by A57, FUNCT_1:12
.= (x * sb1) /. i by A57, PARTFUN1:def_6 ;
dom b2 = n by PARTFUN1:def_2;
then A67: i in dom (b2 * (Ins (sb1,k9,u))) by A63, A64, A60, FUNCT_1:11;
then (b2 * (Ins (sb1,k9,u))) /. i = (b2 * (Ins (sb1,k9,u))) . i by PARTFUN1:def_6
.= b2 . ((Ins (sb1,k9,u)) . i) by A67, FUNCT_1:12
.= b2 . ((Ins (sb1,k9,u)) /. i) by A63, PARTFUN1:def_6
.= b2 . (sb1 /. i) by A62, FINSEQ_5:72
.= b1 . (sb1 /. i) by A3, A58
.= b1 . (sb1 . i) by A55, PARTFUN1:def_6
.= (b1 * sb1) . i by A61, FUNCT_1:12
.= (b1 * sb1) /. i by A61, PARTFUN1:def_6 ;
then A68: yb2 /. i = yb1 /. i by A23, A28, A30, A31, A54, A66
.= (Ins (yb1,k9,a)) /. i by A50, FINSEQ_5:72 ;
thus yb2 . i = yb2 /. i by A32, PARTFUN1:def_6
.= (Ins (yb1,k9,a)) . i by A33, A68, PARTFUN1:def_6 ; ::_thesis: verum
end;
caseA69: i > k ; ::_thesis: yb2 . i = (Ins (yb1,k9,a)) . i
reconsider i1 = i - 1 as Element of NAT by A34, INT_1:3;
A70: (i - 1) + 1 = i + 0 ;
A71: rng sb1 c= n by FINSEQ_1:def_4;
A72: i - 1 <= (len sb2) - 1 by A18, A29, XREAL_1:9;
1 < i by A12, A69, XXREAL_0:2;
then A73: 1 <= i - 1 by A35, A70, NAT_1:13;
then i1 in Seg (len sb1) by A24, A72, FINSEQ_1:1;
then A74: i1 in dom sb1 by FINSEQ_1:def_3;
then A75: sb1 . i1 in rng sb1 by FUNCT_1:def_3;
then A76: i1 in dom (x * sb1) by A5, A74, A71, FUNCT_1:11;
dom b1 = n by PARTFUN1:def_2;
then A77: i1 in dom (b1 * sb1) by A74, A75, A71, FUNCT_1:11;
A78: now__::_thesis:_not_sb1_/._i1_=_u
assume sb1 /. i1 = u ; ::_thesis: contradiction
then sb1 . i1 = u by A74, PARTFUN1:def_6;
then u in rng sb1 by A74, FUNCT_1:def_3;
hence contradiction by A1, A14, PRE_POLY:def_2; ::_thesis: verum
end;
A79: i = i1 + 1 ;
A80: rng (Ins (sb1,k9,u)) c= n by FINSEQ_1:def_4;
A81: i1 + 1 = i + 0 ;
then A82: k9 + 1 <= i1 by A69, NAT_1:13;
A83: i in dom (Ins (sb1,k9,u)) by A18, A24, A28, A29, A45, FINSEQ_1:1;
then A84: (Ins (sb1,k9,u)) . i in rng (Ins (sb1,k9,u)) by FUNCT_1:def_3;
then A85: i in dom (x * (Ins (sb1,k9,u))) by A5, A83, A80, FUNCT_1:11;
then A86: (x * (Ins (sb1,k9,u))) /. i = (x * (Ins (sb1,k9,u))) . i by PARTFUN1:def_6
.= x . ((Ins (sb1,k9,u)) . i) by A85, FUNCT_1:12
.= x . ((Ins (sb1,k9,u)) /. i) by A83, PARTFUN1:def_6
.= x . (sb1 /. i1) by A24, A72, A81, A82, FINSEQ_5:74
.= x . (sb1 . i1) by A74, PARTFUN1:def_6
.= (x * sb1) . i1 by A76, FUNCT_1:12
.= (x * sb1) /. i1 by A76, PARTFUN1:def_6 ;
dom b2 = n by PARTFUN1:def_2;
then A87: i in dom (b2 * (Ins (sb1,k9,u))) by A83, A84, A80, FUNCT_1:11;
then (b2 * (Ins (sb1,k9,u))) /. i = (b2 * (Ins (sb1,k9,u))) . i by PARTFUN1:def_6
.= b2 . ((Ins (sb1,k9,u)) . i) by A87, FUNCT_1:12
.= b2 . ((Ins (sb1,k9,u)) /. i) by A83, PARTFUN1:def_6
.= b2 . (sb1 /. i1) by A24, A72, A81, A82, FINSEQ_5:74
.= b1 . (sb1 /. i1) by A3, A78
.= b1 . (sb1 . i1) by A74, PARTFUN1:def_6
.= (b1 * sb1) . i1 by A77, FUNCT_1:12
.= (b1 * sb1) /. i1 by A77, PARTFUN1:def_6 ;
then A88: yb2 /. i = yb1 /. i1 by A21, A23, A24, A31, A73, A72, A86
.= (Ins (yb1,k9,a)) /. i by A21, A24, A72, A79, A82, FINSEQ_5:74 ;
thus yb2 . i = yb2 /. i by A32, PARTFUN1:def_6
.= (Ins (yb1,k9,a)) . i by A33, A88, PARTFUN1:def_6 ; ::_thesis: verum
end;
end;
end;
hence yb2 . i = (Ins (yb1,k9,a)) . i ; ::_thesis: verum
end;
end;
end;
hence yb2 . i = (Ins (yb1,k9,a)) . i ; ::_thesis: verum
end;
Product (((yb1 | k9) ^ <*a*>) ^ (yb1 /^ k9)) = (Product ((yb1 | k9) ^ <*a*>)) * (Product (yb1 /^ k9)) by GROUP_4:5
.= ((Product (yb1 | k9)) * (Product <*a*>)) * (Product (yb1 /^ k9)) by GROUP_4:5
.= ((Product (yb1 | k9)) * (Product (yb1 /^ k9))) * (Product <*a*>) by GROUP_1:def_3
.= (Product ((yb1 | k9) ^ (yb1 /^ k9))) * (Product <*a*>) by GROUP_4:5
.= (Product yb1) * (Product <*a*>) by RFINSEQ:8
.= (eval (b1,x9)) * a by A22, GROUP_4:9 ;
then Product (Ins (yb1,k9,a)) = (eval (b1,x9)) * a by FINSEQ_5:def_4;
hence eval (b2,x) = a * (eval (b1,x)) by A19, A25, A27, FINSEQ_1:14; ::_thesis: verum
end;
end;
end;
Lm9: for n being Ordinal
for L being non empty non trivial right_complementable associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b1 being bag of n st ex u being set st support b1 = {u} holds
for b2 being bag of n
for x being Function of n,L holds eval ((b1 + b2),x) = (eval (b1,x)) * (eval (b2,x))
proof
let n be Ordinal; ::_thesis: for L being non empty non trivial right_complementable associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b1 being bag of n st ex u being set st support b1 = {u} holds
for b2 being bag of n
for x being Function of n,L holds eval ((b1 + b2),x) = (eval (b1,x)) * (eval (b2,x))
let L be non trivial right_complementable associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr ; ::_thesis: for b1 being bag of n st ex u being set st support b1 = {u} holds
for b2 being bag of n
for x being Function of n,L holds eval ((b1 + b2),x) = (eval (b1,x)) * (eval (b2,x))
let b1 be bag of n; ::_thesis: ( ex u being set st support b1 = {u} implies for b2 being bag of n
for x being Function of n,L holds eval ((b1 + b2),x) = (eval (b1,x)) * (eval (b2,x)) )
assume ex u being set st support b1 = {u} ; ::_thesis: for b2 being bag of n
for x being Function of n,L holds eval ((b1 + b2),x) = (eval (b1,x)) * (eval (b2,x))
then consider u being set such that
A1: support b1 = {u} ;
let b2 be bag of n; ::_thesis: for x being Function of n,L holds eval ((b1 + b2),x) = (eval (b1,x)) * (eval (b2,x))
let x be Function of n,L; ::_thesis: eval ((b1 + b2),x) = (eval (b1,x)) * (eval (b2,x))
A2: support (b1 + b2) = (support b2) \/ {u} by A1, PRE_POLY:38;
A3: for u9 being set st u9 <> u holds
(b1 + b2) . u9 = b2 . u9
proof
let u9 be set ; ::_thesis: ( u9 <> u implies (b1 + b2) . u9 = b2 . u9 )
assume u9 <> u ; ::_thesis: (b1 + b2) . u9 = b2 . u9
then A4: not u9 in support b1 by A1, TARSKI:def_1;
thus (b1 + b2) . u9 = (b1 . u9) + (b2 . u9) by PRE_POLY:def_5
.= 0 + (b2 . u9) by A4, PRE_POLY:def_7
.= b2 . u9 ; ::_thesis: verum
end;
set sb2 = SgmX ((RelIncl n),(support b2));
set sb1b2 = SgmX ((RelIncl n),(support (b1 + b2)));
A5: n c= dom x by FUNCT_2:def_1;
A6: RelIncl n linearly_orders support b2 by Th13;
consider yb1b2 being FinSequence of the carrier of L such that
A7: len yb1b2 = len (SgmX ((RelIncl n),(support (b1 + b2)))) and
A8: eval ((b1 + b2),x) = Product yb1b2 and
A9: for i being Element of NAT st 1 <= i & i <= len yb1b2 holds
yb1b2 /. i = (power L) . (((x * (SgmX ((RelIncl n),(support (b1 + b2))))) /. i),(((b1 + b2) * (SgmX ((RelIncl n),(support (b1 + b2))))) /. i)) by Def2;
consider yb2 being FinSequence of the carrier of L such that
A10: len yb2 = len (SgmX ((RelIncl n),(support b2))) and
A11: eval (b2,x) = Product yb2 and
A12: for i being Element of NAT st 1 <= i & i <= len yb2 holds
yb2 /. i = (power L) . (((x * (SgmX ((RelIncl n),(support b2)))) /. i),((b2 * (SgmX ((RelIncl n),(support b2)))) /. i)) by Def2;
percases ( u in support b2 or not u in support b2 ) ;
supposeA13: u in support b2 ; ::_thesis: eval ((b1 + b2),x) = (eval (b1,x)) * (eval (b2,x))
consider sb2k being Nat such that
A14: dom (SgmX ((RelIncl n),(support b2))) = Seg sb2k by FINSEQ_1:def_2;
A15: for v being set st v in support b2 holds
v in support (b1 + b2)
proof
let v be set ; ::_thesis: ( v in support b2 implies v in support (b1 + b2) )
assume A16: v in support b2 ; ::_thesis: v in support (b1 + b2)
now__::_thesis:_(_(_u_=_v_&_v_in_support_(b1_+_b2)_)_or_(_u_<>_v_&_v_in_support_(b1_+_b2)_)_)
percases ( u = v or u <> v ) ;
case u = v ; ::_thesis: v in support (b1 + b2)
then v in {u} by TARSKI:def_1;
hence v in support (b1 + b2) by A2, XBOOLE_0:def_3; ::_thesis: verum
end;
case u <> v ; ::_thesis: v in support (b1 + b2)
then (b1 + b2) . v = b2 . v by A3;
then (b1 + b2) . v <> 0 by A16, PRE_POLY:def_7;
hence v in support (b1 + b2) by PRE_POLY:def_7; ::_thesis: verum
end;
end;
end;
hence v in support (b1 + b2) ; ::_thesis: verum
end;
A17: for v being set st v in support (b1 + b2) holds
v in support b2
proof
let v be set ; ::_thesis: ( v in support (b1 + b2) implies v in support b2 )
assume A18: v in support (b1 + b2) ; ::_thesis: v in support b2
now__::_thesis:_(_(_v_in_{u}_&_v_in_support_b2_)_or_(_v_in_support_b2_&_v_in_support_b2_)_)
percases ( v in {u} or v in support b2 ) by A2, A18, XBOOLE_0:def_3;
case v in {u} ; ::_thesis: v in support b2
hence v in support b2 by A13, TARSKI:def_1; ::_thesis: verum
end;
case v in support b2 ; ::_thesis: v in support b2
hence v in support b2 ; ::_thesis: verum
end;
end;
end;
hence v in support b2 ; ::_thesis: verum
end;
then A19: len yb1b2 = len yb2 by A7, A10, A15, TARSKI:1;
A20: support (b1 + b2) = support b2 by A17, A15, TARSKI:1;
u in rng (SgmX ((RelIncl n),(support b2))) by A6, A13, PRE_POLY:def_2;
then consider k being Nat such that
A21: k in dom (SgmX ((RelIncl n),(support b2))) and
A22: (SgmX ((RelIncl n),(support b2))) . k = u by FINSEQ_2:10;
reconsider k = k as Element of NAT by ORDINAL1:def_12;
A23: 1 <= k by A21, A14, FINSEQ_1:1;
A24: support b2 c= dom b2 by PRE_POLY:37;
then A25: k in dom (b2 * (SgmX ((RelIncl n),(support b2)))) by A13, A21, A22, FUNCT_1:11;
then A26: (b2 * (SgmX ((RelIncl n),(support b2)))) /. k = (b2 * (SgmX ((RelIncl n),(support b2)))) . k by PARTFUN1:def_6;
then A27: (b2 * (SgmX ((RelIncl n),(support b2)))) /. k = b2 . u by A22, A25, FUNCT_1:12;
A28: rng x c= the carrier of L by RELAT_1:def_19;
consider sb1b2k being Nat such that
A29: dom (SgmX ((RelIncl n),(support (b1 + b2)))) = Seg sb1b2k by FINSEQ_1:def_2;
support (b1 + b2) c= dom (b1 + b2) by PRE_POLY:37;
then A30: k in dom ((b1 + b2) * (SgmX ((RelIncl n),(support b2)))) by A13, A20, A21, A22, FUNCT_1:11;
then A31: ((b1 + b2) * (SgmX ((RelIncl n),(support b2)))) /. k = ((b1 + b2) * (SgmX ((RelIncl n),(support b2)))) . k by PARTFUN1:def_6
.= (b1 + b2) . u by A22, A30, FUNCT_1:12 ;
consider i being Nat such that
A32: dom yb2 = Seg i by FINSEQ_1:def_2;
reconsider sb2k = sb2k, sb1b2k = sb1b2k as Element of NAT by ORDINAL1:def_12;
A33: k <= sb2k by A21, A14, FINSEQ_1:1;
i in NAT by ORDINAL1:def_12;
then A34: len yb2 = i by A32, FINSEQ_1:def_3;
A35: sb2k = len yb2 by A10, A14, FINSEQ_1:def_3;
then A36: k in dom yb2 by A21, A14, FINSEQ_1:def_3;
reconsider bbS = b2 * (SgmX ((RelIncl n),(support b2))) as PartFunc of NAT,NAT ;
A37: bbS /. k = bbS . k by A25, PARTFUN1:def_6;
A38: sb2k = len (SgmX ((RelIncl n),(support b2))) by A14, FINSEQ_1:def_3
.= len (SgmX ((RelIncl n),(support (b1 + b2)))) by A17, A15, TARSKI:1
.= sb1b2k by A29, FINSEQ_1:def_3 ;
then len yb1b2 = sb2k by A7, A29, FINSEQ_1:def_3;
then A39: yb1b2 /. k = (power L) . (((x * (SgmX ((RelIncl n),(support b2)))) /. k),(((b1 + b2) * (SgmX ((RelIncl n),(support b2)))) /. k)) by A9, A20, A23, A33
.= (power L) . (((x * (SgmX ((RelIncl n),(support b2)))) /. k),((b1 . u) + (b2 . u))) by A31, PRE_POLY:def_5
.= ((power L) . (((x * (SgmX ((RelIncl n),(support b2)))) /. k),(b1 . u))) * ((power L) . (((x * (SgmX ((RelIncl n),(support b2)))) /. k),(bbS /. k))) by A26, A27, A37, Th1
.= ((power L) . (((x * (SgmX ((RelIncl n),(support b2)))) /. k),(b1 . u))) * (yb2 /. k) by A12, A23, A33, A35, A37, A26 ;
A40: dom (b1 + b2) = n by PARTFUN1:def_2;
A41: for i9 being Element of NAT st i9 in dom yb2 & i9 <> k holds
yb1b2 /. i9 = yb2 /. i9
proof
rng (SgmX ((RelIncl n),(support (b1 + b2)))) c= dom b2 by A6, A20, A24, PRE_POLY:def_2;
then A42: rng (SgmX ((RelIncl n),(support (b1 + b2)))) c= n by PARTFUN1:def_2;
A43: rng (SgmX ((RelIncl n),(support b2))) c= dom b2 by A6, A24, PRE_POLY:def_2;
let i9 be Element of NAT ; ::_thesis: ( i9 in dom yb2 & i9 <> k implies yb1b2 /. i9 = yb2 /. i9 )
assume that
A44: i9 in dom yb2 and
A45: i9 <> k ; ::_thesis: yb1b2 /. i9 = yb2 /. i9
A46: 1 <= i9 by A32, A44, FINSEQ_1:1;
A47: i9 in dom (SgmX ((RelIncl n),(support b2))) by A10, A32, A34, A44, FINSEQ_1:def_3;
then (SgmX ((RelIncl n),(support b2))) . i9 in rng (SgmX ((RelIncl n),(support b2))) by FUNCT_1:def_3;
then A48: i9 in dom (b2 * (SgmX ((RelIncl n),(support b2)))) by A47, A43, FUNCT_1:11;
then A49: (b2 * (SgmX ((RelIncl n),(support b2)))) /. i9 = (b2 * (SgmX ((RelIncl n),(support b2)))) . i9 by PARTFUN1:def_6
.= b2 . ((SgmX ((RelIncl n),(support b2))) . i9) by A48, FUNCT_1:12
.= b2 . ((SgmX ((RelIncl n),(support b2))) /. i9) by A47, PARTFUN1:def_6 ;
A50: i9 <= len yb2 by A32, A34, A44, FINSEQ_1:1;
A51: i9 in Seg sb1b2k by A38, A35, A44, FINSEQ_1:def_3;
A52: (SgmX ((RelIncl n),(support (b1 + b2)))) /. i9 <> u
proof
assume (SgmX ((RelIncl n),(support (b1 + b2)))) /. i9 = u ; ::_thesis: contradiction
then A53: (SgmX ((RelIncl n),(support (b1 + b2)))) . i9 = u by A29, A51, PARTFUN1:def_6;
A54: SgmX ((RelIncl n),(support (b1 + b2))) is one-to-one by Th13, PRE_POLY:10;
(SgmX ((RelIncl n),(support (b1 + b2)))) . k = u by A17, A15, A22, TARSKI:1;
hence contradiction by A21, A14, A29, A38, A45, A51, A53, A54, FUNCT_1:def_4; ::_thesis: verum
end;
(SgmX ((RelIncl n),(support (b1 + b2)))) . i9 in rng (SgmX ((RelIncl n),(support (b1 + b2)))) by A29, A51, FUNCT_1:def_3;
then A55: i9 in dom ((b1 + b2) * (SgmX ((RelIncl n),(support (b1 + b2))))) by A29, A40, A51, A42, FUNCT_1:11;
then ((b1 + b2) * (SgmX ((RelIncl n),(support (b1 + b2))))) /. i9 = ((b1 + b2) * (SgmX ((RelIncl n),(support (b1 + b2))))) . i9 by PARTFUN1:def_6
.= (b1 + b2) . ((SgmX ((RelIncl n),(support (b1 + b2)))) . i9) by A55, FUNCT_1:12
.= (b1 + b2) . ((SgmX ((RelIncl n),(support (b1 + b2)))) /. i9) by A29, A51, PARTFUN1:def_6 ;
hence yb1b2 /. i9 = (power L) . (((x * (SgmX ((RelIncl n),(support (b1 + b2))))) /. i9),((b1 + b2) . ((SgmX ((RelIncl n),(support (b1 + b2)))) /. i9))) by A9, A19, A46, A50
.= (power L) . (((x * (SgmX ((RelIncl n),(support b2)))) /. i9),(b2 . ((SgmX ((RelIncl n),(support b2))) /. i9))) by A3, A20, A52
.= yb2 /. i9 by A12, A46, A50, A49 ;
::_thesis: verum
end;
A56: support b1 c= dom b1 by PRE_POLY:37;
u in support b1 by A1, TARSKI:def_1;
then A57: u in dom b1 by A56;
A58: dom b1 = n by PARTFUN1:def_2;
then x . u in rng x by A5, A57, FUNCT_1:def_3;
then reconsider xu = x . u as Element of L by A28;
consider a being Element of L such that
A59: a = (power L) . (xu,(b1 . u)) ;
A60: k in dom (x * (SgmX ((RelIncl n),(support b2)))) by A5, A21, A22, A57, A58, FUNCT_1:11;
then (x * (SgmX ((RelIncl n),(support b2)))) . k = x . ((SgmX ((RelIncl n),(support b2))) . k) by FUNCT_1:12;
then yb1b2 /. k = a * (yb2 /. k) by A22, A39, A59, A60, PARTFUN1:def_6;
hence eval ((b1 + b2),x) = a * (Product yb2) by A8, A19, A36, A41, Th6
.= (eval (b1,x)) * (eval (b2,x)) by A1, A11, A59, Th15 ;
::_thesis: verum
end;
supposeA61: not u in support b2 ; ::_thesis: eval ((b1 + b2),x) = (eval (b1,x)) * (eval (b2,x))
A62: support b1 c= dom b1 by PRE_POLY:37;
u in support b1 by A1, TARSKI:def_1;
then A63: u in dom b1 by A62;
A64: rng x c= the carrier of L by RELAT_1:def_19;
dom b1 = n by PARTFUN1:def_2;
then x . u in rng x by A5, A63, FUNCT_1:def_3;
then reconsider xu = x . u as Element of L by A64;
consider a being Element of L such that
A65: a = (power L) . (xu,((b1 + b2) . u)) ;
A66: (b1 + b2) . u = (b1 . u) + (b2 . u) by PRE_POLY:def_5
.= (b1 . u) + 0 by A61, PRE_POLY:def_7 ;
thus eval ((b1 + b2),x) = a * (eval (b2,x)) by A3, A2, A61, A65, Lm8
.= (eval (b1,x)) * (eval (b2,x)) by A1, A66, A65, Th15 ; ::_thesis: verum
end;
end;
end;
theorem Th16: :: POLYNOM2:16
for n being Ordinal
for L being non empty non trivial right_complementable associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b1, b2 being bag of n
for x being Function of n,L holds eval ((b1 + b2),x) = (eval (b1,x)) * (eval (b2,x))
proof
let n be Ordinal; ::_thesis: for L being non empty non trivial right_complementable associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b1, b2 being bag of n
for x being Function of n,L holds eval ((b1 + b2),x) = (eval (b1,x)) * (eval (b2,x))
let L be non trivial right_complementable associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr ; ::_thesis: for b1, b2 being bag of n
for x being Function of n,L holds eval ((b1 + b2),x) = (eval (b1,x)) * (eval (b2,x))
let b1, b2 be bag of n; ::_thesis: for x being Function of n,L holds eval ((b1 + b2),x) = (eval (b1,x)) * (eval (b2,x))
let x be Function of n,L; ::_thesis: eval ((b1 + b2),x) = (eval (b1,x)) * (eval (b2,x))
defpred S1[ Element of NAT ] means for b1 being bag of n st card (support b1) = $1 holds
eval ((b1 + b2),x) = (eval (b1,x)) * (eval (b2,x));
A1: ex k being Element of NAT st card (support b1) = k ;
A2: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A3: S1[k] ; ::_thesis: S1[k + 1]
let b1 be bag of n; ::_thesis: ( card (support b1) = k + 1 implies eval ((b1 + b2),x) = (eval (b1,x)) * (eval (b2,x)) )
assume A4: card (support b1) = k + 1 ; ::_thesis: eval ((b1 + b2),x) = (eval (b1,x)) * (eval (b2,x))
set sgb1 = SgmX ((RelIncl n),(support b1));
set bg = (SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))));
A5: RelIncl n linearly_orders support b1 by Th13;
then SgmX ((RelIncl n),(support b1)) <> {} by A4, CARD_1:27, PRE_POLY:def_2, RELAT_1:38;
then 1 <= len (SgmX ((RelIncl n),(support b1))) by NAT_1:14;
then len (SgmX ((RelIncl n),(support b1))) in Seg (len (SgmX ((RelIncl n),(support b1)))) by FINSEQ_1:1;
then A6: len (SgmX ((RelIncl n),(support b1))) in dom (SgmX ((RelIncl n),(support b1))) by FINSEQ_1:def_3;
then (SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))) = (SgmX ((RelIncl n),(support b1))) . (len (SgmX ((RelIncl n),(support b1)))) by PARTFUN1:def_6;
then (SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))) in rng (SgmX ((RelIncl n),(support b1))) by A6, FUNCT_1:def_3;
then A7: (SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))) in support b1 by A5, PRE_POLY:def_2;
set b19 = b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0);
support b1 c= dom b1 by PRE_POLY:37;
then A8: b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0) = b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))) .--> 0) by A7, FUNCT_7:def_3;
A9: for u being set st u in support b1 holds
u in (support (b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0))) \/ {((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))}
proof
let u be set ; ::_thesis: ( u in support b1 implies u in (support (b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0))) \/ {((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))} )
assume u in support b1 ; ::_thesis: u in (support (b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0))) \/ {((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))}
then A10: b1 . u <> 0 by PRE_POLY:def_7;
percases ( u = (SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))) or u <> (SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))) ) ;
suppose u = (SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))) ; ::_thesis: u in (support (b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0))) \/ {((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))}
then u in {((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))} by TARSKI:def_1;
hence u in (support (b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0))) \/ {((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))} by XBOOLE_0:def_3; ::_thesis: verum
end;
suppose u <> (SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))) ; ::_thesis: u in (support (b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0))) \/ {((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))}
then not u in {((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))} by TARSKI:def_1;
then not u in dom (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))) .--> 0) ;
then (b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0)) . u = b1 . u by A8, FUNCT_4:11;
then u in support (b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0)) by A10, PRE_POLY:def_7;
hence u in (support (b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0))) \/ {((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))} by XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
(SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))) in {((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))} by TARSKI:def_1;
then (SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))) in dom (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))) .--> 0) by FUNCOP_1:13;
then (b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0)) . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))) = (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))) .--> 0) . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))) by A8, FUNCT_4:13;
then A11: (b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0)) . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))) = 0 by FUNCOP_1:72;
then A12: not (SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))) in support (b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0)) by PRE_POLY:def_7;
for u being set st u in (support (b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0))) \/ {((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))} holds
u in support b1
proof
let u be set ; ::_thesis: ( u in (support (b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0))) \/ {((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))} implies u in support b1 )
assume A13: u in (support (b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0))) \/ {((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))} ; ::_thesis: u in support b1
percases ( u in support (b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0)) or u in {((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))} ) by A13, XBOOLE_0:def_3;
supposeA14: u in support (b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0)) ; ::_thesis: u in support b1
then u <> (SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))) by A11, PRE_POLY:def_7;
then not u in {((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))} by TARSKI:def_1;
then not u in dom (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))) .--> 0) ;
then A15: (b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0)) . u = b1 . u by A8, FUNCT_4:11;
(b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0)) . u <> 0 by A14, PRE_POLY:def_7;
hence u in support b1 by A15, PRE_POLY:def_7; ::_thesis: verum
end;
suppose u in {((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))} ; ::_thesis: u in support b1
hence u in support b1 by A7, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
then support b1 = (support (b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0))) \/ {((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))} by A9, TARSKI:1;
then A16: k + 1 = (card (support (b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0)))) + 1 by A4, A12, CARD_2:41;
set m = (EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))));
A17: dom b1 = n by PARTFUN1:def_2;
dom (EmptyBag n) = n by PARTFUN1:def_2;
then A18: (EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))))) = (EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))) .--> (b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))))) by A7, FUNCT_7:def_3;
A19: for u being set st u in support ((EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))))) holds
u in {((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))}
proof
let u be set ; ::_thesis: ( u in support ((EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))))) implies u in {((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))} )
assume u in support ((EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))))) ; ::_thesis: u in {((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))}
then A20: ((EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))))) . u <> 0 by PRE_POLY:def_7;
now__::_thesis:_not_u_<>_(SgmX_((RelIncl_n),(support_b1)))_/._(len_(SgmX_((RelIncl_n),(support_b1))))
assume u <> (SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))) ; ::_thesis: contradiction
then not u in {((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))} by TARSKI:def_1;
then not u in dom (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))) .--> (b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))))) ;
then ((EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))))) . u = (EmptyBag n) . u by A18, FUNCT_4:11;
hence contradiction by A20, PRE_POLY:52; ::_thesis: verum
end;
hence u in {((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))} by TARSKI:def_1; ::_thesis: verum
end;
A21: b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))) <> 0 by A7, PRE_POLY:def_7;
for u being set st u in {((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))} holds
u in support ((EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))))))
proof
let u be set ; ::_thesis: ( u in {((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))} implies u in support ((EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))))) )
(SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))) in {((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))} by TARSKI:def_1;
then (SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))) in dom (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))) .--> (b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))))) by FUNCOP_1:13;
then ((EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))))) . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))) = (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))) .--> (b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))))) . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))) by A18, FUNCT_4:13;
then A22: ((EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))))) . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))) = b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))) by FUNCOP_1:72;
assume u in {((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))} ; ::_thesis: u in support ((EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))))))
then u = (SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))) by TARSKI:def_1;
hence u in support ((EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))))) by A21, A22, PRE_POLY:def_7; ::_thesis: verum
end;
then A23: support ((EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))))) = {((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))} by A19, TARSKI:1;
A24: for u being set st u in n holds
((b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0)) + ((EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))))))) . u = b1 . u
proof
let u be set ; ::_thesis: ( u in n implies ((b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0)) + ((EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))))))) . u = b1 . u )
assume u in n ; ::_thesis: ((b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0)) + ((EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))))))) . u = b1 . u
percases ( u = (SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))) or u <> (SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))) ) ;
supposeA25: u = (SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))) ; ::_thesis: ((b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0)) + ((EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))))))) . u = b1 . u
(SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))) in {((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))} by TARSKI:def_1;
then (SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))) in dom (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))) .--> (b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))))) by FUNCOP_1:13;
then ((EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))))) . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))) = (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))) .--> (b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))))) . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))) by A18, FUNCT_4:13;
then A26: ((EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))))) . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))) = b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))) by FUNCOP_1:72;
u in {((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))} by A25, TARSKI:def_1;
then u in dom (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))) .--> 0) by FUNCOP_1:13;
then A27: (b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0)) . u = (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))) .--> 0) . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))) by A8, A25, FUNCT_4:13;
((b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0)) + ((EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))))))) . u = ((b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0)) . u) + (((EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))))) . u) by PRE_POLY:def_5
.= 0 + (b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))) by A25, A27, A26, FUNCOP_1:72
.= b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))) ;
hence ((b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0)) + ((EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))))))) . u = b1 . u by A25; ::_thesis: verum
end;
suppose u <> (SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))) ; ::_thesis: ((b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0)) + ((EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))))))) . u = b1 . u
then A28: not u in {((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))} by TARSKI:def_1;
then A29: not u in dom (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))) .--> 0) ;
not u in dom (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))) .--> (b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))))) by A28;
then ((EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))))) . u = (EmptyBag n) . u by A18, FUNCT_4:11;
then A30: ((EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))))) . u = 0 by PRE_POLY:52;
((b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0)) + ((EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))))))) . u = ((b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0)) . u) + (((EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))))) . u) by PRE_POLY:def_5
.= b1 . u by A8, A29, A30, FUNCT_4:11 ;
hence ((b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0)) + ((EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))))))) . u = b1 . u ; ::_thesis: verum
end;
end;
end;
A31: dom ((b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0)) + ((EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))))))) = n by PARTFUN1:def_2;
then eval (b1,x) = eval ((((EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))))) + (b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0))),x) by A17, A24, FUNCT_1:2
.= (eval ((b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0)),x)) * (eval (((EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))))),x)) by A23, Lm9 ;
hence (eval (b1,x)) * (eval (b2,x)) = ((eval ((b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0)),x)) * (eval (b2,x))) * (eval (((EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))))),x)) by GROUP_1:def_3
.= (eval (((b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0)) + b2),x)) * (eval (((EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))))),x)) by A3, A16
.= eval ((((EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1)))))))) + ((b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0)) + b2)),x) by A23, Lm9
.= eval ((((b1 +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),0)) + ((EmptyBag n) +* (((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))),(b1 . ((SgmX ((RelIncl n),(support b1))) /. (len (SgmX ((RelIncl n),(support b1))))))))) + b2),x) by PRE_POLY:35
.= eval ((b1 + b2),x) by A31, A17, A24, FUNCT_1:2 ;
::_thesis: verum
end;
A32: S1[ 0 ]
proof
let b1 be bag of n; ::_thesis: ( card (support b1) = 0 implies eval ((b1 + b2),x) = (eval (b1,x)) * (eval (b2,x)) )
assume card (support b1) = 0 ; ::_thesis: eval ((b1 + b2),x) = (eval (b1,x)) * (eval (b2,x))
then support b1 = {} ;
then A33: b1 = EmptyBag n by Th12;
hence eval ((b1 + b2),x) = eval (b2,x) by PRE_POLY:53
.= (1. L) * (eval (b2,x)) by VECTSP_1:def_6
.= (eval (b1,x)) * (eval (b2,x)) by A33, Th14 ;
::_thesis: verum
end;
for k being Element of NAT holds S1[k] from NAT_1:sch_1(A32, A2);
hence eval ((b1 + b2),x) = (eval (b1,x)) * (eval (b2,x)) by A1; ::_thesis: verum
end;
begin
registration
let n be Ordinal;
let L be non empty right_complementable add-associative right_zeroed addLoopStr ;
let p, q be Polynomial of n,L;
clusterp - q -> finite-Support ;
coherence
p - q is finite-Support
proof
p - q = p + (- q) by POLYNOM1:def_6;
hence p - q is finite-Support ; ::_thesis: verum
end;
end;
theorem Th17: :: POLYNOM2:17
for L being non trivial right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr
for n being Ordinal
for p being Polynomial of n,L st Support p = {} holds
p = 0_ (n,L)
proof
let L be non trivial right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr ; ::_thesis: for n being Ordinal
for p being Polynomial of n,L st Support p = {} holds
p = 0_ (n,L)
let n be Ordinal; ::_thesis: for p being Polynomial of n,L st Support p = {} holds
p = 0_ (n,L)
let p be Polynomial of n,L; ::_thesis: ( Support p = {} implies p = 0_ (n,L) )
assume A1: Support p = {} ; ::_thesis: p = 0_ (n,L)
A2: for u being set st u in Bags n holds
p . u = (0_ (n,L)) . u
proof
let u be set ; ::_thesis: ( u in Bags n implies p . u = (0_ (n,L)) . u )
assume A3: u in Bags n ; ::_thesis: p . u = (0_ (n,L)) . u
then reconsider b = u as bag of n ;
p . b = 0. L by A1, A3, POLYNOM1:def_3;
hence p . u = (0_ (n,L)) . u by POLYNOM1:22; ::_thesis: verum
end;
A4: Bags n = dom (0_ (n,L)) by FUNCT_2:def_1;
Bags n = dom p by FUNCT_2:def_1;
hence p = 0_ (n,L) by A4, A2, FUNCT_1:2; ::_thesis: verum
end;
registration
let n be Ordinal;
let L be non trivial right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr ;
let p be Polynomial of n,L;
cluster Support p -> finite ;
coherence
Support p is finite by POLYNOM1:def_4;
end;
theorem Th18: :: POLYNOM2:18
for n being Ordinal
for L being non trivial right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr
for p being Polynomial of n,L holds BagOrder n linearly_orders Support p
proof
let n be Ordinal; ::_thesis: for L being non trivial right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr
for p being Polynomial of n,L holds BagOrder n linearly_orders Support p
let L be non trivial right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr ; ::_thesis: for p being Polynomial of n,L holds BagOrder n linearly_orders Support p
let p be Polynomial of n,L; ::_thesis: BagOrder n linearly_orders Support p
set R = BagOrder n;
BagOrder n is connected by ORDERS_1:def_5;
then A1: BagOrder n is_connected_in field (BagOrder n) by RELAT_2:def_14;
for x being set st x in Bags n holds
x in field (BagOrder n)
proof
let x be set ; ::_thesis: ( x in Bags n implies x in field (BagOrder n) )
assume x in Bags n ; ::_thesis: x in field (BagOrder n)
then reconsider x = x as bag of n ;
EmptyBag n <=' x by PRE_POLY:60;
then [(EmptyBag n),x] in BagOrder n by PRE_POLY:def_14;
then A2: x in rng (BagOrder n) by XTUPLE_0:def_13;
field (BagOrder n) = (dom (BagOrder n)) \/ (rng (BagOrder n)) by RELAT_1:def_6;
then rng (BagOrder n) c= field (BagOrder n) by XBOOLE_1:7;
hence x in field (BagOrder n) by A2; ::_thesis: verum
end;
then A3: Bags n c= field (BagOrder n) by TARSKI:def_3;
then [:(Bags n),(Bags n):] c= [:(field (BagOrder n)),(field (BagOrder n)):] by ZFMISC_1:96;
then reconsider R9 = BagOrder n as Relation of (field (BagOrder n)) by XBOOLE_1:1;
BagOrder n is_reflexive_in field (BagOrder n) by RELAT_2:def_9;
then dom R9 = field (BagOrder n) by ORDERS_1:13;
then A4: R9 is total by PARTFUN1:def_2;
Support p c= field (BagOrder n) by A3, XBOOLE_1:1;
then A5: R9 is_connected_in Support p by A1, A4, Lm2;
A6: BagOrder n is_antisymmetric_in Support p by Lm1;
A7: BagOrder n is_transitive_in Support p by Lm1;
BagOrder n is_reflexive_in Support p by Lm1;
hence BagOrder n linearly_orders Support p by A6, A7, A5, ORDERS_1:def_8; ::_thesis: verum
end;
definition
let n be Ordinal;
let b be Element of Bags n;
funcb @ -> bag of n equals :: POLYNOM2:def 3
b;
correctness
coherence
b is bag of n;
;
end;
:: deftheorem defines @ POLYNOM2:def_3_:_
for n being Ordinal
for b being Element of Bags n holds b @ = b;
definition
let n be Ordinal;
let L be non trivial right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr ;
let p be Polynomial of n,L;
let x be Function of n,L;
func eval (p,x) -> Element of L means :Def4: :: POLYNOM2:def 4
ex y being FinSequence of the carrier of L st
( len y = len (SgmX ((BagOrder n),(Support p))) & it = Sum y & ( for i being Element of NAT st 1 <= i & i <= len y holds
y /. i = ((p * (SgmX ((BagOrder n),(Support p)))) /. i) * (eval ((((SgmX ((BagOrder n),(Support p))) /. i) @),x)) ) );
existence
ex b1 being Element of L ex y being FinSequence of the carrier of L st
( len y = len (SgmX ((BagOrder n),(Support p))) & b1 = Sum y & ( for i being Element of NAT st 1 <= i & i <= len y holds
y /. i = ((p * (SgmX ((BagOrder n),(Support p)))) /. i) * (eval ((((SgmX ((BagOrder n),(Support p))) /. i) @),x)) ) )
proof
set S = SgmX ((BagOrder n),(Support p));
set l = len (SgmX ((BagOrder n),(Support p)));
defpred S1[ Element of NAT , Element of L] means $2 = ((p * (SgmX ((BagOrder n),(Support p)))) /. $1) * (eval ((((SgmX ((BagOrder n),(Support p))) /. $1) @),x));
A1: for k being Element of NAT st k in Seg (len (SgmX ((BagOrder n),(Support p)))) holds
ex x being Element of L st S1[k,x] ;
consider q being FinSequence of the carrier of L such that
A2: ( dom q = Seg (len (SgmX ((BagOrder n),(Support p)))) & ( for m being Element of NAT st m in Seg (len (SgmX ((BagOrder n),(Support p)))) holds
S1[m,q /. m] ) ) from RECDEF_1:sch_17(A1);
take Sum q ; ::_thesis: ex y being FinSequence of the carrier of L st
( len y = len (SgmX ((BagOrder n),(Support p))) & Sum q = Sum y & ( for i being Element of NAT st 1 <= i & i <= len y holds
y /. i = ((p * (SgmX ((BagOrder n),(Support p)))) /. i) * (eval ((((SgmX ((BagOrder n),(Support p))) /. i) @),x)) ) )
A3: len q = len (SgmX ((BagOrder n),(Support p))) by A2, FINSEQ_1:def_3;
now__::_thesis:_for_m_being_Element_of_NAT_st_1_<=_m_&_m_<=_len_q_holds_
q_/._m_=_((p_*_(SgmX_((BagOrder_n),(Support_p))))_/._m)_*_(eval_((((SgmX_((BagOrder_n),(Support_p)))_/._m)_@),x))
let m be Element of NAT ; ::_thesis: ( 1 <= m & m <= len q implies q /. m = ((p * (SgmX ((BagOrder n),(Support p)))) /. m) * (eval ((((SgmX ((BagOrder n),(Support p))) /. m) @),x)) )
assume that
A4: 1 <= m and
A5: m <= len q ; ::_thesis: q /. m = ((p * (SgmX ((BagOrder n),(Support p)))) /. m) * (eval ((((SgmX ((BagOrder n),(Support p))) /. m) @),x))
m in Seg (len q) by A4, A5, FINSEQ_1:1;
hence q /. m = ((p * (SgmX ((BagOrder n),(Support p)))) /. m) * (eval ((((SgmX ((BagOrder n),(Support p))) /. m) @),x)) by A2, A3; ::_thesis: verum
end;
hence ex y being FinSequence of the carrier of L st
( len y = len (SgmX ((BagOrder n),(Support p))) & Sum q = Sum y & ( for i being Element of NAT st 1 <= i & i <= len y holds
y /. i = ((p * (SgmX ((BagOrder n),(Support p)))) /. i) * (eval ((((SgmX ((BagOrder n),(Support p))) /. i) @),x)) ) ) by A3; ::_thesis: verum
end;
uniqueness
for b1, b2 being Element of L st ex y being FinSequence of the carrier of L st
( len y = len (SgmX ((BagOrder n),(Support p))) & b1 = Sum y & ( for i being Element of NAT st 1 <= i & i <= len y holds
y /. i = ((p * (SgmX ((BagOrder n),(Support p)))) /. i) * (eval ((((SgmX ((BagOrder n),(Support p))) /. i) @),x)) ) ) & ex y being FinSequence of the carrier of L st
( len y = len (SgmX ((BagOrder n),(Support p))) & b2 = Sum y & ( for i being Element of NAT st 1 <= i & i <= len y holds
y /. i = ((p * (SgmX ((BagOrder n),(Support p)))) /. i) * (eval ((((SgmX ((BagOrder n),(Support p))) /. i) @),x)) ) ) holds
b1 = b2
proof
let a, c be Element of L; ::_thesis: ( ex y being FinSequence of the carrier of L st
( len y = len (SgmX ((BagOrder n),(Support p))) & a = Sum y & ( for i being Element of NAT st 1 <= i & i <= len y holds
y /. i = ((p * (SgmX ((BagOrder n),(Support p)))) /. i) * (eval ((((SgmX ((BagOrder n),(Support p))) /. i) @),x)) ) ) & ex y being FinSequence of the carrier of L st
( len y = len (SgmX ((BagOrder n),(Support p))) & c = Sum y & ( for i being Element of NAT st 1 <= i & i <= len y holds
y /. i = ((p * (SgmX ((BagOrder n),(Support p)))) /. i) * (eval ((((SgmX ((BagOrder n),(Support p))) /. i) @),x)) ) ) implies a = c )
assume that
A6: ex y1 being FinSequence of the carrier of L st
( len y1 = len (SgmX ((BagOrder n),(Support p))) & a = Sum y1 & ( for i being Element of NAT st 1 <= i & i <= len y1 holds
y1 /. i = ((p * (SgmX ((BagOrder n),(Support p)))) /. i) * (eval ((((SgmX ((BagOrder n),(Support p))) /. i) @),x)) ) ) and
A7: ex y2 being FinSequence of the carrier of L st
( len y2 = len (SgmX ((BagOrder n),(Support p))) & c = Sum y2 & ( for i being Element of NAT st 1 <= i & i <= len y2 holds
y2 /. i = ((p * (SgmX ((BagOrder n),(Support p)))) /. i) * (eval ((((SgmX ((BagOrder n),(Support p))) /. i) @),x)) ) ) ; ::_thesis: a = c
consider y1 being FinSequence of the carrier of L such that
A8: len y1 = len (SgmX ((BagOrder n),(Support p))) and
A9: a = Sum y1 and
A10: for i being Element of NAT st 1 <= i & i <= len y1 holds
y1 /. i = ((p * (SgmX ((BagOrder n),(Support p)))) /. i) * (eval ((((SgmX ((BagOrder n),(Support p))) /. i) @),x)) by A6;
consider y2 being FinSequence of the carrier of L such that
A11: len y2 = len (SgmX ((BagOrder n),(Support p))) and
A12: c = Sum y2 and
A13: for i being Element of NAT st 1 <= i & i <= len y2 holds
y2 /. i = ((p * (SgmX ((BagOrder n),(Support p)))) /. i) * (eval ((((SgmX ((BagOrder n),(Support p))) /. i) @),x)) by A7;
for k being Nat st 1 <= k & k <= len y1 holds
y1 . k = y2 . k
proof
let k be Nat; ::_thesis: ( 1 <= k & k <= len y1 implies y1 . k = y2 . k )
assume that
A14: 1 <= k and
A15: k <= len y1 ; ::_thesis: y1 . k = y2 . k
k in Seg (len y2) by A8, A11, A14, A15, FINSEQ_1:1;
then A16: k in dom y2 by FINSEQ_1:def_3;
A17: k in Seg (len y1) by A14, A15, FINSEQ_1:1;
then k in dom y1 by FINSEQ_1:def_3;
hence y1 . k = y1 /. k by PARTFUN1:def_6
.= ((p * (SgmX ((BagOrder n),(Support p)))) /. k) * (eval ((((SgmX ((BagOrder n),(Support p))) /. k) @),x)) by A10, A14, A15, A17
.= y2 /. k by A8, A11, A13, A14, A15, A17
.= y2 . k by A16, PARTFUN1:def_6 ;
::_thesis: verum
end;
hence a = c by A8, A9, A11, A12, FINSEQ_1:14; ::_thesis: verum
end;
end;
:: deftheorem Def4 defines eval POLYNOM2:def_4_:_
for n being Ordinal
for L being non trivial right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr
for p being Polynomial of n,L
for x being Function of n,L
for b5 being Element of L holds
( b5 = eval (p,x) iff ex y being FinSequence of the carrier of L st
( len y = len (SgmX ((BagOrder n),(Support p))) & b5 = Sum y & ( for i being Element of NAT st 1 <= i & i <= len y holds
y /. i = ((p * (SgmX ((BagOrder n),(Support p)))) /. i) * (eval ((((SgmX ((BagOrder n),(Support p))) /. i) @),x)) ) ) );
theorem Th19: :: POLYNOM2:19
for n being Ordinal
for L being non trivial right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr
for p being Polynomial of n,L
for b being bag of n st Support p = {b} holds
for x being Function of n,L holds eval (p,x) = (p . b) * (eval (b,x))
proof
let n be Ordinal; ::_thesis: for L being non trivial right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr
for p being Polynomial of n,L
for b being bag of n st Support p = {b} holds
for x being Function of n,L holds eval (p,x) = (p . b) * (eval (b,x))
let L be non trivial right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr ; ::_thesis: for p being Polynomial of n,L
for b being bag of n st Support p = {b} holds
for x being Function of n,L holds eval (p,x) = (p . b) * (eval (b,x))
let p be Polynomial of n,L; ::_thesis: for b being bag of n st Support p = {b} holds
for x being Function of n,L holds eval (p,x) = (p . b) * (eval (b,x))
let b be bag of n; ::_thesis: ( Support p = {b} implies for x being Function of n,L holds eval (p,x) = (p . b) * (eval (b,x)) )
reconsider sp = Support p as finite Subset of (Bags n) ;
set sg = SgmX ((BagOrder n),sp);
A1: b in Bags n by PRE_POLY:def_12;
A2: dom p = Bags n by FUNCT_2:def_1;
A3: BagOrder n linearly_orders sp by Th18;
assume Support p = {b} ; ::_thesis: for x being Function of n,L holds eval (p,x) = (p . b) * (eval (b,x))
then A4: rng (SgmX ((BagOrder n),sp)) = {b} by A3, PRE_POLY:def_2;
then A5: b in rng (SgmX ((BagOrder n),sp)) by TARSKI:def_1;
then A6: 1 in dom (SgmX ((BagOrder n),sp)) by FINSEQ_3:31;
then A7: (SgmX ((BagOrder n),sp)) /. 1 = (SgmX ((BagOrder n),sp)) . 1 by PARTFUN1:def_6;
A8: for u being set st u in dom (SgmX ((BagOrder n),sp)) holds
u in {1}
proof
let u be set ; ::_thesis: ( u in dom (SgmX ((BagOrder n),sp)) implies u in {1} )
assume A9: u in dom (SgmX ((BagOrder n),sp)) ; ::_thesis: u in {1}
assume A10: not u in {1} ; ::_thesis: contradiction
reconsider u = u as Element of NAT by A9;
(SgmX ((BagOrder n),sp)) /. u = (SgmX ((BagOrder n),sp)) . u by A9, PARTFUN1:def_6;
then A11: (SgmX ((BagOrder n),sp)) /. u in rng (SgmX ((BagOrder n),sp)) by A9, FUNCT_1:def_3;
A12: u <> 1 by A10, TARSKI:def_1;
A13: 1 < u
proof
consider k being Nat such that
A14: dom (SgmX ((BagOrder n),sp)) = Seg k by FINSEQ_1:def_2;
Seg k = { l where l is Element of NAT : ( 1 <= l & l <= k ) } by FINSEQ_1:def_1;
then ex m9 being Element of NAT st
( m9 = u & 1 <= m9 & m9 <= k ) by A9, A14;
hence 1 < u by A12, XXREAL_0:1; ::_thesis: verum
end;
(SgmX ((BagOrder n),sp)) /. 1 = (SgmX ((BagOrder n),sp)) . 1 by A5, A9, FINSEQ_3:31, PARTFUN1:def_6;
then (SgmX ((BagOrder n),sp)) /. 1 in rng (SgmX ((BagOrder n),sp)) by A6, FUNCT_1:def_3;
then (SgmX ((BagOrder n),sp)) /. 1 = b by A4, TARSKI:def_1
.= (SgmX ((BagOrder n),sp)) /. u by A4, A11, TARSKI:def_1 ;
hence contradiction by A3, A6, A9, A13, PRE_POLY:def_2; ::_thesis: verum
end;
for u being set st u in {1} holds
u in dom (SgmX ((BagOrder n),sp)) by A6, TARSKI:def_1;
then A15: dom (SgmX ((BagOrder n),sp)) = Seg 1 by A8, FINSEQ_1:2, TARSKI:1;
then A16: len (SgmX ((BagOrder n),sp)) = 1 by FINSEQ_1:def_3;
A17: (SgmX ((BagOrder n),sp)) . 1 in rng (SgmX ((BagOrder n),sp)) by A6, FUNCT_1:def_3;
then (SgmX ((BagOrder n),sp)) . 1 = b by A4, TARSKI:def_1;
then 1 in dom (p * (SgmX ((BagOrder n),sp))) by A6, A1, A2, FUNCT_1:11;
then A18: (p * (SgmX ((BagOrder n),sp))) /. 1 = (p * (SgmX ((BagOrder n),sp))) . 1 by PARTFUN1:def_6
.= p . ((SgmX ((BagOrder n),sp)) . 1) by A6, FUNCT_1:13
.= p . b by A4, A17, TARSKI:def_1 ;
1 in dom (SgmX ((BagOrder n),sp)) by A15, FINSEQ_1:2, TARSKI:def_1;
then A19: (SgmX ((BagOrder n),sp)) /. 1 in rng (SgmX ((BagOrder n),sp)) by A7, FUNCT_1:def_3;
let x be Function of n,L; ::_thesis: eval (p,x) = (p . b) * (eval (b,x))
consider y being FinSequence of the carrier of L such that
A20: len y = len (SgmX ((BagOrder n),(Support p))) and
A21: eval (p,x) = Sum y and
A22: for i being Element of NAT st 1 <= i & i <= len y holds
y /. i = ((p * (SgmX ((BagOrder n),(Support p)))) /. i) * (eval ((((SgmX ((BagOrder n),(Support p))) /. i) @),x)) by Def4;
y . 1 = y /. 1 by A20, A16, FINSEQ_4:15
.= ((p * (SgmX ((BagOrder n),sp))) /. 1) * (eval ((((SgmX ((BagOrder n),sp)) /. 1) @),x)) by A20, A22, A16
.= ((p * (SgmX ((BagOrder n),sp))) /. 1) * (eval (b,x)) by A4, A19, TARSKI:def_1 ;
then y = <*((p . b) * (eval (b,x)))*> by A20, A16, A18, FINSEQ_1:40;
hence eval (p,x) = (p . b) * (eval (b,x)) by A21, RLVECT_1:44; ::_thesis: verum
end;
theorem Th20: :: POLYNOM2:20
for n being Ordinal
for L being non trivial right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr
for x being Function of n,L holds eval ((0_ (n,L)),x) = 0. L
proof
let n be Ordinal; ::_thesis: for L being non trivial right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr
for x being Function of n,L holds eval ((0_ (n,L)),x) = 0. L
let L be non trivial right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr ; ::_thesis: for x being Function of n,L holds eval ((0_ (n,L)),x) = 0. L
let x be Function of n,L; ::_thesis: eval ((0_ (n,L)),x) = 0. L
set 0p = 0_ (n,L);
consider y being FinSequence of the carrier of L such that
A1: len y = len (SgmX ((BagOrder n),(Support (0_ (n,L))))) and
A2: Sum y = eval ((0_ (n,L)),x) and
for i being Element of NAT st 1 <= i & i <= len y holds
y /. i = (((0_ (n,L)) * (SgmX ((BagOrder n),(Support (0_ (n,L)))))) /. i) * (eval ((((SgmX ((BagOrder n),(Support (0_ (n,L))))) /. i) @),x)) by Def4;
Support (0_ (n,L)) = {}
proof
set u = the Element of Support (0_ (n,L));
assume Support (0_ (n,L)) <> {} ; ::_thesis: contradiction
then A3: the Element of Support (0_ (n,L)) in Support (0_ (n,L)) ;
then A4: the Element of Support (0_ (n,L)) is Element of Bags n ;
(0_ (n,L)) . the Element of Support (0_ (n,L)) <> 0. L by A3, POLYNOM1:def_3;
hence contradiction by A4, POLYNOM1:22; ::_thesis: verum
end;
then SgmX ((BagOrder n),(Support (0_ (n,L)))) = {} by Th7, Th18;
then y = <*> the carrier of L by A1;
hence eval ((0_ (n,L)),x) = 0. L by A2, RLVECT_1:43; ::_thesis: verum
end;
theorem Th21: :: POLYNOM2:21
for n being Ordinal
for L being non trivial right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr
for x being Function of n,L holds eval ((1_ (n,L)),x) = 1. L
proof
let n be Ordinal; ::_thesis: for L being non trivial right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr
for x being Function of n,L holds eval ((1_ (n,L)),x) = 1. L
let L be non trivial right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr ; ::_thesis: for x being Function of n,L holds eval ((1_ (n,L)),x) = 1. L
let x be Function of n,L; ::_thesis: eval ((1_ (n,L)),x) = 1. L
set 1p = 1_ (n,L);
A1: for u being set st u in {(EmptyBag n)} holds
u in Support (1_ (n,L))
proof
let u be set ; ::_thesis: ( u in {(EmptyBag n)} implies u in Support (1_ (n,L)) )
assume A2: u in {(EmptyBag n)} ; ::_thesis: u in Support (1_ (n,L))
then A3: u = EmptyBag n by TARSKI:def_1;
reconsider u = u as Element of Bags n by A2, TARSKI:def_1;
(1_ (n,L)) . u = 1. L by A3, POLYNOM1:25;
then (1_ (n,L)) . u <> 0. L ;
hence u in Support (1_ (n,L)) by POLYNOM1:def_3; ::_thesis: verum
end;
reconsider s1p = Support (1_ (n,L)) as finite Subset of (Bags n) ;
set sg1p = SgmX ((BagOrder n),s1p);
A4: BagOrder n linearly_orders Support (1_ (n,L)) by Th18;
for u being set st u in Support (1_ (n,L)) holds
u in {(EmptyBag n)}
proof
let u be set ; ::_thesis: ( u in Support (1_ (n,L)) implies u in {(EmptyBag n)} )
assume A5: u in Support (1_ (n,L)) ; ::_thesis: u in {(EmptyBag n)}
then reconsider u = u as Element of Bags n ;
(1_ (n,L)) . u <> 0. L by A5, POLYNOM1:def_3;
then u = EmptyBag n by POLYNOM1:25;
hence u in {(EmptyBag n)} by TARSKI:def_1; ::_thesis: verum
end;
then A6: Support (1_ (n,L)) = {(EmptyBag n)} by A1, TARSKI:1;
then A7: rng (SgmX ((BagOrder n),s1p)) = {(EmptyBag n)} by A4, PRE_POLY:def_2;
then A8: EmptyBag n in rng (SgmX ((BagOrder n),s1p)) by TARSKI:def_1;
then A9: 1 in dom (SgmX ((BagOrder n),s1p)) by FINSEQ_3:31;
then (SgmX ((BagOrder n),s1p)) /. 1 = (SgmX ((BagOrder n),s1p)) . 1 by PARTFUN1:def_6;
then (SgmX ((BagOrder n),s1p)) /. 1 in rng (SgmX ((BagOrder n),s1p)) by A9, FUNCT_1:def_3;
then A10: (SgmX ((BagOrder n),s1p)) /. 1 = EmptyBag n by A7, TARSKI:def_1;
A11: for u being set st u in dom (SgmX ((BagOrder n),s1p)) holds
u in {1}
proof
let u be set ; ::_thesis: ( u in dom (SgmX ((BagOrder n),s1p)) implies u in {1} )
assume A12: u in dom (SgmX ((BagOrder n),s1p)) ; ::_thesis: u in {1}
assume A13: not u in {1} ; ::_thesis: contradiction
reconsider u = u as Element of NAT by A12;
(SgmX ((BagOrder n),s1p)) /. u = (SgmX ((BagOrder n),s1p)) . u by A12, PARTFUN1:def_6;
then A14: (SgmX ((BagOrder n),s1p)) /. u in rng (SgmX ((BagOrder n),s1p)) by A12, FUNCT_1:def_3;
A15: u <> 1 by A13, TARSKI:def_1;
A16: 1 < u
proof
consider k being Nat such that
A17: dom (SgmX ((BagOrder n),s1p)) = Seg k by FINSEQ_1:def_2;
Seg k = { m where m is Element of NAT : ( 1 <= m & m <= k ) } by FINSEQ_1:def_1;
then ex m9 being Element of NAT st
( m9 = u & 1 <= m9 & m9 <= k ) by A12, A17;
hence 1 < u by A15, XXREAL_0:1; ::_thesis: verum
end;
(SgmX ((BagOrder n),s1p)) /. 1 = (SgmX ((BagOrder n),s1p)) . 1 by A8, A12, FINSEQ_3:31, PARTFUN1:def_6;
then (SgmX ((BagOrder n),s1p)) /. 1 in rng (SgmX ((BagOrder n),s1p)) by A9, FUNCT_1:def_3;
then (SgmX ((BagOrder n),s1p)) /. 1 = EmptyBag n by A7, TARSKI:def_1
.= (SgmX ((BagOrder n),s1p)) /. u by A7, A14, TARSKI:def_1 ;
hence contradiction by A4, A9, A12, A16, PRE_POLY:def_2; ::_thesis: verum
end;
A18: dom (1_ (n,L)) = Bags n by FUNCT_2:def_1;
A19: 1 in dom (SgmX ((BagOrder n),s1p)) by A8, FINSEQ_3:31;
A20: (SgmX ((BagOrder n),s1p)) . 1 in rng (SgmX ((BagOrder n),s1p)) by A9, FUNCT_1:def_3;
then (SgmX ((BagOrder n),s1p)) . 1 in {(EmptyBag n)} by A6, A4, PRE_POLY:def_2;
then (SgmX ((BagOrder n),s1p)) . 1 = EmptyBag n by TARSKI:def_1;
then 1 in dom ((1_ (n,L)) * (SgmX ((BagOrder n),s1p))) by A19, A18, FUNCT_1:11;
then A21: ((1_ (n,L)) * (SgmX ((BagOrder n),s1p))) /. 1 = ((1_ (n,L)) * (SgmX ((BagOrder n),s1p))) . 1 by PARTFUN1:def_6
.= (1_ (n,L)) . ((SgmX ((BagOrder n),s1p)) . 1) by A9, FUNCT_1:13
.= (1_ (n,L)) . (EmptyBag n) by A7, A20, TARSKI:def_1
.= 1. L by POLYNOM1:25 ;
consider y being FinSequence of the carrier of L such that
A22: len y = len (SgmX ((BagOrder n),s1p)) and
A23: Sum y = eval ((1_ (n,L)),x) and
A24: for i being Element of NAT st 1 <= i & i <= len y holds
y /. i = (((1_ (n,L)) * (SgmX ((BagOrder n),s1p))) /. i) * (eval ((((SgmX ((BagOrder n),s1p)) /. i) @),x)) by Def4;
for u being set st u in {1} holds
u in dom (SgmX ((BagOrder n),s1p)) by A9, TARSKI:def_1;
then dom (SgmX ((BagOrder n),s1p)) = Seg 1 by A11, FINSEQ_1:2, TARSKI:1;
then A25: len (SgmX ((BagOrder n),s1p)) = 1 by FINSEQ_1:def_3;
then y . 1 = y /. 1 by A22, FINSEQ_4:15
.= (((1_ (n,L)) * (SgmX ((BagOrder n),s1p))) /. 1) * (eval ((((SgmX ((BagOrder n),s1p)) /. 1) @),x)) by A25, A22, A24
.= (((1_ (n,L)) * (SgmX ((BagOrder n),s1p))) /. 1) * (1. L) by A10, Th14
.= ((1_ (n,L)) * (SgmX ((BagOrder n),s1p))) /. 1 by VECTSP_1:def_6 ;
then y = <*(1. L)*> by A25, A22, A21, FINSEQ_1:40;
hence eval ((1_ (n,L)),x) = 1. L by A23, RLVECT_1:44; ::_thesis: verum
end;
theorem Th22: :: POLYNOM2:22
for n being Ordinal
for L being non trivial right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr
for p being Polynomial of n,L
for x being Function of n,L holds eval ((- p),x) = - (eval (p,x))
proof
let n be Ordinal; ::_thesis: for L being non trivial right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr
for p being Polynomial of n,L
for x being Function of n,L holds eval ((- p),x) = - (eval (p,x))
let L be non trivial right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr ; ::_thesis: for p being Polynomial of n,L
for x being Function of n,L holds eval ((- p),x) = - (eval (p,x))
let p be Polynomial of n,L; ::_thesis: for x being Function of n,L holds eval ((- p),x) = - (eval (p,x))
let x be Function of n,L; ::_thesis: eval ((- p),x) = - (eval (p,x))
set mp = - p;
A1: for u being set st u in Support p holds
u in Support (- p)
proof
let u be set ; ::_thesis: ( u in Support p implies u in Support (- p) )
assume A2: u in Support p ; ::_thesis: u in Support (- p)
then reconsider u = u as Element of Bags n ;
reconsider u = u as bag of n ;
A3: p . u <> 0. L by A2, POLYNOM1:def_3;
(- p) . u <> 0. L
proof
assume (- p) . u = 0. L ; ::_thesis: contradiction
then - (- (p . u)) = - (0. L) by POLYNOM1:17;
then p . u = - (0. L) by RLVECT_1:17;
hence contradiction by A3, RLVECT_1:12; ::_thesis: verum
end;
hence u in Support (- p) by POLYNOM1:def_3; ::_thesis: verum
end;
consider ymp being FinSequence of the carrier of L such that
A4: len ymp = len (SgmX ((BagOrder n),(Support (- p)))) and
A5: Sum ymp = eval ((- p),x) and
A6: for i being Element of NAT st 1 <= i & i <= len ymp holds
ymp /. i = (((- p) * (SgmX ((BagOrder n),(Support (- p))))) /. i) * (eval ((((SgmX ((BagOrder n),(Support (- p)))) /. i) @),x)) by Def4;
consider yp being FinSequence of the carrier of L such that
A7: len yp = len (SgmX ((BagOrder n),(Support p))) and
A8: Sum yp = eval (p,x) and
A9: for i being Element of NAT st 1 <= i & i <= len yp holds
yp /. i = ((p * (SgmX ((BagOrder n),(Support p)))) /. i) * (eval ((((SgmX ((BagOrder n),(Support p))) /. i) @),x)) by Def4;
A10: for u being set st u in Support (- p) holds
u in Support p
proof
let u be set ; ::_thesis: ( u in Support (- p) implies u in Support p )
assume A11: u in Support (- p) ; ::_thesis: u in Support p
then reconsider u = u as Element of Bags n ;
reconsider u = u as bag of n ;
(- p) . u <> 0. L by A11, POLYNOM1:def_3;
then - (p . u) <> 0. L by POLYNOM1:17;
then p . u <> 0. L by RLVECT_1:12;
hence u in Support p by POLYNOM1:def_3; ::_thesis: verum
end;
then A12: len ymp = len yp by A1, A7, A4, TARSKI:1;
A13: dom ((- (1. L)) * yp) = dom yp by POLYNOM1:def_1;
consider k being Element of NAT such that
A14: k = len ((- (1. L)) * yp) ;
consider l being Element of NAT such that
A15: l = len yp ;
A16: dom ((- (1. L)) * yp) = Seg k by A14, FINSEQ_1:def_3;
A17: SgmX ((BagOrder n),(Support p)) = SgmX ((BagOrder n),(Support (- p))) by A1, A10, TARSKI:1;
A18: for k being Nat st 1 <= k & k <= len ymp holds
ymp . k = ((- (1. L)) * yp) . k
proof
let k be Nat; ::_thesis: ( 1 <= k & k <= len ymp implies ymp . k = ((- (1. L)) * yp) . k )
assume that
A19: 1 <= k and
A20: k <= len ymp ; ::_thesis: ymp . k = ((- (1. L)) * yp) . k
A21: k <= len ((- (1. L)) * yp) by A12, A13, A14, A16, A20, FINSEQ_1:def_3;
A22: ((- p) * (SgmX ((BagOrder n),(Support p)))) /. k = (- (1. L)) * ((p * (SgmX ((BagOrder n),(Support p)))) /. k)
proof
reconsider b = (SgmX ((BagOrder n),(Support p))) /. k as bag of n ;
k in Seg (len (SgmX ((BagOrder n),(Support p)))) by A7, A12, A19, A20, FINSEQ_1:1;
then A23: k in dom (SgmX ((BagOrder n),(Support p))) by FINSEQ_1:def_3;
A24: dom p = Bags n by FUNCT_2:def_1;
then b in dom p ;
then A25: k in dom (p * (SgmX ((BagOrder n),(Support p)))) by A23, PARTFUN2:3;
A26: dom (- p) = Bags n by FUNCT_2:def_1;
then b in dom (- p) ;
then k in dom ((- p) * (SgmX ((BagOrder n),(Support p)))) by A23, PARTFUN2:3;
hence ((- p) * (SgmX ((BagOrder n),(Support p)))) /. k = (- p) /. b by PARTFUN2:3
.= (- p) . b by A26, PARTFUN1:def_6
.= - (p . b) by POLYNOM1:17
.= - (p /. b) by A24, PARTFUN1:def_6
.= - ((1. L) * (p /. b)) by VECTSP_1:def_6
.= (- (1. L)) * (p /. b) by VECTSP_1:9
.= (- (1. L)) * ((p * (SgmX ((BagOrder n),(Support p)))) /. k) by A25, PARTFUN2:3 ;
::_thesis: verum
end;
A27: k in Seg l by A12, A15, A19, A20, FINSEQ_1:1;
then A28: k in dom yp by A15, FINSEQ_1:def_3;
thus ymp . k = ymp /. k by A19, A20, FINSEQ_4:15
.= ((- (1. L)) * ((p * (SgmX ((BagOrder n),(Support p)))) /. k)) * (eval ((((SgmX ((BagOrder n),(Support p))) /. k) @),x)) by A17, A6, A19, A20, A27, A22
.= (- ((1. L) * ((p * (SgmX ((BagOrder n),(Support p)))) /. k))) * (eval ((((SgmX ((BagOrder n),(Support p))) /. k) @),x)) by VECTSP_1:9
.= (- ((p * (SgmX ((BagOrder n),(Support p)))) /. k)) * (eval ((((SgmX ((BagOrder n),(Support p))) /. k) @),x)) by VECTSP_1:def_6
.= - (((p * (SgmX ((BagOrder n),(Support p)))) /. k) * (eval ((((SgmX ((BagOrder n),(Support p))) /. k) @),x))) by VECTSP_1:9
.= - (yp /. k) by A9, A12, A19, A20, A27
.= - ((1. L) * (yp /. k)) by VECTSP_1:def_6
.= (- (1. L)) * (yp /. k) by VECTSP_1:9
.= ((- (1. L)) * yp) /. k by A28, POLYNOM1:def_1
.= ((- (1. L)) * yp) . k by A19, A21, FINSEQ_4:15 ; ::_thesis: verum
end;
dom yp = Seg l by A15, FINSEQ_1:def_3;
hence eval ((- p),x) = Sum ((- (1. L)) * yp) by A5, A12, A13, A14, A15, A16, A18, FINSEQ_1:6, FINSEQ_1:14
.= (- (1. L)) * (Sum yp) by POLYNOM1:12
.= (- (1. L)) * (eval (p,x)) by A8
.= - ((1. L) * (eval (p,x))) by VECTSP_1:9
.= - (eval (p,x)) by VECTSP_1:def_6 ;
::_thesis: verum
end;
Lm10: for n being Ordinal
for L being non trivial right_complementable Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for p, q being Polynomial of n,L
for x being Function of n,L
for b being bag of n st not b in Support p & Support q = (Support p) \/ {b} & ( for b9 being bag of n st b9 <> b holds
q . b9 = p . b9 ) holds
eval (q,x) = (eval (p,x)) + ((q . b) * (eval (b,x)))
proof
let n be Ordinal; ::_thesis: for L being non trivial right_complementable Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for p, q being Polynomial of n,L
for x being Function of n,L
for b being bag of n st not b in Support p & Support q = (Support p) \/ {b} & ( for b9 being bag of n st b9 <> b holds
q . b9 = p . b9 ) holds
eval (q,x) = (eval (p,x)) + ((q . b) * (eval (b,x)))
let L be non trivial right_complementable Abelian add-associative right_zeroed well-unital distributive doubleLoopStr ; ::_thesis: for p, q being Polynomial of n,L
for x being Function of n,L
for b being bag of n st not b in Support p & Support q = (Support p) \/ {b} & ( for b9 being bag of n st b9 <> b holds
q . b9 = p . b9 ) holds
eval (q,x) = (eval (p,x)) + ((q . b) * (eval (b,x)))
let p, q be Polynomial of n,L; ::_thesis: for x being Function of n,L
for b being bag of n st not b in Support p & Support q = (Support p) \/ {b} & ( for b9 being bag of n st b9 <> b holds
q . b9 = p . b9 ) holds
eval (q,x) = (eval (p,x)) + ((q . b) * (eval (b,x)))
let x be Function of n,L; ::_thesis: for b being bag of n st not b in Support p & Support q = (Support p) \/ {b} & ( for b9 being bag of n st b9 <> b holds
q . b9 = p . b9 ) holds
eval (q,x) = (eval (p,x)) + ((q . b) * (eval (b,x)))
let b be bag of n; ::_thesis: ( not b in Support p & Support q = (Support p) \/ {b} & ( for b9 being bag of n st b9 <> b holds
q . b9 = p . b9 ) implies eval (q,x) = (eval (p,x)) + ((q . b) * (eval (b,x))) )
assume that
A1: not b in Support p and
A2: Support q = (Support p) \/ {b} and
A3: for b9 being bag of n st b9 <> b holds
q . b9 = p . b9 ; ::_thesis: eval (q,x) = (eval (p,x)) + ((q . b) * (eval (b,x)))
set sq = SgmX ((BagOrder n),(Support q));
set sp = SgmX ((BagOrder n),(Support p));
A4: BagOrder n linearly_orders Support q by Th18;
b in {b} by TARSKI:def_1;
then b in Support q by A2, XBOOLE_0:def_3;
then b in rng (SgmX ((BagOrder n),(Support q))) by A4, PRE_POLY:def_2;
then consider k being Nat such that
A5: k in dom (SgmX ((BagOrder n),(Support q))) and
A6: (SgmX ((BagOrder n),(Support q))) . k = b by FINSEQ_2:10;
A7: (SgmX ((BagOrder n),(Support q))) /. k = b by A5, A6, PARTFUN1:def_6;
reconsider b = b as Element of Bags n by PRE_POLY:def_12;
A8: dom (SgmX ((BagOrder n),(Support q))) = Seg (len (SgmX ((BagOrder n),(Support q)))) by FINSEQ_1:def_3;
then A9: k <= len (SgmX ((BagOrder n),(Support q))) by A5, FINSEQ_1:1;
1 <= k by A5, A8, FINSEQ_1:1;
then 1 - 1 <= k - 1 by XREAL_1:9;
then reconsider k9 = k - 1 as Element of NAT by INT_1:3;
A10: k9 + 1 = k + 0 ;
then A11: SgmX ((BagOrder n),(Support q)) = Ins ((SgmX ((BagOrder n),(Support p))),k9,b) by A1, A2, A5, A7, Th11, Th18;
consider yp being FinSequence of the carrier of L such that
A12: len yp = len (SgmX ((BagOrder n),(Support p))) and
A13: Sum yp = eval (p,x) and
A14: for i being Element of NAT st 1 <= i & i <= len yp holds
yp /. i = ((p * (SgmX ((BagOrder n),(Support p)))) /. i) * (eval ((((SgmX ((BagOrder n),(Support p))) /. i) @),x)) by Def4;
consider yq being FinSequence of the carrier of L such that
A15: len yq = len (SgmX ((BagOrder n),(Support q))) and
A16: Sum yq = eval (q,x) and
A17: for i being Element of NAT st 1 <= i & i <= len yq holds
yq /. i = ((q * (SgmX ((BagOrder n),(Support q)))) /. i) * (eval ((((SgmX ((BagOrder n),(Support q))) /. i) @),x)) by Def4;
reconsider b = b as Element of Bags n ;
set ytmp = Ins (yp,k9,((q . b) * (eval (b,x))));
A18: (len (SgmX ((BagOrder n),(Support p)))) + 1 = (card (Support p)) + 1 by Th18, PRE_POLY:11
.= card (Support q) by A1, A2, CARD_2:41
.= len (SgmX ((BagOrder n),(Support q))) by Th18, PRE_POLY:11 ;
then A19: len yq = len (Ins (yp,k9,((q . b) * (eval (b,x))))) by A15, A12, FINSEQ_5:69;
A20: BagOrder n linearly_orders Support p by Th18;
A21: for i being Nat st 1 <= i & i <= len yq holds
yq . i = (Ins (yp,k9,((q . b) * (eval (b,x))))) . i
proof
let i be Nat; ::_thesis: ( 1 <= i & i <= len yq implies yq . i = (Ins (yp,k9,((q . b) * (eval (b,x))))) . i )
assume that
A22: 1 <= i and
A23: i <= len yq ; ::_thesis: yq . i = (Ins (yp,k9,((q . b) * (eval (b,x))))) . i
i in Seg (len yq) by A22, A23, FINSEQ_1:1;
then A24: i in dom yq by FINSEQ_1:def_3;
i in Seg (len (Ins (yp,k9,((q . b) * (eval (b,x)))))) by A19, A22, A23, FINSEQ_1:1;
then A25: i in dom (Ins (yp,k9,((q . b) * (eval (b,x))))) by FINSEQ_1:def_3;
percases ( i = k or i <> k ) ;
supposeA26: i = k ; ::_thesis: yq . i = (Ins (yp,k9,((q . b) * (eval (b,x))))) . i
dom q = Bags n by FUNCT_2:def_1;
then (SgmX ((BagOrder n),(Support q))) . k in dom q by A6, PRE_POLY:def_12;
then A27: k in dom (q * (SgmX ((BagOrder n),(Support q)))) by A5, FUNCT_1:11;
then A28: (q * (SgmX ((BagOrder n),(Support q)))) /. k = (q * (SgmX ((BagOrder n),(Support q)))) . k by PARTFUN1:def_6
.= q . b by A6, A27, FUNCT_1:12 ;
A29: yq /. i = ((q * (SgmX ((BagOrder n),(Support q)))) /. k) * (eval ((((SgmX ((BagOrder n),(Support q))) /. k) @),x)) by A5, A17, A22, A23, A26
.= (q . b) * (eval (b,x)) by A5, A6, A28, PARTFUN1:def_6 ;
A30: k9 <= len yp by A9, A10, A12, A18, XREAL_1:6;
thus (Ins (yp,k9,((q . b) * (eval (b,x))))) . i = (Ins (yp,k9,((q . b) * (eval (b,x))))) /. i by A25, PARTFUN1:def_6
.= (q . b) * (eval (b,x)) by A10, A26, A30, FINSEQ_5:73
.= yq . i by A24, A29, PARTFUN1:def_6 ; ::_thesis: verum
end;
supposeA31: i <> k ; ::_thesis: yq . i = (Ins (yp,k9,((q . b) * (eval (b,x))))) . i
len (Ins ((SgmX ((BagOrder n),(Support p))),k9,b)) = (len (SgmX ((BagOrder n),(Support p)))) + 1 by FINSEQ_5:69;
then A32: dom (Ins ((SgmX ((BagOrder n),(Support p))),k9,b)) = Seg ((len (SgmX ((BagOrder n),(Support p)))) + 1) by FINSEQ_1:def_3;
now__::_thesis:_(_(_i_<_k_&_yq_._i_=_(Ins_(yp,k9,((q_._b)_*_(eval_(b,x)))))_._i_)_or_(_i_>_k_&_yq_._i_=_(Ins_(yp,k9,((q_._b)_*_(eval_(b,x)))))_._i_)_)
percases ( i < k or i > k ) by A31, XXREAL_0:1;
caseA33: i < k ; ::_thesis: yq . i = (Ins (yp,k9,((q . b) * (eval (b,x))))) . i
k9 < k by A10, NAT_1:19;
then k9 < len yq by A9, A15, XXREAL_0:2;
then A34: k9 <= len yp by A15, A12, A18, NAT_1:13;
A35: yp | (Seg k9) is FinSequence by FINSEQ_1:15;
A36: i <= k9 by A10, A33, NAT_1:13;
then i in Seg k9 by A22, FINSEQ_1:1;
then i in dom (yp | (Seg k9)) by A34, A35, FINSEQ_1:17;
then A37: i in dom (yp | k9) by FINSEQ_1:def_15;
A38: k - 1 <= ((len (SgmX ((BagOrder n),(Support p)))) + 1) - 1 by A9, A18, XREAL_1:9;
then A39: i <= len (SgmX ((BagOrder n),(Support p))) by A36, XXREAL_0:2;
then i in Seg (len (SgmX ((BagOrder n),(Support p)))) by A22, FINSEQ_1:1;
then A40: i in dom (SgmX ((BagOrder n),(Support p))) by FINSEQ_1:def_3;
A41: now__::_thesis:_not_(SgmX_((BagOrder_n),(Support_p)))_/._i_=_b
assume (SgmX ((BagOrder n),(Support p))) /. i = b ; ::_thesis: contradiction
then (SgmX ((BagOrder n),(Support p))) . i = b by A40, PARTFUN1:def_6;
then b in rng (SgmX ((BagOrder n),(Support p))) by A40, FUNCT_1:def_3;
hence contradiction by A1, A20, PRE_POLY:def_2; ::_thesis: verum
end;
i < len yq by A9, A15, A33, XXREAL_0:2;
then A42: i <= len yp by A15, A12, A18, NAT_1:13;
A43: (SgmX ((BagOrder n),(Support p))) | (Seg k9) is FinSequence by FINSEQ_1:15;
A44: rng (Ins ((SgmX ((BagOrder n),(Support p))),k9,b)) c= Bags n by FINSEQ_1:def_4;
A45: dom q = Bags n by FUNCT_2:def_1;
A46: rng (SgmX ((BagOrder n),(Support p))) c= Bags n by FINSEQ_1:def_4;
i in Seg k9 by A22, A36, FINSEQ_1:1;
then i in dom ((SgmX ((BagOrder n),(Support p))) | (Seg k9)) by A38, A43, FINSEQ_1:17;
then A47: i in dom ((SgmX ((BagOrder n),(Support p))) | k9) by FINSEQ_1:def_15;
(SgmX ((BagOrder n),(Support p))) . i in rng (SgmX ((BagOrder n),(Support p))) by A40, FUNCT_1:def_3;
then (SgmX ((BagOrder n),(Support p))) . i in Bags n by A46;
then (SgmX ((BagOrder n),(Support p))) . i in dom p by FUNCT_2:def_1;
then A48: i in dom (p * (SgmX ((BagOrder n),(Support p)))) by A40, FUNCT_1:11;
len (SgmX ((BagOrder n),(Support p))) <= (len (SgmX ((BagOrder n),(Support p)))) + 1 by XREAL_1:29;
then i <= (len (SgmX ((BagOrder n),(Support p)))) + 1 by A39, XXREAL_0:2;
then A49: i in dom (Ins ((SgmX ((BagOrder n),(Support p))),k9,b)) by A22, A32, FINSEQ_1:1;
then (Ins ((SgmX ((BagOrder n),(Support p))),k9,b)) . i in rng (Ins ((SgmX ((BagOrder n),(Support p))),k9,b)) by FUNCT_1:def_3;
then A50: i in dom (q * (Ins ((SgmX ((BagOrder n),(Support p))),k9,b))) by A49, A44, A45, FUNCT_1:11;
then A51: (q * (Ins ((SgmX ((BagOrder n),(Support p))),k9,b))) /. i = (q * (Ins ((SgmX ((BagOrder n),(Support p))),k9,b))) . i by PARTFUN1:def_6
.= q . ((Ins ((SgmX ((BagOrder n),(Support p))),k9,b)) . i) by A50, FUNCT_1:12
.= q . ((Ins ((SgmX ((BagOrder n),(Support p))),k9,b)) /. i) by A49, PARTFUN1:def_6
.= q . ((SgmX ((BagOrder n),(Support p))) /. i) by A47, FINSEQ_5:72
.= p . ((SgmX ((BagOrder n),(Support p))) /. i) by A3, A41
.= p . ((SgmX ((BagOrder n),(Support p))) . i) by A40, PARTFUN1:def_6
.= (p * (SgmX ((BagOrder n),(Support p)))) . i by A48, FUNCT_1:12
.= (p * (SgmX ((BagOrder n),(Support p)))) /. i by A48, PARTFUN1:def_6 ;
A52: yq /. i = ((q * (SgmX ((BagOrder n),(Support q)))) /. i) * (eval ((((SgmX ((BagOrder n),(Support q))) /. i) @),x)) by A17, A22, A23, A49
.= ((p * (SgmX ((BagOrder n),(Support p)))) /. i) * (eval ((((SgmX ((BagOrder n),(Support p))) /. i) @),x)) by A11, A47, A51, FINSEQ_5:72
.= yp /. i by A14, A22, A49, A42
.= (Ins (yp,k9,((q . b) * (eval (b,x))))) /. i by A37, FINSEQ_5:72 ;
thus yq . i = yq /. i by A24, PARTFUN1:def_6
.= (Ins (yp,k9,((q . b) * (eval (b,x))))) . i by A25, A52, PARTFUN1:def_6 ; ::_thesis: verum
end;
caseA53: i > k ; ::_thesis: yq . i = (Ins (yp,k9,((q . b) * (eval (b,x))))) . i
then i - 1 > k9 by XREAL_1:9;
then reconsider i1 = i - 1 as Element of NAT by INT_1:3;
A54: (i - 1) + 1 = i + 0 ;
then A55: k9 + 1 <= i1 by A53, NAT_1:13;
A56: dom q = Bags n by FUNCT_2:def_1;
A57: rng (Ins ((SgmX ((BagOrder n),(Support p))),k9,b)) c= Bags n by FINSEQ_1:def_4;
A58: dom p = Bags n by FUNCT_2:def_1;
A59: rng (SgmX ((BagOrder n),(Support p))) c= Bags n by FINSEQ_1:def_4;
A60: i - 1 <= ((len yp) + 1) - 1 by A15, A12, A18, A23, XREAL_1:9;
0 + 1 <= k9 + 1 by XREAL_1:6;
then 1 < i by A53, XXREAL_0:2;
then A61: 1 <= i1 by A54, NAT_1:13;
then i1 in Seg (len (SgmX ((BagOrder n),(Support p)))) by A12, A60, FINSEQ_1:1;
then A62: i1 in dom (SgmX ((BagOrder n),(Support p))) by FINSEQ_1:def_3;
A63: now__::_thesis:_not_(SgmX_((BagOrder_n),(Support_p)))_/._i1_=_b
assume (SgmX ((BagOrder n),(Support p))) /. i1 = b ; ::_thesis: contradiction
then (SgmX ((BagOrder n),(Support p))) . i1 = b by A62, PARTFUN1:def_6;
then b in rng (SgmX ((BagOrder n),(Support p))) by A62, FUNCT_1:def_3;
hence contradiction by A1, A20, PRE_POLY:def_2; ::_thesis: verum
end;
(SgmX ((BagOrder n),(Support p))) . i1 in rng (SgmX ((BagOrder n),(Support p))) by A62, FUNCT_1:def_3;
then A64: i1 in dom (p * (SgmX ((BagOrder n),(Support p)))) by A62, A59, A58, FUNCT_1:11;
A65: i = i1 + 1 ;
A66: i in dom (Ins ((SgmX ((BagOrder n),(Support p))),k9,b)) by A15, A18, A22, A23, A32, FINSEQ_1:1;
then (Ins ((SgmX ((BagOrder n),(Support p))),k9,b)) . i in rng (Ins ((SgmX ((BagOrder n),(Support p))),k9,b)) by FUNCT_1:def_3;
then A67: i in dom (q * (Ins ((SgmX ((BagOrder n),(Support p))),k9,b))) by A66, A57, A56, FUNCT_1:11;
then A68: (q * (Ins ((SgmX ((BagOrder n),(Support p))),k9,b))) /. i = (q * (Ins ((SgmX ((BagOrder n),(Support p))),k9,b))) . i by PARTFUN1:def_6
.= q . ((Ins ((SgmX ((BagOrder n),(Support p))),k9,b)) . i) by A67, FUNCT_1:12
.= q . ((Ins ((SgmX ((BagOrder n),(Support p))),k9,b)) /. i) by A66, PARTFUN1:def_6
.= q . ((SgmX ((BagOrder n),(Support p))) /. i1) by A12, A60, A65, A55, FINSEQ_5:74
.= p . ((SgmX ((BagOrder n),(Support p))) /. i1) by A3, A63
.= p . ((SgmX ((BagOrder n),(Support p))) . i1) by A62, PARTFUN1:def_6
.= (p * (SgmX ((BagOrder n),(Support p)))) . i1 by A64, FUNCT_1:12
.= (p * (SgmX ((BagOrder n),(Support p)))) /. i1 by A64, PARTFUN1:def_6 ;
A69: yq /. i = ((q * (SgmX ((BagOrder n),(Support q)))) /. i) * (eval ((((SgmX ((BagOrder n),(Support q))) /. i) @),x)) by A17, A22, A23, A66
.= ((p * (SgmX ((BagOrder n),(Support p)))) /. i1) * (eval ((((SgmX ((BagOrder n),(Support p))) /. i1) @),x)) by A11, A12, A60, A65, A55, A68, FINSEQ_5:74
.= yp /. i1 by A14, A60, A61
.= (Ins (yp,k9,((q . b) * (eval (b,x))))) /. i by A60, A65, A55, FINSEQ_5:74 ;
thus yq . i = yq /. i by A24, PARTFUN1:def_6
.= (Ins (yp,k9,((q . b) * (eval (b,x))))) . i by A25, A69, PARTFUN1:def_6 ; ::_thesis: verum
end;
end;
end;
hence yq . i = (Ins (yp,k9,((q . b) * (eval (b,x))))) . i ; ::_thesis: verum
end;
end;
end;
Sum (((yp | k9) ^ <*((q . b) * (eval (b,x)))*>) ^ (yp /^ k9)) = (Sum ((yp | k9) ^ <*((q . b) * (eval (b,x)))*>)) + (Sum (yp /^ k9)) by RLVECT_1:41
.= ((Sum (yp | k9)) + (Sum <*((q . b) * (eval (b,x)))*>)) + (Sum (yp /^ k9)) by RLVECT_1:41
.= ((Sum (yp | k9)) + (Sum (yp /^ k9))) + (Sum <*((q . b) * (eval (b,x)))*>) by RLVECT_1:def_3
.= (Sum ((yp | k9) ^ (yp /^ k9))) + (Sum <*((q . b) * (eval (b,x)))*>) by RLVECT_1:41
.= (Sum yp) + (Sum <*((q . b) * (eval (b,x)))*>) by RFINSEQ:8
.= (eval (p,x)) + ((q . b) * (eval (b,x))) by A13, RLVECT_1:44 ;
then Sum (Ins (yp,k9,((q . b) * (eval (b,x))))) = (eval (p,x)) + ((q . b) * (eval (b,x))) by FINSEQ_5:def_4;
hence eval (q,x) = (eval (p,x)) + ((q . b) * (eval (b,x))) by A16, A19, A21, FINSEQ_1:14; ::_thesis: verum
end;
Lm11: for n being Ordinal
for L being non trivial right_complementable Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for p being Polynomial of n,L st ex b being bag of n st Support p = {b} holds
for q being Polynomial of n,L
for x being Function of n,L holds eval ((p + q),x) = (eval (p,x)) + (eval (q,x))
proof
let n be Ordinal; ::_thesis: for L being non trivial right_complementable Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for p being Polynomial of n,L st ex b being bag of n st Support p = {b} holds
for q being Polynomial of n,L
for x being Function of n,L holds eval ((p + q),x) = (eval (p,x)) + (eval (q,x))
let L be non trivial right_complementable Abelian add-associative right_zeroed well-unital distributive doubleLoopStr ; ::_thesis: for p being Polynomial of n,L st ex b being bag of n st Support p = {b} holds
for q being Polynomial of n,L
for x being Function of n,L holds eval ((p + q),x) = (eval (p,x)) + (eval (q,x))
let p be Polynomial of n,L; ::_thesis: ( ex b being bag of n st Support p = {b} implies for q being Polynomial of n,L
for x being Function of n,L holds eval ((p + q),x) = (eval (p,x)) + (eval (q,x)) )
assume ex b being bag of n st Support p = {b} ; ::_thesis: for q being Polynomial of n,L
for x being Function of n,L holds eval ((p + q),x) = (eval (p,x)) + (eval (q,x))
then consider b being bag of n such that
A1: Support p = {b} ;
b in Support p by A1, TARSKI:def_1;
then A2: p . b <> 0. L by POLYNOM1:def_3;
let q be Polynomial of n,L; ::_thesis: for x being Function of n,L holds eval ((p + q),x) = (eval (p,x)) + (eval (q,x))
let x be Function of n,L; ::_thesis: eval ((p + q),x) = (eval (p,x)) + (eval (q,x))
A3: for b9 being bag of n st b9 <> b holds
(p + q) . b9 = q . b9
proof
let b9 be bag of n; ::_thesis: ( b9 <> b implies (p + q) . b9 = q . b9 )
A4: b9 is Element of Bags n by PRE_POLY:def_12;
assume b9 <> b ; ::_thesis: (p + q) . b9 = q . b9
then A5: not b9 in Support p by A1, TARSKI:def_1;
thus (p + q) . b9 = (p . b9) + (q . b9) by POLYNOM1:15
.= (0. L) + (q . b9) by A5, A4, POLYNOM1:def_3
.= q . b9 by RLVECT_1:def_4 ; ::_thesis: verum
end;
set sq = SgmX ((BagOrder n),(Support q));
set spq = SgmX ((BagOrder n),(Support (p + q)));
A6: b is Element of Bags n by PRE_POLY:def_12;
A7: Support (p + q) c= (Support p) \/ (Support q) by POLYNOM1:20;
consider yq being FinSequence of the carrier of L such that
A8: len yq = len (SgmX ((BagOrder n),(Support q))) and
A9: eval (q,x) = Sum yq and
A10: for i being Element of NAT st 1 <= i & i <= len yq holds
yq /. i = ((q * (SgmX ((BagOrder n),(Support q)))) /. i) * (eval ((((SgmX ((BagOrder n),(Support q))) /. i) @),x)) by Def4;
consider ypq being FinSequence of the carrier of L such that
A11: len ypq = len (SgmX ((BagOrder n),(Support (p + q)))) and
A12: eval ((p + q),x) = Sum ypq and
A13: for i being Element of NAT st 1 <= i & i <= len ypq holds
ypq /. i = (((p + q) * (SgmX ((BagOrder n),(Support (p + q))))) /. i) * (eval ((((SgmX ((BagOrder n),(Support (p + q)))) /. i) @),x)) by Def4;
A14: BagOrder n linearly_orders Support q by Th18;
now__::_thesis:_(_(_b_in_Support_q_&_eval_((p_+_q),x)_=_(eval_(p,x))_+_(eval_(q,x))_)_or_(_not_b_in_Support_q_&_eval_((p_+_q),x)_=_(eval_(q,x))_+_(eval_(p,x))_)_)
percases ( b in Support q or not b in Support q ) ;
caseA15: b in Support q ; ::_thesis: eval ((p + q),x) = (eval (p,x)) + (eval (q,x))
now__::_thesis:_(_(_p_._b_=_-_(q_._b)_&_eval_((p_+_q),x)_=_(eval_(q,x))_+_(eval_(p,x))_)_or_(_p_._b_<>_-_(q_._b)_&_eval_((p_+_q),x)_=_(eval_(p,x))_+_(eval_(q,x))_)_)
percases ( p . b = - (q . b) or p . b <> - (q . b) ) ;
caseA16: p . b = - (q . b) ; ::_thesis: eval ((p + q),x) = (eval (q,x)) + (eval (p,x))
A17: for u being set st u in Support q holds
u in (Support (p + q)) \/ {b}
proof
let u be set ; ::_thesis: ( u in Support q implies u in (Support (p + q)) \/ {b} )
assume A18: u in Support q ; ::_thesis: u in (Support (p + q)) \/ {b}
then reconsider u = u as bag of n ;
percases ( u = b or u <> b ) ;
suppose u = b ; ::_thesis: u in (Support (p + q)) \/ {b}
then u in {b} by TARSKI:def_1;
hence u in (Support (p + q)) \/ {b} by XBOOLE_0:def_3; ::_thesis: verum
end;
suppose u <> b ; ::_thesis: u in (Support (p + q)) \/ {b}
then (p + q) . u = q . u by A3;
then A19: (p + q) . u <> 0. L by A18, POLYNOM1:def_3;
u is Element of Bags n by PRE_POLY:def_12;
then u in Support (p + q) by A19, POLYNOM1:def_3;
hence u in (Support (p + q)) \/ {b} by XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
(p + q) . b = (p . b) + (q . b) by POLYNOM1:15
.= 0. L by A16, RLVECT_1:5 ;
then A20: not b in Support (p + q) by POLYNOM1:def_3;
for u being set st u in (Support (p + q)) \/ {b} holds
u in Support q
proof
let u be set ; ::_thesis: ( u in (Support (p + q)) \/ {b} implies u in Support q )
assume A21: u in (Support (p + q)) \/ {b} ; ::_thesis: u in Support q
percases ( u in Support (p + q) or u in {b} ) by A21, XBOOLE_0:def_3;
supposeA22: u in Support (p + q) ; ::_thesis: u in Support q
then not u in {b} by A20, TARSKI:def_1;
hence u in Support q by A1, A7, A22, XBOOLE_0:def_3; ::_thesis: verum
end;
suppose u in {b} ; ::_thesis: u in Support q
hence u in Support q by A15, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
then A23: (Support (p + q)) \/ {b} = Support q by A17, TARSKI:1;
thus eval ((p + q),x) = (eval ((p + q),x)) + (0. L) by RLVECT_1:def_4
.= (eval ((p + q),x)) + (((q . b) * (eval (b,x))) + (- ((q . b) * (eval (b,x))))) by RLVECT_1:5
.= ((eval ((p + q),x)) + ((q . b) * (eval (b,x)))) + (- ((q . b) * (eval (b,x)))) by RLVECT_1:def_3
.= (eval (q,x)) + (- ((q . b) * (eval (b,x)))) by A3, A20, A23, Lm10
.= (eval (q,x)) + ((p . b) * (eval (b,x))) by A16, VECTSP_1:9
.= (eval (q,x)) + (eval (p,x)) by A1, Th19 ; ::_thesis: verum
end;
caseA24: p . b <> - (q . b) ; ::_thesis: eval ((p + q),x) = (eval (p,x)) + (eval (q,x))
(p . b) + (q . b) <> (- (q . b)) + (q . b)
proof
assume A25: (p . b) + (q . b) = (- (q . b)) + (q . b) ; ::_thesis: contradiction
p . b = (p . b) + (0. L) by RLVECT_1:def_4
.= (p . b) + ((q . b) + (- (q . b))) by RLVECT_1:5
.= ((- (q . b)) + (q . b)) + (- (q . b)) by A25, RLVECT_1:def_3
.= (0. L) + (- (q . b)) by RLVECT_1:5
.= - (q . b) by RLVECT_1:def_4 ;
hence contradiction by A24; ::_thesis: verum
end;
then (p . b) + (q . b) <> 0. L by RLVECT_1:5;
then A26: (p + q) . b <> 0. L by POLYNOM1:15;
A27: for u being set st u in Support q holds
u in Support (p + q)
proof
let u be set ; ::_thesis: ( u in Support q implies u in Support (p + q) )
assume A28: u in Support q ; ::_thesis: u in Support (p + q)
then reconsider u = u as bag of n ;
percases ( u = b or u <> b ) ;
suppose u = b ; ::_thesis: u in Support (p + q)
hence u in Support (p + q) by A6, A26, POLYNOM1:def_3; ::_thesis: verum
end;
suppose u <> b ; ::_thesis: u in Support (p + q)
then (p + q) . u = q . u by A3;
then A29: (p + q) . u <> 0. L by A28, POLYNOM1:def_3;
u is Element of Bags n by PRE_POLY:def_12;
hence u in Support (p + q) by A29, POLYNOM1:def_3; ::_thesis: verum
end;
end;
end;
A30: for u being set st u in Support (p + q) holds
u in Support q
proof
let u be set ; ::_thesis: ( u in Support (p + q) implies u in Support q )
assume A31: u in Support (p + q) ; ::_thesis: u in Support q
then reconsider u = u as bag of n ;
percases ( u in Support p or u in Support q ) by A7, A31, XBOOLE_0:def_3;
suppose u in Support p ; ::_thesis: u in Support q
hence u in Support q by A1, A15, TARSKI:def_1; ::_thesis: verum
end;
suppose u in Support q ; ::_thesis: u in Support q
hence u in Support q ; ::_thesis: verum
end;
end;
end;
then A32: Support (p + q) = Support q by A27, TARSKI:1;
A33: len ypq = len yq by A11, A8, A30, A27, TARSKI:1;
consider spqk being Nat such that
A34: dom (SgmX ((BagOrder n),(Support (p + q)))) = Seg spqk by FINSEQ_1:def_2;
b in rng (SgmX ((BagOrder n),(Support q))) by A14, A15, PRE_POLY:def_2;
then consider k being Nat such that
A35: k in dom (SgmX ((BagOrder n),(Support q))) and
A36: (SgmX ((BagOrder n),(Support q))) . k = b by FINSEQ_2:10;
consider sqk being Nat such that
A37: dom (SgmX ((BagOrder n),(Support q))) = Seg sqk by FINSEQ_1:def_2;
reconsider k = k as Element of NAT by ORDINAL1:def_12;
reconsider k = k, sqk = sqk, spqk = spqk as Element of NAT by ORDINAL1:def_12;
A38: 1 <= k by A35, A37, FINSEQ_1:1;
A39: dom (p + q) = Bags n by FUNCT_2:def_1;
then (SgmX ((BagOrder n),(Support q))) . k in dom (p + q) by A36, PRE_POLY:def_12;
then A40: k in dom ((p + q) * (SgmX ((BagOrder n),(Support q)))) by A35, FUNCT_1:11;
then A41: ((p + q) * (SgmX ((BagOrder n),(Support q)))) /. k = ((p + q) * (SgmX ((BagOrder n),(Support q)))) . k by PARTFUN1:def_6
.= (p + q) . b by A36, A40, FUNCT_1:12 ;
A42: k <= sqk by A35, A37, FINSEQ_1:1;
A43: dom q = Bags n by FUNCT_2:def_1;
then (SgmX ((BagOrder n),(Support q))) . k in dom q by A36, PRE_POLY:def_12;
then A44: k in dom (q * (SgmX ((BagOrder n),(Support q)))) by A35, FUNCT_1:11;
then A45: (q * (SgmX ((BagOrder n),(Support q)))) /. k = (q * (SgmX ((BagOrder n),(Support q)))) . k by PARTFUN1:def_6
.= q . b by A36, A44, FUNCT_1:12 ;
consider i being Nat such that
A46: dom yq = Seg i by FINSEQ_1:def_2;
A47: sqk = len (SgmX ((BagOrder n),(Support q))) by A37, FINSEQ_1:def_3
.= len (SgmX ((BagOrder n),(Support (p + q)))) by A30, A27, TARSKI:1
.= spqk by A34, FINSEQ_1:def_3 ;
A48: i in NAT by ORDINAL1:def_12;
then A49: len yq = i by A46, FINSEQ_1:def_3;
A50: for i9 being Element of NAT st i9 in dom yq & i9 <> k holds
ypq /. i9 = yq /. i9
proof
let i9 be Element of NAT ; ::_thesis: ( i9 in dom yq & i9 <> k implies ypq /. i9 = yq /. i9 )
assume that
A51: i9 in dom yq and
A52: i9 <> k ; ::_thesis: ypq /. i9 = yq /. i9
A53: 1 <= i9 by A46, A51, FINSEQ_1:1;
i9 in Seg (len (SgmX ((BagOrder n),(Support (p + q))))) by A11, A33, A51, FINSEQ_1:def_3;
then A54: i9 in dom (SgmX ((BagOrder n),(Support (p + q)))) by FINSEQ_1:def_3;
then (SgmX ((BagOrder n),(Support (p + q)))) /. i9 = (SgmX ((BagOrder n),(Support (p + q)))) . i9 by PARTFUN1:def_6;
then A55: i9 in dom ((p + q) * (SgmX ((BagOrder n),(Support (p + q))))) by A39, A54, FUNCT_1:11;
then A56: ((p + q) * (SgmX ((BagOrder n),(Support (p + q))))) /. i9 = ((p + q) * (SgmX ((BagOrder n),(Support (p + q))))) . i9 by PARTFUN1:def_6
.= (p + q) . ((SgmX ((BagOrder n),(Support (p + q)))) . i9) by A55, FUNCT_1:12
.= (p + q) . ((SgmX ((BagOrder n),(Support (p + q)))) /. i9) by A54, PARTFUN1:def_6 ;
A57: (SgmX ((BagOrder n),(Support (p + q)))) /. i9 <> b
proof
assume (SgmX ((BagOrder n),(Support (p + q)))) /. i9 = b ; ::_thesis: contradiction
then A58: (SgmX ((BagOrder n),(Support (p + q)))) . i9 = b by A54, PARTFUN1:def_6;
A59: SgmX ((BagOrder n),(Support (p + q))) is one-to-one by Th18, PRE_POLY:10;
(SgmX ((BagOrder n),(Support (p + q)))) . k = b by A30, A27, A36, TARSKI:1;
hence contradiction by A35, A37, A34, A47, A52, A54, A58, A59, FUNCT_1:def_4; ::_thesis: verum
end;
A60: i9 in dom (SgmX ((BagOrder n),(Support q))) by A8, A46, A49, A51, FINSEQ_1:def_3;
(SgmX ((BagOrder n),(Support q))) /. i9 = (SgmX ((BagOrder n),(Support q))) . i9 by A37, A34, A47, A54, PARTFUN1:def_6;
then A61: i9 in dom (q * (SgmX ((BagOrder n),(Support q)))) by A43, A60, FUNCT_1:11;
then A62: (q * (SgmX ((BagOrder n),(Support q)))) /. i9 = (q * (SgmX ((BagOrder n),(Support q)))) . i9 by PARTFUN1:def_6
.= q . ((SgmX ((BagOrder n),(Support q))) . i9) by A61, FUNCT_1:12
.= q . ((SgmX ((BagOrder n),(Support q))) /. i9) by A60, PARTFUN1:def_6 ;
A63: i9 <= len yq by A46, A49, A51, FINSEQ_1:1;
hence ypq /. i9 = (((p + q) * (SgmX ((BagOrder n),(Support (p + q))))) /. i9) * (eval ((((SgmX ((BagOrder n),(Support (p + q)))) /. i9) @),x)) by A13, A33, A53
.= (q . ((SgmX ((BagOrder n),(Support q))) /. i9)) * (eval ((((SgmX ((BagOrder n),(Support q))) /. i9) @),x)) by A3, A32, A57, A56
.= yq /. i9 by A10, A53, A63, A62 ;
::_thesis: verum
end;
A64: (SgmX ((BagOrder n),(Support q))) /. k = b by A35, A36, PARTFUN1:def_6;
A65: sqk = len yq by A8, A37, FINSEQ_1:def_3;
then k <= i by A42, A46, A48, FINSEQ_1:def_3;
then A66: k in dom yq by A38, A46, FINSEQ_1:1;
len ypq = sqk by A11, A34, A47, FINSEQ_1:def_3;
then ypq /. k = (((p + q) * (SgmX ((BagOrder n),(Support (p + q))))) /. k) * (eval ((((SgmX ((BagOrder n),(Support (p + q)))) /. k) @),x)) by A13, A38, A42
.= ((p . b) + (q . b)) * (eval (b,x)) by A32, A64, A41, POLYNOM1:15
.= ((p . b) * (eval (b,x))) + (((q * (SgmX ((BagOrder n),(Support q)))) /. k) * (eval ((((SgmX ((BagOrder n),(Support q))) /. k) @),x))) by A64, A45, VECTSP_1:def_7
.= ((p . b) * (eval (b,x))) + (yq /. k) by A10, A38, A42, A65 ;
hence eval ((p + q),x) = ((p . b) * (eval (b,x))) + (Sum yq) by A12, A33, A66, A50, Th5
.= (eval (p,x)) + (eval (q,x)) by A1, A9, Th19 ;
::_thesis: verum
end;
end;
end;
hence eval ((p + q),x) = (eval (p,x)) + (eval (q,x)) ; ::_thesis: verum
end;
caseA67: not b in Support q ; ::_thesis: eval ((p + q),x) = (eval (q,x)) + (eval (p,x))
A68: (p + q) . b = (p . b) + (q . b) by POLYNOM1:15
.= (p . b) + (0. L) by A6, A67, POLYNOM1:def_3
.= p . b by RLVECT_1:def_4 ;
A69: for u being set st u in (Support p) \/ (Support q) holds
u in Support (p + q)
proof
let u be set ; ::_thesis: ( u in (Support p) \/ (Support q) implies u in Support (p + q) )
assume A70: u in (Support p) \/ (Support q) ; ::_thesis: u in Support (p + q)
percases ( u in Support p or u in Support q ) by A70, XBOOLE_0:def_3;
suppose u in Support p ; ::_thesis: u in Support (p + q)
then u = b by A1, TARSKI:def_1;
hence u in Support (p + q) by A6, A2, A68, POLYNOM1:def_3; ::_thesis: verum
end;
supposeA71: u in Support q ; ::_thesis: u in Support (p + q)
then reconsider u = u as bag of n ;
A72: q . u <> 0. L by A71, POLYNOM1:def_3;
(p + q) . u = q . u by A3, A67, A71;
hence u in Support (p + q) by A71, A72, POLYNOM1:def_3; ::_thesis: verum
end;
end;
end;
for u being set st u in Support (p + q) holds
u in (Support p) \/ (Support q) by A7;
then Support (p + q) = {b} \/ (Support q) by A1, A69, TARSKI:1;
hence eval ((p + q),x) = (eval (q,x)) + (((p + q) . b) * (eval (b,x))) by A3, A67, Lm10
.= (eval (q,x)) + (eval (p,x)) by A1, A68, Th19 ;
::_thesis: verum
end;
end;
end;
hence eval ((p + q),x) = (eval (p,x)) + (eval (q,x)) ; ::_thesis: verum
end;
theorem Th23: :: POLYNOM2:23
for n being Ordinal
for L being non trivial right_complementable Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for p, q being Polynomial of n,L
for x being Function of n,L holds eval ((p + q),x) = (eval (p,x)) + (eval (q,x))
proof
let n be Ordinal; ::_thesis: for L being non trivial right_complementable Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for p, q being Polynomial of n,L
for x being Function of n,L holds eval ((p + q),x) = (eval (p,x)) + (eval (q,x))
let L be non trivial right_complementable Abelian add-associative right_zeroed well-unital distributive doubleLoopStr ; ::_thesis: for p, q being Polynomial of n,L
for x being Function of n,L holds eval ((p + q),x) = (eval (p,x)) + (eval (q,x))
let p, q be Polynomial of n,L; ::_thesis: for x being Function of n,L holds eval ((p + q),x) = (eval (p,x)) + (eval (q,x))
let x be Function of n,L; ::_thesis: eval ((p + q),x) = (eval (p,x)) + (eval (q,x))
defpred S1[ Element of NAT ] means for p being Polynomial of n,L st card (Support p) = $1 holds
eval ((p + q),x) = (eval (p,x)) + (eval (q,x));
A1: ex k being Element of NAT st card (Support p) = k ;
A2: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A3: S1[k] ; ::_thesis: S1[k + 1]
let p be Polynomial of n,L; ::_thesis: ( card (Support p) = k + 1 implies eval ((p + q),x) = (eval (p,x)) + (eval (q,x)) )
assume A4: card (Support p) = k + 1 ; ::_thesis: eval ((p + q),x) = (eval (p,x)) + (eval (q,x))
set sgp = SgmX ((BagOrder n),(Support p));
set bg = (SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p))));
A5: BagOrder n linearly_orders Support p by Th18;
then SgmX ((BagOrder n),(Support p)) <> {} by A4, CARD_1:27, PRE_POLY:def_2, RELAT_1:38;
then 1 <= len (SgmX ((BagOrder n),(Support p))) by NAT_1:14;
then len (SgmX ((BagOrder n),(Support p))) in Seg (len (SgmX ((BagOrder n),(Support p)))) by FINSEQ_1:1;
then A6: len (SgmX ((BagOrder n),(Support p))) in dom (SgmX ((BagOrder n),(Support p))) by FINSEQ_1:def_3;
then (SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p)))) = (SgmX ((BagOrder n),(Support p))) . (len (SgmX ((BagOrder n),(Support p)))) by PARTFUN1:def_6;
then (SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p)))) in rng (SgmX ((BagOrder n),(Support p))) by A6, FUNCT_1:def_3;
then A7: (SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p)))) in Support p by A5, PRE_POLY:def_2;
then A8: p . ((SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p))))) <> 0. L by POLYNOM1:def_3;
set m = (0_ (n,L)) +* (((SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p))))),(p . ((SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p)))))));
set p9 = p +* (((SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p))))),(0. L));
reconsider bg = (SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p)))) as bag of n ;
dom p = Bags n by FUNCT_2:def_1;
then A9: p +* (((SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p))))),(0. L)) = p +* (bg .--> (0. L)) by FUNCT_7:def_3;
reconsider p9 = p +* (((SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p))))),(0. L)) as Function of (Bags n), the carrier of L ;
reconsider p9 = p9 as Function of (Bags n),L ;
for u being set st u in Support p9 holds
u in Support p
proof
let u be set ; ::_thesis: ( u in Support p9 implies u in Support p )
assume A10: u in Support p9 ; ::_thesis: u in Support p
then reconsider u = u as Element of Bags n ;
reconsider u = u as bag of n ;
now__::_thesis:_not_u_=_bg
assume A11: u = bg ; ::_thesis: contradiction
then u in {bg} by TARSKI:def_1;
then u in dom (bg .--> (0. L)) by FUNCOP_1:13;
then p9 . u = (bg .--> (0. L)) . bg by A9, A11, FUNCT_4:13;
then p9 . u = 0. L by FUNCOP_1:72;
hence contradiction by A10, POLYNOM1:def_3; ::_thesis: verum
end;
then not u in {bg} by TARSKI:def_1;
then not u in dom (bg .--> (0. L)) ;
then p . u = p9 . u by A9, FUNCT_4:11;
then p . u <> 0. L by A10, POLYNOM1:def_3;
hence u in Support p by POLYNOM1:def_3; ::_thesis: verum
end;
then Support p9 c= Support p by TARSKI:def_3;
then reconsider p9 = p9 as Polynomial of n,L by POLYNOM1:def_4;
A12: dom p = Bags n by FUNCT_2:def_1;
A13: for u being set st u in Support p holds
u in (Support p9) \/ {bg}
proof
let u be set ; ::_thesis: ( u in Support p implies u in (Support p9) \/ {bg} )
assume A14: u in Support p ; ::_thesis: u in (Support p9) \/ {bg}
then reconsider u = u as Element of Bags n ;
A15: p . u <> 0. L by A14, POLYNOM1:def_3;
percases ( u = bg or u <> bg ) ;
suppose u = bg ; ::_thesis: u in (Support p9) \/ {bg}
then u in {bg} by TARSKI:def_1;
hence u in (Support p9) \/ {bg} by XBOOLE_0:def_3; ::_thesis: verum
end;
suppose u <> bg ; ::_thesis: u in (Support p9) \/ {bg}
then not u in {bg} by TARSKI:def_1;
then not u in dom (bg .--> (0. L)) ;
then p9 . u = p . u by A9, FUNCT_4:11;
then u in Support p9 by A15, POLYNOM1:def_3;
hence u in (Support p9) \/ {bg} by XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
bg in {bg} by TARSKI:def_1;
then bg in dom (bg .--> (0. L)) by FUNCOP_1:13;
then p9 . bg = (bg .--> (0. L)) . bg by A9, FUNCT_4:13;
then A16: p9 . bg = 0. L by FUNCOP_1:72;
then A17: not bg in Support p9 by POLYNOM1:def_3;
for u being set st u in (Support p9) \/ {bg} holds
u in Support p
proof
let u be set ; ::_thesis: ( u in (Support p9) \/ {bg} implies u in Support p )
assume A18: u in (Support p9) \/ {bg} ; ::_thesis: u in Support p
percases ( u in Support p9 or u in {bg} ) by A18, XBOOLE_0:def_3;
supposeA19: u in Support p9 ; ::_thesis: u in Support p
then reconsider u = u as Element of Bags n ;
u <> bg by A16, A19, POLYNOM1:def_3;
then not u in {bg} by TARSKI:def_1;
then not u in dom (bg .--> (0. L)) ;
then A20: p9 . u = p . u by A9, FUNCT_4:11;
p9 . u <> 0. L by A19, POLYNOM1:def_3;
hence u in Support p by A20, POLYNOM1:def_3; ::_thesis: verum
end;
suppose u in {bg} ; ::_thesis: u in Support p
hence u in Support p by A7, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
then Support p = (Support p9) \/ {bg} by A13, TARSKI:1;
then A21: k + 1 = (card (Support p9)) + 1 by A4, A17, CARD_2:41;
dom (0_ (n,L)) = Bags n by FUNCT_2:def_1;
then A22: (0_ (n,L)) +* (((SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p))))),(p . ((SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p))))))) = (0_ (n,L)) +* (bg .--> (p . bg)) by FUNCT_7:def_3;
reconsider m = (0_ (n,L)) +* (((SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p))))),(p . ((SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p))))))) as Function of (Bags n), the carrier of L ;
reconsider m = m as Function of (Bags n),L ;
A23: for u being set st u in Support m holds
u in {bg}
proof
let u be set ; ::_thesis: ( u in Support m implies u in {bg} )
assume A24: u in Support m ; ::_thesis: u in {bg}
then reconsider u = u as Element of Bags n ;
A25: m . u <> 0. L by A24, POLYNOM1:def_3;
now__::_thesis:_not_u_<>_bg
assume u <> bg ; ::_thesis: contradiction
then not u in {bg} by TARSKI:def_1;
then not u in dom (bg .--> (p . bg)) ;
then m . u = (0_ (n,L)) . u by A22, FUNCT_4:11;
hence contradiction by A25, POLYNOM1:22; ::_thesis: verum
end;
hence u in {bg} by TARSKI:def_1; ::_thesis: verum
end;
for u being set st u in {bg} holds
u in Support m
proof
let u be set ; ::_thesis: ( u in {bg} implies u in Support m )
bg in {bg} by TARSKI:def_1;
then bg in dom (bg .--> (p . bg)) by FUNCOP_1:13;
then m . bg = (bg .--> (p . bg)) . bg by A22, FUNCT_4:13;
then A26: m . bg = p . bg by FUNCOP_1:72;
assume u in {bg} ; ::_thesis: u in Support m
then u = bg by TARSKI:def_1;
hence u in Support m by A8, A26, POLYNOM1:def_3; ::_thesis: verum
end;
then A27: Support m = {bg} by A23, TARSKI:1;
then reconsider m = m as Polynomial of n,L by POLYNOM1:def_4;
reconsider m = m as Polynomial of n,L ;
A28: for u being set st u in Bags n holds
(p9 + m) . u = p . u
proof
let u be set ; ::_thesis: ( u in Bags n implies (p9 + m) . u = p . u )
assume u in Bags n ; ::_thesis: (p9 + m) . u = p . u
then reconsider u = u as bag of n ;
percases ( u = bg or u <> bg ) ;
supposeA29: u = bg ; ::_thesis: (p9 + m) . u = p . u
bg in {bg} by TARSKI:def_1;
then bg in dom (bg .--> (p . bg)) by FUNCOP_1:13;
then m . bg = (bg .--> (p . bg)) . bg by A22, FUNCT_4:13;
then A30: m . bg = p . bg by FUNCOP_1:72;
u in {bg} by A29, TARSKI:def_1;
then u in dom (bg .--> (0. L)) by FUNCOP_1:13;
then A31: p9 . u = (bg .--> (0. L)) . bg by A9, A29, FUNCT_4:13;
(p9 + m) . u = (p9 . u) + (m . u) by POLYNOM1:15
.= (0. L) + (p . bg) by A29, A31, A30, FUNCOP_1:72
.= p . bg by RLVECT_1:def_4 ;
hence (p9 + m) . u = p . u by A29; ::_thesis: verum
end;
suppose u <> bg ; ::_thesis: (p9 + m) . u = p . u
then A32: not u in {bg} by TARSKI:def_1;
then A33: not u in dom (bg .--> (0. L)) ;
not u in dom (bg .--> (p . bg)) by A32;
then m . u = (0_ (n,L)) . u by A22, FUNCT_4:11;
then A34: m . u = 0. L by POLYNOM1:22;
(p9 + m) . u = (p9 . u) + (m . u) by POLYNOM1:15
.= (p . u) + (0. L) by A9, A33, A34, FUNCT_4:11
.= p . u by RLVECT_1:def_4 ;
hence (p9 + m) . u = p . u ; ::_thesis: verum
end;
end;
end;
A35: dom (p9 + m) = Bags n by FUNCT_2:def_1;
then eval (p,x) = eval ((m + p9),x) by A12, A28, FUNCT_1:2
.= (eval (p9,x)) + (eval (m,x)) by A27, Lm11 ;
hence (eval (p,x)) + (eval (q,x)) = ((eval (p9,x)) + (eval (q,x))) + (eval (m,x)) by RLVECT_1:def_3
.= (eval ((p9 + q),x)) + (eval (m,x)) by A3, A21
.= eval ((m + (p9 + q)),x) by A27, Lm11
.= eval (((p9 + m) + q),x) by POLYNOM1:21
.= eval ((p + q),x) by A35, A12, A28, FUNCT_1:2 ;
::_thesis: verum
end;
A36: S1[ 0 ]
proof
let p be Polynomial of n,L; ::_thesis: ( card (Support p) = 0 implies eval ((p + q),x) = (eval (p,x)) + (eval (q,x)) )
assume card (Support p) = 0 ; ::_thesis: eval ((p + q),x) = (eval (p,x)) + (eval (q,x))
then Support p = {} ;
then A37: p = 0_ (n,L) by Th17;
hence eval ((p + q),x) = eval (q,x) by POLYNOM1:23
.= (0. L) + (eval (q,x)) by RLVECT_1:4
.= (eval (p,x)) + (eval (q,x)) by A37, Th20 ;
::_thesis: verum
end;
for k being Element of NAT holds S1[k] from NAT_1:sch_1(A36, A2);
hence eval ((p + q),x) = (eval (p,x)) + (eval (q,x)) by A1; ::_thesis: verum
end;
theorem :: POLYNOM2:24
for n being Ordinal
for L being non trivial right_complementable Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for p, q being Polynomial of n,L
for x being Function of n,L holds eval ((p - q),x) = (eval (p,x)) - (eval (q,x))
proof
let n be Ordinal; ::_thesis: for L being non trivial right_complementable Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for p, q being Polynomial of n,L
for x being Function of n,L holds eval ((p - q),x) = (eval (p,x)) - (eval (q,x))
let L be non trivial right_complementable Abelian add-associative right_zeroed well-unital distributive doubleLoopStr ; ::_thesis: for p, q being Polynomial of n,L
for x being Function of n,L holds eval ((p - q),x) = (eval (p,x)) - (eval (q,x))
let p, q be Polynomial of n,L; ::_thesis: for x being Function of n,L holds eval ((p - q),x) = (eval (p,x)) - (eval (q,x))
let x be Function of n,L; ::_thesis: eval ((p - q),x) = (eval (p,x)) - (eval (q,x))
thus eval ((p - q),x) = eval ((p + (- q)),x) by POLYNOM1:def_6
.= (eval (p,x)) + (eval ((- q),x)) by Th23
.= (eval (p,x)) + (- (eval (q,x))) by Th22
.= (eval (p,x)) - (eval (q,x)) ; ::_thesis: verum
end;
Lm12: for n being Ordinal
for L being non empty non trivial right_complementable associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for p, q being Polynomial of n,L
for b1, b2 being bag of n st Support p = {b1} & Support q = {b2} holds
for x being Function of n,L holds eval ((p *' q),x) = (eval (p,x)) * (eval (q,x))
proof
let n be Ordinal; ::_thesis: for L being non empty non trivial right_complementable associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for p, q being Polynomial of n,L
for b1, b2 being bag of n st Support p = {b1} & Support q = {b2} holds
for x being Function of n,L holds eval ((p *' q),x) = (eval (p,x)) * (eval (q,x))
let L be non trivial right_complementable associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr ; ::_thesis: for p, q being Polynomial of n,L
for b1, b2 being bag of n st Support p = {b1} & Support q = {b2} holds
for x being Function of n,L holds eval ((p *' q),x) = (eval (p,x)) * (eval (q,x))
let p, q be Polynomial of n,L; ::_thesis: for b1, b2 being bag of n st Support p = {b1} & Support q = {b2} holds
for x being Function of n,L holds eval ((p *' q),x) = (eval (p,x)) * (eval (q,x))
let b1, b2 be bag of n; ::_thesis: ( Support p = {b1} & Support q = {b2} implies for x being Function of n,L holds eval ((p *' q),x) = (eval (p,x)) * (eval (q,x)) )
assume that
A1: Support p = {b1} and
A2: Support q = {b2} ; ::_thesis: for x being Function of n,L holds eval ((p *' q),x) = (eval (p,x)) * (eval (q,x))
consider s being FinSequence of the carrier of L such that
A3: (p *' q) . (b1 + b2) = Sum s and
A4: len s = len (decomp (b1 + b2)) and
A5: for k being Element of NAT st k in dom s holds
ex u1, u2 being bag of n st
( (decomp (b1 + b2)) /. k = <*u1,u2*> & s /. k = (p . u1) * (q . u2) ) by POLYNOM1:def_9;
A6: b1 + b2 is Element of Bags n by PRE_POLY:def_12;
let x be Function of n,L; ::_thesis: eval ((p *' q),x) = (eval (p,x)) * (eval (q,x))
A7: ((p . b1) * (q . b2)) * ((eval (b1,x)) * (eval (b2,x))) = (((p . b1) * (q . b2)) * (eval (b1,x))) * (eval (b2,x)) by GROUP_1:def_3
.= (((p . b1) * (eval (b1,x))) * (q . b2)) * (eval (b2,x)) by GROUP_1:def_3
.= ((p . b1) * (eval (b1,x))) * ((q . b2) * (eval (b2,x))) by GROUP_1:def_3
.= (eval (p,x)) * ((q . b2) * (eval (b2,x))) by A1, Th19
.= (eval (p,x)) * (eval (q,x)) by A2, Th19 ;
A8: for b being bag of n st b <> b2 holds
q . b = 0. L
proof
let b be bag of n; ::_thesis: ( b <> b2 implies q . b = 0. L )
assume b <> b2 ; ::_thesis: q . b = 0. L
then A9: not b in Support q by A2, TARSKI:def_1;
b is Element of Bags n by PRE_POLY:def_12;
hence q . b = 0. L by A9, POLYNOM1:def_3; ::_thesis: verum
end;
A10: for b being bag of n st b <> b1 holds
p . b = 0. L
proof
let b be bag of n; ::_thesis: ( b <> b1 implies p . b = 0. L )
assume b <> b1 ; ::_thesis: p . b = 0. L
then A11: not b in Support p by A1, TARSKI:def_1;
b is Element of Bags n by PRE_POLY:def_12;
hence p . b = 0. L by A11, POLYNOM1:def_3; ::_thesis: verum
end;
A12: for u being set st u in Support (p *' q) holds
u in {(b1 + b2)}
proof
let u be set ; ::_thesis: ( u in Support (p *' q) implies u in {(b1 + b2)} )
assume A13: u in Support (p *' q) ; ::_thesis: u in {(b1 + b2)}
assume A14: not u in {(b1 + b2)} ; ::_thesis: contradiction
reconsider u = u as bag of n by A13;
consider t being FinSequence of the carrier of L such that
A15: (p *' q) . u = Sum t and
A16: len t = len (decomp u) and
A17: for k being Element of NAT st k in dom t holds
ex b19, b29 being bag of n st
( (decomp u) /. k = <*b19,b29*> & t /. k = (p . b19) * (q . b29) ) by POLYNOM1:def_9;
1 <= len t by A16, NAT_1:14;
then A18: 1 in dom t by FINSEQ_3:25;
A19: dom t = Seg (len t) by FINSEQ_1:def_3
.= dom (decomp u) by A16, FINSEQ_1:def_3 ;
A20: for i being Element of NAT st i in dom t holds
t /. i = 0. L
proof
let i be Element of NAT ; ::_thesis: ( i in dom t implies t /. i = 0. L )
consider S being non empty finite Subset of (Bags n) such that
A21: divisors u = SgmX ((BagOrder n),S) and
A22: for b being bag of n holds
( b in S iff b divides u ) by PRE_POLY:def_16;
BagOrder n linearly_orders S by Lm3;
then A23: S = rng (divisors u) by A21, PRE_POLY:def_2;
assume A24: i in dom t ; ::_thesis: t /. i = 0. L
then consider b19, b29 being bag of n such that
A25: (decomp u) /. i = <*b19,b29*> and
A26: t /. i = (p . b19) * (q . b29) by A17;
A27: b19 = (divisors u) /. i by A19, A24, A25, PRE_POLY:70;
A28: i in dom (divisors u) by A19, A24, PRE_POLY:def_17;
then b19 = (divisors u) . i by A27, PARTFUN1:def_6;
then b19 in rng (divisors u) by A28, FUNCT_1:def_3;
then A29: b19 divides u by A22, A23;
percases ( ( b19 = b1 & b29 = b2 ) or b19 <> b1 or b29 <> b2 ) ;
supposeA30: ( b19 = b1 & b29 = b2 ) ; ::_thesis: t /. i = 0. L
b2 = <*b1,b2*> . 2 by FINSEQ_1:44
.= <*b1,(u -' b1)*> . 2 by A19, A24, A25, A27, A30, PRE_POLY:def_17
.= u -' b1 by FINSEQ_1:44 ;
then b1 + b2 = u by A29, A30, PRE_POLY:47;
hence t /. i = 0. L by A14, TARSKI:def_1; ::_thesis: verum
end;
suppose b19 <> b1 ; ::_thesis: t /. i = 0. L
then p . b19 = 0. L by A10;
hence t /. i = 0. L by A26, VECTSP_1:7; ::_thesis: verum
end;
suppose b29 <> b2 ; ::_thesis: t /. i = 0. L
then q . b29 = 0. L by A8;
hence t /. i = 0. L by A26, VECTSP_1:7; ::_thesis: verum
end;
end;
end;
then for i being Element of NAT st i in dom t & i <> 1 holds
t /. i = 0. L ;
then Sum t = t /. 1 by A18, Th3
.= 0. L by A18, A20 ;
hence contradiction by A13, A15, POLYNOM1:def_3; ::_thesis: verum
end;
consider k being Element of NAT such that
A31: k in dom (decomp (b1 + b2)) and
A32: (decomp (b1 + b2)) /. k = <*b1,b2*> by PRE_POLY:69;
A33: dom s = Seg (len s) by FINSEQ_1:def_3
.= dom (decomp (b1 + b2)) by A4, FINSEQ_1:def_3 ;
then consider b19, b29 being bag of n such that
A34: (decomp (b1 + b2)) /. k = <*b19,b29*> and
A35: s /. k = (p . b19) * (q . b29) by A5, A31;
A36: b2 = <*b1,b2*> . 2 by FINSEQ_1:44
.= b29 by A32, A34, FINSEQ_1:44 ;
A37: for k9 being Element of NAT st k9 in dom s & k9 <> k holds
s /. k9 = 0. L
proof
let k9 be Element of NAT ; ::_thesis: ( k9 in dom s & k9 <> k implies s /. k9 = 0. L )
assume that
A38: k9 in dom s and
A39: k9 <> k ; ::_thesis: s /. k9 = 0. L
consider b19, b29 being bag of n such that
A40: (decomp (b1 + b2)) /. k9 = <*b19,b29*> and
A41: s /. k9 = (p . b19) * (q . b29) by A5, A38;
percases ( ( b19 = b1 & b29 = b2 ) or b19 <> b1 or b29 <> b2 ) ;
supposeA42: ( b19 = b1 & b29 = b2 ) ; ::_thesis: s /. k9 = 0. L
(decomp (b1 + b2)) . k9 = (decomp (b1 + b2)) /. k9 by A33, A38, PARTFUN1:def_6
.= (decomp (b1 + b2)) . k by A31, A32, A40, A42, PARTFUN1:def_6 ;
hence s /. k9 = 0. L by A33, A31, A38, A39, FUNCT_1:def_4; ::_thesis: verum
end;
suppose b19 <> b1 ; ::_thesis: s /. k9 = 0. L
then p . b19 = 0. L by A10;
hence s /. k9 = 0. L by A41, VECTSP_1:7; ::_thesis: verum
end;
suppose b29 <> b2 ; ::_thesis: s /. k9 = 0. L
then q . b29 = 0. L by A8;
hence s /. k9 = 0. L by A41, VECTSP_1:7; ::_thesis: verum
end;
end;
end;
b1 = <*b19,b29*> . 1 by A32, A34, FINSEQ_1:44
.= b19 by FINSEQ_1:44 ;
then A43: (p *' q) . (b1 + b2) = (p . b1) * (q . b2) by A3, A33, A31, A35, A36, A37, Th3;
percases ( (p . b1) * (q . b2) = 0. L or (p . b1) * (q . b2) <> 0. L ) ;
supposeA44: (p . b1) * (q . b2) = 0. L ; ::_thesis: eval ((p *' q),x) = (eval (p,x)) * (eval (q,x))
then A45: not b1 + b2 in Support (p *' q) by A43, POLYNOM1:def_3;
Support (p *' q) = {}
proof
set u = the Element of Support (p *' q);
assume A46: Support (p *' q) <> {} ; ::_thesis: contradiction
then A47: the Element of Support (p *' q) in Support (p *' q) ;
the Element of Support (p *' q) in {(b1 + b2)} by A12, A46;
hence contradiction by A45, A47, TARSKI:def_1; ::_thesis: verum
end;
then p *' q = 0_ (n,L) by Th17;
hence eval ((p *' q),x) = 0. L by Th20
.= (eval (p,x)) * (eval (q,x)) by A7, A44, VECTSP_1:7 ;
::_thesis: verum
end;
suppose (p . b1) * (q . b2) <> 0. L ; ::_thesis: eval ((p *' q),x) = (eval (p,x)) * (eval (q,x))
then b1 + b2 in Support (p *' q) by A43, A6, POLYNOM1:def_3;
then for u being set st u in {(b1 + b2)} holds
u in Support (p *' q) by TARSKI:def_1;
then Support (p *' q) = {(b1 + b2)} by A12, TARSKI:1;
hence eval ((p *' q),x) = ((p *' q) . (b1 + b2)) * (eval ((b1 + b2),x)) by Th19
.= (eval (p,x)) * (eval (q,x)) by A43, A7, Th16 ;
::_thesis: verum
end;
end;
end;
Lm13: for n being Ordinal
for L being non empty non trivial right_complementable associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for q being Polynomial of n,L st ex b being bag of n st Support q = {b} holds
for p being Polynomial of n,L
for x being Function of n,L holds eval ((p *' q),x) = (eval (p,x)) * (eval (q,x))
proof
let n be Ordinal; ::_thesis: for L being non empty non trivial right_complementable associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for q being Polynomial of n,L st ex b being bag of n st Support q = {b} holds
for p being Polynomial of n,L
for x being Function of n,L holds eval ((p *' q),x) = (eval (p,x)) * (eval (q,x))
let L be non trivial right_complementable associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr ; ::_thesis: for q being Polynomial of n,L st ex b being bag of n st Support q = {b} holds
for p being Polynomial of n,L
for x being Function of n,L holds eval ((p *' q),x) = (eval (p,x)) * (eval (q,x))
let q be Polynomial of n,L; ::_thesis: ( ex b being bag of n st Support q = {b} implies for p being Polynomial of n,L
for x being Function of n,L holds eval ((p *' q),x) = (eval (p,x)) * (eval (q,x)) )
given b being bag of n such that A1: Support q = {b} ; ::_thesis: for p being Polynomial of n,L
for x being Function of n,L holds eval ((p *' q),x) = (eval (p,x)) * (eval (q,x))
let p be Polynomial of n,L; ::_thesis: for x being Function of n,L holds eval ((p *' q),x) = (eval (p,x)) * (eval (q,x))
let x be Function of n,L; ::_thesis: eval ((p *' q),x) = (eval (p,x)) * (eval (q,x))
defpred S1[ Element of NAT ] means for p being Polynomial of n,L st card (Support p) = $1 holds
eval ((p *' q),x) = (eval (p,x)) * (eval (q,x));
A2: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A3: S1[k] ; ::_thesis: S1[k + 1]
let p be Polynomial of n,L; ::_thesis: ( card (Support p) = k + 1 implies eval ((p *' q),x) = (eval (p,x)) * (eval (q,x)) )
assume A4: card (Support p) = k + 1 ; ::_thesis: eval ((p *' q),x) = (eval (p,x)) * (eval (q,x))
set sgp = SgmX ((BagOrder n),(Support p));
set bg = (SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p))));
A5: BagOrder n linearly_orders Support p by Th18;
then SgmX ((BagOrder n),(Support p)) <> {} by A4, CARD_1:27, PRE_POLY:def_2, RELAT_1:38;
then 1 <= len (SgmX ((BagOrder n),(Support p))) by NAT_1:14;
then len (SgmX ((BagOrder n),(Support p))) in Seg (len (SgmX ((BagOrder n),(Support p)))) by FINSEQ_1:1;
then A6: len (SgmX ((BagOrder n),(Support p))) in dom (SgmX ((BagOrder n),(Support p))) by FINSEQ_1:def_3;
then (SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p)))) = (SgmX ((BagOrder n),(Support p))) . (len (SgmX ((BagOrder n),(Support p)))) by PARTFUN1:def_6;
then (SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p)))) in rng (SgmX ((BagOrder n),(Support p))) by A6, FUNCT_1:def_3;
then A7: (SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p)))) in Support p by A5, PRE_POLY:def_2;
then A8: p . ((SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p))))) <> 0. L by POLYNOM1:def_3;
set m = (0_ (n,L)) +* (((SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p))))),(p . ((SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p)))))));
set p9 = p +* (((SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p))))),(0. L));
reconsider bg = (SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p)))) as bag of n ;
dom p = Bags n by FUNCT_2:def_1;
then A9: p +* (((SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p))))),(0. L)) = p +* (bg .--> (0. L)) by FUNCT_7:def_3;
reconsider p9 = p +* (((SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p))))),(0. L)) as Function of (Bags n), the carrier of L ;
reconsider p9 = p9 as Function of (Bags n),L ;
for u being set st u in Support p9 holds
u in Support p
proof
let u be set ; ::_thesis: ( u in Support p9 implies u in Support p )
assume A10: u in Support p9 ; ::_thesis: u in Support p
then reconsider u = u as Element of Bags n ;
reconsider u = u as bag of n ;
now__::_thesis:_not_u_=_bg
assume A11: u = bg ; ::_thesis: contradiction
then u in {bg} by TARSKI:def_1;
then u in dom (bg .--> (0. L)) by FUNCOP_1:13;
then p9 . u = (bg .--> (0. L)) . bg by A9, A11, FUNCT_4:13;
then p9 . u = 0. L by FUNCOP_1:72;
hence contradiction by A10, POLYNOM1:def_3; ::_thesis: verum
end;
then not u in {bg} by TARSKI:def_1;
then not u in dom (bg .--> (0. L)) ;
then p . u = p9 . u by A9, FUNCT_4:11;
then p . u <> 0. L by A10, POLYNOM1:def_3;
hence u in Support p by POLYNOM1:def_3; ::_thesis: verum
end;
then Support p9 c= Support p by TARSKI:def_3;
then reconsider p9 = p9 as Polynomial of n,L by POLYNOM1:def_4;
A12: dom p = Bags n by FUNCT_2:def_1;
A13: for u being set st u in Support p holds
u in (Support p9) \/ {bg}
proof
let u be set ; ::_thesis: ( u in Support p implies u in (Support p9) \/ {bg} )
assume A14: u in Support p ; ::_thesis: u in (Support p9) \/ {bg}
then reconsider u = u as Element of Bags n ;
A15: p . u <> 0. L by A14, POLYNOM1:def_3;
percases ( u = bg or u <> bg ) ;
suppose u = bg ; ::_thesis: u in (Support p9) \/ {bg}
then u in {bg} by TARSKI:def_1;
hence u in (Support p9) \/ {bg} by XBOOLE_0:def_3; ::_thesis: verum
end;
suppose u <> bg ; ::_thesis: u in (Support p9) \/ {bg}
then not u in {bg} by TARSKI:def_1;
then not u in dom (bg .--> (0. L)) ;
then p9 . u = p . u by A9, FUNCT_4:11;
then u in Support p9 by A15, POLYNOM1:def_3;
hence u in (Support p9) \/ {bg} by XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
bg in {bg} by TARSKI:def_1;
then bg in dom (bg .--> (0. L)) by FUNCOP_1:13;
then p9 . bg = (bg .--> (0. L)) . bg by A9, FUNCT_4:13;
then A16: p9 . bg = 0. L by FUNCOP_1:72;
then A17: not bg in Support p9 by POLYNOM1:def_3;
for u being set st u in (Support p9) \/ {bg} holds
u in Support p
proof
let u be set ; ::_thesis: ( u in (Support p9) \/ {bg} implies u in Support p )
assume A18: u in (Support p9) \/ {bg} ; ::_thesis: u in Support p
percases ( u in Support p9 or u in {bg} ) by A18, XBOOLE_0:def_3;
supposeA19: u in Support p9 ; ::_thesis: u in Support p
then reconsider u = u as Element of Bags n ;
u <> bg by A16, A19, POLYNOM1:def_3;
then not u in {bg} by TARSKI:def_1;
then not u in dom (bg .--> (0. L)) ;
then A20: p9 . u = p . u by A9, FUNCT_4:11;
p9 . u <> 0. L by A19, POLYNOM1:def_3;
hence u in Support p by A20, POLYNOM1:def_3; ::_thesis: verum
end;
suppose u in {bg} ; ::_thesis: u in Support p
hence u in Support p by A7, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
then Support p = (Support p9) \/ {bg} by A13, TARSKI:1;
then A21: k + 1 = (card (Support p9)) + 1 by A4, A17, CARD_2:41;
dom (0_ (n,L)) = Bags n by FUNCT_2:def_1;
then A22: (0_ (n,L)) +* (((SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p))))),(p . ((SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p))))))) = (0_ (n,L)) +* (bg .--> (p . bg)) by FUNCT_7:def_3;
reconsider m = (0_ (n,L)) +* (((SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p))))),(p . ((SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p))))))) as Function of (Bags n), the carrier of L ;
reconsider m = m as Function of (Bags n),L ;
A23: for u being set st u in Support m holds
u in {bg}
proof
let u be set ; ::_thesis: ( u in Support m implies u in {bg} )
assume A24: u in Support m ; ::_thesis: u in {bg}
then reconsider u = u as Element of Bags n ;
A25: m . u <> 0. L by A24, POLYNOM1:def_3;
now__::_thesis:_not_u_<>_bg
assume u <> bg ; ::_thesis: contradiction
then not u in {bg} by TARSKI:def_1;
then not u in dom (bg .--> (p . bg)) ;
then m . u = (0_ (n,L)) . u by A22, FUNCT_4:11;
hence contradiction by A25, POLYNOM1:22; ::_thesis: verum
end;
hence u in {bg} by TARSKI:def_1; ::_thesis: verum
end;
for u being set st u in {bg} holds
u in Support m
proof
let u be set ; ::_thesis: ( u in {bg} implies u in Support m )
bg in {bg} by TARSKI:def_1;
then bg in dom (bg .--> (p . bg)) by FUNCOP_1:13;
then m . bg = (bg .--> (p . bg)) . bg by A22, FUNCT_4:13;
then A26: m . bg = p . bg by FUNCOP_1:72;
assume u in {bg} ; ::_thesis: u in Support m
then u = bg by TARSKI:def_1;
hence u in Support m by A8, A26, POLYNOM1:def_3; ::_thesis: verum
end;
then A27: Support m = {bg} by A23, TARSKI:1;
then reconsider m = m as Polynomial of n,L by POLYNOM1:def_4;
reconsider m = m as Polynomial of n,L ;
A28: for u being set st u in Bags n holds
(p9 + m) . u = p . u
proof
let u be set ; ::_thesis: ( u in Bags n implies (p9 + m) . u = p . u )
assume u in Bags n ; ::_thesis: (p9 + m) . u = p . u
then reconsider u = u as bag of n ;
percases ( u = bg or u <> bg ) ;
supposeA29: u = bg ; ::_thesis: (p9 + m) . u = p . u
bg in {bg} by TARSKI:def_1;
then bg in dom (bg .--> (p . bg)) by FUNCOP_1:13;
then m . bg = (bg .--> (p . bg)) . bg by A22, FUNCT_4:13;
then A30: m . bg = p . bg by FUNCOP_1:72;
u in {bg} by A29, TARSKI:def_1;
then u in dom (bg .--> (0. L)) by FUNCOP_1:13;
then A31: p9 . u = (bg .--> (0. L)) . bg by A9, A29, FUNCT_4:13;
(p9 + m) . u = (p9 . u) + (m . u) by POLYNOM1:15
.= (0. L) + (p . bg) by A29, A31, A30, FUNCOP_1:72
.= p . bg by RLVECT_1:def_4 ;
hence (p9 + m) . u = p . u by A29; ::_thesis: verum
end;
suppose u <> bg ; ::_thesis: (p9 + m) . u = p . u
then A32: not u in {bg} by TARSKI:def_1;
then A33: not u in dom (bg .--> (0. L)) ;
not u in dom (bg .--> (p . bg)) by A32;
then m . u = (0_ (n,L)) . u by A22, FUNCT_4:11;
then A34: m . u = 0. L by POLYNOM1:22;
(p9 + m) . u = (p9 . u) + (m . u) by POLYNOM1:15
.= (p . u) + (0. L) by A9, A33, A34, FUNCT_4:11
.= p . u by RLVECT_1:def_4 ;
hence (p9 + m) . u = p . u ; ::_thesis: verum
end;
end;
end;
A35: dom (p9 + m) = Bags n by FUNCT_2:def_1;
then eval (p,x) = eval ((m + p9),x) by A12, A28, FUNCT_1:2
.= (eval (p9,x)) + (eval (m,x)) by A27, Lm11 ;
hence (eval (p,x)) * (eval (q,x)) = ((eval (p9,x)) * (eval (q,x))) + ((eval (m,x)) * (eval (q,x))) by VECTSP_1:def_7
.= (eval ((p9 *' q),x)) + ((eval (m,x)) * (eval (q,x))) by A3, A21
.= (eval ((p9 *' q),x)) + (eval ((m *' q),x)) by A1, A27, Lm12
.= eval (((p9 *' q) + (m *' q)),x) by Th23
.= eval ((q *' (p9 + m)),x) by POLYNOM1:26
.= eval ((p *' q),x) by A35, A12, A28, FUNCT_1:2 ;
::_thesis: verum
end;
A36: ex k being Element of NAT st card (Support p) = k ;
A37: S1[ 0 ]
proof
let p be Polynomial of n,L; ::_thesis: ( card (Support p) = 0 implies eval ((p *' q),x) = (eval (p,x)) * (eval (q,x)) )
assume card (Support p) = 0 ; ::_thesis: eval ((p *' q),x) = (eval (p,x)) * (eval (q,x))
then Support p = {} ;
then A38: p = 0_ (n,L) by Th17;
hence eval ((p *' q),x) = eval (p,x) by POLYNOM1:28
.= 0. L by A38, Th20
.= (0. L) * (eval (q,x)) by VECTSP_1:7
.= (eval (p,x)) * (eval (q,x)) by A38, Th20 ;
::_thesis: verum
end;
for k being Element of NAT holds S1[k] from NAT_1:sch_1(A37, A2);
hence eval ((p *' q),x) = (eval (p,x)) * (eval (q,x)) by A36; ::_thesis: verum
end;
theorem Th25: :: POLYNOM2:25
for n being Ordinal
for L being non empty non trivial right_complementable associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for p, q being Polynomial of n,L
for x being Function of n,L holds eval ((p *' q),x) = (eval (p,x)) * (eval (q,x))
proof
let n be Ordinal; ::_thesis: for L being non empty non trivial right_complementable associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for p, q being Polynomial of n,L
for x being Function of n,L holds eval ((p *' q),x) = (eval (p,x)) * (eval (q,x))
let L be non trivial right_complementable associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr ; ::_thesis: for p, q being Polynomial of n,L
for x being Function of n,L holds eval ((p *' q),x) = (eval (p,x)) * (eval (q,x))
let p, q be Polynomial of n,L; ::_thesis: for x being Function of n,L holds eval ((p *' q),x) = (eval (p,x)) * (eval (q,x))
let x be Function of n,L; ::_thesis: eval ((p *' q),x) = (eval (p,x)) * (eval (q,x))
defpred S1[ Element of NAT ] means for p being Polynomial of n,L st card (Support p) = $1 holds
eval ((p *' q),x) = (eval (p,x)) * (eval (q,x));
A1: ex k being Element of NAT st card (Support p) = k ;
A2: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A3: S1[k] ; ::_thesis: S1[k + 1]
let p be Polynomial of n,L; ::_thesis: ( card (Support p) = k + 1 implies eval ((p *' q),x) = (eval (p,x)) * (eval (q,x)) )
assume A4: card (Support p) = k + 1 ; ::_thesis: eval ((p *' q),x) = (eval (p,x)) * (eval (q,x))
set sgp = SgmX ((BagOrder n),(Support p));
set bg = (SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p))));
A5: BagOrder n linearly_orders Support p by Th18;
then SgmX ((BagOrder n),(Support p)) <> {} by A4, CARD_1:27, PRE_POLY:def_2, RELAT_1:38;
then 1 <= len (SgmX ((BagOrder n),(Support p))) by NAT_1:14;
then len (SgmX ((BagOrder n),(Support p))) in Seg (len (SgmX ((BagOrder n),(Support p)))) by FINSEQ_1:1;
then A6: len (SgmX ((BagOrder n),(Support p))) in dom (SgmX ((BagOrder n),(Support p))) by FINSEQ_1:def_3;
then (SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p)))) = (SgmX ((BagOrder n),(Support p))) . (len (SgmX ((BagOrder n),(Support p)))) by PARTFUN1:def_6;
then (SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p)))) in rng (SgmX ((BagOrder n),(Support p))) by A6, FUNCT_1:def_3;
then A7: (SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p)))) in Support p by A5, PRE_POLY:def_2;
then A8: p . ((SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p))))) <> 0. L by POLYNOM1:def_3;
set m = (0_ (n,L)) +* (((SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p))))),(p . ((SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p)))))));
set p9 = p +* (((SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p))))),(0. L));
reconsider bg = (SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p)))) as bag of n ;
dom p = Bags n by FUNCT_2:def_1;
then A9: p +* (((SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p))))),(0. L)) = p +* (bg .--> (0. L)) by FUNCT_7:def_3;
reconsider p9 = p +* (((SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p))))),(0. L)) as Function of (Bags n), the carrier of L ;
reconsider p9 = p9 as Function of (Bags n),L ;
for u being set st u in Support p9 holds
u in Support p
proof
let u be set ; ::_thesis: ( u in Support p9 implies u in Support p )
assume A10: u in Support p9 ; ::_thesis: u in Support p
then reconsider u = u as Element of Bags n ;
reconsider u = u as bag of n ;
now__::_thesis:_not_u_=_bg
assume A11: u = bg ; ::_thesis: contradiction
then u in {bg} by TARSKI:def_1;
then u in dom (bg .--> (0. L)) by FUNCOP_1:13;
then p9 . u = (bg .--> (0. L)) . bg by A9, A11, FUNCT_4:13;
then p9 . u = 0. L by FUNCOP_1:72;
hence contradiction by A10, POLYNOM1:def_3; ::_thesis: verum
end;
then not u in {bg} by TARSKI:def_1;
then not u in dom (bg .--> (0. L)) ;
then p . u = p9 . u by A9, FUNCT_4:11;
then p . u <> 0. L by A10, POLYNOM1:def_3;
hence u in Support p by POLYNOM1:def_3; ::_thesis: verum
end;
then Support p9 c= Support p by TARSKI:def_3;
then reconsider p9 = p9 as Polynomial of n,L by POLYNOM1:def_4;
A12: dom p = Bags n by FUNCT_2:def_1;
A13: for u being set st u in Support p holds
u in (Support p9) \/ {bg}
proof
let u be set ; ::_thesis: ( u in Support p implies u in (Support p9) \/ {bg} )
assume A14: u in Support p ; ::_thesis: u in (Support p9) \/ {bg}
then reconsider u = u as Element of Bags n ;
A15: p . u <> 0. L by A14, POLYNOM1:def_3;
percases ( u = bg or u <> bg ) ;
suppose u = bg ; ::_thesis: u in (Support p9) \/ {bg}
then u in {bg} by TARSKI:def_1;
hence u in (Support p9) \/ {bg} by XBOOLE_0:def_3; ::_thesis: verum
end;
suppose u <> bg ; ::_thesis: u in (Support p9) \/ {bg}
then not u in {bg} by TARSKI:def_1;
then not u in dom (bg .--> (0. L)) ;
then p9 . u = p . u by A9, FUNCT_4:11;
then u in Support p9 by A15, POLYNOM1:def_3;
hence u in (Support p9) \/ {bg} by XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
bg in {bg} by TARSKI:def_1;
then bg in dom (bg .--> (0. L)) by FUNCOP_1:13;
then p9 . bg = (bg .--> (0. L)) . bg by A9, FUNCT_4:13;
then A16: p9 . bg = 0. L by FUNCOP_1:72;
then A17: not bg in Support p9 by POLYNOM1:def_3;
for u being set st u in (Support p9) \/ {bg} holds
u in Support p
proof
let u be set ; ::_thesis: ( u in (Support p9) \/ {bg} implies u in Support p )
assume A18: u in (Support p9) \/ {bg} ; ::_thesis: u in Support p
percases ( u in Support p9 or u in {bg} ) by A18, XBOOLE_0:def_3;
supposeA19: u in Support p9 ; ::_thesis: u in Support p
then reconsider u = u as Element of Bags n ;
u <> bg by A16, A19, POLYNOM1:def_3;
then not u in {bg} by TARSKI:def_1;
then not u in dom (bg .--> (0. L)) ;
then A20: p9 . u = p . u by A9, FUNCT_4:11;
p9 . u <> 0. L by A19, POLYNOM1:def_3;
hence u in Support p by A20, POLYNOM1:def_3; ::_thesis: verum
end;
suppose u in {bg} ; ::_thesis: u in Support p
hence u in Support p by A7, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
then Support p = (Support p9) \/ {bg} by A13, TARSKI:1;
then A21: k + 1 = (card (Support p9)) + 1 by A4, A17, CARD_2:41;
dom (0_ (n,L)) = Bags n by FUNCT_2:def_1;
then A22: (0_ (n,L)) +* (((SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p))))),(p . ((SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p))))))) = (0_ (n,L)) +* (bg .--> (p . bg)) by FUNCT_7:def_3;
reconsider m = (0_ (n,L)) +* (((SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p))))),(p . ((SgmX ((BagOrder n),(Support p))) /. (len (SgmX ((BagOrder n),(Support p))))))) as Function of (Bags n), the carrier of L ;
reconsider m = m as Function of (Bags n),L ;
A23: for u being set st u in Support m holds
u in {bg}
proof
let u be set ; ::_thesis: ( u in Support m implies u in {bg} )
assume A24: u in Support m ; ::_thesis: u in {bg}
then reconsider u = u as Element of Bags n ;
A25: m . u <> 0. L by A24, POLYNOM1:def_3;
now__::_thesis:_not_u_<>_bg
assume u <> bg ; ::_thesis: contradiction
then not u in {bg} by TARSKI:def_1;
then not u in dom (bg .--> (p . bg)) ;
then m . u = (0_ (n,L)) . u by A22, FUNCT_4:11;
hence contradiction by A25, POLYNOM1:22; ::_thesis: verum
end;
hence u in {bg} by TARSKI:def_1; ::_thesis: verum
end;
for u being set st u in {bg} holds
u in Support m
proof
let u be set ; ::_thesis: ( u in {bg} implies u in Support m )
bg in {bg} by TARSKI:def_1;
then bg in dom (bg .--> (p . bg)) by FUNCOP_1:13;
then m . bg = (bg .--> (p . bg)) . bg by A22, FUNCT_4:13;
then A26: m . bg = p . bg by FUNCOP_1:72;
assume u in {bg} ; ::_thesis: u in Support m
then u = bg by TARSKI:def_1;
hence u in Support m by A8, A26, POLYNOM1:def_3; ::_thesis: verum
end;
then A27: Support m = {bg} by A23, TARSKI:1;
then reconsider m = m as Polynomial of n,L by POLYNOM1:def_4;
reconsider m = m as Polynomial of n,L ;
A28: for u being set st u in Bags n holds
(p9 + m) . u = p . u
proof
let u be set ; ::_thesis: ( u in Bags n implies (p9 + m) . u = p . u )
assume u in Bags n ; ::_thesis: (p9 + m) . u = p . u
then reconsider u = u as bag of n ;
percases ( u = bg or u <> bg ) ;
supposeA29: u = bg ; ::_thesis: (p9 + m) . u = p . u
bg in {bg} by TARSKI:def_1;
then bg in dom (bg .--> (p . bg)) by FUNCOP_1:13;
then m . bg = (bg .--> (p . bg)) . bg by A22, FUNCT_4:13;
then A30: m . bg = p . bg by FUNCOP_1:72;
u in {bg} by A29, TARSKI:def_1;
then u in dom (bg .--> (0. L)) by FUNCOP_1:13;
then A31: p9 . u = (bg .--> (0. L)) . bg by A9, A29, FUNCT_4:13;
(p9 + m) . u = (p9 . u) + (m . u) by POLYNOM1:15
.= (0. L) + (p . bg) by A29, A31, A30, FUNCOP_1:72
.= p . bg by RLVECT_1:def_4 ;
hence (p9 + m) . u = p . u by A29; ::_thesis: verum
end;
suppose u <> bg ; ::_thesis: (p9 + m) . u = p . u
then A32: not u in {bg} by TARSKI:def_1;
then A33: not u in dom (bg .--> (0. L)) ;
not u in dom (bg .--> (p . bg)) by A32;
then m . u = (0_ (n,L)) . u by A22, FUNCT_4:11;
then A34: m . u = 0. L by POLYNOM1:22;
(p9 + m) . u = (p9 . u) + (m . u) by POLYNOM1:15
.= (p . u) + (0. L) by A9, A33, A34, FUNCT_4:11
.= p . u by RLVECT_1:def_4 ;
hence (p9 + m) . u = p . u ; ::_thesis: verum
end;
end;
end;
A35: dom (p9 + m) = Bags n by FUNCT_2:def_1;
then eval (p,x) = eval ((m + p9),x) by A12, A28, FUNCT_1:2
.= (eval (p9,x)) + (eval (m,x)) by A27, Lm11 ;
hence (eval (p,x)) * (eval (q,x)) = ((eval (p9,x)) * (eval (q,x))) + ((eval (m,x)) * (eval (q,x))) by VECTSP_1:def_7
.= (eval ((p9 *' q),x)) + ((eval (m,x)) * (eval (q,x))) by A3, A21
.= (eval ((p9 *' q),x)) + (eval ((m *' q),x)) by A27, Lm13
.= eval (((p9 *' q) + (m *' q)),x) by Th23
.= eval ((q *' (p9 + m)),x) by POLYNOM1:26
.= eval ((p *' q),x) by A35, A12, A28, FUNCT_1:2 ;
::_thesis: verum
end;
A36: S1[ 0 ]
proof
let p be Polynomial of n,L; ::_thesis: ( card (Support p) = 0 implies eval ((p *' q),x) = (eval (p,x)) * (eval (q,x)) )
assume card (Support p) = 0 ; ::_thesis: eval ((p *' q),x) = (eval (p,x)) * (eval (q,x))
then Support p = {} ;
then A37: p = 0_ (n,L) by Th17;
hence eval ((p *' q),x) = eval (p,x) by POLYNOM1:28
.= 0. L by A37, Th20
.= (0. L) * (eval (q,x)) by VECTSP_1:7
.= (eval (p,x)) * (eval (q,x)) by A37, Th20 ;
::_thesis: verum
end;
for k being Element of NAT holds S1[k] from NAT_1:sch_1(A36, A2);
hence eval ((p *' q),x) = (eval (p,x)) * (eval (q,x)) by A1; ::_thesis: verum
end;
begin
definition
let n be Ordinal;
let L be non trivial right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr ;
let x be Function of n,L;
func Polynom-Evaluation (n,L,x) -> Function of (Polynom-Ring (n,L)),L means :Def5: :: POLYNOM2:def 5
for p being Polynomial of n,L holds it . p = eval (p,x);
existence
ex b1 being Function of (Polynom-Ring (n,L)),L st
for p being Polynomial of n,L holds b1 . p = eval (p,x)
proof
defpred S1[ set , set ] means ex p9 being Polynomial of n,L st
( p9 = $1 & $2 = eval (p9,x) );
A1: now__::_thesis:_for_p_being_set_st_p_in_the_carrier_of_(Polynom-Ring_(n,L))_holds_
ex_y_being_set_st_
(_y_in_the_carrier_of_L_&_S1[p,y]_)
let p be set ; ::_thesis: ( p in the carrier of (Polynom-Ring (n,L)) implies ex y being set st
( y in the carrier of L & S1[p,y] ) )
assume p in the carrier of (Polynom-Ring (n,L)) ; ::_thesis: ex y being set st
( y in the carrier of L & S1[p,y] )
then reconsider p9 = p as Polynomial of n,L by POLYNOM1:def_10;
thus ex y being set st
( y in the carrier of L & S1[p,y] ) ::_thesis: verum
proof
take eval (p9,x) ; ::_thesis: ( eval (p9,x) in the carrier of L & S1[p, eval (p9,x)] )
thus ( eval (p9,x) in the carrier of L & S1[p, eval (p9,x)] ) ; ::_thesis: verum
end;
end;
consider f being Function of the carrier of (Polynom-Ring (n,L)), the carrier of L such that
A2: for x being set st x in the carrier of (Polynom-Ring (n,L)) holds
S1[x,f . x] from FUNCT_2:sch_1(A1);
reconsider f = f as Function of (Polynom-Ring (n,L)),L ;
take f ; ::_thesis: for p being Polynomial of n,L holds f . p = eval (p,x)
let p be Polynomial of n,L; ::_thesis: f . p = eval (p,x)
p in the carrier of (Polynom-Ring (n,L)) by POLYNOM1:def_10;
then ex p9 being Polynomial of n,L st
( p9 = p & f . p = eval (p9,x) ) by A2;
hence f . p = eval (p,x) ; ::_thesis: verum
end;
uniqueness
for b1, b2 being Function of (Polynom-Ring (n,L)),L st ( for p being Polynomial of n,L holds b1 . p = eval (p,x) ) & ( for p being Polynomial of n,L holds b2 . p = eval (p,x) ) holds
b1 = b2
proof
let f, g be Function of (Polynom-Ring (n,L)),L; ::_thesis: ( ( for p being Polynomial of n,L holds f . p = eval (p,x) ) & ( for p being Polynomial of n,L holds g . p = eval (p,x) ) implies f = g )
assume that
A3: for p being Polynomial of n,L holds f . p = eval (p,x) and
A4: for p being Polynomial of n,L holds g . p = eval (p,x) ; ::_thesis: f = g
reconsider f = f, g = g as Function of the carrier of (Polynom-Ring (n,L)), the carrier of L ;
A5: now__::_thesis:_for_p_being_set_st_p_in_the_carrier_of_(Polynom-Ring_(n,L))_holds_
f_._p_=_g_._p
let p be set ; ::_thesis: ( p in the carrier of (Polynom-Ring (n,L)) implies f . p = g . p )
assume p in the carrier of (Polynom-Ring (n,L)) ; ::_thesis: f . p = g . p
then reconsider p9 = p as Polynomial of n,L by POLYNOM1:def_10;
f . p9 = eval (p9,x) by A3
.= g . p9 by A4 ;
hence f . p = g . p ; ::_thesis: verum
end;
A6: dom g = the carrier of (Polynom-Ring (n,L)) by FUNCT_2:def_1;
dom f = the carrier of (Polynom-Ring (n,L)) by FUNCT_2:def_1;
hence f = g by A6, A5, FUNCT_1:2; ::_thesis: verum
end;
end;
:: deftheorem Def5 defines Polynom-Evaluation POLYNOM2:def_5_:_
for n being Ordinal
for L being non trivial right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr
for x being Function of n,L
for b4 being Function of (Polynom-Ring (n,L)),L holds
( b4 = Polynom-Evaluation (n,L,x) iff for p being Polynomial of n,L holds b4 . p = eval (p,x) );
registration
let n be Ordinal;
let L be non empty non trivial right_complementable associative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr ;
cluster Polynom-Ring (n,L) -> well-unital ;
coherence
Polynom-Ring (n,L) is well-unital ;
end;
registration
let n be Ordinal;
let L be non empty non trivial right_complementable associative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr ;
let x be Function of n,L;
cluster Polynom-Evaluation (n,L,x) -> unity-preserving ;
coherence
Polynom-Evaluation (n,L,x) is unity-preserving
proof
set f = Polynom-Evaluation (n,L,x);
thus (Polynom-Evaluation (n,L,x)) . (1_ (Polynom-Ring (n,L))) = (Polynom-Evaluation (n,L,x)) . (1_ (n,L)) by POLYNOM1:31
.= eval ((1_ (n,L)),x) by Def5
.= 1_ L by Th21 ; :: according to GROUP_1:def_13 ::_thesis: verum
end;
end;
registration
let n be Ordinal;
let L be non trivial right_complementable Abelian add-associative right_zeroed well-unital distributive doubleLoopStr ;
let x be Function of n,L;
cluster Polynom-Evaluation (n,L,x) -> additive ;
coherence
Polynom-Evaluation (n,L,x) is additive
proof
set f = Polynom-Evaluation (n,L,x);
for p, q being Element of (Polynom-Ring (n,L)) holds (Polynom-Evaluation (n,L,x)) . (p + q) = ((Polynom-Evaluation (n,L,x)) . p) + ((Polynom-Evaluation (n,L,x)) . q)
proof
let p, q be Element of (Polynom-Ring (n,L)); ::_thesis: (Polynom-Evaluation (n,L,x)) . (p + q) = ((Polynom-Evaluation (n,L,x)) . p) + ((Polynom-Evaluation (n,L,x)) . q)
reconsider p9 = p, q9 = q as Polynomial of n,L by POLYNOM1:def_10;
reconsider p = p, q = q as Element of (Polynom-Ring (n,L)) ;
A1: (Polynom-Evaluation (n,L,x)) . p = eval (p9,x) by Def5;
(Polynom-Evaluation (n,L,x)) . (p + q) = (Polynom-Evaluation (n,L,x)) . (p9 + q9) by POLYNOM1:def_10
.= eval ((p9 + q9),x) by Def5
.= (eval (p9,x)) + (eval (q9,x)) by Th23 ;
hence (Polynom-Evaluation (n,L,x)) . (p + q) = ((Polynom-Evaluation (n,L,x)) . p) + ((Polynom-Evaluation (n,L,x)) . q) by A1, Def5; ::_thesis: verum
end;
hence Polynom-Evaluation (n,L,x) is additive by VECTSP_1:def_20; ::_thesis: verum
end;
end;
registration
let n be Ordinal;
let L be non trivial right_complementable associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr ;
let x be Function of n,L;
cluster Polynom-Evaluation (n,L,x) -> multiplicative ;
coherence
Polynom-Evaluation (n,L,x) is multiplicative
proof
set f = Polynom-Evaluation (n,L,x);
for p, q being Element of (Polynom-Ring (n,L)) holds (Polynom-Evaluation (n,L,x)) . (p * q) = ((Polynom-Evaluation (n,L,x)) . p) * ((Polynom-Evaluation (n,L,x)) . q)
proof
let p, q be Element of (Polynom-Ring (n,L)); ::_thesis: (Polynom-Evaluation (n,L,x)) . (p * q) = ((Polynom-Evaluation (n,L,x)) . p) * ((Polynom-Evaluation (n,L,x)) . q)
reconsider p9 = p, q9 = q as Polynomial of n,L by POLYNOM1:def_10;
reconsider p = p, q = q as Element of (Polynom-Ring (n,L)) ;
A1: (Polynom-Evaluation (n,L,x)) . p = eval (p9,x) by Def5;
(Polynom-Evaluation (n,L,x)) . (p * q) = (Polynom-Evaluation (n,L,x)) . (p9 *' q9) by POLYNOM1:def_10
.= eval ((p9 *' q9),x) by Def5
.= (eval (p9,x)) * (eval (q9,x)) by Th25 ;
hence (Polynom-Evaluation (n,L,x)) . (p * q) = ((Polynom-Evaluation (n,L,x)) . p) * ((Polynom-Evaluation (n,L,x)) . q) by A1, Def5; ::_thesis: verum
end;
hence Polynom-Evaluation (n,L,x) is multiplicative by GROUP_6:def_6; ::_thesis: verum
end;
end;
registration
let n be Ordinal;
let L be non trivial right_complementable associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr ;
let x be Function of n,L;
cluster Polynom-Evaluation (n,L,x) -> RingHomomorphism ;
coherence
Polynom-Evaluation (n,L,x) is RingHomomorphism
proof
thus ( Polynom-Evaluation (n,L,x) is additive & Polynom-Evaluation (n,L,x) is multiplicative & Polynom-Evaluation (n,L,x) is unity-preserving ) ; :: according to QUOFIELD:def_18 ::_thesis: verum
end;
end;