:: SIMPLEX1 semantic presentation
begin
theorem Th1: :: SIMPLEX1:1
for X being set
for R being Relation
for C being Cardinal st ( for x being set st x in X holds
card (Im (R,x)) = C ) holds
card R = (card (R | ((dom R) \ X))) +` (C *` (card X))
proof
let X be set ; ::_thesis: for R being Relation
for C being Cardinal st ( for x being set st x in X holds
card (Im (R,x)) = C ) holds
card R = (card (R | ((dom R) \ X))) +` (C *` (card X))
let R be Relation; ::_thesis: for C being Cardinal st ( for x being set st x in X holds
card (Im (R,x)) = C ) holds
card R = (card (R | ((dom R) \ X))) +` (C *` (card X))
let C be Cardinal; ::_thesis: ( ( for x being set st x in X holds
card (Im (R,x)) = C ) implies card R = (card (R | ((dom R) \ X))) +` (C *` (card X)) )
assume A1: for x being set st x in X holds
card (Im (R,x)) = C ; ::_thesis: card R = (card (R | ((dom R) \ X))) +` (C *` (card X))
set DA = (dom R) \ X;
percases ( X c= dom R or not X c= dom R ) ;
supposeA2: X c= dom R ; ::_thesis: card R = (card (R | ((dom R) \ X))) +` (C *` (card X))
deffunc H1( set ) -> set = Im (R,$1);
consider f being Function such that
A3: ( dom f = X & ( for x being set st x in X holds
f . x = H1(x) ) ) from FUNCT_1:sch_3();
defpred S1[ set , set ] means ex g being Function st
( g = $2 & dom g = f . $1 & rng g = C & g is one-to-one );
A4: for x being set st x in X holds
ex y being set st S1[x,y]
proof
let x be set ; ::_thesis: ( x in X implies ex y being set st S1[x,y] )
assume x in X ; ::_thesis: ex y being set st S1[x,y]
then ( f . x = Im (R,x) & card (Im (R,x)) = C ) by A1, A3;
then f . x,C are_equipotent by CARD_1:def_2;
then consider g being Function such that
A5: ( g is one-to-one & dom g = f . x & rng g = C ) by WELLORD2:def_4;
take g ; ::_thesis: S1[x,g]
thus S1[x,g] by A5; ::_thesis: verum
end;
consider ff being Function such that
A6: ( dom ff = X & ( for x being set st x in X holds
S1[x,ff . x] ) ) from CLASSES1:sch_1(A4);
now__::_thesis:_for_x_being_set_st_x_in_dom_ff_holds_
ff_._x_is_Function
let x be set ; ::_thesis: ( x in dom ff implies ff . x is Function )
assume x in dom ff ; ::_thesis: ff . x is Function
then ex g being Function st
( g = ff . x & dom g = f . x & rng g = C & g is one-to-one ) by A6;
hence ff . x is Function ; ::_thesis: verum
end;
then reconsider ff = ff as Function-yielding Function by FUNCOP_1:def_6;
deffunc H2( set ) -> set = [($1 `1),((ff . ($1 `1)) . ($1 `2))];
consider p being Function such that
A7: ( dom p = R | X & ( for x being set st x in R | X holds
p . x = H2(x) ) ) from FUNCT_1:sch_3();
A8: rng p = [:X,C:]
proof
hereby :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: [:X,C:] c= rng p
let y be set ; ::_thesis: ( y in rng p implies y in [:X,C:] )
assume y in rng p ; ::_thesis: y in [:X,C:]
then consider x being set such that
A9: x in dom p and
A10: p . x = y by FUNCT_1:def_3;
A11: p . x = [(x `1),((ff . (x `1)) . (x `2))] by A7, A9;
A12: x = [(x `1),(x `2)] by A7, A9, MCART_1:21;
then ( x `1 in {(x `1)} & x in R ) by A7, A9, RELAT_1:def_11, TARSKI:def_1;
then A13: x `2 in R .: {(x `1)} by A12, RELAT_1:def_13;
A14: x `1 in X by A7, A9, A12, RELAT_1:def_11;
then consider g being Function such that
A15: g = ff . (x `1) and
A16: dom g = f . (x `1) and
A17: rng g = C and
g is one-to-one by A6;
f . (x `1) = Im (R,(x `1)) by A3, A14;
then x `2 in dom g by A13, A16, RELAT_1:def_16;
then g . (x `2) in C by A17, FUNCT_1:def_3;
hence y in [:X,C:] by A10, A11, A14, A15, ZFMISC_1:87; ::_thesis: verum
end;
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in [:X,C:] or y in rng p )
assume y in [:X,C:] ; ::_thesis: y in rng p
then consider y1, y2 being set such that
A18: y1 in X and
A19: y2 in C and
A20: y = [y1,y2] by ZFMISC_1:def_2;
consider g being Function such that
A21: g = ff . y1 and
A22: dom g = f . y1 and
A23: rng g = C and
g is one-to-one by A6, A18;
consider x2 being set such that
A24: x2 in dom g and
A25: g . x2 = y2 by A19, A23, FUNCT_1:def_3;
A26: H2([y1,x2]) = y by A20, A21, A25;
f . y1 = Im (R,y1) by A3, A18;
then [y1,x2] in R by A22, A24, RELSET_2:9;
then A27: [y1,x2] in R | X by A18, RELAT_1:def_11;
then p . [y1,x2] = H2([y1,x2]) by A7;
hence y in rng p by A7, A26, A27, FUNCT_1:def_3; ::_thesis: verum
end;
now__::_thesis:_for_x1,_x2_being_set_st_x1_in_dom_p_&_x2_in_dom_p_&_p_._x1_=_p_._x2_holds_
x1_=_x2
let x1, x2 be set ; ::_thesis: ( x1 in dom p & x2 in dom p & p . x1 = p . x2 implies x1 = x2 )
assume that
A28: x1 in dom p and
A29: x2 in dom p and
A30: p . x1 = p . x2 ; ::_thesis: x1 = x2
A31: ( p . x1 = H2(x1) & p . x2 = H2(x2) ) by A7, A28, A29;
then A32: x1 `1 = x2 `1 by A30, XTUPLE_0:1;
A33: x1 = [(x1 `1),(x1 `2)] by A7, A28, MCART_1:21;
then x1 in R by A7, A28, RELAT_1:def_11;
then A34: x1 `2 in Im (R,(x1 `1)) by A33, RELSET_2:9;
A35: x2 = [(x2 `1),(x2 `2)] by A7, A29, MCART_1:21;
then x2 in R by A7, A29, RELAT_1:def_11;
then A36: x2 `2 in Im (R,(x2 `1)) by A35, RELSET_2:9;
x2 `1 in X by A7, A29, A35, RELAT_1:def_11;
then consider g2 being Function such that
A37: g2 = ff . (x2 `1) and
dom g2 = f . (x2 `1) and
rng g2 = C and
g2 is one-to-one by A6;
A38: x1 `1 in X by A7, A28, A33, RELAT_1:def_11;
then consider g1 being Function such that
A39: g1 = ff . (x1 `1) and
A40: dom g1 = f . (x1 `1) and
rng g1 = C and
A41: g1 is one-to-one by A6;
A42: f . (x1 `1) = Im (R,(x1 `1)) by A3, A38;
g1 . (x1 `2) = g2 . (x2 `2) by A30, A31, A37, A39, XTUPLE_0:1;
hence x1 = x2 by A32, A35, A36, A33, A34, A37, A39, A40, A41, A42, FUNCT_1:def_4; ::_thesis: verum
end;
then p is one-to-one by FUNCT_1:def_4;
then R | X,[:X,C:] are_equipotent by A7, A8, WELLORD2:def_4;
then A43: card (R | X) = card [:X,C:] by CARD_1:5
.= card [:(card X),C:] by CARD_2:7
.= C *` (card X) by CARD_2:def_2 ;
A44: R | X misses R | ((dom R) \ X)
proof
assume R | X meets R | ((dom R) \ X) ; ::_thesis: contradiction
then consider x being set such that
A45: x in R | X and
A46: x in R | ((dom R) \ X) by XBOOLE_0:3;
consider x1, x2 being set such that
A47: x = [x1,x2] by A45, RELAT_1:def_1;
( x1 in X & x1 in (dom R) \ X ) by A45, A46, A47, RELAT_1:def_11;
hence contradiction by XBOOLE_0:def_5; ::_thesis: verum
end;
((dom R) \ X) \/ X = (dom R) \/ X by XBOOLE_1:39
.= dom R by A2, XBOOLE_1:12 ;
then (R | X) \/ (R | ((dom R) \ X)) = R | (dom R) by RELAT_1:78
.= R ;
hence card R = (card (R | ((dom R) \ X))) +` (C *` (card X)) by A43, A44, CARD_2:35; ::_thesis: verum
end;
suppose not X c= dom R ; ::_thesis: card R = (card (R | ((dom R) \ X))) +` (C *` (card X))
then consider x being set such that
A48: x in X and
A49: not x in dom R by TARSKI:def_3;
Im (R,x) = {}
proof
assume Im (R,x) <> {} ; ::_thesis: contradiction
then consider y being set such that
A50: y in Im (R,x) by XBOOLE_0:def_1;
[x,y] in R by A50, RELSET_2:9;
hence contradiction by A49, XTUPLE_0:def_12; ::_thesis: verum
end;
then A51: C = card {} by A1, A48;
dom R misses X
proof
assume dom R meets X ; ::_thesis: contradiction
then consider x being set such that
A52: x in dom R and
A53: x in X by XBOOLE_0:3;
card (Im (R,x)) = C by A1, A53;
then A54: Im (R,x) is empty by A51;
ex y being set st [x,y] in R by A52, XTUPLE_0:def_12;
hence contradiction by A54, RELSET_2:9; ::_thesis: verum
end;
then A55: (dom R) \ X = dom R by XBOOLE_1:83;
C *` (card X) = 0 by A51, CARD_2:20;
then (card (R | ((dom R) \ X))) +` (C *` (card X)) = card (R | ((dom R) \ X)) by CARD_2:18;
hence card R = (card (R | ((dom R) \ X))) +` (C *` (card X)) by A55, RELAT_1:69; ::_thesis: verum
end;
end;
end;
theorem Th2: :: SIMPLEX1:2
for X being set
for Y being non empty finite set st card X = (card Y) + 1 holds
for f being Function of X,Y st f is onto holds
ex y being set st
( y in Y & card (f " {y}) = 2 & ( for x being set st x in Y & x <> y holds
card (f " {x}) = 1 ) )
proof
let X be set ; ::_thesis: for Y being non empty finite set st card X = (card Y) + 1 holds
for f being Function of X,Y st f is onto holds
ex y being set st
( y in Y & card (f " {y}) = 2 & ( for x being set st x in Y & x <> y holds
card (f " {x}) = 1 ) )
let Y be non empty finite set ; ::_thesis: ( card X = (card Y) + 1 implies for f being Function of X,Y st f is onto holds
ex y being set st
( y in Y & card (f " {y}) = 2 & ( for x being set st x in Y & x <> y holds
card (f " {x}) = 1 ) ) )
assume A1: card X = (card Y) + 1 ; ::_thesis: for f being Function of X,Y st f is onto holds
ex y being set st
( y in Y & card (f " {y}) = 2 & ( for x being set st x in Y & x <> y holds
card (f " {x}) = 1 ) )
reconsider XX = X as non empty finite set by A1;
card Y > 0 ;
then reconsider c1 = (card Y) - 1 as Element of NAT by NAT_1:20;
let f be Function of X,Y; ::_thesis: ( f is onto implies ex y being set st
( y in Y & card (f " {y}) = 2 & ( for x being set st x in Y & x <> y holds
card (f " {x}) = 1 ) ) )
assume A2: f is onto ; ::_thesis: ex y being set st
( y in Y & card (f " {y}) = 2 & ( for x being set st x in Y & x <> y holds
card (f " {x}) = 1 ) )
A3: rng f = Y by A2, FUNCT_2:def_3;
reconsider F = f as Function of XX,Y ;
A4: dom f = X by FUNCT_2:def_1;
ex y being set st
( y in Y & card (F " {y}) > 1 )
proof
assume A5: for y being set st y in Y holds
card (F " {y}) <= 1 ; ::_thesis: contradiction
now__::_thesis:_for_y_being_set_st_y_in_Y_holds_
ex_x_being_set_st_F_"_{y}_=_{x}
let y be set ; ::_thesis: ( y in Y implies ex x being set st F " {y} = {x} )
set fy = F " {y};
assume A6: y in Y ; ::_thesis: ex x being set st F " {y} = {x}
then F " {y} <> {} by A3, FUNCT_1:72;
then card (F " {y}) = 1 by A5, A6, NAT_1:25;
hence ex x being set st F " {y} = {x} by CARD_2:42; ::_thesis: verum
end;
then f is one-to-one by A3, FUNCT_1:74;
then X,Y are_equipotent by A3, A4, WELLORD2:def_4;
then card X = card Y by CARD_1:5;
hence contradiction by A1; ::_thesis: verum
end;
then consider y being set such that
A7: y in Y and
A8: card (F " {y}) > 1 ;
set fy = F " {y};
set fD = F | ((dom f) \ (F " {y}));
take y ; ::_thesis: ( y in Y & card (f " {y}) = 2 & ( for x being set st x in Y & x <> y holds
card (f " {x}) = 1 ) )
A9: 1 + 1 <= card (F " {y}) by A8, NAT_1:13;
dom (F | ((dom f) \ (F " {y}))) = (dom f) \ (F " {y}) by RELAT_1:62, XBOOLE_1:36;
then A10: card (dom (F | ((dom f) \ (F " {y})))) = (card XX) - (card (F " {y})) by A4, CARD_2:44;
set Yy = Y \ {y};
A11: rng (F | ((dom f) \ (F " {y}))) = Y \ {y} by A3, STIRL2_1:54;
then reconsider FD = F | ((dom f) \ (F " {y})) as Function of (dom (F | ((dom f) \ (F " {y})))),(Y \ {y}) by FUNCT_2:1;
card Y = c1 + 1 ;
then A12: card (Y \ {y}) = c1 by A7, STIRL2_1:55;
then c1 c= card (dom (F | ((dom f) \ (F " {y})))) by A11, CARD_1:12;
then (card Y) + (- 1) <= (card Y) + (1 - (card (F " {y}))) by A1, A10, NAT_1:39;
then - 1 <= 1 - (card (F " {y})) by XREAL_1:6;
then card (F " {y}) <= 1 - (- 1) by XREAL_1:11;
hence A13: ( y in Y & card (f " {y}) = 2 ) by A7, A9, XXREAL_0:1; ::_thesis: for x being set st x in Y & x <> y holds
card (f " {x}) = 1
let x be set ; ::_thesis: ( x in Y & x <> y implies card (f " {x}) = 1 )
assume that
A14: x in Y and
A15: x <> y ; ::_thesis: card (f " {x}) = 1
A16: x in rng FD by A11, A14, A15, ZFMISC_1:56;
FD is onto by A11, FUNCT_2:def_3;
then FD is one-to-one by A1, A10, A12, A13, STIRL2_1:60;
then A17: ex z being set st FD " {x} = {z} by A16, FUNCT_1:74;
FD " {x} = f " {x} by A15, STIRL2_1:54;
hence card (f " {x}) = 1 by A17, CARD_1:30; ::_thesis: verum
end;
definition
let X be 1-sorted ;
mode SimplicialComplexStr of X is SimplicialComplexStr of the carrier of X;
mode SimplicialComplex of X is SimplicialComplex of the carrier of X;
end;
definition
let X be 1-sorted ;
let K be SimplicialComplexStr of X;
let A be Subset of K;
func @ A -> Subset of X equals :: SIMPLEX1:def 1
A;
coherence
A is Subset of X
proof
[#] K c= the carrier of X by SIMPLEX0:def_9;
hence A is Subset of X by XBOOLE_1:1; ::_thesis: verum
end;
end;
:: deftheorem defines @ SIMPLEX1:def_1_:_
for X being 1-sorted
for K being SimplicialComplexStr of X
for A being Subset of K holds @ A = A;
definition
let X be 1-sorted ;
let K be SimplicialComplexStr of X;
let A be Subset-Family of K;
func @ A -> Subset-Family of X equals :: SIMPLEX1:def 2
A;
coherence
A is Subset-Family of X
proof
[#] K c= the carrier of X by SIMPLEX0:def_9;
then bool ([#] K) c= bool the carrier of X by ZFMISC_1:67;
hence A is Subset-Family of X by XBOOLE_1:1; ::_thesis: verum
end;
end;
:: deftheorem defines @ SIMPLEX1:def_2_:_
for X being 1-sorted
for K being SimplicialComplexStr of X
for A being Subset-Family of K holds @ A = A;
theorem Th3: :: SIMPLEX1:3
for X being 1-sorted
for K being subset-closed SimplicialComplexStr of X st K is total holds
for S being finite Subset of K st S is simplex-like holds
Complex_of {(@ S)} is SubSimplicialComplex of K
proof
let X be 1-sorted ; ::_thesis: for K being subset-closed SimplicialComplexStr of X st K is total holds
for S being finite Subset of K st S is simplex-like holds
Complex_of {(@ S)} is SubSimplicialComplex of K
let K be subset-closed SimplicialComplexStr of X; ::_thesis: ( K is total implies for S being finite Subset of K st S is simplex-like holds
Complex_of {(@ S)} is SubSimplicialComplex of K )
assume A1: K is total ; ::_thesis: for S being finite Subset of K st S is simplex-like holds
Complex_of {(@ S)} is SubSimplicialComplex of K
let S be finite Subset of K; ::_thesis: ( S is simplex-like implies Complex_of {(@ S)} is SubSimplicialComplex of K )
assume A2: S is simplex-like ; ::_thesis: Complex_of {(@ S)} is SubSimplicialComplex of K
S in the topology of K by A2, PRE_TOPC:def_2;
then A3: {S} c= the topology of K by ZFMISC_1:31;
set C = Complex_of {(@ S)};
A4: [#] (Complex_of {(@ S)}) c= [#] K by A1, SIMPLEX0:def_10;
the_family_of K is subset-closed ;
then the topology of (Complex_of {(@ S)}) c= the topology of K by A3, SIMPLEX0:def_1;
hence Complex_of {(@ S)} is SubSimplicialComplex of K by A4, SIMPLEX0:def_13; ::_thesis: verum
end;
begin
definition
let RLS be non empty RLSStruct ;
let Kr be SimplicialComplexStr of RLS;
func|.Kr.| -> Subset of RLS means :Def3: :: SIMPLEX1:def 3
for x being set holds
( x in it iff ex A being Subset of Kr st
( A is simplex-like & x in conv (@ A) ) );
existence
ex b1 being Subset of RLS st
for x being set holds
( x in b1 iff ex A being Subset of Kr st
( A is simplex-like & x in conv (@ A) ) )
proof
set KC = { (conv (@ A)) where A is Subset of Kr : A is simplex-like } ;
{ (conv (@ A)) where A is Subset of Kr : A is simplex-like } c= bool the carrier of RLS
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { (conv (@ A)) where A is Subset of Kr : A is simplex-like } or x in bool the carrier of RLS )
assume x in { (conv (@ A)) where A is Subset of Kr : A is simplex-like } ; ::_thesis: x in bool the carrier of RLS
then ex A being Subset of Kr st
( x = conv (@ A) & A is simplex-like ) ;
hence x in bool the carrier of RLS ; ::_thesis: verum
end;
then reconsider KC = { (conv (@ A)) where A is Subset of Kr : A is simplex-like } as Subset-Family of RLS ;
take IT = union KC; ::_thesis: for x being set holds
( x in IT iff ex A being Subset of Kr st
( A is simplex-like & x in conv (@ A) ) )
let x be set ; ::_thesis: ( x in IT iff ex A being Subset of Kr st
( A is simplex-like & x in conv (@ A) ) )
hereby ::_thesis: ( ex A being Subset of Kr st
( A is simplex-like & x in conv (@ A) ) implies x in IT )
assume x in IT ; ::_thesis: ex A being Subset of Kr st
( A is simplex-like & x in conv (@ A) )
then consider y being set such that
A1: x in y and
A2: y in KC by TARSKI:def_4;
consider A being Subset of Kr such that
A3: ( y = conv (@ A) & A is simplex-like ) by A2;
take A = A; ::_thesis: ( A is simplex-like & x in conv (@ A) )
thus ( A is simplex-like & x in conv (@ A) ) by A1, A3; ::_thesis: verum
end;
given A being Subset of Kr such that A4: A is simplex-like and
A5: x in conv (@ A) ; ::_thesis: x in IT
conv (@ A) in KC by A4;
hence x in IT by A5, TARSKI:def_4; ::_thesis: verum
end;
uniqueness
for b1, b2 being Subset of RLS st ( for x being set holds
( x in b1 iff ex A being Subset of Kr st
( A is simplex-like & x in conv (@ A) ) ) ) & ( for x being set holds
( x in b2 iff ex A being Subset of Kr st
( A is simplex-like & x in conv (@ A) ) ) ) holds
b1 = b2
proof
let S1, S2 be Subset of RLS; ::_thesis: ( ( for x being set holds
( x in S1 iff ex A being Subset of Kr st
( A is simplex-like & x in conv (@ A) ) ) ) & ( for x being set holds
( x in S2 iff ex A being Subset of Kr st
( A is simplex-like & x in conv (@ A) ) ) ) implies S1 = S2 )
assume that
A6: for x being set holds
( x in S1 iff ex A being Subset of Kr st
( A is simplex-like & x in conv (@ A) ) ) and
A7: for x being set holds
( x in S2 iff ex A being Subset of Kr st
( A is simplex-like & x in conv (@ A) ) ) ; ::_thesis: S1 = S2
now__::_thesis:_for_x_being_set_holds_
(_x_in_S1_iff_x_in_S2_)
let x be set ; ::_thesis: ( x in S1 iff x in S2 )
( x in S1 iff ex A being Subset of Kr st
( A is simplex-like & x in conv (@ A) ) ) by A6;
hence ( x in S1 iff x in S2 ) by A7; ::_thesis: verum
end;
hence S1 = S2 by TARSKI:1; ::_thesis: verum
end;
end;
:: deftheorem Def3 defines |. SIMPLEX1:def_3_:_
for RLS being non empty RLSStruct
for Kr being SimplicialComplexStr of RLS
for b3 being Subset of RLS holds
( b3 = |.Kr.| iff for x being set holds
( x in b3 iff ex A being Subset of Kr st
( A is simplex-like & x in conv (@ A) ) ) );
theorem Th4: :: SIMPLEX1:4
for RLS being non empty RLSStruct
for K1r, K2r being SimplicialComplexStr of RLS st the topology of K1r c= the topology of K2r holds
|.K1r.| c= |.K2r.|
proof
let RLS be non empty RLSStruct ; ::_thesis: for K1r, K2r being SimplicialComplexStr of RLS st the topology of K1r c= the topology of K2r holds
|.K1r.| c= |.K2r.|
let K1r, K2r be SimplicialComplexStr of RLS; ::_thesis: ( the topology of K1r c= the topology of K2r implies |.K1r.| c= |.K2r.| )
assume A1: the topology of K1r c= the topology of K2r ; ::_thesis: |.K1r.| c= |.K2r.|
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in |.K1r.| or x in |.K2r.| )
assume x in |.K1r.| ; ::_thesis: x in |.K2r.|
then consider A being Subset of K1r such that
A2: A is simplex-like and
A3: x in conv (@ A) by Def3;
A4: A in the topology of K1r by A2, PRE_TOPC:def_2;
then A in the topology of K2r by A1;
then reconsider A1 = A as Subset of K2r ;
( @ A = @ A1 & A1 is simplex-like ) by A1, A4, PRE_TOPC:def_2;
hence x in |.K2r.| by A3, Def3; ::_thesis: verum
end;
theorem Th5: :: SIMPLEX1:5
for RLS being non empty RLSStruct
for Kr being SimplicialComplexStr of RLS
for A being Subset of Kr st A is simplex-like holds
conv (@ A) c= |.Kr.|
proof
let RLS be non empty RLSStruct ; ::_thesis: for Kr being SimplicialComplexStr of RLS
for A being Subset of Kr st A is simplex-like holds
conv (@ A) c= |.Kr.|
let Kr be SimplicialComplexStr of RLS; ::_thesis: for A being Subset of Kr st A is simplex-like holds
conv (@ A) c= |.Kr.|
let A be Subset of Kr; ::_thesis: ( A is simplex-like implies conv (@ A) c= |.Kr.| )
assume A1: A is simplex-like ; ::_thesis: conv (@ A) c= |.Kr.|
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in conv (@ A) or x in |.Kr.| )
thus ( not x in conv (@ A) or x in |.Kr.| ) by A1, Def3; ::_thesis: verum
end;
theorem :: SIMPLEX1:6
for x being set
for V being RealLinearSpace
for K being subset-closed SimplicialComplexStr of V holds
( x in |.K.| iff ex A being Subset of K st
( A is simplex-like & x in Int (@ A) ) )
proof
let x be set ; ::_thesis: for V being RealLinearSpace
for K being subset-closed SimplicialComplexStr of V holds
( x in |.K.| iff ex A being Subset of K st
( A is simplex-like & x in Int (@ A) ) )
let V be RealLinearSpace; ::_thesis: for K being subset-closed SimplicialComplexStr of V holds
( x in |.K.| iff ex A being Subset of K st
( A is simplex-like & x in Int (@ A) ) )
let K be subset-closed SimplicialComplexStr of V; ::_thesis: ( x in |.K.| iff ex A being Subset of K st
( A is simplex-like & x in Int (@ A) ) )
hereby ::_thesis: ( ex A being Subset of K st
( A is simplex-like & x in Int (@ A) ) implies x in |.K.| )
assume x in |.K.| ; ::_thesis: ex B1 being Subset of K st
( B1 is simplex-like & x in Int (@ B1) )
then consider A being Subset of K such that
A1: A is simplex-like and
A2: x in conv (@ A) by Def3;
conv (@ A) = union { (Int B) where B is Subset of V : B c= @ A } by RLAFFIN2:8;
then consider IB being set such that
A3: x in IB and
A4: IB in { (Int B) where B is Subset of V : B c= @ A } by A2, TARSKI:def_4;
consider B being Subset of V such that
A5: IB = Int B and
A6: B c= @ A by A4;
reconsider B1 = B as Subset of K by A6, XBOOLE_1:1;
take B1 = B1; ::_thesis: ( B1 is simplex-like & x in Int (@ B1) )
A in the topology of K by A1, PRE_TOPC:def_2;
then not K is void by PENCIL_1:def_4;
hence ( B1 is simplex-like & x in Int (@ B1) ) by A1, A3, A5, A6, MATROID0:1; ::_thesis: verum
end;
given A being Subset of K such that A7: A is simplex-like and
A8: x in Int (@ A) ; ::_thesis: x in |.K.|
x in conv (@ A) by A8, RLAFFIN2:def_1;
hence x in |.K.| by A7, Def3; ::_thesis: verum
end;
theorem Th7: :: SIMPLEX1:7
for RLS being non empty RLSStruct
for Kr being SimplicialComplexStr of RLS holds
( |.Kr.| is empty iff Kr is empty-membered )
proof
let RLS be non empty RLSStruct ; ::_thesis: for Kr being SimplicialComplexStr of RLS holds
( |.Kr.| is empty iff Kr is empty-membered )
let Kr be SimplicialComplexStr of RLS; ::_thesis: ( |.Kr.| is empty iff Kr is empty-membered )
hereby ::_thesis: ( Kr is empty-membered implies |.Kr.| is empty )
assume A1: |.Kr.| is empty ; ::_thesis: not Kr is with_non-empty_element
assume Kr is with_non-empty_element ; ::_thesis: contradiction
then the topology of Kr is with_non-empty_element by SIMPLEX0:def_7;
then consider x being non empty set such that
A2: x in the topology of Kr by SETFAM_1:def_10;
reconsider X = x as Subset of Kr by A2;
( ex y being set st y in conv (@ X) & X is simplex-like ) by A2, PRE_TOPC:def_2, XBOOLE_0:def_1;
hence contradiction by A1, Def3; ::_thesis: verum
end;
assume A3: Kr is empty-membered ; ::_thesis: |.Kr.| is empty
assume not |.Kr.| is empty ; ::_thesis: contradiction
then consider x being set such that
A4: x in |.Kr.| by XBOOLE_0:def_1;
consider A being Subset of Kr such that
A5: ( A is simplex-like & x in conv (@ A) ) by A4, Def3;
( not A is empty & A in the topology of Kr ) by A5, PRE_TOPC:def_2;
then the topology of Kr is with_non-empty_element by SETFAM_1:def_10;
hence contradiction by A3, SIMPLEX0:def_7; ::_thesis: verum
end;
theorem Th8: :: SIMPLEX1:8
for RLS being non empty RLSStruct
for A being Subset of RLS holds |.(Complex_of {A}).| = conv A
proof
let RLS be non empty RLSStruct ; ::_thesis: for A being Subset of RLS holds |.(Complex_of {A}).| = conv A
let A be Subset of RLS; ::_thesis: |.(Complex_of {A}).| = conv A
set C = Complex_of {A};
reconsider A1 = A as Subset of (Complex_of {A}) ;
A1: the topology of (Complex_of {A}) = bool A by SIMPLEX0:4;
hereby :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: conv A c= |.(Complex_of {A}).|
let x be set ; ::_thesis: ( x in |.(Complex_of {A}).| implies x in conv A )
assume x in |.(Complex_of {A}).| ; ::_thesis: x in conv A
then consider S being Subset of (Complex_of {A}) such that
A2: S is simplex-like and
A3: x in conv (@ S) by Def3;
S in the topology of (Complex_of {A}) by A2, PRE_TOPC:def_2;
then conv (@ S) c= conv A by A1, RLAFFIN1:3;
hence x in conv A by A3; ::_thesis: verum
end;
A c= A ;
then ( @ A1 = A & A1 is simplex-like ) by A1, PRE_TOPC:def_2;
hence conv A c= |.(Complex_of {A}).| by Th5; ::_thesis: verum
end;
theorem :: SIMPLEX1:9
for RLS being non empty RLSStruct
for A, B being Subset-Family of RLS holds |.(Complex_of (A \/ B)).| = |.(Complex_of A).| \/ |.(Complex_of B).|
proof
let RLS be non empty RLSStruct ; ::_thesis: for A, B being Subset-Family of RLS holds |.(Complex_of (A \/ B)).| = |.(Complex_of A).| \/ |.(Complex_of B).|
let A, B be Subset-Family of RLS; ::_thesis: |.(Complex_of (A \/ B)).| = |.(Complex_of A).| \/ |.(Complex_of B).|
set CA = Complex_of A;
set CB = Complex_of B;
set CAB = Complex_of (A \/ B);
A1: the topology of (Complex_of A) \/ the topology of (Complex_of B) = the topology of (Complex_of (A \/ B)) by SIMPLEX0:5;
thus |.(Complex_of (A \/ B)).| c= |.(Complex_of A).| \/ |.(Complex_of B).| :: according to XBOOLE_0:def_10 ::_thesis: |.(Complex_of A).| \/ |.(Complex_of B).| c= |.(Complex_of (A \/ B)).|
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in |.(Complex_of (A \/ B)).| or x in |.(Complex_of A).| \/ |.(Complex_of B).| )
assume x in |.(Complex_of (A \/ B)).| ; ::_thesis: x in |.(Complex_of A).| \/ |.(Complex_of B).|
then consider S being Subset of (Complex_of (A \/ B)) such that
A2: S is simplex-like and
A3: x in conv (@ S) by Def3;
A4: S in the topology of (Complex_of (A \/ B)) by A2, PRE_TOPC:def_2;
percases ( S in the topology of (Complex_of A) or S in the topology of (Complex_of B) ) by A1, A4, XBOOLE_0:def_3;
supposeA5: S in the topology of (Complex_of A) ; ::_thesis: x in |.(Complex_of A).| \/ |.(Complex_of B).|
reconsider S1 = S as Subset of (Complex_of A) ;
( @ S = @ S1 & S1 is simplex-like ) by A5, PRE_TOPC:def_2;
then conv (@ S) c= |.(Complex_of A).| by Th5;
hence x in |.(Complex_of A).| \/ |.(Complex_of B).| by A3, XBOOLE_0:def_3; ::_thesis: verum
end;
supposeA6: S in the topology of (Complex_of B) ; ::_thesis: x in |.(Complex_of A).| \/ |.(Complex_of B).|
reconsider S1 = S as Subset of (Complex_of B) ;
( @ S = @ S1 & S1 is simplex-like ) by A6, PRE_TOPC:def_2;
then conv (@ S) c= |.(Complex_of B).| by Th5;
hence x in |.(Complex_of A).| \/ |.(Complex_of B).| by A3, XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
( |.(Complex_of A).| c= |.(Complex_of (A \/ B)).| & |.(Complex_of B).| c= |.(Complex_of (A \/ B)).| ) by A1, Th4, XBOOLE_1:7;
hence |.(Complex_of A).| \/ |.(Complex_of B).| c= |.(Complex_of (A \/ B)).| by XBOOLE_1:8; ::_thesis: verum
end;
begin
definition
let RLS be non empty RLSStruct ;
let Kr be SimplicialComplexStr of RLS;
mode SubdivisionStr of Kr -> SimplicialComplexStr of RLS means :Def4: :: SIMPLEX1:def 4
( |.Kr.| c= |.it.| & ( for A being Subset of it st A is simplex-like holds
ex B being Subset of Kr st
( B is simplex-like & conv (@ A) c= conv (@ B) ) ) );
existence
ex b1 being SimplicialComplexStr of RLS st
( |.Kr.| c= |.b1.| & ( for A being Subset of b1 st A is simplex-like holds
ex B being Subset of Kr st
( B is simplex-like & conv (@ A) c= conv (@ B) ) ) )
proof
take Kr ; ::_thesis: ( |.Kr.| c= |.Kr.| & ( for A being Subset of Kr st A is simplex-like holds
ex B being Subset of Kr st
( B is simplex-like & conv (@ A) c= conv (@ B) ) ) )
thus ( |.Kr.| c= |.Kr.| & ( for A being Subset of Kr st A is simplex-like holds
ex B being Subset of Kr st
( B is simplex-like & conv (@ A) c= conv (@ B) ) ) ) ; ::_thesis: verum
end;
end;
:: deftheorem Def4 defines SubdivisionStr SIMPLEX1:def_4_:_
for RLS being non empty RLSStruct
for Kr, b3 being SimplicialComplexStr of RLS holds
( b3 is SubdivisionStr of Kr iff ( |.Kr.| c= |.b3.| & ( for A being Subset of b3 st A is simplex-like holds
ex B being Subset of Kr st
( B is simplex-like & conv (@ A) c= conv (@ B) ) ) ) );
theorem Th10: :: SIMPLEX1:10
for RLS being non empty RLSStruct
for Kr being SimplicialComplexStr of RLS
for P being SubdivisionStr of Kr holds |.Kr.| = |.P.|
proof
let RLS be non empty RLSStruct ; ::_thesis: for Kr being SimplicialComplexStr of RLS
for P being SubdivisionStr of Kr holds |.Kr.| = |.P.|
let Kr be SimplicialComplexStr of RLS; ::_thesis: for P being SubdivisionStr of Kr holds |.Kr.| = |.P.|
let P be SubdivisionStr of Kr; ::_thesis: |.Kr.| = |.P.|
thus |.Kr.| c= |.P.| by Def4; :: according to XBOOLE_0:def_10 ::_thesis: |.P.| c= |.Kr.|
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in |.P.| or x in |.Kr.| )
assume x in |.P.| ; ::_thesis: x in |.Kr.|
then consider A being Subset of P such that
A1: A is simplex-like and
A2: x in conv (@ A) by Def3;
ex B being Subset of Kr st
( B is simplex-like & conv (@ A) c= conv (@ B) ) by A1, Def4;
hence x in |.Kr.| by A2, Def3; ::_thesis: verum
end;
registration
let RLS be non empty RLSStruct ;
let Kr be with_non-empty_element SimplicialComplexStr of RLS;
cluster -> with_non-empty_element for SubdivisionStr of Kr;
coherence
for b1 being SubdivisionStr of Kr holds b1 is with_non-empty_element
proof
let P be SubdivisionStr of Kr; ::_thesis: P is with_non-empty_element
( not |.Kr.| is empty & |.Kr.| = |.P.| ) by Th7, Th10;
hence P is with_non-empty_element by Th7; ::_thesis: verum
end;
end;
theorem Th11: :: SIMPLEX1:11
for RLS being non empty RLSStruct
for Kr being SimplicialComplexStr of RLS holds Kr is SubdivisionStr of Kr
proof
let RLS be non empty RLSStruct ; ::_thesis: for Kr being SimplicialComplexStr of RLS holds Kr is SubdivisionStr of Kr
let Kr be SimplicialComplexStr of RLS; ::_thesis: Kr is SubdivisionStr of Kr
thus |.Kr.| c= |.Kr.| ; :: according to SIMPLEX1:def_4 ::_thesis: for A being Subset of Kr st A is simplex-like holds
ex B being Subset of Kr st
( B is simplex-like & conv (@ A) c= conv (@ B) )
thus for A being Subset of Kr st A is simplex-like holds
ex B being Subset of Kr st
( B is simplex-like & conv (@ A) c= conv (@ B) ) ; ::_thesis: verum
end;
theorem Th12: :: SIMPLEX1:12
for RLS being non empty RLSStruct
for Kr being SimplicialComplexStr of RLS holds Complex_of the topology of Kr is SubdivisionStr of Kr
proof
let RLS be non empty RLSStruct ; ::_thesis: for Kr being SimplicialComplexStr of RLS holds Complex_of the topology of Kr is SubdivisionStr of Kr
let Kr be SimplicialComplexStr of RLS; ::_thesis: Complex_of the topology of Kr is SubdivisionStr of Kr
set TOP = the topology of Kr;
set C = Complex_of the topology of Kr;
( [#] (Complex_of the topology of Kr) = [#] Kr & [#] Kr c= the carrier of RLS ) by SIMPLEX0:def_9;
then reconsider C = Complex_of the topology of Kr as SimplicialComplexStr of RLS by SIMPLEX0:def_9;
A1: |.Kr.| c= |.C.|
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in |.Kr.| or x in |.C.| )
assume x in |.Kr.| ; ::_thesis: x in |.C.|
then consider A being Subset of Kr such that
A2: A is simplex-like and
A3: x in conv (@ A) by Def3;
reconsider B = A as Subset of C ;
A in the topology of Kr by A2, PRE_TOPC:def_2;
then A in the topology of C by SIMPLEX0:2;
then A4: B is simplex-like by PRE_TOPC:def_2;
@ A = @ B ;
hence x in |.C.| by A3, A4, Def3; ::_thesis: verum
end;
for A being Subset of C st A is simplex-like holds
ex B being Subset of Kr st
( B is simplex-like & conv (@ A) c= conv (@ B) )
proof
let A be Subset of C; ::_thesis: ( A is simplex-like implies ex B being Subset of Kr st
( B is simplex-like & conv (@ A) c= conv (@ B) ) )
assume A is simplex-like ; ::_thesis: ex B being Subset of Kr st
( B is simplex-like & conv (@ A) c= conv (@ B) )
then A in the topology of C by PRE_TOPC:def_2;
then consider B being set such that
A5: A c= B and
A6: B in the topology of Kr by SIMPLEX0:2;
reconsider B = B as Subset of Kr by A6;
take B ; ::_thesis: ( B is simplex-like & conv (@ A) c= conv (@ B) )
thus ( B is simplex-like & conv (@ A) c= conv (@ B) ) by A5, A6, PRE_TOPC:def_2, RLAFFIN1:3; ::_thesis: verum
end;
hence Complex_of the topology of Kr is SubdivisionStr of Kr by A1, Def4; ::_thesis: verum
end;
theorem Th13: :: SIMPLEX1:13
for V being RealLinearSpace
for K being subset-closed SimplicialComplexStr of V
for SF being Subset-Family of K st SF = Sub_of_Fin the topology of K holds
Complex_of SF is SubdivisionStr of K
proof
let V be RealLinearSpace; ::_thesis: for K being subset-closed SimplicialComplexStr of V
for SF being Subset-Family of K st SF = Sub_of_Fin the topology of K holds
Complex_of SF is SubdivisionStr of K
let K be subset-closed SimplicialComplexStr of V; ::_thesis: for SF being Subset-Family of K st SF = Sub_of_Fin the topology of K holds
Complex_of SF is SubdivisionStr of K
set TOP = the topology of K;
let SF be Subset-Family of K; ::_thesis: ( SF = Sub_of_Fin the topology of K implies Complex_of SF is SubdivisionStr of K )
assume A1: SF = Sub_of_Fin the topology of K ; ::_thesis: Complex_of SF is SubdivisionStr of K
set C = Complex_of SF;
( [#] (Complex_of SF) = [#] K & [#] K c= the carrier of V ) by SIMPLEX0:def_9;
then reconsider C = Complex_of SF as SimplicialComplexStr of V by SIMPLEX0:def_9;
A2: the_family_of K is subset-closed ;
then A3: the topology of C = SF by A1, SIMPLEX0:7;
C is SubdivisionStr of K
proof
thus |.K.| c= |.C.| :: according to SIMPLEX1:def_4 ::_thesis: for A being Subset of C st A is simplex-like holds
ex B being Subset of K st
( B is simplex-like & conv (@ A) c= conv (@ B) )
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in |.K.| or x in |.C.| )
assume x in |.K.| ; ::_thesis: x in |.C.|
then consider S being Subset of K such that
A4: S is simplex-like and
A5: x in conv (@ S) by Def3;
reconsider S1 = @ S as non empty Subset of V by A5;
x in { (Sum L) where L is Convex_Combination of S1 : L in ConvexComb V } by A5, CONVEX3:5;
then consider L being Convex_Combination of S1 such that
A6: ( x = Sum L & L in ConvexComb V ) ;
reconsider Carr = Carrier L as non empty Subset of V by CONVEX1:21;
A7: Carr c= S by RLVECT_2:def_6;
then reconsider Carr1 = Carr as Subset of C by XBOOLE_1:1;
S in the topology of K by A4, PRE_TOPC:def_2;
then Carr1 in the topology of K by A2, A7, CLASSES1:def_1;
then Carr1 in the topology of C by A1, A3, COHSP_1:def_3;
then A8: Carr1 is simplex-like by PRE_TOPC:def_2;
reconsider LC = L as Linear_Combination of Carr by RLVECT_2:def_6;
LC is convex ;
then x in { (Sum M) where M is Convex_Combination of Carr : M in ConvexComb V } by A6;
then A9: x in conv Carr by CONVEX3:5;
Carr = @ Carr1 ;
hence x in |.C.| by A8, A9, Def3; ::_thesis: verum
end;
let A be Subset of C; ::_thesis: ( A is simplex-like implies ex B being Subset of K st
( B is simplex-like & conv (@ A) c= conv (@ B) ) )
reconsider B = A as Subset of K ;
assume A is simplex-like ; ::_thesis: ex B being Subset of K st
( B is simplex-like & conv (@ A) c= conv (@ B) )
then A in the topology of C by PRE_TOPC:def_2;
then A10: B is simplex-like by A1, A3, PRE_TOPC:def_2;
@ A = @ B ;
hence ex B being Subset of K st
( B is simplex-like & conv (@ A) c= conv (@ B) ) by A10; ::_thesis: verum
end;
hence Complex_of SF is SubdivisionStr of K ; ::_thesis: verum
end;
theorem Th14: :: SIMPLEX1:14
for RLS being non empty RLSStruct
for Kr being SimplicialComplexStr of RLS
for P1 being SubdivisionStr of Kr
for P2 being SubdivisionStr of P1 holds P2 is SubdivisionStr of Kr
proof
let RLS be non empty RLSStruct ; ::_thesis: for Kr being SimplicialComplexStr of RLS
for P1 being SubdivisionStr of Kr
for P2 being SubdivisionStr of P1 holds P2 is SubdivisionStr of Kr
let Kr be SimplicialComplexStr of RLS; ::_thesis: for P1 being SubdivisionStr of Kr
for P2 being SubdivisionStr of P1 holds P2 is SubdivisionStr of Kr
let P1 be SubdivisionStr of Kr; ::_thesis: for P2 being SubdivisionStr of P1 holds P2 is SubdivisionStr of Kr
let P2 be SubdivisionStr of P1; ::_thesis: P2 is SubdivisionStr of Kr
|.P2.| = |.P1.| by Th10
.= |.Kr.| by Th10 ;
hence |.Kr.| c= |.P2.| ; :: according to SIMPLEX1:def_4 ::_thesis: for A being Subset of P2 st A is simplex-like holds
ex B being Subset of Kr st
( B is simplex-like & conv (@ A) c= conv (@ B) )
let A2 be Subset of P2; ::_thesis: ( A2 is simplex-like implies ex B being Subset of Kr st
( B is simplex-like & conv (@ A2) c= conv (@ B) ) )
assume A2 is simplex-like ; ::_thesis: ex B being Subset of Kr st
( B is simplex-like & conv (@ A2) c= conv (@ B) )
then consider A1 being Subset of P1 such that
A1: A1 is simplex-like and
A2: conv (@ A2) c= conv (@ A1) by Def4;
ex A being Subset of Kr st
( A is simplex-like & conv (@ A1) c= conv (@ A) ) by A1, Def4;
hence ex B being Subset of Kr st
( B is simplex-like & conv (@ A2) c= conv (@ B) ) by A2, XBOOLE_1:1; ::_thesis: verum
end;
registration
let V be RealLinearSpace;
let K be SimplicialComplexStr of V;
cluster subset-closed finite-membered for SubdivisionStr of K;
existence
ex b1 being SubdivisionStr of K st
( b1 is finite-membered & b1 is subset-closed )
proof
reconsider C = Complex_of the topology of K as SubdivisionStr of K by Th12;
reconsider SF = Sub_of_Fin the topology of C as Subset-Family of C by XBOOLE_1:1;
Complex_of SF is SubdivisionStr of C by Th13;
then reconsider CSF = Complex_of SF as SubdivisionStr of K by Th14;
take CSF ; ::_thesis: ( CSF is finite-membered & CSF is subset-closed )
thus ( CSF is finite-membered & CSF is subset-closed ) ; ::_thesis: verum
end;
end;
definition
let V be RealLinearSpace;
let K be SimplicialComplexStr of V;
mode Subdivision of K is subset-closed finite-membered SubdivisionStr of K;
end;
theorem Th15: :: SIMPLEX1:15
for V being RealLinearSpace
for K being with_empty_element SimplicialComplex of V st |.K.| c= [#] K holds
for B being Function of (BOOL the carrier of V), the carrier of V st ( for S being Simplex of K st not S is empty holds
B . S in conv (@ S) ) holds
subdivision (B,K) is SubdivisionStr of K
proof
let V be RealLinearSpace; ::_thesis: for K being with_empty_element SimplicialComplex of V st |.K.| c= [#] K holds
for B being Function of (BOOL the carrier of V), the carrier of V st ( for S being Simplex of K st not S is empty holds
B . S in conv (@ S) ) holds
subdivision (B,K) is SubdivisionStr of K
let K be with_empty_element SimplicialComplex of V; ::_thesis: ( |.K.| c= [#] K implies for B being Function of (BOOL the carrier of V), the carrier of V st ( for S being Simplex of K st not S is empty holds
B . S in conv (@ S) ) holds
subdivision (B,K) is SubdivisionStr of K )
assume A1: |.K.| c= [#] K ; ::_thesis: for B being Function of (BOOL the carrier of V), the carrier of V st ( for S being Simplex of K st not S is empty holds
B . S in conv (@ S) ) holds
subdivision (B,K) is SubdivisionStr of K
let B be Function of (BOOL the carrier of V), the carrier of V; ::_thesis: ( ( for S being Simplex of K st not S is empty holds
B . S in conv (@ S) ) implies subdivision (B,K) is SubdivisionStr of K )
assume A2: for S being Simplex of K st not S is empty holds
B . S in conv (@ S) ; ::_thesis: subdivision (B,K) is SubdivisionStr of K
set P = subdivision (B,K);
defpred S1[ Nat] means for x being set
for A being Simplex of K st x in conv (@ A) & card A = $1 holds
ex S being c=-linear finite simplex-like Subset-Family of K ex BS being Subset of (subdivision (B,K)) st
( BS = B .: S & x in conv (@ BS) & union S c= A );
A3: dom B = BOOL the carrier of V by FUNCT_2:def_1;
A4: for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A5: S1[n] ; ::_thesis: S1[n + 1]
let x be set ; ::_thesis: for A being Simplex of K st x in conv (@ A) & card A = n + 1 holds
ex S being c=-linear finite simplex-like Subset-Family of K ex BS being Subset of (subdivision (B,K)) st
( BS = B .: S & x in conv (@ BS) & union S c= A )
let A be Simplex of K; ::_thesis: ( x in conv (@ A) & card A = n + 1 implies ex S being c=-linear finite simplex-like Subset-Family of K ex BS being Subset of (subdivision (B,K)) st
( BS = B .: S & x in conv (@ BS) & union S c= A ) )
assume that
A6: x in conv (@ A) and
A7: card A = n + 1 ; ::_thesis: ex S being c=-linear finite simplex-like Subset-Family of K ex BS being Subset of (subdivision (B,K)) st
( BS = B .: S & x in conv (@ BS) & union S c= A )
reconsider A1 = @ A as non empty Subset of V by A6;
A8: union {A} = A by ZFMISC_1:25;
A9: for P being Subset of K st P in {A} holds
P is simplex-like by TARSKI:def_1;
then A10: {A} is simplex-like by TOPS_2:def_1;
A11: B . A1 in conv (@ A) by A2;
then reconsider BA = B . A as Element of V ;
A1 = A ;
then A12: A in dom B by A3, ZFMISC_1:56;
A13: B .: {A} = Im (B,A) by RELAT_1:def_16;
then A14: B .: {A} = {BA} by A12, FUNCT_1:59;
( BA in conv A1 & conv A1 c= |.K.| ) by A2, Th5;
then BA in |.K.| ;
then ( [#] (subdivision (B,K)) = [#] K & {BA} is Subset of K ) by A1, SIMPLEX0:def_20, ZFMISC_1:31;
then reconsider BY = B .: {A} as Subset of (subdivision (B,K)) by A12, A13, FUNCT_1:59;
percases ( x = B . A or x <> B . A ) ;
supposeA15: x = B . A ; ::_thesis: ex S being c=-linear finite simplex-like Subset-Family of K ex BS being Subset of (subdivision (B,K)) st
( BS = B .: S & x in conv (@ BS) & union S c= A )
now__::_thesis:_for_x1,_x2_being_set_st_x1_in_{A}_&_x2_in_{A}_holds_
x1,x2_are_c=-comparable
let x1, x2 be set ; ::_thesis: ( x1 in {A} & x2 in {A} implies x1,x2 are_c=-comparable )
assume that
A16: x1 in {A} and
A17: x2 in {A} ; ::_thesis: x1,x2 are_c=-comparable
x1 = A by A16, TARSKI:def_1;
hence x1,x2 are_c=-comparable by A17, TARSKI:def_1; ::_thesis: verum
end;
then reconsider Y = {A} as c=-linear finite simplex-like Subset-Family of K by A9, ORDINAL1:def_8, TOPS_2:def_1;
take Y ; ::_thesis: ex BS being Subset of (subdivision (B,K)) st
( BS = B .: Y & x in conv (@ BS) & union Y c= A )
take BY ; ::_thesis: ( BY = B .: Y & x in conv (@ BY) & union Y c= A )
conv {BA} = {BA} by RLAFFIN1:1;
hence ( BY = B .: Y & x in conv (@ BY) & union Y c= A ) by A14, A15, TARSKI:def_1, ZFMISC_1:25; ::_thesis: verum
end;
suppose x <> B . A ; ::_thesis: ex S being c=-linear finite simplex-like Subset-Family of K ex BS being Subset of (subdivision (B,K)) st
( BS = B .: S & x in conv (@ BS) & union S c= A )
then consider p, w being Element of V, r being Real such that
A18: p in A and
A19: w in conv (A1 \ {p}) and
A20: ( 0 <= r & r < 1 & (r * BA) + ((1 - r) * w) = x ) by A6, A11, RLAFFIN2:26;
( @ (A \ {p}) = A1 \ {p} & card (A \ {p}) = n ) by A7, A18, STIRL2_1:55;
then consider S being c=-linear finite simplex-like Subset-Family of K, BS being Subset of (subdivision (B,K)) such that
A21: BS = B .: S and
A22: w in conv (@ BS) and
A23: union S c= A \ {p} by A5, A19;
set S1 = S \/ {A};
A24: A \ {p} c= A by XBOOLE_1:36;
then A25: union S c= A by A23, XBOOLE_1:1;
for x1, x2 being set st x1 in S \/ {A} & x2 in S \/ {A} holds
x1,x2 are_c=-comparable
proof
let x1, x2 be set ; ::_thesis: ( x1 in S \/ {A} & x2 in S \/ {A} implies x1,x2 are_c=-comparable )
assume A26: ( x1 in S \/ {A} & x2 in S \/ {A} ) ; ::_thesis: x1,x2 are_c=-comparable
percases ( ( x1 in S & x2 in S ) or ( x1 in S & x2 in {A} ) or ( x2 in S & x1 in {A} ) or ( x1 in {A} & x2 in {A} ) ) by A26, XBOOLE_0:def_3;
suppose ( x1 in S & x2 in S ) ; ::_thesis: x1,x2 are_c=-comparable
hence x1,x2 are_c=-comparable by ORDINAL1:def_8; ::_thesis: verum
end;
suppose ( x1 in S & x2 in {A} ) ; ::_thesis: x1,x2 are_c=-comparable
then ( x1 c= union S & x2 = A ) by TARSKI:def_1, ZFMISC_1:74;
then x1 c= x2 by A25, XBOOLE_1:1;
hence x1,x2 are_c=-comparable by XBOOLE_0:def_9; ::_thesis: verum
end;
suppose ( x2 in S & x1 in {A} ) ; ::_thesis: x1,x2 are_c=-comparable
then ( x2 c= union S & x1 = A ) by TARSKI:def_1, ZFMISC_1:74;
then x2 c= x1 by A25, XBOOLE_1:1;
hence x1,x2 are_c=-comparable by XBOOLE_0:def_9; ::_thesis: verum
end;
supposeA27: ( x1 in {A} & x2 in {A} ) ; ::_thesis: x1,x2 are_c=-comparable
then x1 = A by TARSKI:def_1;
hence x1,x2 are_c=-comparable by A27, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
then reconsider S1 = S \/ {A} as c=-linear finite simplex-like Subset-Family of K by A10, ORDINAL1:def_8, TOPS_2:13;
A28: B .: S1 = BS \/ BY by A21, RELAT_1:120;
then reconsider BS1 = B .: S1 as Subset of (subdivision (B,K)) ;
A29: conv (@ BS) c= conv (@ BS1) by A28, RLTOPSP1:20, XBOOLE_1:7;
BA in BY by A14, TARSKI:def_1;
then A30: BA in @ BS1 by A28, XBOOLE_0:def_3;
take S1 ; ::_thesis: ex BS being Subset of (subdivision (B,K)) st
( BS = B .: S1 & x in conv (@ BS) & union S1 c= A )
take BS1 ; ::_thesis: ( BS1 = B .: S1 & x in conv (@ BS1) & union S1 c= A )
A31: @ BS1 c= conv (@ BS1) by CONVEX1:41;
union S1 = (union S) \/ A by A8, ZFMISC_1:78
.= A by A23, A24, XBOOLE_1:1, XBOOLE_1:12 ;
hence ( BS1 = B .: S1 & x in conv (@ BS1) & union S1 c= A ) by A20, A22, A29, A30, A31, RLTOPSP1:def_1; ::_thesis: verum
end;
end;
end;
A32: S1[ 0 ]
proof
let x be set ; ::_thesis: for A being Simplex of K st x in conv (@ A) & card A = 0 holds
ex S being c=-linear finite simplex-like Subset-Family of K ex BS being Subset of (subdivision (B,K)) st
( BS = B .: S & x in conv (@ BS) & union S c= A )
let A be Simplex of K; ::_thesis: ( x in conv (@ A) & card A = 0 implies ex S being c=-linear finite simplex-like Subset-Family of K ex BS being Subset of (subdivision (B,K)) st
( BS = B .: S & x in conv (@ BS) & union S c= A ) )
assume that
A33: x in conv (@ A) and
A34: card A = 0 ; ::_thesis: ex S being c=-linear finite simplex-like Subset-Family of K ex BS being Subset of (subdivision (B,K)) st
( BS = B .: S & x in conv (@ BS) & union S c= A )
not @ A is empty by A33;
hence ex S being c=-linear finite simplex-like Subset-Family of K ex BS being Subset of (subdivision (B,K)) st
( BS = B .: S & x in conv (@ BS) & union S c= A ) by A34; ::_thesis: verum
end;
A35: for n being Nat holds S1[n] from NAT_1:sch_2(A32, A4);
thus |.K.| c= |.(subdivision (B,K)).| :: according to SIMPLEX1:def_4 ::_thesis: for A being Subset of (subdivision (B,K)) st A is simplex-like holds
ex B being Subset of K st
( B is simplex-like & conv (@ A) c= conv (@ B) )
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in |.K.| or x in |.(subdivision (B,K)).| )
assume x in |.K.| ; ::_thesis: x in |.(subdivision (B,K)).|
then consider A being Subset of K such that
A36: A is simplex-like and
A37: x in conv (@ A) by Def3;
reconsider A = A as Simplex of K by A36;
S1[ card A] by A35;
then consider S being c=-linear finite simplex-like Subset-Family of K, BS being Subset of (subdivision (B,K)) such that
A38: BS = B .: S and
A39: x in conv (@ BS) and
union S c= A by A37;
BS is simplex-like by A38, SIMPLEX0:def_20;
then conv (@ BS) c= |.(subdivision (B,K)).| by Th5;
hence x in |.(subdivision (B,K)).| by A39; ::_thesis: verum
end;
for A being Subset of (subdivision (B,K)) st A is simplex-like holds
ex B being Subset of K st
( B is simplex-like & conv (@ A) c= conv (@ B) )
proof
let A be Subset of (subdivision (B,K)); ::_thesis: ( A is simplex-like implies ex B being Subset of K st
( B is simplex-like & conv (@ A) c= conv (@ B) ) )
assume A is simplex-like ; ::_thesis: ex B being Subset of K st
( B is simplex-like & conv (@ A) c= conv (@ B) )
then consider S being c=-linear finite simplex-like Subset-Family of K such that
A40: A = B .: S by SIMPLEX0:def_20;
percases ( S is empty or not S is empty ) ;
supposeA41: S is empty ; ::_thesis: ex B being Subset of K st
( B is simplex-like & conv (@ A) c= conv (@ B) )
take {} K ; ::_thesis: ( {} K is simplex-like & conv (@ A) c= conv (@ ({} K)) )
thus ( {} K is simplex-like & conv (@ A) c= conv (@ ({} K)) ) by A40, A41; ::_thesis: verum
end;
supposeA42: not S is empty ; ::_thesis: ex B being Subset of K st
( B is simplex-like & conv (@ A) c= conv (@ B) )
take U = union S; ::_thesis: ( U is simplex-like & conv (@ A) c= conv (@ U) )
A43: A c= conv (@ U)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in A or x in conv (@ U) )
assume x in A ; ::_thesis: x in conv (@ U)
then consider y being set such that
A44: y in dom B and
A45: y in S and
A46: B . y = x by A40, FUNCT_1:def_6;
reconsider y = y as Simplex of K by A45, TOPS_2:def_1;
y <> {} by A44, ZFMISC_1:56;
then A47: B . y in conv (@ y) by A2;
conv (@ y) c= conv (@ U) by A45, RLTOPSP1:20, ZFMISC_1:74;
hence x in conv (@ U) by A46, A47; ::_thesis: verum
end;
U in S by A42, SIMPLEX0:9;
hence ( U is simplex-like & conv (@ A) c= conv (@ U) ) by A43, CONVEX1:30, TOPS_2:def_1; ::_thesis: verum
end;
end;
end;
hence for A being Subset of (subdivision (B,K)) st A is simplex-like holds
ex B being Subset of K st
( B is simplex-like & conv (@ A) c= conv (@ B) ) ; ::_thesis: verum
end;
registration
let V be RealLinearSpace;
let Kv be non void SimplicialComplex of V;
cluster non void subset-closed finite-membered for SubdivisionStr of Kv;
existence
not for b1 being Subdivision of Kv holds b1 is void
proof
reconsider P = Kv as Subdivision of Kv by Th11;
take P ; ::_thesis: not P is void
thus not P is void ; ::_thesis: verum
end;
end;
begin
definition
let V be RealLinearSpace;
let Kv be non void SimplicialComplex of V;
assume A1: |.Kv.| c= [#] Kv ;
func BCS Kv -> non void Subdivision of Kv equals :Def5: :: SIMPLEX1:def 5
subdivision ((center_of_mass V),Kv);
coherence
subdivision ((center_of_mass V),Kv) is non void Subdivision of Kv
proof
set B = center_of_mass V;
for S being Simplex of Kv st not S is empty holds
(center_of_mass V) . S in conv (@ S) by RLAFFIN2:16;
hence subdivision ((center_of_mass V),Kv) is non void Subdivision of Kv by A1, Th15; ::_thesis: verum
end;
end;
:: deftheorem Def5 defines BCS SIMPLEX1:def_5_:_
for V being RealLinearSpace
for Kv being non void SimplicialComplex of V st |.Kv.| c= [#] Kv holds
BCS Kv = subdivision ((center_of_mass V),Kv);
definition
let n be Nat;
let V be RealLinearSpace;
let Kv be non void SimplicialComplex of V;
assume B1: |.Kv.| c= [#] Kv ;
func BCS (n,Kv) -> non void Subdivision of Kv equals :Def6: :: SIMPLEX1:def 6
subdivision (n,(center_of_mass V),Kv);
coherence
subdivision (n,(center_of_mass V),Kv) is non void Subdivision of Kv
proof
defpred S1[ Nat] means subdivision ($1,(center_of_mass V),Kv) is non void Subdivision of Kv;
A1: for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] )
assume S1[k] ; ::_thesis: S1[k + 1]
then reconsider P = subdivision (k,(center_of_mass V),Kv) as non void Subdivision of Kv ;
A2: ( |.P.| = |.Kv.| & [#] P = [#] Kv ) by Th10, SIMPLEX0:64;
k in NAT by ORDINAL1:def_12;
then subdivision ((k + 1),(center_of_mass V),Kv) = subdivision (1,(center_of_mass V),(subdivision (k,(center_of_mass V),Kv))) by SIMPLEX0:63
.= subdivision ((center_of_mass V),P) by SIMPLEX0:62
.= BCS P by B1, A2, Def5 ;
hence S1[k + 1] by Th14; ::_thesis: verum
end;
Kv = subdivision (0,(center_of_mass V),Kv) by SIMPLEX0:61;
then A3: S1[ 0 ] by Th11;
for k being Nat holds S1[k] from NAT_1:sch_2(A3, A1);
hence subdivision (n,(center_of_mass V),Kv) is non void Subdivision of Kv ; ::_thesis: verum
end;
end;
:: deftheorem Def6 defines BCS SIMPLEX1:def_6_:_
for n being Nat
for V being RealLinearSpace
for Kv being non void SimplicialComplex of V st |.Kv.| c= [#] Kv holds
BCS (n,Kv) = subdivision (n,(center_of_mass V),Kv);
theorem Th16: :: SIMPLEX1:16
for V being RealLinearSpace
for Kv being non void SimplicialComplex of V st |.Kv.| c= [#] Kv holds
BCS (0,Kv) = Kv
proof
let V be RealLinearSpace; ::_thesis: for Kv being non void SimplicialComplex of V st |.Kv.| c= [#] Kv holds
BCS (0,Kv) = Kv
let Kv be non void SimplicialComplex of V; ::_thesis: ( |.Kv.| c= [#] Kv implies BCS (0,Kv) = Kv )
assume |.Kv.| c= [#] Kv ; ::_thesis: BCS (0,Kv) = Kv
hence BCS (0,Kv) = subdivision (0,(center_of_mass V),Kv) by Def6
.= Kv by SIMPLEX0:61 ;
::_thesis: verum
end;
theorem Th17: :: SIMPLEX1:17
for V being RealLinearSpace
for Kv being non void SimplicialComplex of V st |.Kv.| c= [#] Kv holds
BCS (1,Kv) = BCS Kv
proof
let V be RealLinearSpace; ::_thesis: for Kv being non void SimplicialComplex of V st |.Kv.| c= [#] Kv holds
BCS (1,Kv) = BCS Kv
let Kv be non void SimplicialComplex of V; ::_thesis: ( |.Kv.| c= [#] Kv implies BCS (1,Kv) = BCS Kv )
assume A1: |.Kv.| c= [#] Kv ; ::_thesis: BCS (1,Kv) = BCS Kv
hence BCS (1,Kv) = subdivision (1,(center_of_mass V),Kv) by Def6
.= subdivision ((center_of_mass V),Kv) by SIMPLEX0:62
.= BCS Kv by A1, Def5 ;
::_thesis: verum
end;
theorem Th18: :: SIMPLEX1:18
for n being Nat
for V being RealLinearSpace
for Kv being non void SimplicialComplex of V st |.Kv.| c= [#] Kv holds
[#] (BCS (n,Kv)) = [#] Kv
proof
let n be Nat; ::_thesis: for V being RealLinearSpace
for Kv being non void SimplicialComplex of V st |.Kv.| c= [#] Kv holds
[#] (BCS (n,Kv)) = [#] Kv
let V be RealLinearSpace; ::_thesis: for Kv being non void SimplicialComplex of V st |.Kv.| c= [#] Kv holds
[#] (BCS (n,Kv)) = [#] Kv
let Kv be non void SimplicialComplex of V; ::_thesis: ( |.Kv.| c= [#] Kv implies [#] (BCS (n,Kv)) = [#] Kv )
assume |.Kv.| c= [#] Kv ; ::_thesis: [#] (BCS (n,Kv)) = [#] Kv
then BCS (n,Kv) = subdivision (n,(center_of_mass V),Kv) by Def6;
hence [#] (BCS (n,Kv)) = [#] Kv by SIMPLEX0:64; ::_thesis: verum
end;
theorem :: SIMPLEX1:19
for n being Nat
for V being RealLinearSpace
for Kv being non void SimplicialComplex of V st |.Kv.| c= [#] Kv holds
|.(BCS (n,Kv)).| = |.Kv.| by Th10;
theorem Th20: :: SIMPLEX1:20
for n being Nat
for V being RealLinearSpace
for Kv being non void SimplicialComplex of V st |.Kv.| c= [#] Kv holds
BCS ((n + 1),Kv) = BCS (BCS (n,Kv))
proof
let n be Nat; ::_thesis: for V being RealLinearSpace
for Kv being non void SimplicialComplex of V st |.Kv.| c= [#] Kv holds
BCS ((n + 1),Kv) = BCS (BCS (n,Kv))
let V be RealLinearSpace; ::_thesis: for Kv being non void SimplicialComplex of V st |.Kv.| c= [#] Kv holds
BCS ((n + 1),Kv) = BCS (BCS (n,Kv))
let Kv be non void SimplicialComplex of V; ::_thesis: ( |.Kv.| c= [#] Kv implies BCS ((n + 1),Kv) = BCS (BCS (n,Kv)) )
A1: |.(BCS (n,Kv)).| = |.Kv.| by Th10;
assume A2: |.Kv.| c= [#] Kv ; ::_thesis: BCS ((n + 1),Kv) = BCS (BCS (n,Kv))
then A3: [#] (BCS (n,Kv)) = [#] Kv by Th18;
n in NAT by ORDINAL1:def_12;
then subdivision ((1 + n),(center_of_mass V),Kv) = subdivision (1,(center_of_mass V),(subdivision (n,(center_of_mass V),Kv))) by SIMPLEX0:63
.= subdivision (1,(center_of_mass V),(BCS (n,Kv))) by A2, Def6
.= BCS (1,(BCS (n,Kv))) by A2, A3, A1, Def6
.= BCS (BCS (n,Kv)) by A2, A3, A1, Th17 ;
hence BCS ((n + 1),Kv) = BCS (BCS (n,Kv)) by A2, Def6; ::_thesis: verum
end;
theorem Th21: :: SIMPLEX1:21
for V being RealLinearSpace
for Kv being non void SimplicialComplex of V st |.Kv.| c= [#] Kv & degree Kv <= 0 holds
TopStruct(# the carrier of Kv, the topology of Kv #) = BCS Kv
proof
let V be RealLinearSpace; ::_thesis: for Kv being non void SimplicialComplex of V st |.Kv.| c= [#] Kv & degree Kv <= 0 holds
TopStruct(# the carrier of Kv, the topology of Kv #) = BCS Kv
let Kv be non void SimplicialComplex of V; ::_thesis: ( |.Kv.| c= [#] Kv & degree Kv <= 0 implies TopStruct(# the carrier of Kv, the topology of Kv #) = BCS Kv )
reconsider o = 1 as ext-real number ;
assume that
A1: |.Kv.| c= [#] Kv and
A2: degree Kv <= 0 ; ::_thesis: TopStruct(# the carrier of Kv, the topology of Kv #) = BCS Kv
set B = center_of_mass V;
set BC = BCS Kv;
A3: BCS Kv = subdivision ((center_of_mass V),Kv) by A1, Def5;
then A4: [#] (BCS Kv) = [#] Kv by SIMPLEX0:def_20;
A5: dom (center_of_mass V) = (bool the carrier of V) \ {{}} by FUNCT_2:def_1;
A6: ( 0 + o = 0 + 1 & (degree Kv) + o <= 0 + o ) by A2, XXREAL_3:35, XXREAL_3:def_2;
A7: the topology of (BCS Kv) c= the topology of Kv
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the topology of (BCS Kv) or x in the topology of Kv )
assume x in the topology of (BCS Kv) ; ::_thesis: x in the topology of Kv
then reconsider X = x as Simplex of (BCS Kv) by PRE_TOPC:def_2;
reconsider X1 = X as Subset of Kv by A4;
consider S being c=-linear finite simplex-like Subset-Family of Kv such that
A8: X = (center_of_mass V) .: S by A3, SIMPLEX0:def_20;
A9: (center_of_mass V) .: S = (center_of_mass V) .: (S /\ (dom (center_of_mass V))) by RELAT_1:112;
percases ( X is empty or not X is empty ) ;
suppose X is empty ; ::_thesis: x in the topology of Kv
then X1 is simplex-like ;
hence x in the topology of Kv by PRE_TOPC:def_2; ::_thesis: verum
end;
supposeA10: not X is empty ; ::_thesis: x in the topology of Kv
then not S is empty by A8;
then union S in S by SIMPLEX0:9;
then reconsider U = union S as Simplex of Kv by TOPS_2:def_1;
A11: not U is empty
proof
assume A12: U is empty ; ::_thesis: contradiction
S /\ (dom (center_of_mass V)) is empty
proof
assume not S /\ (dom (center_of_mass V)) is empty ; ::_thesis: contradiction
then consider y being set such that
A13: y in S /\ (dom (center_of_mass V)) by XBOOLE_0:def_1;
y in S by A13, XBOOLE_0:def_4;
then A14: y c= U by ZFMISC_1:74;
y <> {} by A13, ZFMISC_1:56;
hence contradiction by A12, A14; ::_thesis: verum
end;
hence contradiction by A8, A9, A10; ::_thesis: verum
end;
then A15: @ U in dom (center_of_mass V) by A5, ZFMISC_1:56;
card U <= (degree Kv) + 1 by SIMPLEX0:24;
then A16: card U <= 1 by A6, XXREAL_0:2;
card U >= 1 by A11, NAT_1:14;
then A17: card U = 1 by A16, XXREAL_0:1;
then consider u being set such that
A18: {u} = @ U by CARD_2:42;
u in {u} by TARSKI:def_1;
then reconsider u = u as Element of V by A18;
A19: S /\ (dom (center_of_mass V)) c= {U}
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in S /\ (dom (center_of_mass V)) or y in {U} )
assume A20: y in S /\ (dom (center_of_mass V)) ; ::_thesis: y in {U}
then y in S by XBOOLE_0:def_4;
then A21: y c= U by ZFMISC_1:74;
y <> {} by A20, ZFMISC_1:56;
then y = U by A18, A21, ZFMISC_1:33;
hence y in {U} by TARSKI:def_1; ::_thesis: verum
end;
not S /\ (dom (center_of_mass V)) is empty by A8, A9, A10;
then S /\ (dom (center_of_mass V)) = {U} by A19, ZFMISC_1:33;
then X = Im ((center_of_mass V),U) by A8, A9, RELAT_1:def_16
.= {((center_of_mass V) . U)} by A15, FUNCT_1:59
.= {((1 / 1) * (Sum {u}))} by A17, A18, RLAFFIN2:def_2
.= {(Sum {u})} by RLVECT_1:def_8
.= U by A18, RLVECT_2:9 ;
hence x in the topology of Kv by PRE_TOPC:def_2; ::_thesis: verum
end;
end;
end;
the topology of Kv c= the topology of (BCS Kv)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the topology of Kv or x in the topology of (BCS Kv) )
assume x in the topology of Kv ; ::_thesis: x in the topology of (BCS Kv)
then reconsider X = x as Simplex of Kv by PRE_TOPC:def_2;
reconsider X1 = X as Subset of (BCS Kv) by A4;
percases ( X is empty or not X is empty ) ;
suppose X is empty ; ::_thesis: x in the topology of (BCS Kv)
then X1 is simplex-like ;
hence x in the topology of (BCS Kv) by PRE_TOPC:def_2; ::_thesis: verum
end;
supposeA22: not X is empty ; ::_thesis: x in the topology of (BCS Kv)
for Y being Subset of Kv st Y in {X} holds
Y is simplex-like by TARSKI:def_1;
then reconsider XX = {X} as finite simplex-like Subset-Family of Kv by TOPS_2:def_1;
now__::_thesis:_for_x,_y_being_set_st_x_in_XX_&_y_in_XX_holds_
x,y_are_c=-comparable
let x, y be set ; ::_thesis: ( x in XX & y in XX implies x,y are_c=-comparable )
assume that
A23: x in XX and
A24: y in XX ; ::_thesis: x,y are_c=-comparable
x = X by A23, TARSKI:def_1;
hence x,y are_c=-comparable by A24, TARSKI:def_1; ::_thesis: verum
end;
then A25: XX is c=-linear by ORDINAL1:def_8;
card X <= (degree Kv) + 1 by SIMPLEX0:24;
then A26: card X <= 1 by A6, XXREAL_0:2;
card X >= 1 by A22, NAT_1:14;
then A27: card X = 1 by A26, XXREAL_0:1;
then consider u being set such that
A28: @ X = {u} by CARD_2:42;
A29: @ X in dom (center_of_mass V) by A5, A22, ZFMISC_1:56;
u in {u} by TARSKI:def_1;
then reconsider u = u as Element of V by A28;
(center_of_mass V) .: XX = Im ((center_of_mass V),X) by RELAT_1:def_16
.= {((center_of_mass V) . X)} by A29, FUNCT_1:59
.= {((1 / 1) * (Sum {u}))} by A27, A28, RLAFFIN2:def_2
.= {(Sum {u})} by RLVECT_1:def_8
.= X1 by A28, RLVECT_2:9 ;
then X1 is simplex-like by A3, A25, SIMPLEX0:def_20;
hence x in the topology of (BCS Kv) by PRE_TOPC:def_2; ::_thesis: verum
end;
end;
end;
hence TopStruct(# the carrier of Kv, the topology of Kv #) = BCS Kv by A3, A4, A7, XBOOLE_0:def_10; ::_thesis: verum
end;
theorem Th22: :: SIMPLEX1:22
for n being Nat
for V being RealLinearSpace
for Kv being non void SimplicialComplex of V st n > 0 & |.Kv.| c= [#] Kv & degree Kv <= 0 holds
TopStruct(# the carrier of Kv, the topology of Kv #) = BCS (n,Kv)
proof
let n be Nat; ::_thesis: for V being RealLinearSpace
for Kv being non void SimplicialComplex of V st n > 0 & |.Kv.| c= [#] Kv & degree Kv <= 0 holds
TopStruct(# the carrier of Kv, the topology of Kv #) = BCS (n,Kv)
let V be RealLinearSpace; ::_thesis: for Kv being non void SimplicialComplex of V st n > 0 & |.Kv.| c= [#] Kv & degree Kv <= 0 holds
TopStruct(# the carrier of Kv, the topology of Kv #) = BCS (n,Kv)
let Kv be non void SimplicialComplex of V; ::_thesis: ( n > 0 & |.Kv.| c= [#] Kv & degree Kv <= 0 implies TopStruct(# the carrier of Kv, the topology of Kv #) = BCS (n,Kv) )
assume that
A1: n > 0 and
A2: |.Kv.| c= [#] Kv and
A3: degree Kv <= 0 ; ::_thesis: TopStruct(# the carrier of Kv, the topology of Kv #) = BCS (n,Kv)
defpred S1[ Nat] means ( $1 > 0 implies ( TopStruct(# the carrier of Kv, the topology of Kv #) = BCS ($1,Kv) & degree (BCS ($1,Kv)) <= 0 ) );
A4: for n being Nat st S1[n] holds
S1[n + 1]
proof
not {} in dom (center_of_mass V) by ZFMISC_1:56;
then A5: dom (center_of_mass V) is with_non-empty_elements by SETFAM_1:def_8;
let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A6: S1[n] ; ::_thesis: S1[n + 1]
assume n + 1 > 0 ; ::_thesis: ( TopStruct(# the carrier of Kv, the topology of Kv #) = BCS ((n + 1),Kv) & degree (BCS ((n + 1),Kv)) <= 0 )
percases ( n = 0 or n > 0 ) ;
supposeA7: n = 0 ; ::_thesis: ( TopStruct(# the carrier of Kv, the topology of Kv #) = BCS ((n + 1),Kv) & degree (BCS ((n + 1),Kv)) <= 0 )
A8: degree (subdivision ((center_of_mass V),Kv)) <= degree Kv by A5, SIMPLEX0:52;
BCS ((n + 1),Kv) = BCS Kv by A2, A7, Th17;
hence ( TopStruct(# the carrier of Kv, the topology of Kv #) = BCS ((n + 1),Kv) & degree (BCS ((n + 1),Kv)) <= 0 ) by A2, A3, A8, Def5, Th21; ::_thesis: verum
end;
supposeA9: n > 0 ; ::_thesis: ( TopStruct(# the carrier of Kv, the topology of Kv #) = BCS ((n + 1),Kv) & degree (BCS ((n + 1),Kv)) <= 0 )
A10: |.Kv.| = |.(BCS (n,Kv)).| by Th10;
[#] Kv = [#] (BCS (n,Kv)) by A6, A9;
then BCS (n,Kv) = BCS (BCS (n,Kv)) by A2, A6, A9, A10, Th21;
hence ( TopStruct(# the carrier of Kv, the topology of Kv #) = BCS ((n + 1),Kv) & degree (BCS ((n + 1),Kv)) <= 0 ) by A2, A6, A9, Th20; ::_thesis: verum
end;
end;
end;
A11: S1[ 0 ] ;
for n being Nat holds S1[n] from NAT_1:sch_2(A11, A4);
hence TopStruct(# the carrier of Kv, the topology of Kv #) = BCS (n,Kv) by A1; ::_thesis: verum
end;
theorem Th23: :: SIMPLEX1:23
for n being Nat
for V being RealLinearSpace
for Kv being non void SimplicialComplex of V
for Sv being non void SubSimplicialComplex of Kv st |.Kv.| c= [#] Kv & |.Sv.| c= [#] Sv holds
BCS (n,Sv) is SubSimplicialComplex of BCS (n,Kv)
proof
let n be Nat; ::_thesis: for V being RealLinearSpace
for Kv being non void SimplicialComplex of V
for Sv being non void SubSimplicialComplex of Kv st |.Kv.| c= [#] Kv & |.Sv.| c= [#] Sv holds
BCS (n,Sv) is SubSimplicialComplex of BCS (n,Kv)
let V be RealLinearSpace; ::_thesis: for Kv being non void SimplicialComplex of V
for Sv being non void SubSimplicialComplex of Kv st |.Kv.| c= [#] Kv & |.Sv.| c= [#] Sv holds
BCS (n,Sv) is SubSimplicialComplex of BCS (n,Kv)
let Kv be non void SimplicialComplex of V; ::_thesis: for Sv being non void SubSimplicialComplex of Kv st |.Kv.| c= [#] Kv & |.Sv.| c= [#] Sv holds
BCS (n,Sv) is SubSimplicialComplex of BCS (n,Kv)
let S be non void SubSimplicialComplex of Kv; ::_thesis: ( |.Kv.| c= [#] Kv & |.S.| c= [#] S implies BCS (n,S) is SubSimplicialComplex of BCS (n,Kv) )
assume ( |.Kv.| c= [#] Kv & |.S.| c= [#] S ) ; ::_thesis: BCS (n,S) is SubSimplicialComplex of BCS (n,Kv)
then ( BCS (n,S) = subdivision (n,(center_of_mass V),S) & BCS (n,Kv) = subdivision (n,(center_of_mass V),Kv) ) by Def6;
hence BCS (n,S) is SubSimplicialComplex of BCS (n,Kv) by SIMPLEX0:65; ::_thesis: verum
end;
Lm1: for n being Nat holds card n = n
proof
let n be Nat; ::_thesis: card n = n
card n = card (card n) ;
hence card n = n by CARD_1:40; ::_thesis: verum
end;
theorem Th24: :: SIMPLEX1:24
for n being Nat
for V being RealLinearSpace
for Kv being non void SimplicialComplex of V st |.Kv.| c= [#] Kv holds
Vertices Kv c= Vertices (BCS (n,Kv))
proof
let n be Nat; ::_thesis: for V being RealLinearSpace
for Kv being non void SimplicialComplex of V st |.Kv.| c= [#] Kv holds
Vertices Kv c= Vertices (BCS (n,Kv))
let V be RealLinearSpace; ::_thesis: for Kv being non void SimplicialComplex of V st |.Kv.| c= [#] Kv holds
Vertices Kv c= Vertices (BCS (n,Kv))
let Kv be non void SimplicialComplex of V; ::_thesis: ( |.Kv.| c= [#] Kv implies Vertices Kv c= Vertices (BCS (n,Kv)) )
set S = Skeleton_of (Kv,0);
assume A1: |.Kv.| c= [#] Kv ; ::_thesis: Vertices Kv c= Vertices (BCS (n,Kv))
percases ( n = 0 or n > 0 ) ;
suppose n = 0 ; ::_thesis: Vertices Kv c= Vertices (BCS (n,Kv))
hence Vertices Kv c= Vertices (BCS (n,Kv)) by A1, Th16; ::_thesis: verum
end;
supposeA2: n > 0 ; ::_thesis: Vertices Kv c= Vertices (BCS (n,Kv))
the topology of (Skeleton_of (Kv,0)) c= the topology of Kv by SIMPLEX0:def_13;
then |.(Skeleton_of (Kv,0)).| c= |.Kv.| by Th4;
then A3: |.(Skeleton_of (Kv,0)).| c= [#] (Skeleton_of (Kv,0)) by A1, XBOOLE_1:1;
then ( degree (Skeleton_of (Kv,0)) <= 0 & BCS (n,(Skeleton_of (Kv,0))) is SubSimplicialComplex of BCS (n,Kv) ) by A1, Th23, SIMPLEX0:44;
then Skeleton_of (Kv,0) is SubSimplicialComplex of BCS (n,Kv) by A2, A3, Th22;
then A4: Vertices (Skeleton_of (Kv,0)) c= Vertices (BCS (n,Kv)) by SIMPLEX0:31;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Vertices Kv or x in Vertices (BCS (n,Kv)) )
assume A5: x in Vertices Kv ; ::_thesis: x in Vertices (BCS (n,Kv))
then reconsider v = x as Element of Kv ;
v is vertex-like by A5, SIMPLEX0:def_4;
then consider A being Subset of Kv such that
A6: A is simplex-like and
A7: v in A by SIMPLEX0:def_3;
reconsider vv = {v} as Subset of Kv by A7, ZFMISC_1:31;
{v} c= A by A7, ZFMISC_1:31;
then vv is simplex-like by A6, MATROID0:1;
then A8: vv in the topology of Kv by PRE_TOPC:def_2;
( card vv = 1 & card 1 = 1 ) by Lm1, CARD_1:30;
then vv in the_subsets_with_limited_card (1, the topology of Kv) by A8, SIMPLEX0:def_2;
then vv in the topology of (Skeleton_of (Kv,0)) by SIMPLEX0:2;
then reconsider vv = vv as Simplex of (Skeleton_of (Kv,0)) by PRE_TOPC:def_2;
A9: v in vv by TARSKI:def_1;
reconsider v = v as Element of (Skeleton_of (Kv,0)) ;
v is vertex-like by A9, SIMPLEX0:def_3;
then v in Vertices (Skeleton_of (Kv,0)) by SIMPLEX0:def_4;
hence x in Vertices (BCS (n,Kv)) by A4; ::_thesis: verum
end;
end;
end;
registration
let n be Nat;
let V be RealLinearSpace;
let K be non void total SimplicialComplex of V;
cluster BCS (n,K) -> non void total ;
coherence
BCS (n,K) is total
proof
A1: [#] K = [#] V by SIMPLEX0:def_10;
then |.K.| c= [#] K ;
then [#] (BCS (n,K)) = [#] V by A1, Th18;
hence BCS (n,K) is total by SIMPLEX0:def_10; ::_thesis: verum
end;
end;
registration
let n be Nat;
let V be RealLinearSpace;
let K be non void finite-vertices total SimplicialComplex of V;
cluster BCS (n,K) -> non void finite-vertices ;
coherence
BCS (n,K) is finite-vertices
proof
defpred S1[ Nat] means BCS (n,K) is finite-vertices ;
[#] K = [#] V by SIMPLEX0:def_10;
then A1: |.K.| c= [#] K ;
A2: for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A3: S1[n] ; ::_thesis: S1[n + 1]
[#] (BCS (n,K)) = [#] V by SIMPLEX0:def_10;
then A4: |.(BCS (n,K)).| c= [#] (BCS (n,K)) ;
BCS ((n + 1),K) = BCS (BCS (n,K)) by A1, Th20
.= subdivision ((center_of_mass V),(BCS (n,K))) by A4, Def5 ;
hence S1[n + 1] by A3; ::_thesis: verum
end;
A5: S1[ 0 ] by A1, Th16;
for n being Nat holds S1[n] from NAT_1:sch_2(A5, A2);
hence BCS (n,K) is finite-vertices ; ::_thesis: verum
end;
end;
begin
definition
let V be RealLinearSpace;
let K be SimplicialComplexStr of V;
attrK is affinely-independent means :Def7: :: SIMPLEX1:def 7
for A being Subset of K st A is simplex-like holds
@ A is affinely-independent ;
end;
:: deftheorem Def7 defines affinely-independent SIMPLEX1:def_7_:_
for V being RealLinearSpace
for K being SimplicialComplexStr of V holds
( K is affinely-independent iff for A being Subset of K st A is simplex-like holds
@ A is affinely-independent );
definition
let RLS be non empty RLSStruct ;
let Kr be SimplicialComplexStr of RLS;
attrKr is simplex-join-closed means :Def8: :: SIMPLEX1:def 8
for A, B being Subset of Kr st A is simplex-like & B is simplex-like holds
(conv (@ A)) /\ (conv (@ B)) = conv (@ (A /\ B));
end;
:: deftheorem Def8 defines simplex-join-closed SIMPLEX1:def_8_:_
for RLS being non empty RLSStruct
for Kr being SimplicialComplexStr of RLS holds
( Kr is simplex-join-closed iff for A, B being Subset of Kr st A is simplex-like & B is simplex-like holds
(conv (@ A)) /\ (conv (@ B)) = conv (@ (A /\ B)) );
registration
let V be RealLinearSpace;
cluster empty-membered -> affinely-independent for SimplicialComplexStr of the carrier of V;
coherence
for b1 being SimplicialComplexStr of V st b1 is empty-membered holds
b1 is affinely-independent
proof
let K be SimplicialComplexStr of V; ::_thesis: ( K is empty-membered implies K is affinely-independent )
assume K is empty-membered ; ::_thesis: K is affinely-independent
then A1: the topology of K is empty-membered by SIMPLEX0:def_7;
let A be Subset of K; :: according to SIMPLEX1:def_7 ::_thesis: ( A is simplex-like implies @ A is affinely-independent )
assume A is simplex-like ; ::_thesis: @ A is affinely-independent
then A in the topology of K by PRE_TOPC:def_2;
then A is empty by A1, SETFAM_1:def_10;
hence @ A is affinely-independent ; ::_thesis: verum
end;
let F be affinely-independent Subset-Family of V;
cluster Complex_of F -> affinely-independent ;
coherence
Complex_of F is affinely-independent
proof
let A be Subset of (Complex_of F); :: according to SIMPLEX1:def_7 ::_thesis: ( A is simplex-like implies @ A is affinely-independent )
assume A is simplex-like ; ::_thesis: @ A is affinely-independent
then A in subset-closed_closure_of F by PRE_TOPC:def_2;
then consider x being set such that
A2: A c= x and
A3: x in F by SIMPLEX0:2;
x is affinely-independent Subset of V by A3, RLAFFIN1:def_5;
hence @ A is affinely-independent by A2, RLAFFIN1:43; ::_thesis: verum
end;
end;
registration
let RLS be non empty RLSStruct ;
cluster empty-membered -> simplex-join-closed for SimplicialComplexStr of the carrier of RLS;
coherence
for b1 being SimplicialComplexStr of RLS st b1 is empty-membered holds
b1 is simplex-join-closed
proof
let K be SimplicialComplexStr of RLS; ::_thesis: ( K is empty-membered implies K is simplex-join-closed )
assume K is empty-membered ; ::_thesis: K is simplex-join-closed
then A1: the topology of K is empty-membered by SIMPLEX0:def_7;
let A, B be Subset of K; :: according to SIMPLEX1:def_8 ::_thesis: ( A is simplex-like & B is simplex-like implies (conv (@ A)) /\ (conv (@ B)) = conv (@ (A /\ B)) )
assume that
A2: A is simplex-like and
A3: B is simplex-like ; ::_thesis: (conv (@ A)) /\ (conv (@ B)) = conv (@ (A /\ B))
A in the topology of K by A2, PRE_TOPC:def_2;
then A4: A is empty by A1, SETFAM_1:def_10;
B in the topology of K by A3, PRE_TOPC:def_2;
then B is empty by A1, SETFAM_1:def_10;
hence (conv (@ A)) /\ (conv (@ B)) = conv (@ (A /\ B)) by A4; ::_thesis: verum
end;
end;
registration
let V be RealLinearSpace;
let I be affinely-independent Subset of V;
cluster Complex_of {I} -> simplex-join-closed ;
coherence
Complex_of {I} is simplex-join-closed
proof
set C = Complex_of {I};
let A, B be Subset of (Complex_of {I}); :: according to SIMPLEX1:def_8 ::_thesis: ( A is simplex-like & B is simplex-like implies (conv (@ A)) /\ (conv (@ B)) = conv (@ (A /\ B)) )
assume that
A1: A is simplex-like and
A2: B is simplex-like ; ::_thesis: (conv (@ A)) /\ (conv (@ B)) = conv (@ (A /\ B))
A3: the topology of (Complex_of {I}) = bool I by SIMPLEX0:4;
A4: B in the topology of (Complex_of {I}) by A2, PRE_TOPC:def_2;
A5: A /\ B c= A by XBOOLE_1:17;
A6: @ A is affinely-independent by A1, Def7;
A7: conv (@ B) c= Affin (@ B) by RLAFFIN1:65;
A8: conv (@ A) c= Affin (@ A) by RLAFFIN1:65;
A9: A in the topology of (Complex_of {I}) by A1, PRE_TOPC:def_2;
then A10: Affin (@ A) c= Affin I by A3, RLAFFIN1:52;
A11: @ (A /\ B) is affinely-independent by A1, Def7;
thus (conv (@ A)) /\ (conv (@ B)) c= conv (@ (A /\ B)) :: according to XBOOLE_0:def_10 ::_thesis: conv (@ (A /\ B)) c= (conv (@ A)) /\ (conv (@ B))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (conv (@ A)) /\ (conv (@ B)) or x in conv (@ (A /\ B)) )
set IAB = I \ (@ (A /\ B));
A12: @ (A /\ B) = I /\ (@ (A /\ B)) by A3, A5, A9, XBOOLE_1:1, XBOOLE_1:28
.= I \ (I \ (@ (A /\ B))) by XBOOLE_1:48 ;
assume A13: x in (conv (@ A)) /\ (conv (@ B)) ; ::_thesis: x in conv (@ (A /\ B))
then A14: x in conv (@ A) by XBOOLE_0:def_4;
then A15: x |-- (@ A) = x |-- I by A3, A8, A9, RLAFFIN1:77;
x in conv (@ B) by A13, XBOOLE_0:def_4;
then A16: x |-- (@ B) = x |-- I by A3, A4, A7, RLAFFIN1:77;
A17: ( Carrier (x |-- (@ A)) c= A & Carrier (x |-- (@ B)) c= B ) by RLVECT_2:def_6;
A18: for y being set st y in I \ (@ (A /\ B)) holds
(x |-- I) . y = 0
proof
let y be set ; ::_thesis: ( y in I \ (@ (A /\ B)) implies (x |-- I) . y = 0 )
assume A19: y in I \ (@ (A /\ B)) ; ::_thesis: (x |-- I) . y = 0
then not y in A /\ B by XBOOLE_0:def_5;
then ( not y in Carrier (x |-- (@ A)) or not y in Carrier (x |-- (@ B)) ) by A17, XBOOLE_0:def_4;
hence (x |-- I) . y = 0 by A15, A16, A19; ::_thesis: verum
end;
A20: x in Affin (@ A) by A8, A14;
A21: now__::_thesis:_for_v_being_Element_of_V_st_v_in_@_(A_/\_B)_holds_
0_<=_(x_|--_(@_(A_/\_B)))_._v
let v be Element of V; ::_thesis: ( v in @ (A /\ B) implies 0 <= (x |-- (@ (A /\ B))) . v )
assume v in @ (A /\ B) ; ::_thesis: 0 <= (x |-- (@ (A /\ B))) . v
0 <= (x |-- (@ A)) . v by A6, A14, RLAFFIN1:71;
hence 0 <= (x |-- (@ (A /\ B))) . v by A10, A12, A15, A18, A20, RLAFFIN1:75; ::_thesis: verum
end;
x in Affin (@ (A /\ B)) by A10, A12, A18, A20, RLAFFIN1:75;
hence x in conv (@ (A /\ B)) by A11, A21, RLAFFIN1:73; ::_thesis: verum
end;
( conv (@ (A /\ B)) c= conv (@ A) & conv (@ (A /\ B)) c= conv (@ B) ) by RLTOPSP1:20, XBOOLE_1:17;
hence conv (@ (A /\ B)) c= (conv (@ A)) /\ (conv (@ B)) by XBOOLE_1:19; ::_thesis: verum
end;
end;
registration
let V be RealLinearSpace;
cluster1 -element affinely-independent for Element of bool the carrier of V;
existence
ex b1 being Subset of V st
( b1 is 1 -element & b1 is affinely-independent )
proof
take { the Element of V} ; ::_thesis: ( { the Element of V} is 1 -element & { the Element of V} is affinely-independent )
thus ( { the Element of V} is 1 -element & { the Element of V} is affinely-independent ) ; ::_thesis: verum
end;
end;
registration
let V be RealLinearSpace;
cluster subset-closed finite-membered finite-vertices with_non-empty_element total affinely-independent simplex-join-closed for SimplicialComplexStr of the carrier of V;
existence
ex b1 being SimplicialComplex of V st
( b1 is with_non-empty_element & b1 is finite-vertices & b1 is affinely-independent & b1 is simplex-join-closed & b1 is total )
proof
set v = the Element of V;
take Complex_of {{ the Element of V}} ; ::_thesis: ( Complex_of {{ the Element of V}} is with_non-empty_element & Complex_of {{ the Element of V}} is finite-vertices & Complex_of {{ the Element of V}} is affinely-independent & Complex_of {{ the Element of V}} is simplex-join-closed & Complex_of {{ the Element of V}} is total )
thus ( Complex_of {{ the Element of V}} is with_non-empty_element & Complex_of {{ the Element of V}} is finite-vertices & Complex_of {{ the Element of V}} is affinely-independent & Complex_of {{ the Element of V}} is simplex-join-closed & Complex_of {{ the Element of V}} is total ) by SIMPLEX0:def_7; ::_thesis: verum
end;
end;
registration
let V be RealLinearSpace;
let K be affinely-independent SimplicialComplexStr of V;
cluster -> affinely-independent for SubSimplicialComplex of K;
coherence
for b1 being SubSimplicialComplex of K holds b1 is affinely-independent
proof
let S be SubSimplicialComplex of K; ::_thesis: S is affinely-independent
let A be Subset of S; :: according to SIMPLEX1:def_7 ::_thesis: ( A is simplex-like implies @ A is affinely-independent )
assume A is simplex-like ; ::_thesis: @ A is affinely-independent
then A1: A in the topology of S by PRE_TOPC:def_2;
A2: the topology of S c= the topology of K by SIMPLEX0:def_13;
then A in the topology of K by A1;
then reconsider A1 = A as Subset of K ;
A1 is simplex-like by A1, A2, PRE_TOPC:def_2;
then @ A1 is affinely-independent by Def7;
hence @ A is affinely-independent ; ::_thesis: verum
end;
end;
registration
let V be RealLinearSpace;
let K be simplex-join-closed SimplicialComplexStr of V;
cluster -> simplex-join-closed for SubSimplicialComplex of K;
coherence
for b1 being SubSimplicialComplex of K holds b1 is simplex-join-closed
proof
let S be SubSimplicialComplex of K; ::_thesis: S is simplex-join-closed
A1: the topology of S c= the topology of K by SIMPLEX0:def_13;
let A, B be Subset of S; :: according to SIMPLEX1:def_8 ::_thesis: ( A is simplex-like & B is simplex-like implies (conv (@ A)) /\ (conv (@ B)) = conv (@ (A /\ B)) )
assume that
A2: A is simplex-like and
A3: B is simplex-like ; ::_thesis: (conv (@ A)) /\ (conv (@ B)) = conv (@ (A /\ B))
A4: A in the topology of S by A2, PRE_TOPC:def_2;
then A5: A in the topology of K by A1;
A6: B in the topology of S by A3, PRE_TOPC:def_2;
then B in the topology of K by A1;
then reconsider A1 = A, B1 = B as Subset of K by A5;
A7: A1 is simplex-like by A1, A4, PRE_TOPC:def_2;
B1 is simplex-like by A1, A6, PRE_TOPC:def_2;
then (conv (@ A1)) /\ (conv (@ B1)) = conv (@ (A1 /\ B1)) by A7, Def8;
hence (conv (@ A)) /\ (conv (@ B)) = conv (@ (A /\ B)) ; ::_thesis: verum
end;
end;
theorem Th25: :: SIMPLEX1:25
for V being RealLinearSpace
for K being subset-closed SimplicialComplexStr of V holds
( K is simplex-join-closed iff for A, B being Subset of K st A is simplex-like & B is simplex-like & Int (@ A) meets Int (@ B) holds
A = B )
proof
let V be RealLinearSpace; ::_thesis: for K being subset-closed SimplicialComplexStr of V holds
( K is simplex-join-closed iff for A, B being Subset of K st A is simplex-like & B is simplex-like & Int (@ A) meets Int (@ B) holds
A = B )
let K be subset-closed SimplicialComplexStr of V; ::_thesis: ( K is simplex-join-closed iff for A, B being Subset of K st A is simplex-like & B is simplex-like & Int (@ A) meets Int (@ B) holds
A = B )
hereby ::_thesis: ( ( for A, B being Subset of K st A is simplex-like & B is simplex-like & Int (@ A) meets Int (@ B) holds
A = B ) implies K is simplex-join-closed )
assume A1: K is simplex-join-closed ; ::_thesis: for A, B being Subset of K st A is simplex-like & B is simplex-like & Int (@ A) meets Int (@ B) holds
not A <> B
let A, B be Subset of K; ::_thesis: ( A is simplex-like & B is simplex-like & Int (@ A) meets Int (@ B) implies not A <> B )
assume that
A2: ( A is simplex-like & B is simplex-like ) and
A3: Int (@ A) meets Int (@ B) ; ::_thesis: not A <> B
A4: (conv (@ A)) /\ (conv (@ B)) = conv (@ (A /\ B)) by A1, A2, Def8;
assume A <> B ; ::_thesis: contradiction
then A5: ( A /\ B <> A or A /\ B <> B ) ;
A6: ( A /\ B c= A & A /\ B c= B ) by XBOOLE_1:17;
consider x being set such that
A7: x in Int (@ A) and
A8: x in Int (@ B) by A3, XBOOLE_0:3;
( Int (@ A) c= conv (@ A) & Int (@ B) c= conv (@ B) ) by RLAFFIN2:5;
then A9: x in (conv (@ A)) /\ (conv (@ B)) by A7, A8, XBOOLE_0:def_4;
percases ( A /\ B c< A or A /\ B c< B ) by A5, A6, XBOOLE_0:def_8;
suppose A /\ B c< A ; ::_thesis: contradiction
then conv (@ (A /\ B)) misses Int (@ A) by RLAFFIN2:7;
hence contradiction by A4, A7, A9, XBOOLE_0:3; ::_thesis: verum
end;
suppose A /\ B c< B ; ::_thesis: contradiction
then conv (@ (A /\ B)) misses Int (@ B) by RLAFFIN2:7;
hence contradiction by A4, A8, A9, XBOOLE_0:3; ::_thesis: verum
end;
end;
end;
assume A10: for A, B being Subset of K st A is simplex-like & B is simplex-like & Int (@ A) meets Int (@ B) holds
A = B ; ::_thesis: K is simplex-join-closed
let A, B be Subset of K; :: according to SIMPLEX1:def_8 ::_thesis: ( A is simplex-like & B is simplex-like implies (conv (@ A)) /\ (conv (@ B)) = conv (@ (A /\ B)) )
assume that
A11: A is simplex-like and
A12: B is simplex-like ; ::_thesis: (conv (@ A)) /\ (conv (@ B)) = conv (@ (A /\ B))
A13: (conv (@ A)) /\ (conv (@ B)) c= conv (@ (A /\ B))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (conv (@ A)) /\ (conv (@ B)) or x in conv (@ (A /\ B)) )
A14: the_family_of K is subset-closed ;
assume A15: x in (conv (@ A)) /\ (conv (@ B)) ; ::_thesis: x in conv (@ (A /\ B))
then x in conv (@ A) by XBOOLE_0:def_4;
then x in union { (Int a) where a is Subset of V : a c= @ A } by RLAFFIN2:8;
then consider Ia being set such that
A16: x in Ia and
A17: Ia in { (Int a) where a is Subset of V : a c= @ A } by TARSKI:def_4;
consider a being Subset of V such that
A18: Ia = Int a and
A19: a c= @ A by A17;
x in conv (@ B) by A15, XBOOLE_0:def_4;
then x in union { (Int b) where b is Subset of V : b c= @ B } by RLAFFIN2:8;
then consider Ib being set such that
A20: x in Ib and
A21: Ib in { (Int b) where b is Subset of V : b c= @ B } by TARSKI:def_4;
consider b being Subset of V such that
A22: Ib = Int b and
A23: b c= @ B by A21;
reconsider a1 = a, b1 = b as Subset of K by A19, A23, XBOOLE_1:1;
A in the topology of K by A11, PRE_TOPC:def_2;
then a1 in the topology of K by A14, A19, CLASSES1:def_1;
then A24: a1 is simplex-like by PRE_TOPC:def_2;
B in the topology of K by A12, PRE_TOPC:def_2;
then b1 in the topology of K by A14, A23, CLASSES1:def_1;
then A25: b1 is simplex-like by PRE_TOPC:def_2;
Int (@ a1) meets Int (@ b1) by A16, A18, A20, A22, XBOOLE_0:3;
then a1 = b1 by A10, A24, A25;
then a c= @ (A /\ B) by A19, A23, XBOOLE_1:19;
then A26: conv a c= conv (@ (A /\ B)) by RLAFFIN1:3;
x in conv a by A16, A18, RLAFFIN2:def_1;
hence x in conv (@ (A /\ B)) by A26; ::_thesis: verum
end;
( conv (@ (A /\ B)) c= conv (@ A) & conv (@ (A /\ B)) c= conv (@ B) ) by RLAFFIN1:3, XBOOLE_1:17;
then conv (@ (A /\ B)) c= (conv (@ A)) /\ (conv (@ B)) by XBOOLE_1:19;
hence (conv (@ A)) /\ (conv (@ B)) = conv (@ (A /\ B)) by A13, XBOOLE_0:def_10; ::_thesis: verum
end;
registration
let V be RealLinearSpace;
let Ka be non void affinely-independent SimplicialComplex of V;
let S be Simplex of Ka;
cluster @ S -> affinely-independent ;
coherence
@ S is affinely-independent by Def7;
end;
theorem Th26: :: SIMPLEX1:26
for V being RealLinearSpace
for Ks being simplex-join-closed SimplicialComplex of V
for As, Bs being Subset of Ks st As is simplex-like & Bs is simplex-like & Int (@ As) meets conv (@ Bs) holds
As c= Bs
proof
let V be RealLinearSpace; ::_thesis: for Ks being simplex-join-closed SimplicialComplex of V
for As, Bs being Subset of Ks st As is simplex-like & Bs is simplex-like & Int (@ As) meets conv (@ Bs) holds
As c= Bs
let Ks be simplex-join-closed SimplicialComplex of V; ::_thesis: for As, Bs being Subset of Ks st As is simplex-like & Bs is simplex-like & Int (@ As) meets conv (@ Bs) holds
As c= Bs
let As, Bs be Subset of Ks; ::_thesis: ( As is simplex-like & Bs is simplex-like & Int (@ As) meets conv (@ Bs) implies As c= Bs )
assume that
A1: As is simplex-like and
A2: Bs is simplex-like and
A3: Int (@ As) meets conv (@ Bs) ; ::_thesis: As c= Bs
consider x being set such that
A4: x in Int (@ As) and
A5: x in conv (@ Bs) by A3, XBOOLE_0:3;
x in union { (Int b) where b is Subset of V : b c= @ Bs } by A5, RLAFFIN2:8;
then consider Ib being set such that
A6: x in Ib and
A7: Ib in { (Int b) where b is Subset of V : b c= @ Bs } by TARSKI:def_4;
consider b being Subset of V such that
A8: Ib = Int b and
A9: b c= @ Bs by A7;
reconsider b1 = b as Subset of Ks by A9, XBOOLE_1:1;
As in the topology of Ks by A1, PRE_TOPC:def_2;
then not Ks is void by PENCIL_1:def_4;
then A10: b1 is simplex-like by A2, A9, MATROID0:1;
Int (@ As) meets Int (@ b1) by A4, A6, A8, XBOOLE_0:3;
hence As c= Bs by A1, A9, A10, Th25; ::_thesis: verum
end;
theorem :: SIMPLEX1:27
for V being RealLinearSpace
for Ks being simplex-join-closed SimplicialComplex of V
for As, Bs being Subset of Ks st As is simplex-like & @ As is affinely-independent & Bs is simplex-like holds
( Int (@ As) c= conv (@ Bs) iff As c= Bs )
proof
let V be RealLinearSpace; ::_thesis: for Ks being simplex-join-closed SimplicialComplex of V
for As, Bs being Subset of Ks st As is simplex-like & @ As is affinely-independent & Bs is simplex-like holds
( Int (@ As) c= conv (@ Bs) iff As c= Bs )
let Ks be simplex-join-closed SimplicialComplex of V; ::_thesis: for As, Bs being Subset of Ks st As is simplex-like & @ As is affinely-independent & Bs is simplex-like holds
( Int (@ As) c= conv (@ Bs) iff As c= Bs )
let As, Bs be Subset of Ks; ::_thesis: ( As is simplex-like & @ As is affinely-independent & Bs is simplex-like implies ( Int (@ As) c= conv (@ Bs) iff As c= Bs ) )
assume that
A1: As is simplex-like and
A2: @ As is affinely-independent and
A3: Bs is simplex-like ; ::_thesis: ( Int (@ As) c= conv (@ Bs) iff As c= Bs )
As in the topology of Ks by A1, PRE_TOPC:def_2;
then A4: not Ks is void by PENCIL_1:def_4;
percases ( As is empty or not As is empty ) ;
supposeA5: As is empty ; ::_thesis: ( Int (@ As) c= conv (@ Bs) iff As c= Bs )
then Int (@ As) is empty ;
hence ( Int (@ As) c= conv (@ Bs) iff As c= Bs ) by A5, XBOOLE_1:2; ::_thesis: verum
end;
suppose not As is empty ; ::_thesis: ( Int (@ As) c= conv (@ Bs) iff As c= Bs )
then not Int (@ As) is empty by A1, A2, A4, RLAFFIN2:20;
then consider x being set such that
A6: x in Int (@ As) by XBOOLE_0:def_1;
hereby ::_thesis: ( As c= Bs implies Int (@ As) c= conv (@ Bs) )
assume Int (@ As) c= conv (@ Bs) ; ::_thesis: As c= Bs
then x in conv (@ Bs) by A6;
then x in union { (Int b) where b is Subset of V : b c= @ Bs } by RLAFFIN2:8;
then consider Ib being set such that
A7: x in Ib and
A8: Ib in { (Int b) where b is Subset of V : b c= @ Bs } by TARSKI:def_4;
consider b being Subset of V such that
A9: Ib = Int b and
A10: b c= @ Bs by A8;
reconsider b1 = b as Subset of Ks by A10, XBOOLE_1:1;
A11: b1 is simplex-like by A3, A4, A10, MATROID0:1;
Int (@ As) meets Int (@ b1) by A6, A7, A9, XBOOLE_0:3;
hence As c= Bs by A1, A10, A11, Th25; ::_thesis: verum
end;
assume As c= Bs ; ::_thesis: Int (@ As) c= conv (@ Bs)
then ( Int (@ As) c= conv (@ As) & conv (@ As) c= conv (@ Bs) ) by RLAFFIN1:3, RLAFFIN2:5;
hence Int (@ As) c= conv (@ Bs) by XBOOLE_1:1; ::_thesis: verum
end;
end;
end;
theorem Th28: :: SIMPLEX1:28
for V being RealLinearSpace
for Ka being non void affinely-independent SimplicialComplex of V st |.Ka.| c= [#] Ka holds
BCS Ka is affinely-independent
proof
let V be RealLinearSpace; ::_thesis: for Ka being non void affinely-independent SimplicialComplex of V st |.Ka.| c= [#] Ka holds
BCS Ka is affinely-independent
let Ka be non void affinely-independent SimplicialComplex of V; ::_thesis: ( |.Ka.| c= [#] Ka implies BCS Ka is affinely-independent )
set P = BCS Ka;
set B = center_of_mass V;
assume |.Ka.| c= [#] Ka ; ::_thesis: BCS Ka is affinely-independent
then A1: BCS Ka = subdivision ((center_of_mass V),Ka) by Def5;
let A be Subset of (BCS Ka); :: according to SIMPLEX1:def_7 ::_thesis: ( A is simplex-like implies @ A is affinely-independent )
assume A is simplex-like ; ::_thesis: @ A is affinely-independent
then consider S being c=-linear finite simplex-like Subset-Family of Ka such that
A2: A = (center_of_mass V) .: S by A1, SIMPLEX0:def_20;
percases ( S is empty or not S is empty ) ;
suppose S is empty ; ::_thesis: @ A is affinely-independent
then A = {} by A2;
hence @ A is affinely-independent ; ::_thesis: verum
end;
supposeA3: not S is empty ; ::_thesis: @ A is affinely-independent
( S c= bool (union S) & bool (@ (union S)) c= bool the carrier of V ) by ZFMISC_1:67, ZFMISC_1:82;
then reconsider s = S as c=-linear finite Subset-Family of V by XBOOLE_1:1;
union S in S by A3, SIMPLEX0:9;
then union S is simplex-like by TOPS_2:def_1;
then @ (union S) is affinely-independent ;
then union s is affinely-independent ;
hence @ A is affinely-independent by A2, RLAFFIN2:29; ::_thesis: verum
end;
end;
end;
registration
let V be RealLinearSpace;
let Ka be non void total affinely-independent SimplicialComplex of V;
cluster BCS Ka -> non void affinely-independent ;
coherence
BCS Ka is affinely-independent
proof
[#] Ka = the carrier of V by SIMPLEX0:def_10;
then |.Ka.| c= [#] Ka ;
hence BCS Ka is affinely-independent by Th28; ::_thesis: verum
end;
let n be Nat;
cluster BCS (n,Ka) -> non void affinely-independent ;
coherence
BCS (n,Ka) is affinely-independent
proof
defpred S1[ Nat] means BCS (V,Ka) is affinely-independent ;
[#] Ka = [#] V by SIMPLEX0:def_10;
then A1: |.Ka.| c= [#] Ka ;
A2: for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A3: S1[n] ; ::_thesis: S1[n + 1]
BCS ((n + 1),Ka) = BCS (BCS (n,Ka)) by A1, Th20;
hence S1[n + 1] by A3; ::_thesis: verum
end;
A4: S1[ 0 ] by A1, Th16;
for n being Nat holds S1[n] from NAT_1:sch_2(A4, A2);
hence BCS (n,Ka) is affinely-independent ; ::_thesis: verum
end;
end;
registration
let V be RealLinearSpace;
let Kas be non void affinely-independent simplex-join-closed SimplicialComplex of V;
cluster(center_of_mass V) | the topology of Kas -> one-to-one ;
coherence
(center_of_mass V) | the topology of Kas is one-to-one
proof
now__::_thesis:_for_x1,_x2_being_set_st_x1_in_dom_((center_of_mass_V)_|_the_topology_of_Kas)_&_x2_in_dom_((center_of_mass_V)_|_the_topology_of_Kas)_&_((center_of_mass_V)_|_the_topology_of_Kas)_._x1_=_((center_of_mass_V)_|_the_topology_of_Kas)_._x2_holds_
x1_=_x2
set B = center_of_mass V;
set T = the topology of Kas;
let x1, x2 be set ; ::_thesis: ( x1 in dom ((center_of_mass V) | the topology of Kas) & x2 in dom ((center_of_mass V) | the topology of Kas) & ((center_of_mass V) | the topology of Kas) . x1 = ((center_of_mass V) | the topology of Kas) . x2 implies x1 = x2 )
set BT = (center_of_mass V) | the topology of Kas;
assume that
A1: x1 in dom ((center_of_mass V) | the topology of Kas) and
A2: x2 in dom ((center_of_mass V) | the topology of Kas) and
A3: ((center_of_mass V) | the topology of Kas) . x1 = ((center_of_mass V) | the topology of Kas) . x2 ; ::_thesis: x1 = x2
A4: ( ((center_of_mass V) | the topology of Kas) . x1 = (center_of_mass V) . x1 & ((center_of_mass V) | the topology of Kas) . x2 = (center_of_mass V) . x2 ) by A1, A2, FUNCT_1:47;
dom ((center_of_mass V) | the topology of Kas) = (dom (center_of_mass V)) /\ the topology of Kas by RELAT_1:61;
then ( x1 in the topology of Kas & x2 in the topology of Kas ) by A1, A2, XBOOLE_0:def_4;
then reconsider A1 = x1, A2 = x2 as Simplex of Kas by PRE_TOPC:def_2;
not A1 is empty by A1, ZFMISC_1:56;
then A5: (center_of_mass V) . A1 in conv (@ A1) by RLAFFIN2:16;
not A2 is empty by A2, ZFMISC_1:56;
then (center_of_mass V) . A2 in conv (@ A2) by RLAFFIN2:16;
then A6: (center_of_mass V) . A1 in (conv (@ A1)) /\ (conv (@ A2)) by A3, A4, A5, XBOOLE_0:def_4;
A7: ( (conv (@ A1)) /\ (conv (@ A2)) = conv (@ (A1 /\ A2)) & conv (@ (A1 /\ A2)) c= Affin (@ (A1 /\ A2)) ) by Def8, RLAFFIN1:65;
then A1 /\ A2 = A1 by A6, RLAFFIN2:21, XBOOLE_1:17;
hence x1 = x2 by A3, A4, A6, A7, RLAFFIN2:21, XBOOLE_1:17; ::_thesis: verum
end;
hence (center_of_mass V) | the topology of Kas is one-to-one by FUNCT_1:def_4; ::_thesis: verum
end;
end;
theorem Th29: :: SIMPLEX1:29
for V being RealLinearSpace
for Kas being non void affinely-independent simplex-join-closed SimplicialComplex of V st |.Kas.| c= [#] Kas holds
BCS Kas is simplex-join-closed
proof
let V be RealLinearSpace; ::_thesis: for Kas being non void affinely-independent simplex-join-closed SimplicialComplex of V st |.Kas.| c= [#] Kas holds
BCS Kas is simplex-join-closed
let Kas be non void affinely-independent simplex-join-closed SimplicialComplex of V; ::_thesis: ( |.Kas.| c= [#] Kas implies BCS Kas is simplex-join-closed )
set B = center_of_mass V;
set BC = BCS Kas;
defpred S1[ Nat] means for S1, S2 being c=-linear finite simplex-like Subset-Family of Kas
for A1, A2 being Simplex of (BCS Kas) st A1 = (center_of_mass V) .: S1 & A2 = (center_of_mass V) .: S2 & card (union S1) <= $1 & card (union S2) <= $1 & Int (@ A1) meets Int (@ A2) holds
A1 = A2;
assume A1: |.Kas.| c= [#] Kas ; ::_thesis: BCS Kas is simplex-join-closed
then A2: BCS Kas = subdivision ((center_of_mass V),Kas) by Def5;
A3: BCS Kas is affinely-independent by A1, Th28;
A4: dom (center_of_mass V) = (bool the carrier of V) \ {{}} by FUNCT_2:def_1;
A5: for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A6: S1[n] ; ::_thesis: S1[n + 1]
let S1, S2 be c=-linear finite simplex-like Subset-Family of Kas; ::_thesis: for A1, A2 being Simplex of (BCS Kas) st A1 = (center_of_mass V) .: S1 & A2 = (center_of_mass V) .: S2 & card (union S1) <= n + 1 & card (union S2) <= n + 1 & Int (@ A1) meets Int (@ A2) holds
A1 = A2
let A1, A2 be Simplex of (BCS Kas); ::_thesis: ( A1 = (center_of_mass V) .: S1 & A2 = (center_of_mass V) .: S2 & card (union S1) <= n + 1 & card (union S2) <= n + 1 & Int (@ A1) meets Int (@ A2) implies A1 = A2 )
assume that
A7: A1 = (center_of_mass V) .: S1 and
A8: A2 = (center_of_mass V) .: S2 and
A9: card (union S1) <= n + 1 and
card (union S2) <= n + 1 and
A10: Int (@ A1) meets Int (@ A2) ; ::_thesis: A1 = A2
A11: ( union S2 in S2 or S2 is empty ) by SIMPLEX0:9;
then A12: union S2 is simplex-like by TOPS_2:def_1, ZFMISC_1:2;
set U = union S1;
( S2 c= bool (union S2) & bool (@ (union S2)) c= bool the carrier of V ) by ZFMISC_1:67, ZFMISC_1:82;
then A13: S2 is Subset-Family of V by XBOOLE_1:1;
A14: ( union S1 in S1 or S1 is empty ) by SIMPLEX0:9;
then A15: union S1 is simplex-like by TOPS_2:def_1, ZFMISC_1:2;
( S1 c= bool (union S1) & bool (@ (union S1)) c= bool the carrier of V ) by ZFMISC_1:67, ZFMISC_1:82;
then A16: S1 is Subset-Family of V by XBOOLE_1:1;
then A17: Int ((center_of_mass V) .: S1) c= Int (@ (union S1)) by A15, RLAFFIN2:30;
Int (@ (union S1)) meets Int ((center_of_mass V) .: S2) by A7, A8, A10, A15, A16, RLAFFIN2:30, XBOOLE_1:63;
then Int (@ (union S1)) meets Int (@ (union S2)) by A13, A12, RLAFFIN2:30, XBOOLE_1:63;
then A18: union S1 = union S2 by A12, A15, Th25;
percases ( card (union S1) <= n or card (union S1) = n + 1 ) by A9, NAT_1:8;
suppose card (union S1) <= n ; ::_thesis: A1 = A2
hence A1 = A2 by A6, A7, A8, A10, A18; ::_thesis: verum
end;
supposeA19: card (union S1) = n + 1 ; ::_thesis: A1 = A2
then A20: not @ (union S1) is empty ;
then A21: union S1 in dom (center_of_mass V) by A4, ZFMISC_1:56;
then A22: (center_of_mass V) . (union S1) in @ A1 by A7, A14, A19, FUNCT_1:def_6, ZFMISC_1:2;
then reconsider Bu = (center_of_mass V) . (union S1) as Element of V ;
A23: {Bu} c= @ A1 by A22, ZFMISC_1:31;
A24: (center_of_mass V) . (union S1) in @ A2 by A8, A11, A18, A19, A21, FUNCT_1:def_6, ZFMISC_1:2;
then A25: {Bu} c= @ A2 by ZFMISC_1:31;
A26: Bu in {Bu} by ZFMISC_1:31;
A27: conv {Bu} = {Bu} by RLAFFIN1:1;
consider x being set such that
A28: x in Int (@ A1) and
A29: x in Int (@ A2) by A10, XBOOLE_0:3;
reconsider x = x as Element of V by A28;
percases ( ( A1 = {Bu} & A2 = {Bu} ) or ( A1 = {Bu} & A2 <> {Bu} ) or ( A1 <> {Bu} & A2 = {Bu} ) or ( A1 <> {Bu} & A2 <> {Bu} ) ) ;
suppose ( A1 = {Bu} & A2 = {Bu} ) ; ::_thesis: A1 = A2
hence A1 = A2 ; ::_thesis: verum
end;
supposeA30: ( A1 = {Bu} & A2 <> {Bu} ) ; ::_thesis: A1 = A2
then ( {Bu} c< @ A2 & Int (@ A1) = @ A1 ) by A25, RLAFFIN2:6, XBOOLE_0:def_8;
hence A1 = A2 by A27, A28, A29, A30, RLAFFIN2:def_1; ::_thesis: verum
end;
supposeA31: ( A1 <> {Bu} & A2 = {Bu} ) ; ::_thesis: A1 = A2
then ( {Bu} c< @ A1 & Int (@ A2) = @ A2 ) by A23, RLAFFIN2:6, XBOOLE_0:def_8;
hence A1 = A2 by A27, A28, A29, A31, RLAFFIN2:def_1; ::_thesis: verum
end;
suppose ( A1 <> {Bu} & A2 <> {Bu} ) ; ::_thesis: A1 = A2
then {Bu} c< @ A1 by A23, XBOOLE_0:def_8;
then A32: Bu <> x by A26, A27, A28, RLAFFIN2:def_1;
( S1 \ {(union S1)} c= S1 & S2 \ {(union S1)} c= S2 ) by XBOOLE_1:36;
then reconsider s1u = S1 \ {(union S1)}, s2u = S2 \ {(union S1)} as c=-linear finite simplex-like Subset-Family of Kas by TOPS_2:11;
A33: S1 c= the topology of Kas
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in S1 or x in the topology of Kas )
assume A34: x in S1 ; ::_thesis: x in the topology of Kas
then reconsider A = x as Subset of Kas ;
A is simplex-like by A34, TOPS_2:def_1;
hence x in the topology of Kas by PRE_TOPC:def_2; ::_thesis: verum
end;
[#] Kas c= the carrier of V by SIMPLEX0:def_9;
then bool the carrier of Kas c= bool the carrier of V by ZFMISC_1:67;
then reconsider S1U = s1u, S2U = s2u as Subset-Family of V by XBOOLE_1:1;
set Bu1 = x |-- (@ A1);
set Bu2 = x |-- (@ A2);
set BT = (center_of_mass V) | the topology of Kas;
A35: S1 \ {(union S1)} c= S1 by XBOOLE_1:36;
A36: {(union S1)} c= S1 by A14, A19, ZFMISC_1:2, ZFMISC_1:31;
A37: union s2u c= union S1 by A18, XBOOLE_1:36, ZFMISC_1:77;
union s2u <> union S1
proof
assume A38: union s2u = union S1 ; ::_thesis: contradiction
then union s2u in s2u by A20, SIMPLEX0:9, ZFMISC_1:2;
hence contradiction by A38, ZFMISC_1:56; ::_thesis: verum
end;
then A39: union s2u c< union S1 by A37, XBOOLE_0:def_8;
then consider xS2U being set such that
A40: xS2U in @ (union S1) and
A41: not xS2U in union S2U by XBOOLE_0:6;
reconsider xS2U = xS2U as Element of V by A40;
union S2U c= (union S1) \ {xS2U} by A37, A41, ZFMISC_1:34;
then A42: conv (union S2U) c= conv (@ ((union S1) \ {xS2U})) by RLAFFIN1:3;
A43: x in conv (@ A1) by A28, RLAFFIN2:def_1;
then A44: (x |-- (@ A1)) . Bu <= 1 by A3, RLAFFIN1:71;
A45: (x |-- (@ A1)) . Bu < 1
proof
assume (x |-- (@ A1)) . Bu >= 1 ; ::_thesis: contradiction
then (x |-- (@ A1)) . Bu = 1 by A44, XXREAL_0:1;
hence contradiction by A3, A32, A43, RLAFFIN1:72; ::_thesis: verum
end;
conv (@ A1) c= Affin (@ A1) by RLAFFIN1:65;
then A46: x = Sum (x |-- (@ A1)) by A3, A43, RLAFFIN1:def_7;
then Bu in Carrier (x |-- (@ A1)) by A3, A22, A28, A43, RLAFFIN1:71, RLAFFIN2:11;
then A47: (x |-- (@ A1)) . Bu <> 0 by RLVECT_2:19;
x |-- (@ A1) is convex by A3, A43, RLAFFIN1:71;
then consider p1 being Element of V such that
A48: p1 in conv ((@ A1) \ {Bu}) and
A49: x = (((x |-- (@ A1)) . Bu) * Bu) + ((1 - ((x |-- (@ A1)) . Bu)) * p1) and
((1 / ((x |-- (@ A1)) . Bu)) * x) + ((1 - (1 / ((x |-- (@ A1)) . Bu))) * p1) = Bu by A32, A46, A47, RLAFFIN2:1;
A50: p1 in Int ((@ A1) \ {Bu}) by A3, A22, A28, A48, A49, RLAFFIN2:14;
A51: {Bu} = Im ((center_of_mass V),(union S1)) by A21, FUNCT_1:59
.= (center_of_mass V) .: {(union S1)} by RELAT_1:def_16 ;
then A52: A1 \ {Bu} = (((center_of_mass V) | the topology of Kas) .: S1) \ ((center_of_mass V) .: {(union S1)}) by A33, A7, RELAT_1:129
.= (((center_of_mass V) | the topology of Kas) .: S1) \ (((center_of_mass V) | the topology of Kas) .: {(union S1)}) by A33, A36, RELAT_1:129, XBOOLE_1:1
.= ((center_of_mass V) | the topology of Kas) .: (S1 \ {(union S1)}) by FUNCT_1:64
.= (center_of_mass V) .: (S1 \ {(union S1)}) by A35, A33, RELAT_1:129, XBOOLE_1:1 ;
then conv ((@ A1) \ {Bu}) c= conv (union S1U) by CONVEX1:30, RLAFFIN2:17;
then A53: p1 in conv (union S1U) by A48;
card (union s2u) < n + 1 by A19, A39, CARD_2:48;
then A54: card (union s2u) <= n by NAT_1:13;
A55: union s1u c= union S1 by XBOOLE_1:36, ZFMISC_1:77;
A56: x in conv (@ A2) by A29, RLAFFIN2:def_1;
then A57: (x |-- (@ A2)) . Bu <= 1 by A3, RLAFFIN1:71;
A58: (x |-- (@ A2)) . Bu < 1
proof
assume (x |-- (@ A2)) . Bu >= 1 ; ::_thesis: contradiction
then (x |-- (@ A2)) . Bu = 1 by A57, XXREAL_0:1;
hence contradiction by A3, A32, A56, RLAFFIN1:72; ::_thesis: verum
end;
conv (@ A2) c= Affin (@ A2) by RLAFFIN1:65;
then A59: x = Sum (x |-- (@ A2)) by A3, A56, RLAFFIN1:def_7;
then Bu in Carrier (x |-- (@ A2)) by A3, A24, A29, A56, RLAFFIN1:71, RLAFFIN2:11;
then A60: (x |-- (@ A2)) . Bu <> 0 by RLVECT_2:19;
x |-- (@ A2) is convex by A3, A56, RLAFFIN1:71;
then consider p2 being Element of V such that
A61: p2 in conv ((@ A2) \ {Bu}) and
A62: x = (((x |-- (@ A2)) . Bu) * Bu) + ((1 - ((x |-- (@ A2)) . Bu)) * p2) and
((1 / ((x |-- (@ A2)) . Bu)) * x) + ((1 - (1 / ((x |-- (@ A2)) . Bu))) * p2) = Bu by A32, A59, A60, RLAFFIN2:1;
A63: p2 in Int ((@ A2) \ {Bu}) by A3, A24, A29, A61, A62, RLAFFIN2:14;
@ (union S1) is non empty finite Subset of V by A19;
then A64: Bu in Int (@ (union S1)) by A15, RLAFFIN2:20;
then A65: Bu in conv (@ (union S1)) by RLAFFIN2:def_1;
A66: S2 c= the topology of Kas
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in S2 or x in the topology of Kas )
assume A67: x in S2 ; ::_thesis: x in the topology of Kas
then reconsider A = x as Subset of Kas ;
A is simplex-like by A67, TOPS_2:def_1;
hence x in the topology of Kas by PRE_TOPC:def_2; ::_thesis: verum
end;
union s1u <> union S1
proof
assume A68: union s1u = union S1 ; ::_thesis: contradiction
then union s1u in s1u by A20, SIMPLEX0:9, ZFMISC_1:2;
hence contradiction by A68, ZFMISC_1:56; ::_thesis: verum
end;
then A69: union s1u c< union S1 by A55, XBOOLE_0:def_8;
then consider xS1U being set such that
A70: xS1U in @ (union S1) and
A71: not xS1U in union S1U by XBOOLE_0:6;
reconsider xS1U = xS1U as Element of V by A70;
union S1U c= (union S1) \ {xS1U} by A55, A71, ZFMISC_1:34;
then A72: conv (union S1U) c= conv (@ ((union S1) \ {xS1U})) by RLAFFIN1:3;
( (union S1) \ {xS1U} c= union S1 & (union S1) \ {xS1U} <> union S1 ) by A70, XBOOLE_1:36, ZFMISC_1:56;
then (union S1) \ {xS1U} c< union S1 by XBOOLE_0:def_8;
then A73: not Bu in conv (@ ((union S1) \ {xS1U})) by A64, RLAFFIN2:def_1;
card (union s1u) < n + 1 by A19, A69, CARD_2:48;
then A74: card (union s1u) <= n by NAT_1:13;
( (union S1) \ {xS2U} c= union S1 & (union S1) \ {xS2U} <> union S1 ) by A40, XBOOLE_1:36, ZFMISC_1:56;
then (union S1) \ {xS2U} c< union S1 by XBOOLE_0:def_8;
then A75: not Bu in conv (@ ((union S1) \ {xS2U})) by A64, RLAFFIN2:def_1;
A76: {(union S1)} c= S2 by A11, A18, A19, ZFMISC_1:2, ZFMISC_1:31;
A77: S2 \ {(union S1)} c= S2 by XBOOLE_1:36;
A78: A2 \ {Bu} = (((center_of_mass V) | the topology of Kas) .: S2) \ ((center_of_mass V) .: {(union S1)}) by A66, A8, A51, RELAT_1:129
.= (((center_of_mass V) | the topology of Kas) .: S2) \ (((center_of_mass V) | the topology of Kas) .: {(union S1)}) by A76, A66, RELAT_1:129, XBOOLE_1:1
.= ((center_of_mass V) | the topology of Kas) .: (S2 \ {(union S1)}) by FUNCT_1:64
.= (center_of_mass V) .: (S2 \ {(union S1)}) by A66, A77, RELAT_1:129, XBOOLE_1:1 ;
then conv ((@ A2) \ {Bu}) c= conv (union S2U) by CONVEX1:30, RLAFFIN2:17;
then A79: p2 in conv (union S2U) by A61;
x in conv (@ (union S1)) by A7, A17, A28, RLAFFIN2:def_1;
then p2 = p1 by A15, A45, A49, A58, A62, A65, A42, A53, A75, A72, A73, A79, RLAFFIN2:2;
then A80: Int ((@ A1) \ {Bu}) meets Int ((@ A2) \ {Bu}) by A50, A63, XBOOLE_0:3;
( (@ A1) \ {Bu} = @ (A1 \ {Bu}) & (@ A2) \ {Bu} = @ (A2 \ {Bu}) ) ;
then A1 \ {Bu} = A2 \ {Bu} by A6, A54, A52, A74, A78, A80;
hence A1 = (A2 \ {Bu}) \/ {Bu} by A22, ZFMISC_1:116
.= A2 by A24, ZFMISC_1:116 ;
::_thesis: verum
end;
end;
end;
end;
end;
A81: S1[ 0 ]
proof
let S1, S2 be c=-linear finite simplex-like Subset-Family of Kas; ::_thesis: for A1, A2 being Simplex of (BCS Kas) st A1 = (center_of_mass V) .: S1 & A2 = (center_of_mass V) .: S2 & card (union S1) <= 0 & card (union S2) <= 0 & Int (@ A1) meets Int (@ A2) holds
A1 = A2
let A1, A2 be Simplex of (BCS Kas); ::_thesis: ( A1 = (center_of_mass V) .: S1 & A2 = (center_of_mass V) .: S2 & card (union S1) <= 0 & card (union S2) <= 0 & Int (@ A1) meets Int (@ A2) implies A1 = A2 )
assume that
A82: A1 = (center_of_mass V) .: S1 and
A2 = (center_of_mass V) .: S2 and
A83: card (union S1) <= 0 and
card (union S2) <= 0 and
A84: Int (@ A1) meets Int (@ A2) ; ::_thesis: A1 = A2
not Int (@ A1) is empty by A84, XBOOLE_1:65;
then not A1 is empty ;
then consider y being set such that
A85: y in A1 by XBOOLE_0:def_1;
consider x being set such that
A86: x in dom (center_of_mass V) and
A87: x in S1 and
(center_of_mass V) . x = y by A82, A85, FUNCT_1:def_6;
A88: x <> {} by A86, ZFMISC_1:56;
union S1 is empty by A83;
then x c= {} by A87, ZFMISC_1:74;
hence A1 = A2 by A88; ::_thesis: verum
end;
A89: for n being Nat holds S1[n] from NAT_1:sch_2(A81, A5);
now__::_thesis:_for_A1,_A2_being_Subset_of_(BCS_Kas)_st_A1_is_simplex-like_&_A2_is_simplex-like_&_Int_(@_A1)_meets_Int_(@_A2)_holds_
A1_=_A2
let A1, A2 be Subset of (BCS Kas); ::_thesis: ( A1 is simplex-like & A2 is simplex-like & Int (@ A1) meets Int (@ A2) implies A1 = A2 )
assume that
A90: A1 is simplex-like and
A91: A2 is simplex-like and
A92: Int (@ A1) meets Int (@ A2) ; ::_thesis: A1 = A2
consider S1 being c=-linear finite simplex-like Subset-Family of Kas such that
A93: A1 = (center_of_mass V) .: S1 by A2, A90, SIMPLEX0:def_20;
consider S2 being c=-linear finite simplex-like Subset-Family of Kas such that
A94: A2 = (center_of_mass V) .: S2 by A2, A91, SIMPLEX0:def_20;
( card (union S1) <= card (union S2) or card (union S2) <= card (union S1) ) ;
hence A1 = A2 by A89, A90, A91, A92, A93, A94; ::_thesis: verum
end;
hence BCS Kas is simplex-join-closed by Th25; ::_thesis: verum
end;
registration
let V be RealLinearSpace;
let K be non void total affinely-independent simplex-join-closed SimplicialComplex of V;
cluster BCS K -> non void simplex-join-closed ;
coherence
BCS K is simplex-join-closed
proof
[#] K = the carrier of V by SIMPLEX0:def_10;
then |.K.| c= [#] K ;
hence BCS K is simplex-join-closed by Th29; ::_thesis: verum
end;
let n be Nat;
cluster BCS (n,K) -> non void simplex-join-closed ;
coherence
BCS (n,K) is simplex-join-closed
proof
defpred S1[ Nat] means ( BCS (V,K) is simplex-join-closed & BCS (V,K) is affinely-independent );
[#] K = [#] V by SIMPLEX0:def_10;
then A1: |.K.| c= [#] K ;
A2: for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A3: S1[n] ; ::_thesis: S1[n + 1]
BCS ((n + 1),K) = BCS (BCS (n,K)) by A1, Th20;
hence S1[n + 1] by A3; ::_thesis: verum
end;
A4: S1[ 0 ] by A1, Th16;
for n being Nat holds S1[n] from NAT_1:sch_2(A4, A2);
hence BCS (n,K) is simplex-join-closed ; ::_thesis: verum
end;
end;
theorem Th30: :: SIMPLEX1:30
for V being RealLinearSpace
for Kv being non void SimplicialComplex of V st |.Kv.| c= [#] Kv & ( for n being Nat st n <= degree Kv holds
ex S being Simplex of Kv st
( card S = n + 1 & @ S is affinely-independent ) ) holds
degree Kv = degree (BCS Kv)
proof
let V be RealLinearSpace; ::_thesis: for Kv being non void SimplicialComplex of V st |.Kv.| c= [#] Kv & ( for n being Nat st n <= degree Kv holds
ex S being Simplex of Kv st
( card S = n + 1 & @ S is affinely-independent ) ) holds
degree Kv = degree (BCS Kv)
let Kv be non void SimplicialComplex of V; ::_thesis: ( |.Kv.| c= [#] Kv & ( for n being Nat st n <= degree Kv holds
ex S being Simplex of Kv st
( card S = n + 1 & @ S is affinely-independent ) ) implies degree Kv = degree (BCS Kv) )
assume that
A1: |.Kv.| c= [#] Kv and
A2: for n being Nat st n <= degree Kv holds
ex S being Simplex of Kv st
( card S = n + 1 & @ S is affinely-independent ) ; ::_thesis: degree Kv = degree (BCS Kv)
A3: dom (center_of_mass V) = (bool the carrier of V) \ {{}} by FUNCT_2:def_1;
A4: for n being Nat st n <= degree Kv holds
ex S being Subset of Kv st
( S is simplex-like & card S = n + 1 & BOOL S c= dom (center_of_mass V) & (center_of_mass V) .: (BOOL S) is Subset of Kv & (center_of_mass V) | (BOOL S) is one-to-one )
proof
let n be Nat; ::_thesis: ( n <= degree Kv implies ex S being Subset of Kv st
( S is simplex-like & card S = n + 1 & BOOL S c= dom (center_of_mass V) & (center_of_mass V) .: (BOOL S) is Subset of Kv & (center_of_mass V) | (BOOL S) is one-to-one ) )
assume n <= degree Kv ; ::_thesis: ex S being Subset of Kv st
( S is simplex-like & card S = n + 1 & BOOL S c= dom (center_of_mass V) & (center_of_mass V) .: (BOOL S) is Subset of Kv & (center_of_mass V) | (BOOL S) is one-to-one )
then consider S being Simplex of Kv such that
A5: card S = n + 1 and
A6: @ S is affinely-independent by A2;
take S ; ::_thesis: ( S is simplex-like & card S = n + 1 & BOOL S c= dom (center_of_mass V) & (center_of_mass V) .: (BOOL S) is Subset of Kv & (center_of_mass V) | (BOOL S) is one-to-one )
thus ( S is simplex-like & card S = n + 1 ) by A5; ::_thesis: ( BOOL S c= dom (center_of_mass V) & (center_of_mass V) .: (BOOL S) is Subset of Kv & (center_of_mass V) | (BOOL S) is one-to-one )
A7: the topology of (Complex_of {(@ S)}) = bool S by SIMPLEX0:4;
reconsider SS = {(@ S)} as affinely-independent Subset-Family of V by A6;
A8: (center_of_mass V) .: (BOOL S) c= conv (@ S)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in (center_of_mass V) .: (BOOL S) or y in conv (@ S) )
assume y in (center_of_mass V) .: (BOOL S) ; ::_thesis: y in conv (@ S)
then consider x being set such that
A9: x in dom (center_of_mass V) and
A10: ( x in BOOL S & (center_of_mass V) . x = y ) by FUNCT_1:def_6;
reconsider x = x as non empty Subset of V by A9, ZFMISC_1:56;
( conv x c= conv (@ S) & y in conv x ) by A10, RLAFFIN2:16, RLTOPSP1:20;
hence y in conv (@ S) ; ::_thesis: verum
end;
bool (@ S) c= bool the carrier of V by ZFMISC_1:67;
hence BOOL S c= dom (center_of_mass V) by A3, XBOOLE_1:33; ::_thesis: ( (center_of_mass V) .: (BOOL S) is Subset of Kv & (center_of_mass V) | (BOOL S) is one-to-one )
conv (@ S) c= |.Kv.| by Th5;
then conv (@ S) c= [#] Kv by A1, XBOOLE_1:1;
hence (center_of_mass V) .: (BOOL S) is Subset of Kv by A8, XBOOLE_1:1; ::_thesis: (center_of_mass V) | (BOOL S) is one-to-one
( ((center_of_mass V) | (bool S)) | (BOOL S) = (center_of_mass V) | (BOOL S) & Complex_of SS is SimplicialComplex of V ) by RELAT_1:74;
hence (center_of_mass V) | (BOOL S) is one-to-one by A6, A7, FUNCT_1:52; ::_thesis: verum
end;
not {} in dom (center_of_mass V) by ZFMISC_1:56;
then A11: dom (center_of_mass V) is with_non-empty_elements by SETFAM_1:def_8;
BCS Kv = subdivision ((center_of_mass V),Kv) by A1, Def5;
hence degree Kv = degree (BCS Kv) by A4, A11, SIMPLEX0:53; ::_thesis: verum
end;
theorem Th31: :: SIMPLEX1:31
for V being RealLinearSpace
for Ka being non void affinely-independent SimplicialComplex of V st |.Ka.| c= [#] Ka holds
degree Ka = degree (BCS Ka)
proof
let V be RealLinearSpace; ::_thesis: for Ka being non void affinely-independent SimplicialComplex of V st |.Ka.| c= [#] Ka holds
degree Ka = degree (BCS Ka)
let Ka be non void affinely-independent SimplicialComplex of V; ::_thesis: ( |.Ka.| c= [#] Ka implies degree Ka = degree (BCS Ka) )
A1: for n being Nat st n <= degree Ka holds
ex S being Simplex of Ka st
( card S = n + 1 & @ S is affinely-independent )
proof
let n be Nat; ::_thesis: ( n <= degree Ka implies ex S being Simplex of Ka st
( card S = n + 1 & @ S is affinely-independent ) )
reconsider N = n as ext-real number ;
set S = the Simplex of n,Ka;
assume n <= degree Ka ; ::_thesis: ex S being Simplex of Ka st
( card S = n + 1 & @ S is affinely-independent )
then A2: card the Simplex of n,Ka = N + 1 by SIMPLEX0:def_18;
( N + 1 = n + 1 & @ the Simplex of n,Ka is affinely-independent ) by XXREAL_3:def_2;
hence ex S being Simplex of Ka st
( card S = n + 1 & @ S is affinely-independent ) by A2; ::_thesis: verum
end;
assume |.Ka.| c= [#] Ka ; ::_thesis: degree Ka = degree (BCS Ka)
hence degree Ka = degree (BCS Ka) by A1, Th30; ::_thesis: verum
end;
theorem Th32: :: SIMPLEX1:32
for n being Nat
for V being RealLinearSpace
for Ka being non void affinely-independent SimplicialComplex of V st |.Ka.| c= [#] Ka holds
degree Ka = degree (BCS (n,Ka))
proof
let n be Nat; ::_thesis: for V being RealLinearSpace
for Ka being non void affinely-independent SimplicialComplex of V st |.Ka.| c= [#] Ka holds
degree Ka = degree (BCS (n,Ka))
let V be RealLinearSpace; ::_thesis: for Ka being non void affinely-independent SimplicialComplex of V st |.Ka.| c= [#] Ka holds
degree Ka = degree (BCS (n,Ka))
let Ka be non void affinely-independent SimplicialComplex of V; ::_thesis: ( |.Ka.| c= [#] Ka implies degree Ka = degree (BCS (n,Ka)) )
defpred S1[ Nat] means ( degree Ka = degree (BCS ($1,Ka)) & not BCS ($1,Ka) is void & BCS ($1,Ka) is affinely-independent );
assume A1: |.Ka.| c= [#] Ka ; ::_thesis: degree Ka = degree (BCS (n,Ka))
A2: for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A3: S1[n] ; ::_thesis: S1[n + 1]
A4: [#] (BCS (n,Ka)) = [#] Ka by A1, Th18;
( BCS ((n + 1),Ka) = BCS (BCS (n,Ka)) & |.(BCS (n,Ka)).| = |.Ka.| ) by A1, Th10, Th20;
hence S1[n + 1] by A1, A3, A4, Th28, Th31; ::_thesis: verum
end;
A5: S1[ 0 ] by A1, Th16;
for n being Nat holds S1[n] from NAT_1:sch_2(A5, A2);
hence degree Ka = degree (BCS (n,Ka)) ; ::_thesis: verum
end;
theorem Th33: :: SIMPLEX1:33
for V being RealLinearSpace
for Kas being non void affinely-independent simplex-join-closed SimplicialComplex of V
for S being simplex-like Subset-Family of Kas st S is with_non-empty_elements holds
card S = card ((center_of_mass V) .: S)
proof
let V be RealLinearSpace; ::_thesis: for Kas being non void affinely-independent simplex-join-closed SimplicialComplex of V
for S being simplex-like Subset-Family of Kas st S is with_non-empty_elements holds
card S = card ((center_of_mass V) .: S)
let Kas be non void affinely-independent simplex-join-closed SimplicialComplex of V; ::_thesis: for S being simplex-like Subset-Family of Kas st S is with_non-empty_elements holds
card S = card ((center_of_mass V) .: S)
set B = center_of_mass V;
set T = the topology of Kas;
let S be simplex-like Subset-Family of Kas; ::_thesis: ( S is with_non-empty_elements implies card S = card ((center_of_mass V) .: S) )
assume A1: S is with_non-empty_elements ; ::_thesis: card S = card ((center_of_mass V) .: S)
A2: not {} in S by A1;
[#] Kas c= the carrier of V by SIMPLEX0:def_9;
then bool the carrier of Kas c= bool the carrier of V by ZFMISC_1:67;
then ( dom (center_of_mass V) = (bool the carrier of V) \ {{}} & S c= bool the carrier of V ) by FUNCT_2:def_1, XBOOLE_1:1;
then A3: dom ((center_of_mass V) | S) = S by A2, RELAT_1:62, ZFMISC_1:34;
S c= the topology of Kas
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in S or x in the topology of Kas )
assume x in S ; ::_thesis: x in the topology of Kas
then x is Simplex of Kas by TOPS_2:def_1;
hence x in the topology of Kas by PRE_TOPC:def_2; ::_thesis: verum
end;
then ((center_of_mass V) | the topology of Kas) | S = (center_of_mass V) | S by RELAT_1:74;
then A4: (center_of_mass V) | S is one-to-one by FUNCT_1:52;
(center_of_mass V) .: S = rng ((center_of_mass V) | S) by RELAT_1:115;
then S,(center_of_mass V) .: S are_equipotent by A3, A4, WELLORD2:def_4;
hence card S = card ((center_of_mass V) .: S) by CARD_1:5; ::_thesis: verum
end;
theorem Th34: :: SIMPLEX1:34
for V being RealLinearSpace
for Kas being non void affinely-independent simplex-join-closed SimplicialComplex of V
for S1, S2 being simplex-like Subset-Family of Kas st |.Kas.| c= [#] Kas & S1 is with_non-empty_elements & (center_of_mass V) .: S2 is Simplex of (BCS Kas) & (center_of_mass V) .: S1 c= (center_of_mass V) .: S2 holds
( S1 c= S2 & S2 is c=-linear )
proof
let V be RealLinearSpace; ::_thesis: for Kas being non void affinely-independent simplex-join-closed SimplicialComplex of V
for S1, S2 being simplex-like Subset-Family of Kas st |.Kas.| c= [#] Kas & S1 is with_non-empty_elements & (center_of_mass V) .: S2 is Simplex of (BCS Kas) & (center_of_mass V) .: S1 c= (center_of_mass V) .: S2 holds
( S1 c= S2 & S2 is c=-linear )
let Kas be non void affinely-independent simplex-join-closed SimplicialComplex of V; ::_thesis: for S1, S2 being simplex-like Subset-Family of Kas st |.Kas.| c= [#] Kas & S1 is with_non-empty_elements & (center_of_mass V) .: S2 is Simplex of (BCS Kas) & (center_of_mass V) .: S1 c= (center_of_mass V) .: S2 holds
( S1 c= S2 & S2 is c=-linear )
set B = center_of_mass V;
set BK = BCS Kas;
let S1, S2 be simplex-like Subset-Family of Kas; ::_thesis: ( |.Kas.| c= [#] Kas & S1 is with_non-empty_elements & (center_of_mass V) .: S2 is Simplex of (BCS Kas) & (center_of_mass V) .: S1 c= (center_of_mass V) .: S2 implies ( S1 c= S2 & S2 is c=-linear ) )
assume that
A1: |.Kas.| c= [#] Kas and
A2: S1 is with_non-empty_elements and
A3: (center_of_mass V) .: S2 is Simplex of (BCS Kas) and
A4: (center_of_mass V) .: S1 c= (center_of_mass V) .: S2 ; ::_thesis: ( S1 c= S2 & S2 is c=-linear )
BCS Kas = subdivision ((center_of_mass V),Kas) by A1, Def5;
then consider W2 being c=-linear finite simplex-like Subset-Family of Kas such that
A5: (center_of_mass V) .: S2 = (center_of_mass V) .: W2 by A3, SIMPLEX0:def_20;
reconsider s2 = S2 \ {{}} as simplex-like Subset-Family of Kas by TOPS_2:11, XBOOLE_1:36;
set TK = the topology of Kas;
set BTK = (center_of_mass V) | the topology of Kas;
A6: dom ((center_of_mass V) | the topology of Kas) = (dom (center_of_mass V)) /\ the topology of Kas by RELAT_1:61;
A7: s2 c= the topology of Kas by SIMPLEX0:14;
A8: dom (center_of_mass V) = (bool the carrier of V) \ {{}} by FUNCT_2:def_1;
then (@ S2) \ {{}} c= dom (center_of_mass V) by XBOOLE_1:33;
then s2 c= (dom (center_of_mass V)) /\ the topology of Kas by A7, XBOOLE_1:19;
then A9: s2 c= dom ((center_of_mass V) | the topology of Kas) by RELAT_1:61;
W2 /\ (dom (center_of_mass V)) c= W2 by XBOOLE_1:17;
then reconsider w2 = W2 /\ (dom (center_of_mass V)) as c=-linear finite simplex-like Subset-Family of Kas by TOPS_2:11, XBOOLE_1:1;
A10: w2 c= the topology of Kas by SIMPLEX0:14;
then A11: ( (center_of_mass V) .: W2 = (center_of_mass V) .: (W2 /\ (dom (center_of_mass V))) & (center_of_mass V) .: w2 = ((center_of_mass V) | the topology of Kas) .: w2 ) by RELAT_1:112, RELAT_1:129;
W2 /\ (dom (center_of_mass V)) c= dom (center_of_mass V) by XBOOLE_1:17;
then A12: w2 c= dom ((center_of_mass V) | the topology of Kas) by A6, A10, XBOOLE_1:19;
S2 c= the topology of Kas by SIMPLEX0:14;
then (center_of_mass V) .: S2 = ((center_of_mass V) | the topology of Kas) .: S2 by RELAT_1:129;
then A13: w2 c= S2 by A5, A11, A12, FUNCT_1:87;
A14: S1 c= the topology of Kas by SIMPLEX0:14;
S2 /\ (dom (center_of_mass V)) = ((@ S2) /\ (bool the carrier of V)) \ {{}} by A8, XBOOLE_1:49
.= s2 by XBOOLE_1:28 ;
then A15: (center_of_mass V) .: S2 = (center_of_mass V) .: s2 by RELAT_1:112;
then ((center_of_mass V) | the topology of Kas) .: s2 = (center_of_mass V) .: S2 by A7, RELAT_1:129;
then A16: s2 c= w2 by A5, A11, A9, FUNCT_1:87;
( @ S1 c= bool the carrier of V & not {} in S1 ) by A2;
then S1 c= dom (center_of_mass V) by A8, ZFMISC_1:34;
then A17: S1 c= dom ((center_of_mass V) | the topology of Kas) by A6, A14, XBOOLE_1:19;
(center_of_mass V) .: S1 = ((center_of_mass V) | the topology of Kas) .: S1 by A14, RELAT_1:129;
then S1 c= w2 by A4, A5, A11, A17, FUNCT_1:87;
hence S1 c= S2 by A13, XBOOLE_1:1; ::_thesis: S2 is c=-linear
let x be set ; :: according to ORDINAL1:def_8 ::_thesis: for b1 being set holds
( not x in S2 or not b1 in S2 or x,b1 are_c=-comparable )
let y be set ; ::_thesis: ( not x in S2 or not y in S2 or x,y are_c=-comparable )
assume A18: ( x in S2 & y in S2 ) ; ::_thesis: x,y are_c=-comparable
(center_of_mass V) .: s2 = ((center_of_mass V) | the topology of Kas) .: s2 by A7, RELAT_1:129;
then w2 c= s2 by A5, A11, A12, A15, FUNCT_1:87;
then A19: s2 = w2 by A16, XBOOLE_0:def_10;
percases ( x is empty or y is empty or ( not x is empty & not y is empty ) ) ;
suppose ( x is empty or y is empty ) ; ::_thesis: x,y are_c=-comparable
then ( x c= y or y c= x ) by XBOOLE_1:2;
hence x,y are_c=-comparable by XBOOLE_0:def_9; ::_thesis: verum
end;
suppose ( not x is empty & not y is empty ) ; ::_thesis: x,y are_c=-comparable
then ( x in w2 & y in w2 ) by A18, A19, ZFMISC_1:56;
hence x,y are_c=-comparable by ORDINAL1:def_8; ::_thesis: verum
end;
end;
end;
theorem Th35: :: SIMPLEX1:35
for n being Nat
for V being RealLinearSpace
for Aff being finite affinely-independent Subset of V
for Bf being finite Subset of V
for S being finite Subset-Family of V st S is with_non-empty_elements & union S c= Aff & ((card S) + n) + 1 <= card Aff holds
( ( Bf is Simplex of n + (card S), BCS (Complex_of {Aff}) & (center_of_mass V) .: S c= Bf ) iff ex T being finite Subset-Family of V st
( T misses S & T \/ S is c=-linear & T \/ S is with_non-empty_elements & card T = n + 1 & union T c= Aff & Bf = ((center_of_mass V) .: S) \/ ((center_of_mass V) .: T) ) )
proof
let n be Nat; ::_thesis: for V being RealLinearSpace
for Aff being finite affinely-independent Subset of V
for Bf being finite Subset of V
for S being finite Subset-Family of V st S is with_non-empty_elements & union S c= Aff & ((card S) + n) + 1 <= card Aff holds
( ( Bf is Simplex of n + (card S), BCS (Complex_of {Aff}) & (center_of_mass V) .: S c= Bf ) iff ex T being finite Subset-Family of V st
( T misses S & T \/ S is c=-linear & T \/ S is with_non-empty_elements & card T = n + 1 & union T c= Aff & Bf = ((center_of_mass V) .: S) \/ ((center_of_mass V) .: T) ) )
let V be RealLinearSpace; ::_thesis: for Aff being finite affinely-independent Subset of V
for Bf being finite Subset of V
for S being finite Subset-Family of V st S is with_non-empty_elements & union S c= Aff & ((card S) + n) + 1 <= card Aff holds
( ( Bf is Simplex of n + (card S), BCS (Complex_of {Aff}) & (center_of_mass V) .: S c= Bf ) iff ex T being finite Subset-Family of V st
( T misses S & T \/ S is c=-linear & T \/ S is with_non-empty_elements & card T = n + 1 & union T c= Aff & Bf = ((center_of_mass V) .: S) \/ ((center_of_mass V) .: T) ) )
let Aff be finite affinely-independent Subset of V; ::_thesis: for Bf being finite Subset of V
for S being finite Subset-Family of V st S is with_non-empty_elements & union S c= Aff & ((card S) + n) + 1 <= card Aff holds
( ( Bf is Simplex of n + (card S), BCS (Complex_of {Aff}) & (center_of_mass V) .: S c= Bf ) iff ex T being finite Subset-Family of V st
( T misses S & T \/ S is c=-linear & T \/ S is with_non-empty_elements & card T = n + 1 & union T c= Aff & Bf = ((center_of_mass V) .: S) \/ ((center_of_mass V) .: T) ) )
let Bf be finite Subset of V; ::_thesis: for S being finite Subset-Family of V st S is with_non-empty_elements & union S c= Aff & ((card S) + n) + 1 <= card Aff holds
( ( Bf is Simplex of n + (card S), BCS (Complex_of {Aff}) & (center_of_mass V) .: S c= Bf ) iff ex T being finite Subset-Family of V st
( T misses S & T \/ S is c=-linear & T \/ S is with_non-empty_elements & card T = n + 1 & union T c= Aff & Bf = ((center_of_mass V) .: S) \/ ((center_of_mass V) .: T) ) )
let S be finite Subset-Family of V; ::_thesis: ( S is with_non-empty_elements & union S c= Aff & ((card S) + n) + 1 <= card Aff implies ( ( Bf is Simplex of n + (card S), BCS (Complex_of {Aff}) & (center_of_mass V) .: S c= Bf ) iff ex T being finite Subset-Family of V st
( T misses S & T \/ S is c=-linear & T \/ S is with_non-empty_elements & card T = n + 1 & union T c= Aff & Bf = ((center_of_mass V) .: S) \/ ((center_of_mass V) .: T) ) ) )
set B = center_of_mass V;
set U = union S;
assume that
A1: S is with_non-empty_elements and
A2: union S c= Aff and
A3: ((card S) + n) + 1 <= card Aff ; ::_thesis: ( ( Bf is Simplex of n + (card S), BCS (Complex_of {Aff}) & (center_of_mass V) .: S c= Bf ) iff ex T being finite Subset-Family of V st
( T misses S & T \/ S is c=-linear & T \/ S is with_non-empty_elements & card T = n + 1 & union T c= Aff & Bf = ((center_of_mass V) .: S) \/ ((center_of_mass V) .: T) ) )
set C = Complex_of {Aff};
reconsider c = card Aff as ext-real number ;
set BTC = (center_of_mass V) | the topology of (Complex_of {Aff});
set BC = BCS (Complex_of {Aff});
A4: the topology of (Complex_of {Aff}) = bool Aff by SIMPLEX0:4;
A5: degree (Complex_of {Aff}) = c - 1 by SIMPLEX0:26
.= (card Aff) + (- 1) by XXREAL_3:def_2 ;
reconsider c = (card S) + n as ext-real number ;
A6: |.(Complex_of {Aff}).| c= [#] (Complex_of {Aff}) ;
then A7: BCS (Complex_of {Aff}) = subdivision ((center_of_mass V),(Complex_of {Aff})) by Def5;
(card S) + n <= (card Aff) - 1 by A3, XREAL_1:19;
then A8: (card S) + n <= degree (BCS (Complex_of {Aff})) by A5, A6, Th31;
hereby ::_thesis: ( ex T being finite Subset-Family of V st
( T misses S & T \/ S is c=-linear & T \/ S is with_non-empty_elements & card T = n + 1 & union T c= Aff & Bf = ((center_of_mass V) .: S) \/ ((center_of_mass V) .: T) ) implies ( Bf is Simplex of n + (card S), BCS (Complex_of {Aff}) & (center_of_mass V) .: S c= Bf ) )
A9: S c= the topology of (Complex_of {Aff})
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in S or x in the topology of (Complex_of {Aff}) )
assume x in S ; ::_thesis: x in the topology of (Complex_of {Aff})
then x c= union S by ZFMISC_1:74;
then x c= Aff by A2, XBOOLE_1:1;
hence x in the topology of (Complex_of {Aff}) by A4; ::_thesis: verum
end;
then A10: (center_of_mass V) .: S = ((center_of_mass V) | the topology of (Complex_of {Aff})) .: S by RELAT_1:129;
( dom (center_of_mass V) = (bool the carrier of V) \ {{}} & not {} in S ) by A1, FUNCT_2:def_1;
then ( dom ((center_of_mass V) | the topology of (Complex_of {Aff})) = (dom (center_of_mass V)) /\ the topology of (Complex_of {Aff}) & S c= dom (center_of_mass V) ) by RELAT_1:61, ZFMISC_1:34;
then A11: S c= dom ((center_of_mass V) | the topology of (Complex_of {Aff})) by A9, XBOOLE_1:19;
assume that
A12: Bf is Simplex of n + (card S), BCS (Complex_of {Aff}) and
A13: (center_of_mass V) .: S c= Bf ; ::_thesis: ex T being finite Subset-Family of V st
( T misses S & T \/ S is c=-linear & T \/ S is with_non-empty_elements & card T = n + 1 & union T c= Aff & Bf = ((center_of_mass V) .: S) \/ ((center_of_mass V) .: T) )
consider a being c=-linear finite simplex-like Subset-Family of (Complex_of {Aff}) such that
A14: Bf = (center_of_mass V) .: a by A7, A12, SIMPLEX0:def_20;
a /\ (dom (center_of_mass V)) c= a by XBOOLE_1:17;
then reconsider AA = a /\ (dom (center_of_mass V)) as c=-linear finite simplex-like Subset-Family of (Complex_of {Aff}) by TOPS_2:11, XBOOLE_1:1;
A15: (center_of_mass V) .: S c= (center_of_mass V) .: AA by A13, A14, RELAT_1:112;
reconsider T = AA \ S as Subset-Family of V ;
A16: AA c= the topology of (Complex_of {Aff}) by SIMPLEX0:14;
then A17: (center_of_mass V) .: AA = ((center_of_mass V) | the topology of (Complex_of {Aff})) .: AA by RELAT_1:129;
A18: S \/ T = AA \/ S by XBOOLE_1:39
.= AA by A10, A11, A15, A17, FUNCT_1:87, XBOOLE_1:12 ;
T c= AA by XBOOLE_1:36;
then A19: T c= bool Aff by A4, A16, XBOOLE_1:1;
A20: not {} in AA by ZFMISC_1:56;
then ( (center_of_mass V) .: a = (center_of_mass V) .: (a /\ (dom (center_of_mass V))) & AA is with_non-empty_elements ) by RELAT_1:112, SETFAM_1:def_8;
then A21: card Bf = card AA by A14, Th33;
A22: Bf = (center_of_mass V) .: AA by A14, RELAT_1:112
.= ((center_of_mass V) .: S) \/ ((center_of_mass V) .: T) by A18, RELAT_1:120 ;
reconsider T = T as finite Subset-Family of V ;
take T = T; ::_thesis: ( T misses S & T \/ S is c=-linear & T \/ S is with_non-empty_elements & card T = n + 1 & union T c= Aff & Bf = ((center_of_mass V) .: S) \/ ((center_of_mass V) .: T) )
card Bf = c + 1 by A8, A12, SIMPLEX0:def_18
.= ((card S) + n) + 1 by XXREAL_3:def_2 ;
then ( union (bool Aff) = Aff & (card S) + (card (AA \ S)) = ((card S) + n) + 1 ) by A18, A21, CARD_2:40, XBOOLE_1:79, ZFMISC_1:81;
hence ( T misses S & T \/ S is c=-linear & T \/ S is with_non-empty_elements & card T = n + 1 & union T c= Aff & Bf = ((center_of_mass V) .: S) \/ ((center_of_mass V) .: T) ) by A18, A19, A20, A22, SETFAM_1:def_8, XBOOLE_1:79, ZFMISC_1:77; ::_thesis: verum
end;
given T being finite Subset-Family of V such that A23: T misses S and
A24: ( T \/ S is c=-linear & T \/ S is with_non-empty_elements ) and
A25: card T = n + 1 and
A26: union T c= Aff and
A27: Bf = ((center_of_mass V) .: S) \/ ((center_of_mass V) .: T) ; ::_thesis: ( Bf is Simplex of n + (card S), BCS (Complex_of {Aff}) & (center_of_mass V) .: S c= Bf )
reconsider TS = T \/ S as Subset-Family of (Complex_of {Aff}) ;
reconsider t = T as finite Subset-Family of V ;
A28: card TS = (card t) + (card S) by A23, CARD_2:40
.= ((card S) + n) + 1 by A25 ;
union (T \/ S) = (union T) \/ (union S) by ZFMISC_1:78;
then union (T \/ S) c= Aff by A2, A26, XBOOLE_1:8;
then ( T \/ S c= bool (union (T \/ S)) & bool (union (T \/ S)) c= bool Aff ) by ZFMISC_1:67, ZFMISC_1:82;
then A29: T \/ S c= the topology of (Complex_of {Aff}) by A4, XBOOLE_1:1;
A30: TS is simplex-like
proof
let a be Subset of (Complex_of {Aff}); :: according to TOPS_2:def_1 ::_thesis: ( not a in TS or not a is dependent )
thus ( not a in TS or not a is dependent ) by A29, PRE_TOPC:def_2; ::_thesis: verum
end;
[#] (BCS (Complex_of {Aff})) = [#] (Complex_of {Aff}) by A7, SIMPLEX0:def_20;
then reconsider BTS = (center_of_mass V) .: TS as Simplex of (BCS (Complex_of {Aff})) by A7, A24, A30, SIMPLEX0:def_20;
card TS = card ((center_of_mass V) .: TS) by A24, A30, Th33;
then A31: card BTS = c + 1 by A28, XXREAL_3:def_2;
BTS = Bf by A27, RELAT_1:120;
hence ( Bf is Simplex of n + (card S), BCS (Complex_of {Aff}) & (center_of_mass V) .: S c= Bf ) by A8, A27, A31, SIMPLEX0:def_18, XBOOLE_1:7; ::_thesis: verum
end;
theorem Th36: :: SIMPLEX1:36
for V being RealLinearSpace
for Aff being finite affinely-independent Subset of V
for Sf being c=-linear finite finite-membered Subset-Family of V st Sf is with_non-empty_elements & union Sf c= Aff holds
( (center_of_mass V) .: Sf is Simplex of (card (union Sf)) - 1, BCS (Complex_of {Aff}) iff for n being Nat st 0 < n & n <= card (union Sf) holds
ex x being set st
( x in Sf & card x = n ) )
proof
let V be RealLinearSpace; ::_thesis: for Aff being finite affinely-independent Subset of V
for Sf being c=-linear finite finite-membered Subset-Family of V st Sf is with_non-empty_elements & union Sf c= Aff holds
( (center_of_mass V) .: Sf is Simplex of (card (union Sf)) - 1, BCS (Complex_of {Aff}) iff for n being Nat st 0 < n & n <= card (union Sf) holds
ex x being set st
( x in Sf & card x = n ) )
let Aff be finite affinely-independent Subset of V; ::_thesis: for Sf being c=-linear finite finite-membered Subset-Family of V st Sf is with_non-empty_elements & union Sf c= Aff holds
( (center_of_mass V) .: Sf is Simplex of (card (union Sf)) - 1, BCS (Complex_of {Aff}) iff for n being Nat st 0 < n & n <= card (union Sf) holds
ex x being set st
( x in Sf & card x = n ) )
let Sf be c=-linear finite finite-membered Subset-Family of V; ::_thesis: ( Sf is with_non-empty_elements & union Sf c= Aff implies ( (center_of_mass V) .: Sf is Simplex of (card (union Sf)) - 1, BCS (Complex_of {Aff}) iff for n being Nat st 0 < n & n <= card (union Sf) holds
ex x being set st
( x in Sf & card x = n ) ) )
reconsider N = 0 as Nat ;
set U = union Sf;
assume that
A1: Sf is with_non-empty_elements and
A2: union Sf c= Aff ; ::_thesis: ( (center_of_mass V) .: Sf is Simplex of (card (union Sf)) - 1, BCS (Complex_of {Aff}) iff for n being Nat st 0 < n & n <= card (union Sf) holds
ex x being set st
( x in Sf & card x = n ) )
set B = center_of_mass V;
set C = Complex_of {Aff};
reconsider s = Sf as c=-linear finite Subset-Family of (Complex_of {Aff}) ;
A3: the topology of (Complex_of {Aff}) = bool Aff by SIMPLEX0:4;
card (union Sf) c= card Aff by A2, CARD_1:11;
then card (union Sf) <= card Aff by NAT_1:39;
then A4: ( N - 1 <= (card (union Sf)) - 1 & (card (union Sf)) - 1 <= (card Aff) - 1 ) by XREAL_1:9;
( Sf c= bool (union Sf) & bool (union Sf) c= bool Aff ) by A2, ZFMISC_1:67, ZFMISC_1:82;
then A5: s c= the topology of (Complex_of {Aff}) by A3, XBOOLE_1:1;
A6: s is simplex-like
proof
let a be Subset of (Complex_of {Aff}); :: according to TOPS_2:def_1 ::_thesis: ( not a in s or not a is dependent )
assume a in s ; ::_thesis: not a is dependent
hence not a is dependent by A5, PRE_TOPC:def_2; ::_thesis: verum
end;
then A7: card s = card ((center_of_mass V) .: Sf) by A1, Th33;
card Sf c= card (union Sf) by A1, SIMPLEX0:10;
then A8: card Sf <= card (union Sf) by NAT_1:39;
set BC = BCS (Complex_of {Aff});
reconsider c = card Aff as ext-real number ;
A9: degree (Complex_of {Aff}) = c - 1 by SIMPLEX0:26
.= (card Aff) + (- 1) by XXREAL_3:def_2 ;
A10: |.(Complex_of {Aff}).| c= [#] (Complex_of {Aff}) ;
then A11: BCS (Complex_of {Aff}) = subdivision ((center_of_mass V),(Complex_of {Aff})) by Def5;
then [#] (BCS (Complex_of {Aff})) = [#] (Complex_of {Aff}) by SIMPLEX0:def_20;
then reconsider BS = (center_of_mass V) .: Sf as Subset of (BCS (Complex_of {Aff})) ;
A12: N - 1 <= (card Aff) - 1 by XREAL_1:9;
A13: degree (BCS (Complex_of {Aff})) = degree (Complex_of {Aff}) by A10, Th31;
thus ( (center_of_mass V) .: Sf is Simplex of (card (union Sf)) - 1, BCS (Complex_of {Aff}) implies for n being Nat st 0 < n & n <= card (union Sf) holds
ex x being set st
( x in Sf & card x = n ) ) ::_thesis: ( ( for n being Nat st 0 < n & n <= card (union Sf) holds
ex x being set st
( x in Sf & card x = n ) ) implies (center_of_mass V) .: Sf is Simplex of (card (union Sf)) - 1, BCS (Complex_of {Aff}) )
proof
assume (center_of_mass V) .: Sf is Simplex of (card (union Sf)) - 1, BCS (Complex_of {Aff}) ; ::_thesis: for n being Nat st 0 < n & n <= card (union Sf) holds
ex x being set st
( x in Sf & card x = n )
then reconsider BS = (center_of_mass V) .: Sf as Simplex of (card (union Sf)) - 1, BCS (Complex_of {Aff}) ;
reconsider c1 = (card (union Sf)) - 1 as ext-real number ;
let n be Nat; ::_thesis: ( 0 < n & n <= card (union Sf) implies ex x being set st
( x in Sf & card x = n ) )
reconsider s = Sf as Subset-Family of (union Sf) by ZFMISC_1:82;
defpred S1[ Nat] means ( $1 < card Sf implies ex x being finite set st
( x in Sf & card x = (card Sf) - $1 ) );
assume that
A14: 0 < n and
A15: n <= card (union Sf) ; ::_thesis: ex x being set st
( x in Sf & card x = n )
A16: (card Sf) - 0 > (card Sf) - n by A14, XREAL_1:6;
A17: card BS = c1 + 1 by A4, A9, A13, SIMPLEX0:def_18
.= ((card (union Sf)) - 1) + 1 by XXREAL_3:def_2
.= card (union Sf) ;
then A18: not Sf is empty by A14, A15;
then consider s1 being Subset-Family of (union Sf) such that
A19: s c= s1 and
( s1 is with_non-empty_elements & s1 is c=-linear ) and
A20: card (union Sf) = card s1 and
A21: for Z being set st Z in s1 & card Z <> 1 holds
ex x being set st
( x in Z & Z \ {x} in s1 ) by A1, SIMPLEX0:9, SIMPLEX0:13;
card (union Sf) = card Sf by A1, A6, A17, Th33;
then A22: s = s1 by A19, A20, CARD_FIN:1;
A23: for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A24: S1[n] ; ::_thesis: S1[n + 1]
assume A25: n + 1 < card Sf ; ::_thesis: ex x being finite set st
( x in Sf & card x = (card Sf) - (n + 1) )
then consider X being finite set such that
A26: X in Sf and
A27: card X = (card Sf) - n by A24, NAT_1:13;
A28: (n + 1) - n < (card Sf) - n by A25, XREAL_1:9;
then consider x being set such that
A29: ( x in X & X \ {x} in Sf ) by A21, A22, A26, A27;
reconsider C = (card X) - 1 as Element of NAT by A27, A28, NAT_1:20;
take X \ {x} ; ::_thesis: ( X \ {x} in Sf & card (X \ {x}) = (card Sf) - (n + 1) )
card X = C + 1 ;
hence ( X \ {x} in Sf & card (X \ {x}) = (card Sf) - (n + 1) ) by A27, A29, STIRL2_1:55; ::_thesis: verum
end;
A30: S1[ 0 ] by A7, A17, A18, SIMPLEX0:9;
A31: for n being Nat holds S1[n] from NAT_1:sch_2(A30, A23);
(card Sf) - n is Nat by A7, A15, A17, NAT_1:21;
then ex x being finite set st
( x in Sf & card x = (card Sf) - ((card Sf) - n) ) by A16, A31;
hence ex x being set st
( x in Sf & card x = n ) ; ::_thesis: verum
end;
assume A32: for n being Nat st 0 < n & n <= card (union Sf) holds
ex x being set st
( x in Sf & card x = n ) ; ::_thesis: (center_of_mass V) .: Sf is Simplex of (card (union Sf)) - 1, BCS (Complex_of {Aff})
percases ( union Sf is empty or not union Sf is empty ) ;
supposeA33: union Sf is empty ; ::_thesis: (center_of_mass V) .: Sf is Simplex of (card (union Sf)) - 1, BCS (Complex_of {Aff})
reconsider O = - 1 as ext-real number ;
A34: ( O <= degree (BCS (Complex_of {Aff})) & 0 = O + 1 ) by A9, A10, A12, Th31, SIMPLEX0:def_12, XXREAL_3:7;
Sf is empty by A1, A33;
then A35: BS is empty ;
then card BS = 0 ;
hence (center_of_mass V) .: Sf is Simplex of (card (union Sf)) - 1, BCS (Complex_of {Aff}) by A33, A34, A35, SIMPLEX0:def_18; ::_thesis: verum
end;
supposeA36: not union Sf is empty ; ::_thesis: (center_of_mass V) .: Sf is Simplex of (card (union Sf)) - 1, BCS (Complex_of {Aff})
reconsider c1 = (card (union Sf)) - 1 as ext-real number ;
consider x being set such that
A37: x in Sf and
card x = card (union Sf) by A32, A36;
defpred S1[ set , set ] means card $2 = $1;
A38: for x being set st x in Seg (card (union Sf)) holds
ex y being set st
( y in Sf & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in Seg (card (union Sf)) implies ex y being set st
( y in Sf & S1[x,y] ) )
assume A39: x in Seg (card (union Sf)) ; ::_thesis: ex y being set st
( y in Sf & S1[x,y] )
reconsider n = x as Nat by A39;
( 0 < n & n <= card (union Sf) ) by A39, FINSEQ_1:1;
hence ex y being set st
( y in Sf & S1[x,y] ) by A32; ::_thesis: verum
end;
consider f being Function of (Seg (card (union Sf))),Sf such that
A40: for x being set st x in Seg (card (union Sf)) holds
S1[x,f . x] from FUNCT_2:sch_1(A38);
now__::_thesis:_for_x1,_x2_being_set_st_x1_in_dom_f_&_x2_in_dom_f_&_f_._x1_=_f_._x2_holds_
x1_=_x2
let x1, x2 be set ; ::_thesis: ( x1 in dom f & x2 in dom f & f . x1 = f . x2 implies x1 = x2 )
assume that
A41: x1 in dom f and
A42: ( x2 in dom f & f . x1 = f . x2 ) ; ::_thesis: x1 = x2
thus x1 = card (f . x1) by A40, A41
.= x2 by A40, A42 ; ::_thesis: verum
end;
then A43: ( rng f c= Sf & f is one-to-one ) by FUNCT_1:def_4;
dom f = Seg (card (union Sf)) by A37, FUNCT_2:def_1;
then ( card (Seg (card (union Sf))) = card (union Sf) & card (Seg (card (union Sf))) c= card Sf ) by A43, CARD_1:10, FINSEQ_1:57;
then card (union Sf) <= card Sf by NAT_1:39;
then A44: card BS = card (union Sf) by A7, A8, XXREAL_0:1;
( BS is Simplex of (BCS (Complex_of {Aff})) & c1 + 1 = ((card (union Sf)) - 1) + 1 ) by A6, A11, SIMPLEX0:def_20, XXREAL_3:def_2;
hence (center_of_mass V) .: Sf is Simplex of (card (union Sf)) - 1, BCS (Complex_of {Aff}) by A4, A9, A13, A44, SIMPLEX0:def_18; ::_thesis: verum
end;
end;
end;
Lm2: for V being RealLinearSpace
for S being finite finite-membered Subset-Family of V st S is c=-linear & S is with_non-empty_elements & card S = card (union S) holds
for A being non empty finite Subset of V st A misses union S & (union S) \/ A is affinely-independent holds
((center_of_mass V) .: S) \/ ((center_of_mass V) .: {((union S) \/ A)}) is Simplex of card S, BCS (Complex_of {((union S) \/ A)})
proof
let V be RealLinearSpace; ::_thesis: for S being finite finite-membered Subset-Family of V st S is c=-linear & S is with_non-empty_elements & card S = card (union S) holds
for A being non empty finite Subset of V st A misses union S & (union S) \/ A is affinely-independent holds
((center_of_mass V) .: S) \/ ((center_of_mass V) .: {((union S) \/ A)}) is Simplex of card S, BCS (Complex_of {((union S) \/ A)})
let S be finite finite-membered Subset-Family of V; ::_thesis: ( S is c=-linear & S is with_non-empty_elements & card S = card (union S) implies for A being non empty finite Subset of V st A misses union S & (union S) \/ A is affinely-independent holds
((center_of_mass V) .: S) \/ ((center_of_mass V) .: {((union S) \/ A)}) is Simplex of card S, BCS (Complex_of {((union S) \/ A)}) )
assume that
A1: S is c=-linear and
A2: S is with_non-empty_elements and
A3: card S = card (union S) ; ::_thesis: for A being non empty finite Subset of V st A misses union S & (union S) \/ A is affinely-independent holds
((center_of_mass V) .: S) \/ ((center_of_mass V) .: {((union S) \/ A)}) is Simplex of card S, BCS (Complex_of {((union S) \/ A)})
set U = union S;
set B = center_of_mass V;
let A be non empty finite Subset of V; ::_thesis: ( A misses union S & (union S) \/ A is affinely-independent implies ((center_of_mass V) .: S) \/ ((center_of_mass V) .: {((union S) \/ A)}) is Simplex of card S, BCS (Complex_of {((union S) \/ A)}) )
assume that
A4: A misses union S and
A5: (union S) \/ A is affinely-independent ; ::_thesis: ((center_of_mass V) .: S) \/ ((center_of_mass V) .: {((union S) \/ A)}) is Simplex of card S, BCS (Complex_of {((union S) \/ A)})
reconsider UA = (union S) \/ A as finite affinely-independent Subset of V by A5;
set C = Complex_of {UA};
reconsider SUA = S \/ {UA} as Subset-Family of (Complex_of {UA}) ;
A6: union S c= UA by XBOOLE_1:7;
A7: the topology of (Complex_of {UA}) = bool UA by SIMPLEX0:4;
A8: SUA c= the topology of (Complex_of {UA})
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in SUA or x in the topology of (Complex_of {UA}) )
assume x in SUA ; ::_thesis: x in the topology of (Complex_of {UA})
then ( x in S or x in {UA} ) by XBOOLE_0:def_3;
then ( x c= union S or x = UA ) by TARSKI:def_1, ZFMISC_1:74;
then x c= UA by A6, XBOOLE_1:1;
hence x in the topology of (Complex_of {UA}) by A7; ::_thesis: verum
end;
A9: SUA is simplex-like
proof
let A be Subset of (Complex_of {UA}); :: according to TOPS_2:def_1 ::_thesis: ( not A in SUA or not A is dependent )
assume A in SUA ; ::_thesis: not A is dependent
hence not A is dependent by A8, PRE_TOPC:def_2; ::_thesis: verum
end;
set BC = BCS (Complex_of {UA});
A10: |.(Complex_of {UA}).| c= [#] (Complex_of {UA}) ;
then A11: BCS (Complex_of {UA}) = subdivision ((center_of_mass V),(Complex_of {UA})) by Def5;
then [#] (BCS (Complex_of {UA})) = [#] (Complex_of {UA}) by SIMPLEX0:def_20;
then reconsider BSUA = (center_of_mass V) .: SUA as Subset of (BCS (Complex_of {UA})) ;
SUA is c=-linear
proof
let x, y be set ; :: according to ORDINAL1:def_8 ::_thesis: ( not x in SUA or not y in SUA or x,y are_c=-comparable )
assume A12: ( x in SUA & y in SUA ) ; ::_thesis: x,y are_c=-comparable
percases ( ( x in S & y in S ) or ( x in S & y in {UA} ) or ( x in {UA} & y in S ) or ( x in {UA} & y in {UA} ) ) by A12, XBOOLE_0:def_3;
suppose ( x in S & y in S ) ; ::_thesis: x,y are_c=-comparable
hence x,y are_c=-comparable by A1, ORDINAL1:def_8; ::_thesis: verum
end;
suppose ( x in S & y in {UA} ) ; ::_thesis: x,y are_c=-comparable
then ( x c= union S & y = UA ) by TARSKI:def_1, ZFMISC_1:74;
then x c= y by A6, XBOOLE_1:1;
hence x,y are_c=-comparable by XBOOLE_0:def_9; ::_thesis: verum
end;
suppose ( x in {UA} & y in S ) ; ::_thesis: x,y are_c=-comparable
then ( x = UA & y c= union S ) by TARSKI:def_1, ZFMISC_1:74;
then y c= x by A6, XBOOLE_1:1;
hence x,y are_c=-comparable by XBOOLE_0:def_9; ::_thesis: verum
end;
supposeA13: ( x in {UA} & y in {UA} ) ; ::_thesis: x,y are_c=-comparable
then x = UA by TARSKI:def_1;
hence x,y are_c=-comparable by A13, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
then reconsider BSUA = BSUA as Simplex of (BCS (Complex_of {UA})) by A9, A11, SIMPLEX0:def_20;
reconsider c = card UA as ext-real number ;
A14: degree (Complex_of {UA}) = c - 1 by SIMPLEX0:26
.= (card UA) + (- 1) by XXREAL_3:def_2 ;
A15: UA <> union S
proof
assume UA = union S ; ::_thesis: contradiction
then A c= union S by XBOOLE_1:7;
hence contradiction by A4, XBOOLE_1:67; ::_thesis: verum
end;
not UA in S
proof
assume UA in S ; ::_thesis: contradiction
then UA c= union S by ZFMISC_1:74;
hence contradiction by A6, A15, XBOOLE_0:def_10; ::_thesis: verum
end;
then A16: card SUA = (card S) + 1 by CARD_2:41;
union S c< UA by A6, A15, XBOOLE_0:def_8;
then card (union S) < card UA by CARD_2:48;
then (card (union S)) + 1 <= card UA by NAT_1:13;
then A17: card (union S) <= (card UA) - 1 by XREAL_1:19;
reconsider c = card S as ext-real number ;
card BSUA = card SUA by A2, A9, Th33;
then A18: card BSUA = c + 1 by A16, XXREAL_3:def_2;
degree (BCS (Complex_of {UA})) = degree (Complex_of {UA}) by A10, Th31;
then BSUA is Simplex of card S, BCS (Complex_of {UA}) by A3, A14, A17, A18, SIMPLEX0:def_18;
hence ((center_of_mass V) .: S) \/ ((center_of_mass V) .: {((union S) \/ A)}) is Simplex of card S, BCS (Complex_of {((union S) \/ A)}) by RELAT_1:120; ::_thesis: verum
end;
theorem Th37: :: SIMPLEX1:37
for V being RealLinearSpace
for S being finite Subset-Family of V st S is c=-linear & S is with_non-empty_elements & card S = card (union S) holds
for Af, Bf being finite Subset of V st not Af is empty & Af misses union S & (union S) \/ Af is affinely-independent & (union S) \/ Af c= Bf holds
((center_of_mass V) .: S) \/ ((center_of_mass V) .: {((union S) \/ Af)}) is Simplex of card S, BCS (Complex_of {Bf})
proof
let V be RealLinearSpace; ::_thesis: for S being finite Subset-Family of V st S is c=-linear & S is with_non-empty_elements & card S = card (union S) holds
for Af, Bf being finite Subset of V st not Af is empty & Af misses union S & (union S) \/ Af is affinely-independent & (union S) \/ Af c= Bf holds
((center_of_mass V) .: S) \/ ((center_of_mass V) .: {((union S) \/ Af)}) is Simplex of card S, BCS (Complex_of {Bf})
let S be finite Subset-Family of V; ::_thesis: ( S is c=-linear & S is with_non-empty_elements & card S = card (union S) implies for Af, Bf being finite Subset of V st not Af is empty & Af misses union S & (union S) \/ Af is affinely-independent & (union S) \/ Af c= Bf holds
((center_of_mass V) .: S) \/ ((center_of_mass V) .: {((union S) \/ Af)}) is Simplex of card S, BCS (Complex_of {Bf}) )
assume that
A1: ( S is c=-linear & S is with_non-empty_elements ) and
A2: card S = card (union S) ; ::_thesis: for Af, Bf being finite Subset of V st not Af is empty & Af misses union S & (union S) \/ Af is affinely-independent & (union S) \/ Af c= Bf holds
((center_of_mass V) .: S) \/ ((center_of_mass V) .: {((union S) \/ Af)}) is Simplex of card S, BCS (Complex_of {Bf})
set U = union S;
set b = center_of_mass V;
let A, B be finite Subset of V; ::_thesis: ( not A is empty & A misses union S & (union S) \/ A is affinely-independent & (union S) \/ A c= B implies ((center_of_mass V) .: S) \/ ((center_of_mass V) .: {((union S) \/ A)}) is Simplex of card S, BCS (Complex_of {B}) )
assume that
A3: not A is empty and
A4: ( A misses union S & (union S) \/ A is affinely-independent ) and
A5: (union S) \/ A c= B ; ::_thesis: ((center_of_mass V) .: S) \/ ((center_of_mass V) .: {((union S) \/ A)}) is Simplex of card S, BCS (Complex_of {B})
reconsider UA = (union S) \/ A as finite Subset of V by A5;
dom (center_of_mass V) = (bool the carrier of V) \ {{}} by FUNCT_2:def_1;
then UA in dom (center_of_mass V) by A3, ZFMISC_1:56;
then A6: {((center_of_mass V) . UA)} = Im ((center_of_mass V),UA) by FUNCT_1:59
.= (center_of_mass V) .: {UA} by RELAT_1:def_16 ;
set CA = Complex_of {UA};
set CB = Complex_of {B};
{UA} is_finer_than {B}
proof
let x be set ; :: according to SETFAM_1:def_2 ::_thesis: ( not x in {UA} or ex b1 being set st
( b1 in {B} & x c= b1 ) )
assume x in {UA} ; ::_thesis: ex b1 being set st
( b1 in {B} & x c= b1 )
then A7: x = UA by TARSKI:def_1;
B in {B} by TARSKI:def_1;
hence ex b1 being set st
( b1 in {B} & x c= b1 ) by A5, A7; ::_thesis: verum
end;
then Complex_of {UA} is SubSimplicialComplex of Complex_of {B} by SIMPLEX0:30;
then A8: subdivision ((center_of_mass V),(Complex_of {UA})) is SubSimplicialComplex of subdivision ((center_of_mass V),(Complex_of {B})) by SIMPLEX0:58;
|.(Complex_of {UA}).| c= [#] (Complex_of {UA}) ;
then A9: subdivision ((center_of_mass V),(Complex_of {UA})) = BCS (Complex_of {UA}) by Def5;
|.(Complex_of {B}).| c= [#] (Complex_of {B}) ;
then A10: BCS (Complex_of {UA}) is SubSimplicialComplex of BCS (Complex_of {B}) by A8, A9, Def5;
S is finite-membered
proof
let x be set ; :: according to FINSET_1:def_6 ::_thesis: ( not x in S or x is finite )
assume x in S ; ::_thesis: x is finite
then A11: x c= union S by ZFMISC_1:74;
union S is finite by A2;
hence x is finite by A11; ::_thesis: verum
end;
then ((center_of_mass V) .: S) \/ ((center_of_mass V) .: {UA}) is Simplex of card S, BCS (Complex_of {UA}) by A1, A2, A3, A4, Lm2;
hence ((center_of_mass V) .: S) \/ ((center_of_mass V) .: {((union S) \/ A)}) is Simplex of card S, BCS (Complex_of {B}) by A6, A10, SIMPLEX0:49; ::_thesis: verum
end;
theorem Th38: :: SIMPLEX1:38
for V being RealLinearSpace
for Sf being c=-linear finite finite-membered Subset-Family of V st Sf is with_non-empty_elements & card Sf = card (union Sf) holds
for v being Element of V st not v in union Sf & (union Sf) \/ {v} is affinely-independent holds
{ S1 where S1 is Simplex of card Sf, BCS (Complex_of {((union Sf) \/ {v})}) : (center_of_mass V) .: Sf c= S1 } = {(((center_of_mass V) .: Sf) \/ ((center_of_mass V) .: {((union Sf) \/ {v})}))}
proof
let V be RealLinearSpace; ::_thesis: for Sf being c=-linear finite finite-membered Subset-Family of V st Sf is with_non-empty_elements & card Sf = card (union Sf) holds
for v being Element of V st not v in union Sf & (union Sf) \/ {v} is affinely-independent holds
{ S1 where S1 is Simplex of card Sf, BCS (Complex_of {((union Sf) \/ {v})}) : (center_of_mass V) .: Sf c= S1 } = {(((center_of_mass V) .: Sf) \/ ((center_of_mass V) .: {((union Sf) \/ {v})}))}
let S be c=-linear finite finite-membered Subset-Family of V; ::_thesis: ( S is with_non-empty_elements & card S = card (union S) implies for v being Element of V st not v in union S & (union S) \/ {v} is affinely-independent holds
{ S1 where S1 is Simplex of card S, BCS (Complex_of {((union S) \/ {v})}) : (center_of_mass V) .: S c= S1 } = {(((center_of_mass V) .: S) \/ ((center_of_mass V) .: {((union S) \/ {v})}))} )
assume that
A1: S is with_non-empty_elements and
A2: card S = card (union S) ; ::_thesis: for v being Element of V st not v in union S & (union S) \/ {v} is affinely-independent holds
{ S1 where S1 is Simplex of card S, BCS (Complex_of {((union S) \/ {v})}) : (center_of_mass V) .: S c= S1 } = {(((center_of_mass V) .: S) \/ ((center_of_mass V) .: {((union S) \/ {v})}))}
set U = union S;
set B = center_of_mass V;
let v be Element of V; ::_thesis: ( not v in union S & (union S) \/ {v} is affinely-independent implies { S1 where S1 is Simplex of card S, BCS (Complex_of {((union S) \/ {v})}) : (center_of_mass V) .: S c= S1 } = {(((center_of_mass V) .: S) \/ ((center_of_mass V) .: {((union S) \/ {v})}))} )
assume that
A3: not v in union S and
A4: (union S) \/ {v} is affinely-independent ; ::_thesis: { S1 where S1 is Simplex of card S, BCS (Complex_of {((union S) \/ {v})}) : (center_of_mass V) .: S c= S1 } = {(((center_of_mass V) .: S) \/ ((center_of_mass V) .: {((union S) \/ {v})}))}
reconsider Uv = (union S) \/ {v} as finite affinely-independent Subset of V by A4;
set CUv = Complex_of {Uv};
set BC = BCS (Complex_of {Uv});
set SS = { S1 where S1 is Simplex of card S, BCS (Complex_of {((union S) \/ {v})}) : (center_of_mass V) .: S c= S1 } ;
set TT = {(((center_of_mass V) .: S) \/ ((center_of_mass V) .: {((union S) \/ {v})}))};
A5: union S c= Uv by XBOOLE_1:7;
hereby :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: {(((center_of_mass V) .: S) \/ ((center_of_mass V) .: {((union S) \/ {v})}))} c= { S1 where S1 is Simplex of card S, BCS (Complex_of {((union S) \/ {v})}) : (center_of_mass V) .: S c= S1 }
let x be set ; ::_thesis: ( x in { S1 where S1 is Simplex of card S, BCS (Complex_of {((union S) \/ {v})}) : (center_of_mass V) .: S c= S1 } implies x in {(((center_of_mass V) .: S) \/ ((center_of_mass V) .: {((union S) \/ {v})}))} )
reconsider n = 0 as Nat ;
assume x in { S1 where S1 is Simplex of card S, BCS (Complex_of {((union S) \/ {v})}) : (center_of_mass V) .: S c= S1 } ; ::_thesis: x in {(((center_of_mass V) .: S) \/ ((center_of_mass V) .: {((union S) \/ {v})}))}
then consider S1 being Simplex of card S, BCS (Complex_of {Uv}) such that
A6: x = S1 and
A7: (center_of_mass V) .: S c= S1 ;
((card S) + n) + 1 <= card Uv by A2, A3, CARD_2:41;
then consider T being finite Subset-Family of V such that
A8: T misses S and
A9: ( T \/ S is c=-linear & T \/ S is with_non-empty_elements ) and
A10: card T = n + 1 and
A11: union T c= Uv and
A12: @ S1 = ((center_of_mass V) .: S) \/ ((center_of_mass V) .: T) by A1, A5, A7, Th35;
A13: ex x being set st T = {x} by A10, CARD_2:42;
A14: union (T \/ S) = (union T) \/ (union S) by ZFMISC_1:78;
T \/ S is finite-membered
proof
let x be set ; :: according to FINSET_1:def_6 ::_thesis: ( not x in T \/ S or x is finite )
assume x in T \/ S ; ::_thesis: x is finite
then x c= union (T \/ S) by ZFMISC_1:74;
hence x is finite by A11, A14; ::_thesis: verum
end;
then reconsider TS = T \/ S as finite finite-membered Subset-Family of V ;
union (T \/ S) c= Uv by A5, A11, A14, XBOOLE_1:8;
then A15: card (union TS) c= card Uv by CARD_1:11;
card TS = (card S) + 1 by A8, A10, CARD_2:40;
then A16: card TS = card Uv by A2, A3, CARD_2:41;
card TS c= card (union TS) by A9, SIMPLEX0:10;
then card (union TS) = card TS by A15, A16, XBOOLE_0:def_10;
then A17: union TS = Uv by A5, A11, A14, A16, CARD_FIN:1, XBOOLE_1:8;
A18: union S c= union (T \/ S) by A14, XBOOLE_1:7;
A19: not union TS in S
proof
assume union TS in S ; ::_thesis: contradiction
then union TS c= union S by ZFMISC_1:74;
then A20: union S = Uv by A17, A18, XBOOLE_0:def_10;
v in {v} by TARSKI:def_1;
hence contradiction by A3, A20, XBOOLE_0:def_3; ::_thesis: verum
end;
not T is empty by A10;
then union TS in TS by A9, SIMPLEX0:9;
then union TS in T by A19, XBOOLE_0:def_3;
then T = {Uv} by A13, A17, TARSKI:def_1;
hence x in {(((center_of_mass V) .: S) \/ ((center_of_mass V) .: {((union S) \/ {v})}))} by A6, A12, TARSKI:def_1; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(((center_of_mass V) .: S) \/ ((center_of_mass V) .: {((union S) \/ {v})}))} or x in { S1 where S1 is Simplex of card S, BCS (Complex_of {((union S) \/ {v})}) : (center_of_mass V) .: S c= S1 } )
assume x in {(((center_of_mass V) .: S) \/ ((center_of_mass V) .: {((union S) \/ {v})}))} ; ::_thesis: x in { S1 where S1 is Simplex of card S, BCS (Complex_of {((union S) \/ {v})}) : (center_of_mass V) .: S c= S1 }
then A21: x = ((center_of_mass V) .: S) \/ ((center_of_mass V) .: {Uv}) by TARSKI:def_1;
( (center_of_mass V) .: S c= ((center_of_mass V) .: S) \/ ((center_of_mass V) .: {Uv}) & ((center_of_mass V) .: S) \/ ((center_of_mass V) .: {Uv}) is Simplex of card S, BCS (Complex_of {Uv}) ) by A1, A2, A3, Th37, XBOOLE_1:7, ZFMISC_1:50;
hence x in { S1 where S1 is Simplex of card S, BCS (Complex_of {((union S) \/ {v})}) : (center_of_mass V) .: S c= S1 } by A21; ::_thesis: verum
end;
theorem Th39: :: SIMPLEX1:39
for V being RealLinearSpace
for Sf being c=-linear finite finite-membered Subset-Family of V st Sf is with_non-empty_elements & (card Sf) + 1 = card (union Sf) & union Sf is affinely-independent holds
card { S1 where S1 is Simplex of card Sf, BCS (Complex_of {(union Sf)}) : (center_of_mass V) .: Sf c= S1 } = 2
proof
let V be RealLinearSpace; ::_thesis: for Sf being c=-linear finite finite-membered Subset-Family of V st Sf is with_non-empty_elements & (card Sf) + 1 = card (union Sf) & union Sf is affinely-independent holds
card { S1 where S1 is Simplex of card Sf, BCS (Complex_of {(union Sf)}) : (center_of_mass V) .: Sf c= S1 } = 2
let S be c=-linear finite finite-membered Subset-Family of V; ::_thesis: ( S is with_non-empty_elements & (card S) + 1 = card (union S) & union S is affinely-independent implies card { S1 where S1 is Simplex of card S, BCS (Complex_of {(union S)}) : (center_of_mass V) .: S c= S1 } = 2 )
assume that
A1: S is with_non-empty_elements and
A2: (card S) + 1 = card (union S) and
A3: union S is affinely-independent ; ::_thesis: card { S1 where S1 is Simplex of card S, BCS (Complex_of {(union S)}) : (center_of_mass V) .: S c= S1 } = 2
set B = center_of_mass V;
reconsider U = union S as finite affinely-independent Subset of V by A3;
reconsider s = S as Subset-Family of U by ZFMISC_1:82;
A4: not S is empty by A2, ZFMISC_1:2;
then consider ss1 being Subset-Family of U such that
A5: s c= ss1 and
A6: ( ss1 is with_non-empty_elements & ss1 is c=-linear ) and
A7: card ss1 = card U and
A8: for X being set st X in ss1 & card X <> 1 holds
ex x being set st
( x in X & X \ {x} in ss1 ) by A1, SIMPLEX0:9, SIMPLEX0:13;
card (ss1 \ s) = ((card S) + 1) - (card S) by A2, A5, A7, CARD_2:44;
then consider x being set such that
A9: ss1 \ s = {x} by CARD_2:42;
reconsider c = card U as ext-real number ;
set CU = Complex_of {U};
set TC = the topology of (Complex_of {U});
A10: the topology of (Complex_of {U}) = bool U by SIMPLEX0:4;
then reconsider ss = ss1 as Subset-Family of (Complex_of {U}) by XBOOLE_1:1;
set BC = BCS (Complex_of {U});
reconsider cc = (card U) - 1 as ext-real number ;
A11: |.(Complex_of {U}).| c= [#] (Complex_of {U}) ;
then A12: BCS (Complex_of {U}) = subdivision ((center_of_mass V),(Complex_of {U})) by Def5;
then A13: [#] (BCS (Complex_of {U})) = [#] (Complex_of {U}) by SIMPLEX0:def_20;
then reconsider Bss = (center_of_mass V) .: ss as Subset of (BCS (Complex_of {U})) ;
A14: ss is simplex-like
proof
let A be Subset of (Complex_of {U}); :: according to TOPS_2:def_1 ::_thesis: ( not A in ss or not A is dependent )
assume A in ss ; ::_thesis: not A is dependent
hence not A is dependent by A10, PRE_TOPC:def_2; ::_thesis: verum
end;
then A15: card Bss = card U by A6, A7, Th33;
then A16: card Bss = cc + 1 by A2, XXREAL_3:def_2;
A17: x in {x} by TARSKI:def_1;
then A18: x in ss1 by A9, XBOOLE_0:def_5;
A19: not x in s by A9, A17, XBOOLE_0:def_5;
reconsider x = x as finite Subset of V by A9, A17, XBOOLE_1:1;
degree (Complex_of {U}) = c - 1 by SIMPLEX0:26
.= (card U) + (- 1) by XXREAL_3:def_2 ;
then A20: cc = degree (BCS (Complex_of {U})) by A11, Th31;
Bss is simplex-like by A6, A12, A14, SIMPLEX0:def_20;
then A21: Bss is Simplex of (card U) - 1, BCS (Complex_of {U}) by A2, A16, A20, SIMPLEX0:def_18;
x <> {} by A6, A18;
then reconsider c1 = (card x) - 1 as Element of NAT by NAT_1:20;
ex xm being set st
( ( xm in s or xm = {} ) & card xm = (card x) - 1 & ( for y being set st y in s & y c= x holds
y c= xm ) )
proof
percases ( card x = 1 or card x <> 1 ) ;
supposeA22: card x = 1 ; ::_thesis: ex xm being set st
( ( xm in s or xm = {} ) & card xm = (card x) - 1 & ( for y being set st y in s & y c= x holds
y c= xm ) )
then A23: ex z being set st x = {z} by CARD_2:42;
take xm = {} ; ::_thesis: ( ( xm in s or xm = {} ) & card xm = (card x) - 1 & ( for y being set st y in s & y c= x holds
y c= xm ) )
thus ( ( xm in s or xm = {} ) & card xm = (card x) - 1 ) by A22; ::_thesis: for y being set st y in s & y c= x holds
y c= xm
let y be set ; ::_thesis: ( y in s & y c= x implies y c= xm )
assume that
A24: y in s and
A25: y c= x ; ::_thesis: y c= xm
y <> x by A9, A17, A24, XBOOLE_0:def_5;
hence y c= xm by A23, A25, ZFMISC_1:33; ::_thesis: verum
end;
suppose card x <> 1 ; ::_thesis: ex xm being set st
( ( xm in s or xm = {} ) & card xm = (card x) - 1 & ( for y being set st y in s & y c= x holds
y c= xm ) )
then consider z being set such that
A26: z in x and
A27: x \ {z} in ss1 by A8, A18;
take xm = x \ {z}; ::_thesis: ( ( xm in s or xm = {} ) & card xm = (card x) - 1 & ( for y being set st y in s & y c= x holds
y c= xm ) )
A28: x = xm \/ {z} by A26, ZFMISC_1:116;
xm in s
proof
assume not xm in s ; ::_thesis: contradiction
then xm in ss1 \ s by A27, XBOOLE_0:def_5;
then xm = x by A9, TARSKI:def_1;
hence contradiction by A26, ZFMISC_1:56; ::_thesis: verum
end;
hence ( xm in s or xm = {} ) ; ::_thesis: ( card xm = (card x) - 1 & ( for y being set st y in s & y c= x holds
y c= xm ) )
card x = c1 + 1 ;
hence card xm = (card x) - 1 by A26, STIRL2_1:55; ::_thesis: for y being set st y in s & y c= x holds
y c= xm
let y be set ; ::_thesis: ( y in s & y c= x implies y c= xm )
assume that
A29: y in s and
A30: y c= x ; ::_thesis: y c= xm
assume A31: not y c= xm ; ::_thesis: contradiction
xm,y are_c=-comparable by A5, A6, A27, A29, ORDINAL1:def_8;
then xm c= y by A31, XBOOLE_0:def_9;
hence contradiction by A19, A28, A29, A30, A31, ZFMISC_1:138; ::_thesis: verum
end;
end;
end;
then consider xm being set such that
A32: ( xm in s or xm = {} ) and
A33: card xm = (card x) - 1 and
A34: for y being set st y in s & y c= x holds
y c= xm ;
A35: U in S by A4, SIMPLEX0:9;
then ( union ss1 c= U & U c= union ss ) by A5, ZFMISC_1:74;
then A36: union ss = U by XBOOLE_0:def_10;
x c< U by A9, A17, A19, A35, XBOOLE_0:def_8;
then card x < card U by CARD_2:48;
then (card x) + 1 <= card U by NAT_1:13;
then consider xM being set such that
A37: xM in ss and
A38: card xM = (card x) + 1 by A6, A36, A21, Th36;
reconsider xm = xm as finite Subset of V by A32, XBOOLE_1:2;
reconsider xM = xM as finite Subset of V by A37;
A39: not xM c= xm
proof
assume xM c= xm ; ::_thesis: contradiction
then (card x) + 1 <= (card x) + (- 1) by A33, A38, NAT_1:43;
hence contradiction by XREAL_1:6; ::_thesis: verum
end;
A40: xM in s
proof
assume not xM in s ; ::_thesis: contradiction
then xM in ss \ s by A37, XBOOLE_0:def_5;
then xM = x by A9, TARSKI:def_1;
hence contradiction by A38; ::_thesis: verum
end;
then ( xm,xM are_c=-comparable or xm c= xM ) by A32, ORDINAL1:def_8, XBOOLE_1:2;
then A41: xm c= xM by A39, XBOOLE_0:def_9;
then card (xM \ xm) = (card xM) - (card xm) by CARD_2:44;
then consider x1, x2 being set such that
A42: x1 <> x2 and
A43: xM \ xm = {x1,x2} by A33, A38, CARD_2:60;
A44: x1 in {x1,x2} by TARSKI:def_2;
A45: x2 in {x1,x2} by TARSKI:def_2;
then reconsider x1 = x1, x2 = x2 as Element of V by A43, A44;
set xm1 = xm \/ {x1};
set xm2 = xm \/ {x2};
reconsider S1 = S \/ {(xm \/ {x1})}, S2 = S \/ {(xm \/ {x2})} as Subset-Family of (Complex_of {U}) ;
reconsider BS1 = (center_of_mass V) .: S1, BS2 = (center_of_mass V) .: S2 as Subset of (BCS (Complex_of {U})) by A13;
A46: BS1 = ((center_of_mass V) .: S) \/ ((center_of_mass V) .: {(xm \/ {x1})}) by RELAT_1:120;
A47: not x1 in xm by A43, A44, XBOOLE_0:def_5;
then A48: card (xm \/ {x1}) = (card xm) + 1 by CARD_2:41;
A49: not xm \/ {x1} in S
proof
assume A50: xm \/ {x1} in S ; ::_thesis: contradiction
then x,xm \/ {x1} are_c=-comparable by A5, A6, A18, ORDINAL1:def_8;
then ( x c= xm \/ {x1} or xm \/ {x1} c= x ) by XBOOLE_0:def_9;
hence contradiction by A19, A33, A48, A50, CARD_FIN:1; ::_thesis: verum
end;
not x2 in xm by A43, A45, XBOOLE_0:def_5;
then A51: card (xm \/ {x2}) = (card xm) + 1 by CARD_2:41;
A52: not xm \/ {x2} in S
proof
assume A53: xm \/ {x2} in S ; ::_thesis: contradiction
then x,xm \/ {x2} are_c=-comparable by A5, A6, A18, ORDINAL1:def_8;
then ( x c= xm \/ {x2} or xm \/ {x2} c= x ) by XBOOLE_0:def_9;
hence contradiction by A19, A33, A51, A53, CARD_FIN:1; ::_thesis: verum
end;
x2 in xM by A43, A45, XBOOLE_0:def_5;
then {x2} c= xM by ZFMISC_1:31;
then A54: xm \/ {x2} c= xM by A41, XBOOLE_1:8;
A55: S2 c= bool U
proof
let A be set ; :: according to TARSKI:def_3 ::_thesis: ( not A in S2 or A in bool U )
assume A56: A in S2 ; ::_thesis: A in bool U
percases ( A in S or A in {(xm \/ {x2})} ) by A56, XBOOLE_0:def_3;
suppose A in S ; ::_thesis: A in bool U
then A c= U by ZFMISC_1:74;
hence A in bool U ; ::_thesis: verum
end;
suppose A in {(xm \/ {x2})} ; ::_thesis: A in bool U
then A = xm \/ {x2} by TARSKI:def_1;
then A c= U by A37, A54, XBOOLE_1:1;
hence A in bool U ; ::_thesis: verum
end;
end;
end;
A57: S2 is simplex-like
proof
let A be Subset of (Complex_of {U}); :: according to TOPS_2:def_1 ::_thesis: ( not A in S2 or not A is dependent )
assume A in S2 ; ::_thesis: not A is dependent
hence not A is dependent by A10, A55, PRE_TOPC:def_2; ::_thesis: verum
end;
then card BS2 = card S2 by A1, Th33;
then A58: card BS2 = (card S) + 1 by A52, CARD_2:41;
x1 in xM by A43, A44, XBOOLE_0:def_5;
then {x1} c= xM by ZFMISC_1:31;
then A59: xm \/ {x1} c= xM by A41, XBOOLE_1:8;
A60: S1 c= bool U
proof
let A be set ; :: according to TARSKI:def_3 ::_thesis: ( not A in S1 or A in bool U )
assume A61: A in S1 ; ::_thesis: A in bool U
percases ( A in S or A in {(xm \/ {x1})} ) by A61, XBOOLE_0:def_3;
suppose A in S ; ::_thesis: A in bool U
then A c= U by ZFMISC_1:74;
hence A in bool U ; ::_thesis: verum
end;
suppose A in {(xm \/ {x1})} ; ::_thesis: A in bool U
then A = xm \/ {x1} by TARSKI:def_1;
then A c= U by A37, A59, XBOOLE_1:1;
hence A in bool U ; ::_thesis: verum
end;
end;
end;
then A62: BS1 = ((center_of_mass V) | the topology of (Complex_of {U})) .: S1 by A10, RELAT_1:129;
A63: S1 is simplex-like
proof
let A be Subset of (Complex_of {U}); :: according to TOPS_2:def_1 ::_thesis: ( not A in S1 or not A is dependent )
assume A in S1 ; ::_thesis: not A is dependent
hence not A is dependent by A10, A60, PRE_TOPC:def_2; ::_thesis: verum
end;
then card BS1 = card S1 by A1, Th33;
then A64: card BS1 = (card S) + 1 by A49, CARD_2:41;
A65: ( xm c= xm \/ {x1} & xm c= xm \/ {x2} ) by XBOOLE_1:7;
A66: for y1 being set st y1 in S holds
( y1,xm \/ {x1} are_c=-comparable & y1,xm \/ {x2} are_c=-comparable )
proof
let y1 be set ; ::_thesis: ( y1 in S implies ( y1,xm \/ {x1} are_c=-comparable & y1,xm \/ {x2} are_c=-comparable ) )
assume A67: y1 in S ; ::_thesis: ( y1,xm \/ {x1} are_c=-comparable & y1,xm \/ {x2} are_c=-comparable )
then A68: xM,y1 are_c=-comparable by A40, ORDINAL1:def_8;
percases ( xM c= y1 or xM = y1 or ( y1 c= xM & xM <> y1 ) ) by A68, XBOOLE_0:def_9;
suppose ( xM c= y1 or xM = y1 ) ; ::_thesis: ( y1,xm \/ {x1} are_c=-comparable & y1,xm \/ {x2} are_c=-comparable )
then ( xm \/ {x1} c= y1 & xm \/ {x2} c= y1 ) by A54, A59, XBOOLE_1:1;
hence ( y1,xm \/ {x1} are_c=-comparable & y1,xm \/ {x2} are_c=-comparable ) by XBOOLE_0:def_9; ::_thesis: verum
end;
supposeA69: ( y1 c= xM & xM <> y1 ) ; ::_thesis: ( y1,xm \/ {x1} are_c=-comparable & y1,xm \/ {x2} are_c=-comparable )
then reconsider y1 = y1 as finite set ;
A70: y1 c< xM by A69, XBOOLE_0:def_8;
A71: not x c= y1
proof
A72: card y1 < card xM by A70, CARD_2:48;
assume A73: x c= y1 ; ::_thesis: contradiction
then card x <= card y1 by NAT_1:43;
then card x = card y1 by A38, A72, NAT_1:9;
hence contradiction by A19, A67, A73, CARD_FIN:1; ::_thesis: verum
end;
x in ss by A9, A17, XBOOLE_0:def_5;
then y1,x are_c=-comparable by A5, A6, A67, ORDINAL1:def_8;
then y1 c= x by A71, XBOOLE_0:def_9;
then y1 c= xm by A34, A67;
then ( y1 c= xm \/ {x1} & y1 c= xm \/ {x2} ) by A65, XBOOLE_1:1;
hence ( y1,xm \/ {x1} are_c=-comparable & y1,xm \/ {x2} are_c=-comparable ) by XBOOLE_0:def_9; ::_thesis: verum
end;
end;
end;
S1 is c=-linear
proof
let y1, y2 be set ; :: according to ORDINAL1:def_8 ::_thesis: ( not y1 in S1 or not y2 in S1 or y1,y2 are_c=-comparable )
assume that
A74: y1 in S1 and
A75: y2 in S1 ; ::_thesis: y1,y2 are_c=-comparable
( y1 in S or y1 in {(xm \/ {x1})} ) by A74, XBOOLE_0:def_3;
then A76: ( y1 in S or y1 = xm \/ {x1} ) by TARSKI:def_1;
( y2 in S or y2 in {(xm \/ {x1})} ) by A75, XBOOLE_0:def_3;
then ( y2 in S or y2 = xm \/ {x1} ) by TARSKI:def_1;
hence y1,y2 are_c=-comparable by A66, A76, ORDINAL1:def_8; ::_thesis: verum
end;
then BS1 is simplex-like by A12, A63, SIMPLEX0:def_20;
then A77: BS1 is Simplex of (card U) - 1, BCS (Complex_of {U}) by A2, A15, A16, A20, A64, SIMPLEX0:def_18;
set SS = { S3 where S3 is Simplex of card S, BCS (Complex_of {(union S)}) : (center_of_mass V) .: S c= S3 } ;
(center_of_mass V) .: S c= ((center_of_mass V) .: S) \/ ((center_of_mass V) .: {(xm \/ {x1})}) by XBOOLE_1:7;
then A78: BS1 in { S3 where S3 is Simplex of card S, BCS (Complex_of {(union S)}) : (center_of_mass V) .: S c= S3 } by A2, A46, A77;
A79: BS2 = ((center_of_mass V) .: S) \/ ((center_of_mass V) .: {(xm \/ {x2})}) by RELAT_1:120;
A80: { S3 where S3 is Simplex of card S, BCS (Complex_of {(union S)}) : (center_of_mass V) .: S c= S3 } c= {BS1,BS2}
proof
let w be set ; :: according to TARSKI:def_3 ::_thesis: ( not w in { S3 where S3 is Simplex of card S, BCS (Complex_of {(union S)}) : (center_of_mass V) .: S c= S3 } or w in {BS1,BS2} )
reconsider n = 0 as Nat ;
assume w in { S3 where S3 is Simplex of card S, BCS (Complex_of {(union S)}) : (center_of_mass V) .: S c= S3 } ; ::_thesis: w in {BS1,BS2}
then consider W being Simplex of card S, BCS (Complex_of {U}) such that
A81: w = W and
A82: (center_of_mass V) .: S c= W ;
((card S) + n) + 1 <= card U by A2;
then consider T being finite Subset-Family of V such that
A83: T misses S and
A84: ( T \/ S is c=-linear & T \/ S is with_non-empty_elements ) and
A85: card T = n + 1 and
A86: union T c= U and
A87: @ W = ((center_of_mass V) .: S) \/ ((center_of_mass V) .: T) by A1, A82, Th35;
consider x3 being set such that
A88: {x3} = T by A85, CARD_2:42;
A89: x3 in T by A88, TARSKI:def_1;
then A90: not x3 in S by A83, XBOOLE_0:3;
A91: x3 c= union T by A89, ZFMISC_1:74;
A92: x3 in T \/ S by A89, XBOOLE_0:def_3;
reconsider x3 = x3 as finite Subset of U by A86, A91, XBOOLE_1:1;
A93: not xM c= x3
proof
consider x4 being set such that
A94: x4 in ss and
A95: card x4 = card x3 by A6, A36, A21, A84, A92, Th36, NAT_1:43;
assume xM c= x3 ; ::_thesis: contradiction
then (card x) + 1 <= card x3 by A38, NAT_1:43;
then x <> x4 by A95, NAT_1:13;
then not x4 in {x} by TARSKI:def_1;
then A96: x4 in s by A9, A94, XBOOLE_0:def_5;
then x4 in S \/ T by XBOOLE_0:def_3;
then x3,x4 are_c=-comparable by A84, A92, ORDINAL1:def_8;
then ( x3 c= x4 or x4 c= x3 ) by XBOOLE_0:def_9;
hence contradiction by A90, A95, A96, CARD_FIN:1; ::_thesis: verum
end;
A97: ( xm c= x3 & xm <> x3 )
proof
percases ( xm = {} or xm in s ) by A32;
suppose xm = {} ; ::_thesis: ( xm c= x3 & xm <> x3 )
hence ( xm c= x3 & xm <> x3 ) by A84, A92, XBOOLE_1:2; ::_thesis: verum
end;
supposeA98: xm in s ; ::_thesis: ( xm c= x3 & xm <> x3 )
A99: not x3 c= xm
proof
assume x3 c= xm ; ::_thesis: contradiction
then A100: card x3 <= card xm by NAT_1:43;
consider x4 being set such that
A101: x4 in ss and
A102: card x4 = card x3 by A6, A36, A21, A84, A92, Th36, NAT_1:43;
(card xm) + 1 = card x by A33;
then card x <> card x3 by A100, NAT_1:13;
then not x4 in {x} by A102, TARSKI:def_1;
then A103: x4 in s by A9, A101, XBOOLE_0:def_5;
then x4 in S \/ T by XBOOLE_0:def_3;
then x3,x4 are_c=-comparable by A84, A92, ORDINAL1:def_8;
then ( x3 c= x4 or x4 c= x3 ) by XBOOLE_0:def_9;
hence contradiction by A90, A102, A103, CARD_FIN:1; ::_thesis: verum
end;
xm in T \/ S by A98, XBOOLE_0:def_3;
then xm,x3 are_c=-comparable by A84, A92, ORDINAL1:def_8;
hence ( xm c= x3 & xm <> x3 ) by A99, XBOOLE_0:def_9; ::_thesis: verum
end;
end;
end;
then A104: x3 = x3 \/ xm by XBOOLE_1:12;
xM in S \/ T by A40, XBOOLE_0:def_3;
then xM,x3 are_c=-comparable by A84, A92, ORDINAL1:def_8;
then x3 c= xM by A93, XBOOLE_0:def_9;
then A105: x3 \ xm c= xM \ xm by XBOOLE_1:33;
A106: xM = xm \/ xM by A41, XBOOLE_1:12;
A107: x3 \ xm <> xM \ xm
proof
assume x3 \ xm = xM \ xm ; ::_thesis: contradiction
then x3 = (xM \ xm) \/ xm by A104, XBOOLE_1:39;
hence contradiction by A93, A106, XBOOLE_1:39; ::_thesis: verum
end;
A108: x3 \ xm <> {}
proof
assume x3 \ xm = {} ; ::_thesis: contradiction
then x3 c= xm by XBOOLE_1:37;
hence contradiction by A97, XBOOLE_0:def_10; ::_thesis: verum
end;
x3 \/ xm = (x3 \ xm) \/ xm by XBOOLE_1:39;
then ( x3 = xm \/ {x1} or x3 = xm \/ {x2} ) by A43, A104, A105, A107, A108, ZFMISC_1:36;
hence w in {BS1,BS2} by A79, A46, A81, A87, A88, TARSKI:def_2; ::_thesis: verum
end;
A109: BS2 = ((center_of_mass V) | the topology of (Complex_of {U})) .: S2 by A10, A55, RELAT_1:129;
A110: BS1 <> BS2
proof
assume A111: BS1 = BS2 ; ::_thesis: contradiction
then BS1 \ BS2 = {} by XBOOLE_1:37;
then ((center_of_mass V) | the topology of (Complex_of {U})) .: (S1 \ S2) = {} by A109, A62, FUNCT_1:64;
then A112: dom ((center_of_mass V) | the topology of (Complex_of {U})) misses S1 \ S2 by RELAT_1:118;
BS2 \ BS1 = {} by A111, XBOOLE_1:37;
then ((center_of_mass V) | the topology of (Complex_of {U})) .: (S2 \ S1) = {} by A109, A62, FUNCT_1:64;
then A113: dom ((center_of_mass V) | the topology of (Complex_of {U})) misses S2 \ S1 by RELAT_1:118;
A114: dom ((center_of_mass V) | the topology of (Complex_of {U})) = (dom (center_of_mass V)) /\ the topology of (Complex_of {U}) by RELAT_1:61;
xm \/ {x1} in {(xm \/ {x1})} by TARSKI:def_1;
then A115: xm \/ {x1} in S1 by XBOOLE_0:def_3;
A116: dom (center_of_mass V) = (bool the carrier of V) \ {{}} by FUNCT_2:def_1;
A117: S1 \ S2 c= S1 by XBOOLE_1:36;
then not {} in S1 \ S2 by A1;
then A118: S1 \ S2 c= dom (center_of_mass V) by A116, ZFMISC_1:34;
A119: S2 \ S1 c= S2 by XBOOLE_1:36;
then not {} in S2 \ S1 by A1;
then A120: S2 \ S1 c= dom (center_of_mass V) by A116, ZFMISC_1:34;
S1 \ S2 c= bool U by A60, A117, XBOOLE_1:1;
then S1 \ S2 c= dom ((center_of_mass V) | the topology of (Complex_of {U})) by A10, A114, A118, XBOOLE_1:19;
then A121: S1 \ S2 = {} by A112, XBOOLE_1:67;
S2 \ S1 c= bool U by A55, A119, XBOOLE_1:1;
then S2 \ S1 c= dom ((center_of_mass V) | the topology of (Complex_of {U})) by A10, A114, A120, XBOOLE_1:19;
then S1 = S2 by A113, A121, XBOOLE_1:32, XBOOLE_1:67;
then xm \/ {x1} in {(xm \/ {x2})} by A49, A115, XBOOLE_0:def_3;
then A122: xm \/ {x1} = xm \/ {x2} by TARSKI:def_1;
x1 in {x1} by TARSKI:def_1;
then x1 in xm \/ {x1} by XBOOLE_0:def_3;
then x1 in {x2} by A47, A122, XBOOLE_0:def_3;
hence contradiction by A42, TARSKI:def_1; ::_thesis: verum
end;
S2 is c=-linear
proof
let y1, y2 be set ; :: according to ORDINAL1:def_8 ::_thesis: ( not y1 in S2 or not y2 in S2 or y1,y2 are_c=-comparable )
assume that
A123: y1 in S2 and
A124: y2 in S2 ; ::_thesis: y1,y2 are_c=-comparable
( y1 in S or y1 in {(xm \/ {x2})} ) by A123, XBOOLE_0:def_3;
then A125: ( y1 in S or y1 = xm \/ {x2} ) by TARSKI:def_1;
( y2 in S or y2 in {(xm \/ {x2})} ) by A124, XBOOLE_0:def_3;
then ( y2 in S or y2 = xm \/ {x2} ) by TARSKI:def_1;
hence y1,y2 are_c=-comparable by A66, A125, ORDINAL1:def_8; ::_thesis: verum
end;
then BS2 is simplex-like by A12, A57, SIMPLEX0:def_20;
then A126: BS2 is Simplex of (card U) - 1, BCS (Complex_of {U}) by A2, A15, A16, A20, A58, SIMPLEX0:def_18;
(center_of_mass V) .: S c= ((center_of_mass V) .: S) \/ ((center_of_mass V) .: {(xm \/ {x2})}) by XBOOLE_1:7;
then BS2 in { S3 where S3 is Simplex of card S, BCS (Complex_of {(union S)}) : (center_of_mass V) .: S c= S3 } by A2, A79, A126;
then {BS1,BS2} c= { S3 where S3 is Simplex of card S, BCS (Complex_of {(union S)}) : (center_of_mass V) .: S c= S3 } by A78, ZFMISC_1:32;
then { S3 where S3 is Simplex of card S, BCS (Complex_of {(union S)}) : (center_of_mass V) .: S c= S3 } = {BS1,BS2} by A80, XBOOLE_0:def_10;
hence card { S1 where S1 is Simplex of card S, BCS (Complex_of {(union S)}) : (center_of_mass V) .: S c= S1 } = 2 by A110, CARD_2:57; ::_thesis: verum
end;
theorem Th40: :: SIMPLEX1:40
for V being RealLinearSpace
for K being non void total affinely-independent simplex-join-closed SimplicialComplex of V
for Aff being finite affinely-independent Subset of V
for B being Subset of V st Aff is Simplex of K holds
( B is Simplex of (BCS (Complex_of {Aff})) iff ( B is Simplex of (BCS K) & conv B c= conv Aff ) )
proof
let V be RealLinearSpace; ::_thesis: for K being non void total affinely-independent simplex-join-closed SimplicialComplex of V
for Aff being finite affinely-independent Subset of V
for B being Subset of V st Aff is Simplex of K holds
( B is Simplex of (BCS (Complex_of {Aff})) iff ( B is Simplex of (BCS K) & conv B c= conv Aff ) )
let K be non void total affinely-independent simplex-join-closed SimplicialComplex of V; ::_thesis: for Aff being finite affinely-independent Subset of V
for B being Subset of V st Aff is Simplex of K holds
( B is Simplex of (BCS (Complex_of {Aff})) iff ( B is Simplex of (BCS K) & conv B c= conv Aff ) )
let Aff be finite affinely-independent Subset of V; ::_thesis: for B being Subset of V st Aff is Simplex of K holds
( B is Simplex of (BCS (Complex_of {Aff})) iff ( B is Simplex of (BCS K) & conv B c= conv Aff ) )
let B be Subset of V; ::_thesis: ( Aff is Simplex of K implies ( B is Simplex of (BCS (Complex_of {Aff})) iff ( B is Simplex of (BCS K) & conv B c= conv Aff ) ) )
set Bag = center_of_mass V;
set C = Complex_of {Aff};
A1: the topology of (Complex_of {Aff}) = bool Aff by SIMPLEX0:4;
assume Aff is Simplex of K ; ::_thesis: ( B is Simplex of (BCS (Complex_of {Aff})) iff ( B is Simplex of (BCS K) & conv B c= conv Aff ) )
then reconsider s = Aff as Simplex of K ;
A2: [#] K = the carrier of V by SIMPLEX0:def_10;
then |.K.| c= [#] K ;
then A3: subdivision ((center_of_mass V),K) = BCS K by Def5;
@ s is affinely-independent ;
then A4: Complex_of {Aff} is SubSimplicialComplex of K by Th3;
then the topology of (Complex_of {Aff}) c= the topology of K by SIMPLEX0:def_13;
then A5: |.(Complex_of {Aff}).| c= |.K.| by Th4;
[#] (Complex_of {Aff}) = [#] V ;
then A6: subdivision ((center_of_mass V),(Complex_of {Aff})) = BCS (Complex_of {Aff}) by A5, Def5;
then BCS (Complex_of {Aff}) is SubSimplicialComplex of BCS K by A3, A4, SIMPLEX0:58;
then A7: the topology of (BCS (Complex_of {Aff})) c= the topology of (BCS K) by SIMPLEX0:def_13;
hereby ::_thesis: ( B is Simplex of (BCS K) & conv B c= conv Aff implies B is Simplex of (BCS (Complex_of {Aff})) )
assume B is Simplex of (BCS (Complex_of {Aff})) ; ::_thesis: ( B is Simplex of (BCS K) & conv B c= conv Aff )
then reconsider A = B as Simplex of (BCS (Complex_of {Aff})) ;
A in the topology of (BCS (Complex_of {Aff})) by PRE_TOPC:def_2;
then A in the topology of (BCS K) by A7;
then reconsider a = A as Simplex of (BCS K) by PRE_TOPC:def_2;
( |.(BCS (Complex_of {Aff})).| = |.(Complex_of {Aff}).| & conv (@ A) c= |.(BCS (Complex_of {Aff})).| ) by Th5, Th10;
then conv (@ a) c= conv Aff by Th8;
hence ( B is Simplex of (BCS K) & conv B c= conv Aff ) ; ::_thesis: verum
end;
assume that
A8: B is Simplex of (BCS K) and
A9: conv B c= conv Aff ; ::_thesis: B is Simplex of (BCS (Complex_of {Aff}))
reconsider A = B as Simplex of (BCS K) by A8;
consider SS being c=-linear finite simplex-like Subset-Family of K such that
A10: B = (center_of_mass V) .: SS by A3, A8, SIMPLEX0:def_20;
reconsider ss = SS as c=-linear finite Subset-Family of (Complex_of {Aff}) by A2;
[#] (subdivision ((center_of_mass V),(Complex_of {Aff}))) = [#] (Complex_of {Aff}) by SIMPLEX0:def_20;
then reconsider Bss = (center_of_mass V) .: ss as Subset of (BCS (Complex_of {Aff})) by A5, Def5;
A11: dom (center_of_mass V) = (bool the carrier of V) \ {{}} by FUNCT_2:def_1;
ss is simplex-like
proof
let a be Subset of (Complex_of {Aff}); :: according to TOPS_2:def_1 ::_thesis: ( not a in ss or not a is dependent )
assume A12: a in ss ; ::_thesis: not a is dependent
reconsider aK = a as Simplex of K by A12, TOPS_2:def_1;
percases ( aK is empty or not aK is empty ) ;
suppose aK is empty ; ::_thesis: not a is dependent
hence not a is dependent ; ::_thesis: verum
end;
supposeA13: not aK is empty ; ::_thesis: not a is dependent
then aK in dom (center_of_mass V) by A11, ZFMISC_1:56;
then A14: (center_of_mass V) . aK in A by A10, A12, FUNCT_1:def_6;
A15: (center_of_mass V) . aK in Int (@ aK) by A13, RLAFFIN2:20;
A c= conv (@ A) by RLAFFIN1:2;
then (center_of_mass V) . aK in conv (@ A) by A14;
then Int (@ aK) meets conv (@ s) by A9, A15, XBOOLE_0:3;
then aK c= Aff by Th26;
hence not a is dependent by A1, PRE_TOPC:def_2; ::_thesis: verum
end;
end;
end;
then Bss is simplex-like by A6, SIMPLEX0:def_20;
hence B is Simplex of (BCS (Complex_of {Aff})) by A10; ::_thesis: verum
end;
theorem Th41: :: SIMPLEX1:41
for n being Nat
for V being RealLinearSpace
for K being non void total affinely-independent simplex-join-closed SimplicialComplex of V
for Af being finite Subset of V
for Sk being finite simplex-like Subset-Family of K st Sk is with_non-empty_elements & (card Sk) + n <= degree K holds
( ( Af is Simplex of n + (card Sk), BCS K & (center_of_mass V) .: Sk c= Af ) iff ex Tk being finite simplex-like Subset-Family of K st
( Tk misses Sk & Tk \/ Sk is c=-linear & Tk \/ Sk is with_non-empty_elements & card Tk = n + 1 & Af = ((center_of_mass V) .: Sk) \/ ((center_of_mass V) .: Tk) ) )
proof
let n be Nat; ::_thesis: for V being RealLinearSpace
for K being non void total affinely-independent simplex-join-closed SimplicialComplex of V
for Af being finite Subset of V
for Sk being finite simplex-like Subset-Family of K st Sk is with_non-empty_elements & (card Sk) + n <= degree K holds
( ( Af is Simplex of n + (card Sk), BCS K & (center_of_mass V) .: Sk c= Af ) iff ex Tk being finite simplex-like Subset-Family of K st
( Tk misses Sk & Tk \/ Sk is c=-linear & Tk \/ Sk is with_non-empty_elements & card Tk = n + 1 & Af = ((center_of_mass V) .: Sk) \/ ((center_of_mass V) .: Tk) ) )
let V be RealLinearSpace; ::_thesis: for K being non void total affinely-independent simplex-join-closed SimplicialComplex of V
for Af being finite Subset of V
for Sk being finite simplex-like Subset-Family of K st Sk is with_non-empty_elements & (card Sk) + n <= degree K holds
( ( Af is Simplex of n + (card Sk), BCS K & (center_of_mass V) .: Sk c= Af ) iff ex Tk being finite simplex-like Subset-Family of K st
( Tk misses Sk & Tk \/ Sk is c=-linear & Tk \/ Sk is with_non-empty_elements & card Tk = n + 1 & Af = ((center_of_mass V) .: Sk) \/ ((center_of_mass V) .: Tk) ) )
let K be non void total affinely-independent simplex-join-closed SimplicialComplex of V; ::_thesis: for Af being finite Subset of V
for Sk being finite simplex-like Subset-Family of K st Sk is with_non-empty_elements & (card Sk) + n <= degree K holds
( ( Af is Simplex of n + (card Sk), BCS K & (center_of_mass V) .: Sk c= Af ) iff ex Tk being finite simplex-like Subset-Family of K st
( Tk misses Sk & Tk \/ Sk is c=-linear & Tk \/ Sk is with_non-empty_elements & card Tk = n + 1 & Af = ((center_of_mass V) .: Sk) \/ ((center_of_mass V) .: Tk) ) )
let Af be finite Subset of V; ::_thesis: for Sk being finite simplex-like Subset-Family of K st Sk is with_non-empty_elements & (card Sk) + n <= degree K holds
( ( Af is Simplex of n + (card Sk), BCS K & (center_of_mass V) .: Sk c= Af ) iff ex Tk being finite simplex-like Subset-Family of K st
( Tk misses Sk & Tk \/ Sk is c=-linear & Tk \/ Sk is with_non-empty_elements & card Tk = n + 1 & Af = ((center_of_mass V) .: Sk) \/ ((center_of_mass V) .: Tk) ) )
let Sk be finite simplex-like Subset-Family of K; ::_thesis: ( Sk is with_non-empty_elements & (card Sk) + n <= degree K implies ( ( Af is Simplex of n + (card Sk), BCS K & (center_of_mass V) .: Sk c= Af ) iff ex Tk being finite simplex-like Subset-Family of K st
( Tk misses Sk & Tk \/ Sk is c=-linear & Tk \/ Sk is with_non-empty_elements & card Tk = n + 1 & Af = ((center_of_mass V) .: Sk) \/ ((center_of_mass V) .: Tk) ) ) )
set B = center_of_mass V;
set BK = BCS K;
assume that
A1: Sk is with_non-empty_elements and
A2: (card Sk) + n <= degree K ; ::_thesis: ( ( Af is Simplex of n + (card Sk), BCS K & (center_of_mass V) .: Sk c= Af ) iff ex Tk being finite simplex-like Subset-Family of K st
( Tk misses Sk & Tk \/ Sk is c=-linear & Tk \/ Sk is with_non-empty_elements & card Tk = n + 1 & Af = ((center_of_mass V) .: Sk) \/ ((center_of_mass V) .: Tk) ) )
reconsider nc = n + (card Sk) as ext-real number ;
A3: (nc + 1) - 1 = nc by XXREAL_3:22;
A4: [#] K = the carrier of V by SIMPLEX0:def_10;
then A5: |.K.| c= [#] K ;
then A6: subdivision ((center_of_mass V),K) = BCS K by Def5;
A7: nc <= degree (BCS K) by A2, A5, Th31;
hereby ::_thesis: ( ex Tk being finite simplex-like Subset-Family of K st
( Tk misses Sk & Tk \/ Sk is c=-linear & Tk \/ Sk is with_non-empty_elements & card Tk = n + 1 & Af = ((center_of_mass V) .: Sk) \/ ((center_of_mass V) .: Tk) ) implies ( Af is Simplex of n + (card Sk), BCS K & (center_of_mass V) .: Sk c= Af ) )
assume that
A8: Af is Simplex of n + (card Sk), BCS K and
A9: (center_of_mass V) .: Sk c= Af ; ::_thesis: ex R being finite simplex-like Subset-Family of K st
( R misses Sk & R \/ Sk is c=-linear & R \/ Sk is with_non-empty_elements & card R = n + 1 & Af = ((center_of_mass V) .: Sk) \/ ((center_of_mass V) .: R) )
consider T being c=-linear finite simplex-like Subset-Family of K such that
A10: Af = (center_of_mass V) .: T by A6, A8, SIMPLEX0:def_20;
( union T is empty or union T in T ) by SIMPLEX0:9, ZFMISC_1:2;
then A11: union T is simplex-like by TOPS_2:def_1;
then @ (union T) is affinely-independent ;
then reconsider UT = union T as finite affinely-independent Subset of V ;
UT = union (@ T) ;
then conv Af c= conv UT by A10, CONVEX1:30, RLAFFIN2:17;
then reconsider s1 = Af as Simplex of (BCS (Complex_of {UT})) by A8, A11, Th40;
card Af = nc + 1 by A7, A8, SIMPLEX0:def_18;
then A12: s1 is Simplex of n + (card Sk), BCS (Complex_of {UT}) by A3, SIMPLEX0:48;
set C = Complex_of {UT};
reconsider cT = card UT as ext-real number ;
|.(Complex_of {UT}).| c= [#] (Complex_of {UT}) ;
then A13: degree (Complex_of {UT}) = degree (BCS (Complex_of {UT})) by Th31;
( degree (Complex_of {UT}) = cT - 1 & card s1 <= (degree (BCS (Complex_of {UT}))) + 1 ) by SIMPLEX0:24, SIMPLEX0:26;
then card s1 <= card UT by A13, XXREAL_3:22;
then nc + 1 <= card UT by A7, A8, SIMPLEX0:def_18;
then A14: ((card Sk) + n) + 1 <= card UT by XXREAL_3:def_2;
( the_family_of K is subset-closed & UT in the topology of K ) by A11, PRE_TOPC:def_2;
then A15: bool UT c= the topology of K by SIMPLEX0:1;
union (@ Sk) c= union T by A1, A5, A8, A9, A10, Th34, ZFMISC_1:77;
then consider R being finite Subset-Family of V such that
A16: ( R misses Sk & R \/ Sk is c=-linear & R \/ Sk is with_non-empty_elements & card R = n + 1 ) and
A17: union R c= UT and
A18: Af = ((center_of_mass V) .: Sk) \/ ((center_of_mass V) .: R) by A1, A9, A12, A14, Th35;
reconsider R = R as Subset-Family of K by A4;
( R c= bool (union R) & bool (union R) c= bool UT ) by A17, SIMPLEX0:1, ZFMISC_1:82;
then R c= bool UT by XBOOLE_1:1;
then R c= the topology of K by A15, XBOOLE_1:1;
then reconsider R = R as finite simplex-like Subset-Family of K by SIMPLEX0:14;
take R = R; ::_thesis: ( R misses Sk & R \/ Sk is c=-linear & R \/ Sk is with_non-empty_elements & card R = n + 1 & Af = ((center_of_mass V) .: Sk) \/ ((center_of_mass V) .: R) )
thus ( R misses Sk & R \/ Sk is c=-linear & R \/ Sk is with_non-empty_elements & card R = n + 1 & Af = ((center_of_mass V) .: Sk) \/ ((center_of_mass V) .: R) ) by A16, A18; ::_thesis: verum
end;
given T being finite simplex-like Subset-Family of K such that A19: T misses Sk and
A20: ( T \/ Sk is c=-linear & T \/ Sk is with_non-empty_elements ) and
A21: card T = n + 1 and
A22: Af = ((center_of_mass V) .: Sk) \/ ((center_of_mass V) .: T) ; ::_thesis: ( Af is Simplex of n + (card Sk), BCS K & (center_of_mass V) .: Sk c= Af )
set ST = Sk \/ T;
[#] K = [#] (BCS K) by A6, SIMPLEX0:def_20;
then reconsider BST = (center_of_mass V) .: (Sk \/ T) as Subset of (BCS K) by SIMPLEX0:def_10;
A23: Sk \/ T is simplex-like by TOPS_2:13;
then reconsider BST = BST as Simplex of (BCS K) by A6, A20, SIMPLEX0:def_20;
card (Sk \/ T) = (card Sk) + (card T) by A19, CARD_2:40;
then card BST = ((card Sk) + n) + 1 by A20, A21, A23, Th33;
then ( ((center_of_mass V) .: Sk) \/ ((center_of_mass V) .: T) = (center_of_mass V) .: (Sk \/ T) & card BST = nc + 1 ) by RELAT_1:120, XXREAL_3:def_2;
hence ( Af is Simplex of n + (card Sk), BCS K & (center_of_mass V) .: Sk c= Af ) by A3, A22, SIMPLEX0:48, XBOOLE_1:7; ::_thesis: verum
end;
theorem Th42: :: SIMPLEX1:42
for V being RealLinearSpace
for K being non void total affinely-independent simplex-join-closed SimplicialComplex of V
for Sk being finite simplex-like Subset-Family of K
for Ak being Simplex of K st Sk is c=-linear & Sk is with_non-empty_elements & card Sk = card (union Sk) & union Sk c= Ak & card Ak = (card Sk) + 1 holds
{ S1 where S1 is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= S1 & conv (@ S1) c= conv (@ Ak) ) } = {(((center_of_mass V) .: Sk) \/ ((center_of_mass V) .: {Ak}))}
proof
let V be RealLinearSpace; ::_thesis: for K being non void total affinely-independent simplex-join-closed SimplicialComplex of V
for Sk being finite simplex-like Subset-Family of K
for Ak being Simplex of K st Sk is c=-linear & Sk is with_non-empty_elements & card Sk = card (union Sk) & union Sk c= Ak & card Ak = (card Sk) + 1 holds
{ S1 where S1 is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= S1 & conv (@ S1) c= conv (@ Ak) ) } = {(((center_of_mass V) .: Sk) \/ ((center_of_mass V) .: {Ak}))}
let K be non void total affinely-independent simplex-join-closed SimplicialComplex of V; ::_thesis: for Sk being finite simplex-like Subset-Family of K
for Ak being Simplex of K st Sk is c=-linear & Sk is with_non-empty_elements & card Sk = card (union Sk) & union Sk c= Ak & card Ak = (card Sk) + 1 holds
{ S1 where S1 is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= S1 & conv (@ S1) c= conv (@ Ak) ) } = {(((center_of_mass V) .: Sk) \/ ((center_of_mass V) .: {Ak}))}
let Sk be finite simplex-like Subset-Family of K; ::_thesis: for Ak being Simplex of K st Sk is c=-linear & Sk is with_non-empty_elements & card Sk = card (union Sk) & union Sk c= Ak & card Ak = (card Sk) + 1 holds
{ S1 where S1 is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= S1 & conv (@ S1) c= conv (@ Ak) ) } = {(((center_of_mass V) .: Sk) \/ ((center_of_mass V) .: {Ak}))}
let Ak be Simplex of K; ::_thesis: ( Sk is c=-linear & Sk is with_non-empty_elements & card Sk = card (union Sk) & union Sk c= Ak & card Ak = (card Sk) + 1 implies { S1 where S1 is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= S1 & conv (@ S1) c= conv (@ Ak) ) } = {(((center_of_mass V) .: Sk) \/ ((center_of_mass V) .: {Ak}))} )
set B = center_of_mass V;
assume that
A1: ( Sk is c=-linear & Sk is with_non-empty_elements ) and
A2: card Sk = card (union Sk) and
A3: union Sk c= Ak and
A4: card Ak = (card Sk) + 1 ; ::_thesis: { S1 where S1 is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= S1 & conv (@ S1) c= conv (@ Ak) ) } = {(((center_of_mass V) .: Sk) \/ ((center_of_mass V) .: {Ak}))}
card (Ak \ (union Sk)) = ((card Sk) + 1) - (card Sk) by A2, A3, A4, CARD_2:44
.= 1 ;
then consider v being set such that
A5: Ak \ (union Sk) = {v} by CARD_2:42;
reconsider Ak1 = @ Ak as finite affinely-independent Subset of V ;
set C = Complex_of {Ak1};
reconsider c = card Ak as ext-real number ;
A6: degree (Complex_of {Ak1}) = c - 1 by SIMPLEX0:26
.= (card Ak) + (- 1) by XXREAL_3:def_2
.= card Sk by A4 ;
reconsider Sk1 = @ Sk as c=-linear finite finite-membered Subset-Family of V by A1;
set XX = { W where W is Simplex of card Sk, BCS (Complex_of {Ak1}) : (center_of_mass V) .: Sk c= W } ;
set YY = { W where W is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= W & conv (@ W) c= conv (@ Ak) ) } ;
[#] K = the carrier of V by SIMPLEX0:def_10;
then |.K.| c= [#] K ;
then A7: subdivision ((center_of_mass V),K) = BCS K by Def5;
A8: Complex_of {Ak1} is SubSimplicialComplex of K by Th3;
then the topology of (Complex_of {Ak1}) c= the topology of K by SIMPLEX0:def_13;
then A9: |.(Complex_of {Ak1}).| c= |.K.| by Th4;
A10: [#] (Complex_of {Ak1}) = [#] V ;
then A11: degree (Complex_of {Ak1}) = degree (BCS (Complex_of {Ak1})) by A9, Th31;
subdivision ((center_of_mass V),(Complex_of {Ak1})) = BCS (Complex_of {Ak1}) by A9, A10, Def5;
then BCS (Complex_of {Ak1}) is SubSimplicialComplex of BCS K by A7, A8, SIMPLEX0:58;
then A12: degree (BCS (Complex_of {Ak1})) <= degree (BCS K) by SIMPLEX0:32;
A13: { W where W is Simplex of card Sk, BCS (Complex_of {Ak1}) : (center_of_mass V) .: Sk c= W } c= { W where W is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= W & conv (@ W) c= conv (@ Ak) ) }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { W where W is Simplex of card Sk, BCS (Complex_of {Ak1}) : (center_of_mass V) .: Sk c= W } or x in { W where W is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= W & conv (@ W) c= conv (@ Ak) ) } )
assume x in { W where W is Simplex of card Sk, BCS (Complex_of {Ak1}) : (center_of_mass V) .: Sk c= W } ; ::_thesis: x in { W where W is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= W & conv (@ W) c= conv (@ Ak) ) }
then consider W being Simplex of card Sk, BCS (Complex_of {Ak1}) such that
A14: ( x = W & (center_of_mass V) .: Sk1 c= W ) ;
W = @ W ;
then reconsider w = W as Simplex of (BCS K) by Th40;
card W = (degree (BCS (Complex_of {Ak1}))) + 1 by A6, A11, SIMPLEX0:def_18;
then A15: w is Simplex of card Sk, BCS K by A6, A11, A12, SIMPLEX0:def_18;
( conv (@ W) c= conv (@ Ak) & @ w = w ) by Th40;
hence x in { W where W is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= W & conv (@ W) c= conv (@ Ak) ) } by A14, A15; ::_thesis: verum
end;
A16: [#] (subdivision ((center_of_mass V),(Complex_of {Ak1}))) = [#] (Complex_of {Ak1}) by SIMPLEX0:def_20;
A17: { W where W is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= W & conv (@ W) c= conv (@ Ak) ) } c= { W where W is Simplex of card Sk, BCS (Complex_of {Ak1}) : (center_of_mass V) .: Sk c= W }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { W where W is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= W & conv (@ W) c= conv (@ Ak) ) } or x in { W where W is Simplex of card Sk, BCS (Complex_of {Ak1}) : (center_of_mass V) .: Sk c= W } )
reconsider c1 = card Sk as ext-real number ;
assume x in { W where W is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= W & conv (@ W) c= conv (@ Ak) ) } ; ::_thesis: x in { W where W is Simplex of card Sk, BCS (Complex_of {Ak1}) : (center_of_mass V) .: Sk c= W }
then consider W being Simplex of card Sk, BCS K such that
A18: ( W = x & (center_of_mass V) .: Sk c= W ) and
A19: conv (@ W) c= conv (@ Ak) ;
reconsider w = @ W as Subset of (BCS (Complex_of {Ak1})) by A9, A16, Def5;
reconsider cW = card W as ext-real number ;
card W = c1 + 1 by A6, A11, A12, SIMPLEX0:def_18
.= (card Sk) + 1 by XXREAL_3:def_2 ;
then card Sk = (card W) + (- 1) ;
then A20: card Sk = cW - 1 by XXREAL_3:def_2;
w is simplex-like by A19, Th40;
then w is Simplex of card Sk, BCS (Complex_of {Ak1}) by A20, SIMPLEX0:48;
hence x in { W where W is Simplex of card Sk, BCS (Complex_of {Ak1}) : (center_of_mass V) .: Sk c= W } by A18; ::_thesis: verum
end;
v in {v} by TARSKI:def_1;
then A21: ( v in Ak1 & not v in union Sk ) by A5, XBOOLE_0:def_5;
Ak = Ak \/ (union Sk) by A3, XBOOLE_1:12
.= {v} \/ (union Sk1) by A5, XBOOLE_1:39 ;
then { W where W is Simplex of card Sk, BCS (Complex_of {Ak1}) : (center_of_mass V) .: Sk c= W } = {(((center_of_mass V) .: Sk1) \/ ((center_of_mass V) .: {Ak}))} by A1, A2, A21, Th38;
hence { S1 where S1 is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= S1 & conv (@ S1) c= conv (@ Ak) ) } = {(((center_of_mass V) .: Sk) \/ ((center_of_mass V) .: {Ak}))} by A13, A17, XBOOLE_0:def_10; ::_thesis: verum
end;
theorem Th43: :: SIMPLEX1:43
for V being RealLinearSpace
for K being non void total affinely-independent simplex-join-closed SimplicialComplex of V
for Sk being finite simplex-like Subset-Family of K st Sk is c=-linear & Sk is with_non-empty_elements & (card Sk) + 1 = card (union Sk) holds
card { S1 where S1 is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= S1 & conv (@ S1) c= conv (@ (union Sk)) ) } = 2
proof
let V be RealLinearSpace; ::_thesis: for K being non void total affinely-independent simplex-join-closed SimplicialComplex of V
for Sk being finite simplex-like Subset-Family of K st Sk is c=-linear & Sk is with_non-empty_elements & (card Sk) + 1 = card (union Sk) holds
card { S1 where S1 is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= S1 & conv (@ S1) c= conv (@ (union Sk)) ) } = 2
let K be non void total affinely-independent simplex-join-closed SimplicialComplex of V; ::_thesis: for Sk being finite simplex-like Subset-Family of K st Sk is c=-linear & Sk is with_non-empty_elements & (card Sk) + 1 = card (union Sk) holds
card { S1 where S1 is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= S1 & conv (@ S1) c= conv (@ (union Sk)) ) } = 2
let Sk be finite simplex-like Subset-Family of K; ::_thesis: ( Sk is c=-linear & Sk is with_non-empty_elements & (card Sk) + 1 = card (union Sk) implies card { S1 where S1 is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= S1 & conv (@ S1) c= conv (@ (union Sk)) ) } = 2 )
set B = center_of_mass V;
assume that
A1: ( Sk is c=-linear & Sk is with_non-empty_elements ) and
A2: (card Sk) + 1 = card (union Sk) ; ::_thesis: card { S1 where S1 is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= S1 & conv (@ S1) c= conv (@ (union Sk)) ) } = 2
not Sk is empty by A2, ZFMISC_1:2;
then union Sk in Sk by A1, SIMPLEX0:9;
then reconsider U = union Sk as Simplex of K by TOPS_2:def_1;
reconsider Sk1 = @ Sk as c=-linear finite finite-membered Subset-Family of V by A1;
reconsider c = card U as ext-real number ;
@ U = union Sk1 ;
then reconsider U1 = union Sk1 as finite affinely-independent Subset of V ;
set C = Complex_of {U1};
A3: degree (Complex_of {U1}) = c - 1 by SIMPLEX0:26
.= (card U) + (- 1) by XXREAL_3:def_2
.= card Sk by A2 ;
set YY = { W where W is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= W & conv (@ W) c= conv (@ (union Sk)) ) } ;
[#] K = the carrier of V by SIMPLEX0:def_10;
then |.K.| c= [#] K ;
then A4: subdivision ((center_of_mass V),K) = BCS K by Def5;
set XX = { W where W is Simplex of card Sk, BCS (Complex_of {U1}) : (center_of_mass V) .: Sk c= W } ;
A5: @ U = U1 ;
then A6: Complex_of {U1} is SubSimplicialComplex of K by Th3;
then the topology of (Complex_of {U1}) c= the topology of K by SIMPLEX0:def_13;
then A7: |.(Complex_of {U1}).| c= |.K.| by Th4;
A8: [#] (Complex_of {U1}) = [#] V ;
then A9: degree (Complex_of {U1}) = degree (BCS (Complex_of {U1})) by A7, Th31;
subdivision ((center_of_mass V),(Complex_of {U1})) = BCS (Complex_of {U1}) by A7, A8, Def5;
then BCS (Complex_of {U1}) is SubSimplicialComplex of BCS K by A4, A6, SIMPLEX0:58;
then A10: degree (BCS (Complex_of {U1})) <= degree (BCS K) by SIMPLEX0:32;
A11: { W where W is Simplex of card Sk, BCS (Complex_of {U1}) : (center_of_mass V) .: Sk c= W } c= { W where W is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= W & conv (@ W) c= conv (@ (union Sk)) ) }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { W where W is Simplex of card Sk, BCS (Complex_of {U1}) : (center_of_mass V) .: Sk c= W } or x in { W where W is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= W & conv (@ W) c= conv (@ (union Sk)) ) } )
assume x in { W where W is Simplex of card Sk, BCS (Complex_of {U1}) : (center_of_mass V) .: Sk c= W } ; ::_thesis: x in { W where W is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= W & conv (@ W) c= conv (@ (union Sk)) ) }
then consider W being Simplex of card Sk, BCS (Complex_of {U1}) such that
A12: ( x = W & (center_of_mass V) .: Sk1 c= W ) ;
W = @ W ;
then reconsider w = W as Simplex of (BCS K) by A5, Th40;
card W = (degree (BCS (Complex_of {U1}))) + 1 by A3, A9, SIMPLEX0:def_18;
then A13: w is Simplex of card Sk, BCS K by A3, A9, A10, SIMPLEX0:def_18;
( conv (@ W) c= conv (@ U) & @ w = w ) by Th40;
hence x in { W where W is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= W & conv (@ W) c= conv (@ (union Sk)) ) } by A12, A13; ::_thesis: verum
end;
A14: [#] (subdivision ((center_of_mass V),(Complex_of {U1}))) = [#] (Complex_of {U1}) by SIMPLEX0:def_20;
A15: { W where W is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= W & conv (@ W) c= conv (@ (union Sk)) ) } c= { W where W is Simplex of card Sk, BCS (Complex_of {U1}) : (center_of_mass V) .: Sk c= W }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { W where W is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= W & conv (@ W) c= conv (@ (union Sk)) ) } or x in { W where W is Simplex of card Sk, BCS (Complex_of {U1}) : (center_of_mass V) .: Sk c= W } )
reconsider c1 = card Sk as ext-real number ;
assume x in { W where W is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= W & conv (@ W) c= conv (@ (union Sk)) ) } ; ::_thesis: x in { W where W is Simplex of card Sk, BCS (Complex_of {U1}) : (center_of_mass V) .: Sk c= W }
then consider W being Simplex of card Sk, BCS K such that
A16: ( W = x & (center_of_mass V) .: Sk c= W ) and
A17: conv (@ W) c= conv (@ U) ;
reconsider w = @ W as Subset of (BCS (Complex_of {U1})) by A7, A14, Def5;
reconsider cW = card W as ext-real number ;
card W = c1 + 1 by A3, A9, A10, SIMPLEX0:def_18
.= (card Sk) + 1 by XXREAL_3:def_2 ;
then card Sk = (card W) + (- 1) ;
then A18: card Sk = cW - 1 by XXREAL_3:def_2;
w is simplex-like by A17, Th40;
then w is Simplex of card Sk, BCS (Complex_of {U1}) by A18, SIMPLEX0:48;
hence x in { W where W is Simplex of card Sk, BCS (Complex_of {U1}) : (center_of_mass V) .: Sk c= W } by A16; ::_thesis: verum
end;
card { W where W is Simplex of card Sk, BCS (Complex_of {U1}) : (center_of_mass V) .: Sk c= W } = 2 by A1, A2, Th39;
hence card { S1 where S1 is Simplex of card Sk, BCS K : ( (center_of_mass V) .: Sk c= S1 & conv (@ S1) c= conv (@ (union Sk)) ) } = 2 by A11, A15, XBOOLE_0:def_10; ::_thesis: verum
end;
theorem Th44: :: SIMPLEX1:44
for n being Nat
for V being RealLinearSpace
for K being non void total affinely-independent simplex-join-closed SimplicialComplex of V
for Af being finite Subset of V st K is Subdivision of Complex_of {Af} & card Af = n + 1 & degree K = n & ( for S being Simplex of n - 1,K
for X being set st X = { S1 where S1 is Simplex of n,K : S c= S1 } holds
( ( conv (@ S) meets Int Af implies card X = 2 ) & ( conv (@ S) misses Int Af implies card X = 1 ) ) ) holds
for S being Simplex of n - 1, BCS K
for X being set st X = { S1 where S1 is Simplex of n, BCS K : S c= S1 } holds
( ( conv (@ S) meets Int Af implies card X = 2 ) & ( conv (@ S) misses Int Af implies card X = 1 ) )
proof
let n be Nat; ::_thesis: for V being RealLinearSpace
for K being non void total affinely-independent simplex-join-closed SimplicialComplex of V
for Af being finite Subset of V st K is Subdivision of Complex_of {Af} & card Af = n + 1 & degree K = n & ( for S being Simplex of n - 1,K
for X being set st X = { S1 where S1 is Simplex of n,K : S c= S1 } holds
( ( conv (@ S) meets Int Af implies card X = 2 ) & ( conv (@ S) misses Int Af implies card X = 1 ) ) ) holds
for S being Simplex of n - 1, BCS K
for X being set st X = { S1 where S1 is Simplex of n, BCS K : S c= S1 } holds
( ( conv (@ S) meets Int Af implies card X = 2 ) & ( conv (@ S) misses Int Af implies card X = 1 ) )
let V be RealLinearSpace; ::_thesis: for K being non void total affinely-independent simplex-join-closed SimplicialComplex of V
for Af being finite Subset of V st K is Subdivision of Complex_of {Af} & card Af = n + 1 & degree K = n & ( for S being Simplex of n - 1,K
for X being set st X = { S1 where S1 is Simplex of n,K : S c= S1 } holds
( ( conv (@ S) meets Int Af implies card X = 2 ) & ( conv (@ S) misses Int Af implies card X = 1 ) ) ) holds
for S being Simplex of n - 1, BCS K
for X being set st X = { S1 where S1 is Simplex of n, BCS K : S c= S1 } holds
( ( conv (@ S) meets Int Af implies card X = 2 ) & ( conv (@ S) misses Int Af implies card X = 1 ) )
let K be non void total affinely-independent simplex-join-closed SimplicialComplex of V; ::_thesis: for Af being finite Subset of V st K is Subdivision of Complex_of {Af} & card Af = n + 1 & degree K = n & ( for S being Simplex of n - 1,K
for X being set st X = { S1 where S1 is Simplex of n,K : S c= S1 } holds
( ( conv (@ S) meets Int Af implies card X = 2 ) & ( conv (@ S) misses Int Af implies card X = 1 ) ) ) holds
for S being Simplex of n - 1, BCS K
for X being set st X = { S1 where S1 is Simplex of n, BCS K : S c= S1 } holds
( ( conv (@ S) meets Int Af implies card X = 2 ) & ( conv (@ S) misses Int Af implies card X = 1 ) )
let A be finite Subset of V; ::_thesis: ( K is Subdivision of Complex_of {A} & card A = n + 1 & degree K = n & ( for S being Simplex of n - 1,K
for X being set st X = { S1 where S1 is Simplex of n,K : S c= S1 } holds
( ( conv (@ S) meets Int A implies card X = 2 ) & ( conv (@ S) misses Int A implies card X = 1 ) ) ) implies for S being Simplex of n - 1, BCS K
for X being set st X = { S1 where S1 is Simplex of n, BCS K : S c= S1 } holds
( ( conv (@ S) meets Int A implies card X = 2 ) & ( conv (@ S) misses Int A implies card X = 1 ) ) )
assume that
A1: K is Subdivision of Complex_of {A} and
A2: card A = n + 1 and
A3: degree K = n and
A4: for S being Simplex of n - 1,K
for X being set st X = { S1 where S1 is Simplex of n,K : S c= S1 } holds
( ( conv (@ S) meets Int A implies card X = 2 ) & ( conv (@ S) misses Int A implies card X = 1 ) ) ; ::_thesis: for S being Simplex of n - 1, BCS K
for X being set st X = { S1 where S1 is Simplex of n, BCS K : S c= S1 } holds
( ( conv (@ S) meets Int A implies card X = 2 ) & ( conv (@ S) misses Int A implies card X = 1 ) )
|.(Complex_of {A}).| = conv A by Th8;
then A5: |.K.| = conv A by A1, Th10;
A6: K is finite-degree by A3, SIMPLEX0:def_12;
A7: A is affinely-independent
proof
consider a being Subset of K such that
A8: a is simplex-like and
A9: card a = (degree K) + 1 by A6, SIMPLEX0:def_12;
conv (@ a) c= conv A by A5, A8, Th5;
then A10: Affin (@ a) c= Affin A by RLAFFIN1:68;
card A = card a by A2, A3, A9, XXREAL_3:def_2;
hence A is affinely-independent by A8, A10, RLAFFIN1:80; ::_thesis: verum
end;
set B = center_of_mass V;
reconsider Z = 0 as Nat ;
set TK = TopStruct(# the carrier of K, the topology of K #);
reconsider n1 = n - 1 as ext-real number ;
let S be Simplex of n - 1, BCS K; ::_thesis: for X being set st X = { S1 where S1 is Simplex of n, BCS K : S c= S1 } holds
( ( conv (@ S) meets Int A implies card X = 2 ) & ( conv (@ S) misses Int A implies card X = 1 ) )
let X be set ; ::_thesis: ( X = { S1 where S1 is Simplex of n, BCS K : S c= S1 } implies ( ( conv (@ S) meets Int A implies card X = 2 ) & ( conv (@ S) misses Int A implies card X = 1 ) ) )
assume A11: X = { S1 where S1 is Simplex of n, BCS K : S c= S1 } ; ::_thesis: ( ( conv (@ S) meets Int A implies card X = 2 ) & ( conv (@ S) misses Int A implies card X = 1 ) )
[#] K = the carrier of V by SIMPLEX0:def_10;
then A12: |.K.| c= [#] K ;
then A13: degree K = degree (BCS K) by Th31;
then A14: ( n + (- 1) >= - 1 & n - 1 <= degree (BCS K) ) by A3, XREAL_1:31, XREAL_1:146;
then A15: card S = n1 + 1 by SIMPLEX0:def_18;
then A16: card S = (n - 1) + 1 by XXREAL_3:def_2;
A17: BCS K = subdivision ((center_of_mass V),K) by A12, Def5;
percases ( n = 0 or n > 0 ) ;
supposeA18: n = 0 ; ::_thesis: ( ( conv (@ S) meets Int A implies card X = 2 ) & ( conv (@ S) misses Int A implies card X = 1 ) )
then A19: TopStruct(# the carrier of K, the topology of K #) = BCS K by A3, A12, Th21;
then S in the topology of K by PRE_TOPC:def_2;
then reconsider s = S as Simplex of K by PRE_TOPC:def_2;
reconsider s = s as Simplex of n - 1,K by A3, A15, A18, SIMPLEX0:def_18;
set XX = { W where W is Simplex of n,K : s c= W } ;
A20: @ S = @ s ;
then A21: ( conv (@ S) meets Int A implies card { W where W is Simplex of n,K : s c= W } = 2 ) by A4;
A22: { W where W is Simplex of n,K : s c= W } c= X
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { W where W is Simplex of n,K : s c= W } or x in X )
assume x in { W where W is Simplex of n,K : s c= W } ; ::_thesis: x in X
then consider W being Simplex of n,K such that
A23: ( x = W & S c= W ) ;
W in the topology of (BCS K) by A19, PRE_TOPC:def_2;
then reconsider w = W as Simplex of (BCS K) by PRE_TOPC:def_2;
card W = (degree K) + 1 by A3, SIMPLEX0:def_18;
then w is Simplex of n, BCS K by A3, A13, SIMPLEX0:def_18;
hence x in X by A11, A23; ::_thesis: verum
end;
A24: X c= { W where W is Simplex of n,K : s c= W }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in X or x in { W where W is Simplex of n,K : s c= W } )
assume x in X ; ::_thesis: x in { W where W is Simplex of n,K : s c= W }
then consider W being Simplex of n, BCS K such that
A25: ( x = W & S c= W ) by A11;
W in the topology of K by A19, PRE_TOPC:def_2;
then reconsider w = W as Simplex of K by PRE_TOPC:def_2;
card W = (degree (BCS K)) + 1 by A3, A13, SIMPLEX0:def_18;
then w is Simplex of n,K by A3, A13, SIMPLEX0:def_18;
hence x in { W where W is Simplex of n,K : s c= W } by A25; ::_thesis: verum
end;
( conv (@ S) misses Int A implies card { W where W is Simplex of n,K : s c= W } = 1 ) by A4, A20;
hence ( ( conv (@ S) meets Int A implies card X = 2 ) & ( conv (@ S) misses Int A implies card X = 1 ) ) by A22, A24, A21, XBOOLE_0:def_10; ::_thesis: verum
end;
supposeA26: n > 0 ; ::_thesis: ( ( conv (@ S) meets Int A implies card X = 2 ) & ( conv (@ S) misses Int A implies card X = 1 ) )
consider SS being c=-linear finite simplex-like Subset-Family of K such that
A27: S = (center_of_mass V) .: SS by A17, SIMPLEX0:def_20;
SS \ {{}} c= SS by XBOOLE_1:36;
then reconsider SS1 = SS \ {{}} as c=-linear finite simplex-like Subset-Family of K by TOPS_2:11;
A28: ( SS1 c= bool (@ (union SS1)) & bool (@ (union SS1)) c= bool the carrier of V ) by ZFMISC_1:67, ZFMISC_1:82;
A29: not {} in SS1 by ZFMISC_1:56;
then A30: SS1 is with_non-empty_elements by SETFAM_1:def_8;
A31: dom (center_of_mass V) = (bool the carrier of V) \ {{}} by FUNCT_2:def_1;
then A32: SS /\ (dom (center_of_mass V)) = (SS /\ (bool the carrier of V)) \ {{}} by XBOOLE_1:49
.= SS1 /\ (bool the carrier of V) by XBOOLE_1:49
.= SS1 by A28, XBOOLE_1:1, XBOOLE_1:28 ;
then A33: (center_of_mass V) .: SS = (center_of_mass V) .: SS1 by RELAT_1:112;
then A34: card SS1 = n by A16, A27, A30, Th33;
A35: S = (center_of_mass V) .: SS1 by A27, A32, RELAT_1:112;
A36: card SS1 = card S by A27, A30, A33, Th33;
then A37: not SS1 is empty by A16, A26;
then A38: union SS1 in SS1 by SIMPLEX0:9;
then reconsider U = union SS1 as Simplex of K by TOPS_2:def_1;
card SS1 c= card U by A30, SIMPLEX0:10;
then A39: n <= card U by A16, A36, NAT_1:39;
card U <= (degree K) + 1 by SIMPLEX0:24;
then A40: card U <= n + 1 by A3, XXREAL_3:def_2;
A41: conv (@ U) c= conv A by A5, Th5;
SS1 c= bool the carrier of V by A28, XBOOLE_1:1;
then A42: conv (@ S) c= conv (@ U) by A35, CONVEX1:30, RLAFFIN2:17;
percases ( card U = n or card U = n + 1 ) by A39, A40, NAT_1:9;
supposeA43: card U = n ; ::_thesis: ( ( conv (@ S) meets Int A implies card X = 2 ) & ( conv (@ S) misses Int A implies card X = 1 ) )
set XX = { W where W is Simplex of n,K : U c= W } ;
A44: U is Simplex of n - 1,K by A13, A14, A15, A16, A43, SIMPLEX0:def_18;
hereby ::_thesis: ( conv (@ S) misses Int A implies card X = 1 )
assume conv (@ S) meets Int A ; ::_thesis: card X = 2
then conv (@ U) meets Int A by A42, XBOOLE_1:63;
then A45: card { W where W is Simplex of n,K : U c= W } = 2 by A4, A44;
consider w1, w2 being set such that
A46: w1 <> w2 and
A47: { W where W is Simplex of n,K : U c= W } = {w1,w2} by A45, CARD_2:60;
w2 in { W where W is Simplex of n,K : U c= W } by A47, TARSKI:def_2;
then consider W2 being Simplex of n,K such that
A48: w2 = W2 and
A49: U c= W2 ;
A50: ( SS1 is with_non-empty_elements & S = (center_of_mass V) .: SS1 ) by A27, A29, A32, RELAT_1:112, SETFAM_1:def_8;
w1 in { W where W is Simplex of n,K : U c= W } by A47, TARSKI:def_2;
then consider W1 being Simplex of n,K such that
A51: w1 = W1 and
A52: U c= W1 ;
A53: card W1 = (degree K) + 1 by A3, SIMPLEX0:def_18;
then A54: card W1 = n + 1 by A3, XXREAL_3:def_2;
then { W where W is Simplex of n, BCS K : ( S c= W & conv (@ W) c= conv (@ W1) ) } = {(S \/ ((center_of_mass V) .: {W1}))} by A16, A27, A30, A33, A36, A43, A52, Th42;
then S \/ ((center_of_mass V) .: {W1}) in { W where W is Simplex of n, BCS K : ( S c= W & conv (@ W) c= conv (@ W1) ) } by TARSKI:def_1;
then A55: ex R being Simplex of n, BCS K st
( R = S \/ ((center_of_mass V) .: {W1}) & S c= R & conv (@ R) c= conv (@ W1) ) ;
A56: S \/ ((center_of_mass V) .: {W1}) <> S \/ ((center_of_mass V) .: {W2})
proof
for A being Subset of K st A in {W1} holds
A is simplex-like by TARSKI:def_1;
then {W1} is simplex-like by TOPS_2:def_1;
then A57: SS1 \/ {W1} is simplex-like by TOPS_2:13;
A58: ( S \/ ((center_of_mass V) .: {W1}) = (center_of_mass V) .: (SS1 \/ {W1}) & S \/ ((center_of_mass V) .: {W2}) = (center_of_mass V) .: (SS1 \/ {W2}) ) by A35, RELAT_1:120;
W1 in {W1} by TARSKI:def_1;
then A59: W1 in SS1 \/ {W1} by XBOOLE_0:def_3;
for A being Subset of K st A in {W2} holds
A is simplex-like by TARSKI:def_1;
then {W2} is simplex-like by TOPS_2:def_1;
then A60: SS1 \/ {W2} is simplex-like by TOPS_2:13;
assume A61: S \/ ((center_of_mass V) .: {W1}) = S \/ ((center_of_mass V) .: {W2}) ; ::_thesis: contradiction
not W1 is empty by A3, A53;
then SS1 \/ {W1} c= SS1 \/ {W2} by A12, A30, A55, A60, A58, A57, A61, Th34;
then ( W1 in SS1 or W1 in {W2} ) by A59, XBOOLE_0:def_3;
then W1 c= U by A46, A48, A51, TARSKI:def_1, ZFMISC_1:74;
then W1 = U by A52, XBOOLE_0:def_10;
hence contradiction by A43, A54; ::_thesis: verum
end;
A62: (card SS1) + Z <= degree K by A3, A16, A27, A30, A33, Th33;
A63: X c= {(S \/ ((center_of_mass V) .: {W1})),(S \/ ((center_of_mass V) .: {W2}))}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in X or x in {(S \/ ((center_of_mass V) .: {W1})),(S \/ ((center_of_mass V) .: {W2}))} )
A64: ( n + 1 = (degree K) + 1 & n = ((degree K) + 1) - 1 ) by A3, XXREAL_3:22, XXREAL_3:def_2;
assume x in X ; ::_thesis: x in {(S \/ ((center_of_mass V) .: {W1})),(S \/ ((center_of_mass V) .: {W2}))}
then consider W being Simplex of n, BCS K such that
A65: x = W and
A66: S c= W by A11;
consider T being finite simplex-like Subset-Family of K such that
A67: T misses SS1 and
A68: ( T \/ SS1 is c=-linear & T \/ SS1 is with_non-empty_elements ) and
A69: card T = Z + 1 and
A70: @ W = ((center_of_mass V) .: SS1) \/ ((center_of_mass V) .: T) by A16, A36, A50, A62, A66, Th41;
consider t being set such that
A71: T = {t} by A69, CARD_2:42;
set TS = T \/ SS1;
A72: card (T \/ SS1) = n + 1 by A16, A36, A67, A69, CARD_2:40;
A73: union (T \/ SS1) in T \/ SS1 by A68, A71, SIMPLEX0:9;
T \/ SS1 is simplex-like by TOPS_2:13;
then reconsider UTS = union (T \/ SS1) as Simplex of K by A73, TOPS_2:def_1;
card (T \/ SS1) c= card UTS by A68, SIMPLEX0:10;
then A74: card (T \/ SS1) <= card UTS by NAT_1:39;
UTS in T
proof
assume not UTS in T ; ::_thesis: contradiction
then UTS in SS1 by A73, XBOOLE_0:def_3;
then card UTS <= card U by NAT_1:43, ZFMISC_1:74;
hence contradiction by A43, A72, A74, NAT_1:13; ::_thesis: verum
end;
then A75: UTS = t by A71, TARSKI:def_1;
card UTS <= (degree K) + 1 by SIMPLEX0:24;
then card UTS <= n + 1 by A3, XXREAL_3:def_2;
then card UTS = n + 1 by A72, A74, XXREAL_0:1;
then A76: UTS is Simplex of n,K by A64, SIMPLEX0:48;
U c= UTS by XBOOLE_1:7, ZFMISC_1:77;
then UTS in { W where W is Simplex of n,K : U c= W } by A76;
then ( W = ((center_of_mass V) .: SS1) \/ ((center_of_mass V) .: {W1}) or W = ((center_of_mass V) .: SS1) \/ ((center_of_mass V) .: {W2}) ) by A47, A48, A51, A70, A71, A75, TARSKI:def_2;
hence x in {(S \/ ((center_of_mass V) .: {W1})),(S \/ ((center_of_mass V) .: {W2}))} by A35, A65, TARSKI:def_2; ::_thesis: verum
end;
card W2 = (degree K) + 1 by A3, SIMPLEX0:def_18;
then card W2 = n + 1 by A3, XXREAL_3:def_2;
then { W where W is Simplex of n, BCS K : ( S c= W & conv (@ W) c= conv (@ W2) ) } = {(S \/ ((center_of_mass V) .: {W2}))} by A16, A30, A35, A36, A43, A49, Th42;
then S \/ ((center_of_mass V) .: {W2}) in { W where W is Simplex of n, BCS K : ( S c= W & conv (@ W) c= conv (@ W2) ) } by TARSKI:def_1;
then ex R being Simplex of n, BCS K st
( R = S \/ ((center_of_mass V) .: {W2}) & S c= R & conv (@ R) c= conv (@ W2) ) ;
then A77: S \/ ((center_of_mass V) .: {W2}) in X by A11;
S \/ ((center_of_mass V) .: {W1}) in X by A11, A55;
then {(S \/ ((center_of_mass V) .: {W1})),(S \/ ((center_of_mass V) .: {W2}))} c= X by A77, ZFMISC_1:32;
then X = {(S \/ ((center_of_mass V) .: {W1})),(S \/ ((center_of_mass V) .: {W2}))} by A63, XBOOLE_0:def_10;
hence card X = 2 by A56, CARD_2:57; ::_thesis: verum
end;
A78: ( conv (@ S) c= conv A & not A is empty ) by A2, A41, A42, XBOOLE_1:1;
assume conv (@ S) misses Int A ; ::_thesis: card X = 1
then consider BB being Subset of V such that
A79: BB c< A and
A80: conv (@ S) c= conv BB by A7, A78, RLAFFIN2:23;
A81: BB c= A by A79, XBOOLE_0:def_8;
then reconsider B1 = BB as finite Subset of V ;
card B1 < n + 1 by A2, A79, CARD_2:48;
then A82: card B1 <= n by NAT_1:13;
Affin (@ S) c= Affin BB by A80, RLAFFIN1:68;
then n <= card B1 by A16, RLAFFIN1:79;
then card B1 = n by A82, XXREAL_0:1;
then card (A \ BB) = (n + 1) - n by A2, A81, CARD_2:44;
then consider ab being set such that
A83: A \ BB = {ab} by CARD_2:42;
not U is empty by A26, A43;
then @ U in dom (center_of_mass V) by A31, ZFMISC_1:56;
then A84: ( S c= conv (@ S) & (center_of_mass V) . U in @ S ) by A35, A38, FUNCT_1:def_6, RLAFFIN1:2;
then (center_of_mass V) . U in conv (@ S) ;
then A85: (center_of_mass V) . U in conv (@ U) by A42;
set BUU = ((center_of_mass V) . U) |-- (@ U);
@ U c= conv (@ U) by RLAFFIN1:2;
then A86: @ U c= conv A by A41, XBOOLE_1:1;
A87: ab in {ab} by TARSKI:def_1;
then reconsider ab = ab as Element of V by A83;
A88: ( SS1 is with_non-empty_elements & S = (center_of_mass V) .: SS1 ) by A27, A29, A32, RELAT_1:112, SETFAM_1:def_8;
A89: conv (@ U) c= Affin (@ U) by RLAFFIN1:65;
then sum (((center_of_mass V) . U) |-- (@ U)) = 1 by A85, RLAFFIN1:def_7;
then consider F being FinSequence of REAL , G being FinSequence of the carrier of V such that
A90: ((Sum (((center_of_mass V) . U) |-- (@ U))) |-- A) . ab = Sum F and
A91: len G = len F and
G is one-to-one and
A92: rng G = Carrier (((center_of_mass V) . U) |-- (@ U)) and
A93: for n being Nat st n in dom F holds
F . n = ((((center_of_mass V) . U) |-- (@ U)) . (G . n)) * (((G . n) |-- A) . ab) by A7, A86, RLAFFIN2:3;
A94: dom G = dom F by A91, FINSEQ_3:29;
U c= conv B1
proof
A95: Carrier (((center_of_mass V) . U) |-- (@ U)) c= U by RLVECT_2:def_6;
A96: now__::_thesis:_for_i_being_Nat_st_i_in_dom_F_holds_
0_<=_F_._i
let i be Nat; ::_thesis: ( i in dom F implies 0 <= F . i )
assume A97: i in dom F ; ::_thesis: 0 <= F . i
A98: F . i = ((((center_of_mass V) . U) |-- (@ U)) . (G . i)) * (((G . i) |-- A) . ab) by A93, A97;
A99: G . i in rng G by A94, A97, FUNCT_1:def_3;
then G . i in U by A92, A95;
then A100: ((G . i) |-- A) . ab >= 0 by A7, A86, RLAFFIN1:71;
(((center_of_mass V) . U) |-- (@ U)) . (G . i) = 1 / (card U) by A92, A95, A99, RLAFFIN2:18;
hence 0 <= F . i by A98, A100; ::_thesis: verum
end;
(center_of_mass V) . U in conv (@ S) by A84;
then A101: (center_of_mass V) . U in conv BB by A80;
assume not U c= conv B1 ; ::_thesis: contradiction
then consider t being set such that
A102: t in U and
A103: not t in conv B1 by TARSKI:def_3;
A104: (t |-- A) . ab > 0
proof
A \ {ab} c= B1
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in A \ {ab} or x in B1 )
assume x in A \ {ab} ; ::_thesis: x in B1
then ( x in A & not x in {ab} ) by XBOOLE_0:def_5;
hence x in B1 by A83, XBOOLE_0:def_5; ::_thesis: verum
end;
then A105: conv (A \ {ab}) c= conv B1 by RLAFFIN1:3;
assume A106: (t |-- A) . ab <= 0 ; ::_thesis: contradiction
(t |-- A) . ab >= 0 by A7, A86, A102, RLAFFIN1:71;
then for x being set st x in {ab} holds
(t |-- A) . x = 0 by A106, TARSKI:def_1;
then t in conv (A \ {ab}) by A7, A86, A102, RLAFFIN1:76;
hence contradiction by A103, A105; ::_thesis: verum
end;
A107: (((center_of_mass V) . U) |-- (@ U)) . t = 1 / (card U) by A102, RLAFFIN2:18;
then t in Carrier (((center_of_mass V) . U) |-- (@ U)) by A102;
then consider n being set such that
A108: n in dom G and
A109: G . n = t by A92, FUNCT_1:def_3;
reconsider n = n as Nat by A108;
F . n = ((((center_of_mass V) . U) |-- (@ U)) . t) * ((t |-- A) . ab) by A93, A94, A108, A109;
then 0 < Sum F by A94, A96, A102, A104, A107, A108, RVSUM_1:85;
then A110: (((center_of_mass V) . U) |-- A) . ab > 0 by A85, A89, A90, RLAFFIN1:def_7;
Carrier (((center_of_mass V) . U) |-- BB) c= BB by RLVECT_2:def_6;
then A111: not ab in Carrier (((center_of_mass V) . U) |-- BB) by A83, A87, XBOOLE_0:def_5;
conv BB c= Affin BB by RLAFFIN1:65;
then ((center_of_mass V) . U) |-- A = ((center_of_mass V) . U) |-- BB by A7, A81, A101, RLAFFIN1:77;
hence contradiction by A111, A110; ::_thesis: verum
end;
then conv (@ U) c= conv B1 by CONVEX1:30;
then conv (@ U) misses Int A by A79, RLAFFIN2:7, XBOOLE_1:63;
then card { W where W is Simplex of n,K : U c= W } = 1 by A4, A44;
then consider w1 being set such that
A112: { W where W is Simplex of n,K : U c= W } = {w1} by CARD_2:42;
w1 in { W where W is Simplex of n,K : U c= W } by A112, TARSKI:def_1;
then consider W1 being Simplex of n,K such that
A113: w1 = W1 and
A114: U c= W1 ;
A115: (card SS1) + Z <= degree K by A3, A16, A27, A30, A33, Th33;
A116: X c= {(S \/ ((center_of_mass V) .: {W1}))}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in X or x in {(S \/ ((center_of_mass V) .: {W1}))} )
A117: n + 1 = (degree K) + 1 by A3, XXREAL_3:def_2;
assume x in X ; ::_thesis: x in {(S \/ ((center_of_mass V) .: {W1}))}
then consider W being Simplex of n, BCS K such that
A118: x = W and
A119: S c= W by A11;
consider T being finite simplex-like Subset-Family of K such that
A120: T misses SS1 and
A121: ( T \/ SS1 is c=-linear & T \/ SS1 is with_non-empty_elements ) and
A122: card T = Z + 1 and
A123: @ W = ((center_of_mass V) .: SS1) \/ ((center_of_mass V) .: T) by A16, A36, A88, A115, A119, Th41;
consider t being set such that
A124: T = {t} by A122, CARD_2:42;
set TS = T \/ SS1;
A125: card (T \/ SS1) = n + 1 by A16, A36, A120, A122, CARD_2:40;
A126: union (T \/ SS1) in T \/ SS1 by A121, A124, SIMPLEX0:9;
T \/ SS1 is simplex-like by TOPS_2:13;
then reconsider UTS = union (T \/ SS1) as Simplex of K by A126, TOPS_2:def_1;
card (T \/ SS1) c= card UTS by A121, SIMPLEX0:10;
then A127: card (T \/ SS1) <= card UTS by NAT_1:39;
UTS in T
proof
assume not UTS in T ; ::_thesis: contradiction
then UTS in SS1 by A126, XBOOLE_0:def_3;
then card UTS <= card U by NAT_1:43, ZFMISC_1:74;
hence contradiction by A43, A125, A127, NAT_1:13; ::_thesis: verum
end;
then A128: UTS = t by A124, TARSKI:def_1;
card UTS <= (degree K) + 1 by SIMPLEX0:24;
then card UTS <= n + 1 by A3, XXREAL_3:def_2;
then ( card UTS = n + 1 & SS1 c= T \/ SS1 ) by A125, A127, XBOOLE_1:7, XXREAL_0:1;
then ( U c= UTS & UTS is Simplex of n,K ) by A3, A117, SIMPLEX0:def_18, ZFMISC_1:77;
then UTS in { W where W is Simplex of n,K : U c= W } ;
then W = ((center_of_mass V) .: SS1) \/ ((center_of_mass V) .: {W1}) by A112, A113, A123, A124, A128, TARSKI:def_1;
hence x in {(S \/ ((center_of_mass V) .: {W1}))} by A35, A118, TARSKI:def_1; ::_thesis: verum
end;
card W1 = (degree K) + 1 by A3, SIMPLEX0:def_18;
then card W1 = n + 1 by A3, XXREAL_3:def_2;
then { W where W is Simplex of n, BCS K : ( S c= W & conv (@ W) c= conv (@ W1) ) } = {(S \/ ((center_of_mass V) .: {W1}))} by A16, A27, A30, A33, A36, A43, A114, Th42;
then S \/ ((center_of_mass V) .: {W1}) in { W where W is Simplex of n, BCS K : ( S c= W & conv (@ W) c= conv (@ W1) ) } by TARSKI:def_1;
then ex R being Simplex of n, BCS K st
( R = S \/ ((center_of_mass V) .: {W1}) & S c= R & conv (@ R) c= conv (@ W1) ) ;
then S \/ ((center_of_mass V) .: {W1}) in X by A11;
then X = {(S \/ ((center_of_mass V) .: {W1}))} by A116, ZFMISC_1:33;
hence card X = 1 by CARD_1:30; ::_thesis: verum
end;
supposeA129: card U = n + 1 ; ::_thesis: ( ( conv (@ S) meets Int A implies card X = 2 ) & ( conv (@ S) misses Int A implies card X = 1 ) )
A130: conv (@ S) meets Int A
proof
not U is empty by A129;
then @ U in dom (center_of_mass V) by A31, ZFMISC_1:56;
then A131: ( S c= conv (@ S) & (center_of_mass V) . U in @ S ) by A35, A38, FUNCT_1:def_6, RLAFFIN1:2;
then (center_of_mass V) . U in conv (@ S) ;
then A132: (center_of_mass V) . U in conv (@ U) by A42;
set BUU = ((center_of_mass V) . U) |-- (@ U);
assume A133: conv (@ S) misses Int A ; ::_thesis: contradiction
( conv (@ S) c= conv A & not A is empty ) by A2, A41, A42, XBOOLE_1:1;
then consider BB being Subset of V such that
A134: BB c< A and
A135: conv (@ S) c= conv BB by A7, A133, RLAFFIN2:23;
A136: BB c= A by A134, XBOOLE_0:def_8;
then reconsider B1 = BB as finite Subset of V ;
Affin (@ S) c= Affin BB by A135, RLAFFIN1:68;
then A137: n <= card B1 by A16, RLAFFIN1:79;
A138: card B1 < n + 1 by A2, A134, CARD_2:48;
then card B1 <= n by NAT_1:13;
then card B1 = n by A137, XXREAL_0:1;
then card (A \ BB) = (n + 1) - n by A2, A136, CARD_2:44;
then consider ab being set such that
A139: A \ BB = {ab} by CARD_2:42;
@ U c= conv (@ U) by RLAFFIN1:2;
then A140: @ U c= conv A by A41, XBOOLE_1:1;
A141: ab in {ab} by TARSKI:def_1;
then reconsider ab = ab as Element of V by A139;
A142: conv (@ U) c= Affin (@ U) by RLAFFIN1:65;
then sum (((center_of_mass V) . U) |-- (@ U)) = 1 by A132, RLAFFIN1:def_7;
then consider F being FinSequence of REAL , G being FinSequence of the carrier of V such that
A143: ((Sum (((center_of_mass V) . U) |-- (@ U))) |-- A) . ab = Sum F and
A144: len G = len F and
G is one-to-one and
A145: rng G = Carrier (((center_of_mass V) . U) |-- (@ U)) and
A146: for n being Nat st n in dom F holds
F . n = ((((center_of_mass V) . U) |-- (@ U)) . (G . n)) * (((G . n) |-- A) . ab) by A7, A140, RLAFFIN2:3;
A147: dom G = dom F by A144, FINSEQ_3:29;
A148: A \ {ab} = A /\ BB by A139, XBOOLE_1:48
.= BB by A136, XBOOLE_1:28 ;
U c= conv B1
proof
A149: Carrier (((center_of_mass V) . U) |-- (@ U)) c= U by RLVECT_2:def_6;
A150: now__::_thesis:_for_i_being_Nat_st_i_in_dom_F_holds_
0_<=_F_._i
let i be Nat; ::_thesis: ( i in dom F implies 0 <= F . i )
assume A151: i in dom F ; ::_thesis: 0 <= F . i
A152: F . i = ((((center_of_mass V) . U) |-- (@ U)) . (G . i)) * (((G . i) |-- A) . ab) by A146, A151;
A153: G . i in rng G by A147, A151, FUNCT_1:def_3;
then G . i in U by A145, A149;
then A154: ((G . i) |-- A) . ab >= 0 by A7, A140, RLAFFIN1:71;
(((center_of_mass V) . U) |-- (@ U)) . (G . i) = 1 / (card U) by A145, A149, A153, RLAFFIN2:18;
hence 0 <= F . i by A152, A154; ::_thesis: verum
end;
(center_of_mass V) . U in conv (@ S) by A131;
then A155: (center_of_mass V) . U in conv BB by A135;
assume not U c= conv B1 ; ::_thesis: contradiction
then consider t being set such that
A156: t in U and
A157: not t in conv B1 by TARSKI:def_3;
U c= conv (@ U) by RLAFFIN1:2;
then A158: t in conv (@ U) by A156;
A159: (t |-- A) . ab > 0
proof
assume A160: (t |-- A) . ab <= 0 ; ::_thesis: contradiction
(t |-- A) . ab >= 0 by A7, A41, A158, RLAFFIN1:71;
then for y being set st y in A \ B1 holds
(t |-- A) . y = 0 by A139, A160, TARSKI:def_1;
hence contradiction by A7, A41, A139, A148, A157, A158, RLAFFIN1:76; ::_thesis: verum
end;
A161: (((center_of_mass V) . U) |-- (@ U)) . t = 1 / (card U) by A156, RLAFFIN2:18;
then t in Carrier (((center_of_mass V) . U) |-- (@ U)) by A156;
then consider n being set such that
A162: n in dom G and
A163: G . n = t by A145, FUNCT_1:def_3;
reconsider n = n as Nat by A162;
F . n = ((((center_of_mass V) . U) |-- (@ U)) . t) * ((t |-- A) . ab) by A146, A147, A162, A163;
then 0 < Sum F by A147, A150, A156, A159, A161, A162, RVSUM_1:85;
then A164: (((center_of_mass V) . U) |-- A) . ab > 0 by A132, A142, A143, RLAFFIN1:def_7;
Carrier (((center_of_mass V) . U) |-- BB) c= BB by RLVECT_2:def_6;
then A165: not ab in Carrier (((center_of_mass V) . U) |-- BB) by A139, A141, XBOOLE_0:def_5;
conv BB c= Affin BB by RLAFFIN1:65;
then ((center_of_mass V) . U) |-- A = ((center_of_mass V) . U) |-- BB by A7, A136, A155, RLAFFIN1:77;
hence contradiction by A165, A164; ::_thesis: verum
end;
then conv (@ U) c= conv B1 by CONVEX1:30;
then Affin (@ U) c= Affin B1 by RLAFFIN1:68;
hence contradiction by A129, A138, RLAFFIN1:79; ::_thesis: verum
end;
set XX = { S1 where S1 is Simplex of n, BCS K : ( S c= S1 & conv (@ S1) c= conv (@ U) ) } ;
A166: card { S1 where S1 is Simplex of n, BCS K : ( S c= S1 & conv (@ S1) c= conv (@ U) ) } = 2 by A16, A30, A35, A36, A129, Th43;
consider w1, w2 being set such that
w1 <> w2 and
A167: { S1 where S1 is Simplex of n, BCS K : ( S c= S1 & conv (@ S1) c= conv (@ U) ) } = {w1,w2} by A166, CARD_2:60;
w2 in { S1 where S1 is Simplex of n, BCS K : ( S c= S1 & conv (@ S1) c= conv (@ U) ) } by A167, TARSKI:def_2;
then consider W2 being Simplex of n, BCS K such that
A168: w2 = W2 and
A169: S c= W2 and
conv (@ W2) c= conv (@ U) ;
w1 in { S1 where S1 is Simplex of n, BCS K : ( S c= S1 & conv (@ S1) c= conv (@ U) ) } by A167, TARSKI:def_2;
then consider W1 being Simplex of n, BCS K such that
A170: w1 = W1 and
A171: S c= W1 and
conv (@ W1) c= conv (@ U) ;
A172: W1 in X by A11, A171;
A173: X c= { S1 where S1 is Simplex of n, BCS K : ( S c= S1 & conv (@ S1) c= conv (@ U) ) }
proof
let w be set ; :: according to TARSKI:def_3 ::_thesis: ( not w in X or w in { S1 where S1 is Simplex of n, BCS K : ( S c= S1 & conv (@ S1) c= conv (@ U) ) } )
assume w in X ; ::_thesis: w in { S1 where S1 is Simplex of n, BCS K : ( S c= S1 & conv (@ S1) c= conv (@ U) ) }
then consider W being Simplex of n, BCS K such that
A174: w = W and
A175: S c= W by A11;
(card SS1) + Z <= degree K by A3, A16, A27, A30, A33, Th33;
then consider T being finite simplex-like Subset-Family of K such that
T misses SS1 and
A176: ( T \/ SS1 is c=-linear & T \/ SS1 is with_non-empty_elements ) and
card T = Z + 1 and
A177: @ W = ((center_of_mass V) .: SS1) \/ ((center_of_mass V) .: T) by A27, A30, A33, A34, A175, Th41;
reconsider TS = T \/ SS1 as finite simplex-like Subset-Family of K by TOPS_2:13;
A178: W = (center_of_mass V) .: (@ TS) by A177, RELAT_1:120;
union TS in TS by A37, A176, SIMPLEX0:9;
then reconsider UTS = union TS as Simplex of K by TOPS_2:def_1;
card UTS <= (degree K) + 1 by SIMPLEX0:24;
then A179: card UTS <= n + 1 by A3, XXREAL_3:def_2;
A180: U c= union TS by XBOOLE_1:7, ZFMISC_1:77;
then n + 1 <= card UTS by A129, NAT_1:43;
then UTS = U by A129, A179, A180, CARD_FIN:1, XXREAL_0:1;
then conv (@ W) c= conv (@ U) by A178, CONVEX1:30, RLAFFIN2:17;
hence w in { S1 where S1 is Simplex of n, BCS K : ( S c= S1 & conv (@ S1) c= conv (@ U) ) } by A174, A175; ::_thesis: verum
end;
W2 in X by A11, A169;
then { S1 where S1 is Simplex of n, BCS K : ( S c= S1 & conv (@ S1) c= conv (@ U) ) } c= X by A167, A170, A168, A172, ZFMISC_1:32;
hence ( ( conv (@ S) meets Int A implies card X = 2 ) & ( conv (@ S) misses Int A implies card X = 1 ) ) by A130, A166, A173, XBOOLE_0:def_10; ::_thesis: verum
end;
end;
end;
end;
end;
theorem Th45: :: SIMPLEX1:45
for X being set
for n, k being Nat
for V being RealLinearSpace
for Aff being finite affinely-independent Subset of V
for S being Simplex of n - 1, BCS (k,(Complex_of {Aff})) st card Aff = n + 1 & X = { S1 where S1 is Simplex of n, BCS (k,(Complex_of {Aff})) : S c= S1 } holds
( ( conv (@ S) meets Int Aff implies card X = 2 ) & ( conv (@ S) misses Int Aff implies card X = 1 ) )
proof
let X be set ; ::_thesis: for n, k being Nat
for V being RealLinearSpace
for Aff being finite affinely-independent Subset of V
for S being Simplex of n - 1, BCS (k,(Complex_of {Aff})) st card Aff = n + 1 & X = { S1 where S1 is Simplex of n, BCS (k,(Complex_of {Aff})) : S c= S1 } holds
( ( conv (@ S) meets Int Aff implies card X = 2 ) & ( conv (@ S) misses Int Aff implies card X = 1 ) )
let n, k be Nat; ::_thesis: for V being RealLinearSpace
for Aff being finite affinely-independent Subset of V
for S being Simplex of n - 1, BCS (k,(Complex_of {Aff})) st card Aff = n + 1 & X = { S1 where S1 is Simplex of n, BCS (k,(Complex_of {Aff})) : S c= S1 } holds
( ( conv (@ S) meets Int Aff implies card X = 2 ) & ( conv (@ S) misses Int Aff implies card X = 1 ) )
let V be RealLinearSpace; ::_thesis: for Aff being finite affinely-independent Subset of V
for S being Simplex of n - 1, BCS (k,(Complex_of {Aff})) st card Aff = n + 1 & X = { S1 where S1 is Simplex of n, BCS (k,(Complex_of {Aff})) : S c= S1 } holds
( ( conv (@ S) meets Int Aff implies card X = 2 ) & ( conv (@ S) misses Int Aff implies card X = 1 ) )
let Aff be finite affinely-independent Subset of V; ::_thesis: for S being Simplex of n - 1, BCS (k,(Complex_of {Aff})) st card Aff = n + 1 & X = { S1 where S1 is Simplex of n, BCS (k,(Complex_of {Aff})) : S c= S1 } holds
( ( conv (@ S) meets Int Aff implies card X = 2 ) & ( conv (@ S) misses Int Aff implies card X = 1 ) )
let S be Simplex of n - 1, BCS (k,(Complex_of {Aff})); ::_thesis: ( card Aff = n + 1 & X = { S1 where S1 is Simplex of n, BCS (k,(Complex_of {Aff})) : S c= S1 } implies ( ( conv (@ S) meets Int Aff implies card X = 2 ) & ( conv (@ S) misses Int Aff implies card X = 1 ) ) )
assume A1: card Aff = n + 1 ; ::_thesis: ( not X = { S1 where S1 is Simplex of n, BCS (k,(Complex_of {Aff})) : S c= S1 } or ( ( conv (@ S) meets Int Aff implies card X = 2 ) & ( conv (@ S) misses Int Aff implies card X = 1 ) ) )
set C = Complex_of {Aff};
reconsider cA = card Aff as ext-real number ;
A2: cA - 1 = (card Aff) + (- 1) by XXREAL_3:def_2
.= n by A1 ;
then A3: degree (Complex_of {Aff}) = n by SIMPLEX0:26;
defpred S1[ Nat] means for S being Simplex of n - 1, BCS ($1,(Complex_of {Aff}))
for X being set st X = { S1 where S1 is Simplex of n, BCS ($1,(Complex_of {Aff})) : S c= S1 } holds
( ( conv (@ S) meets Int Aff implies card X = 2 ) & ( conv (@ S) misses Int Aff implies card X = 1 ) );
A4: ( [#] (Complex_of {Aff}) = [#] V & |.(Complex_of {Aff}).| c= [#] V ) ;
A5: S1[ 0 ]
proof
reconsider n1 = n - 1 as ext-real number ;
A6: the topology of (Complex_of {Aff}) = bool Aff by SIMPLEX0:4;
Aff in bool Aff by ZFMISC_1:def_1;
then reconsider A1 = Aff as finite Simplex of (Complex_of {Aff}) by A6, PRE_TOPC:def_2;
A7: BCS (0,(Complex_of {Aff})) = Complex_of {Aff} by A4, Th16;
let S be Simplex of n - 1, BCS (0,(Complex_of {Aff})); ::_thesis: for X being set st X = { S1 where S1 is Simplex of n, BCS (0,(Complex_of {Aff})) : S c= S1 } holds
( ( conv (@ S) meets Int Aff implies card X = 2 ) & ( conv (@ S) misses Int Aff implies card X = 1 ) )
let X be set ; ::_thesis: ( X = { S1 where S1 is Simplex of n, BCS (0,(Complex_of {Aff})) : S c= S1 } implies ( ( conv (@ S) meets Int Aff implies card X = 2 ) & ( conv (@ S) misses Int Aff implies card X = 1 ) ) )
assume A8: X = { S1 where S1 is Simplex of n, BCS (0,(Complex_of {Aff})) : S c= S1 } ; ::_thesis: ( ( conv (@ S) meets Int Aff implies card X = 2 ) & ( conv (@ S) misses Int Aff implies card X = 1 ) )
A9: X c= {Aff}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in X or x in {Aff} )
reconsider N = n as ext-real number ;
assume x in X ; ::_thesis: x in {Aff}
then consider U being Simplex of n, Complex_of {Aff} such that
A10: x = U and
S c= U by A7, A8;
A11: U in the topology of (Complex_of {Aff}) by PRE_TOPC:def_2;
card U = N + 1 by A3, SIMPLEX0:def_18
.= n + 1 by XXREAL_3:def_2 ;
then Aff = U by A1, A6, A11, CARD_FIN:1;
hence x in {Aff} by A10, TARSKI:def_1; ::_thesis: verum
end;
A12: S in bool Aff by A6, A7, PRE_TOPC:def_2;
A1 is Simplex of n, Complex_of {Aff} by A2, SIMPLEX0:48;
then Aff in X by A7, A8, A12;
then A13: X = {Aff} by A9, ZFMISC_1:33;
( n + (- 1) >= - 1 & n - 1 <= degree (Complex_of {Aff}) ) by A3, XREAL_1:31, XREAL_1:146;
then card S = n1 + 1 by A7, SIMPLEX0:def_18
.= (n - 1) + 1 by XXREAL_3:def_2 ;
then S <> Aff by A1;
then S c< Aff by A12, XBOOLE_0:def_8;
hence ( ( conv (@ S) meets Int Aff implies card X = 2 ) & ( conv (@ S) misses Int Aff implies card X = 1 ) ) by A13, CARD_1:30, RLAFFIN2:7; ::_thesis: verum
end;
A14: for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A15: S1[k] ; ::_thesis: S1[k + 1]
A16: ( degree (BCS (k,(Complex_of {Aff}))) = n & BCS ((k + 1),(Complex_of {Aff})) = BCS (BCS (k,(Complex_of {Aff}))) ) by A3, A4, Th20, Th32;
let S be Simplex of n - 1, BCS ((k + 1),(Complex_of {Aff})); ::_thesis: for X being set st X = { S1 where S1 is Simplex of n, BCS ((k + 1),(Complex_of {Aff})) : S c= S1 } holds
( ( conv (@ S) meets Int Aff implies card X = 2 ) & ( conv (@ S) misses Int Aff implies card X = 1 ) )
let X be set ; ::_thesis: ( X = { S1 where S1 is Simplex of n, BCS ((k + 1),(Complex_of {Aff})) : S c= S1 } implies ( ( conv (@ S) meets Int Aff implies card X = 2 ) & ( conv (@ S) misses Int Aff implies card X = 1 ) ) )
assume A17: X = { S1 where S1 is Simplex of n, BCS ((k + 1),(Complex_of {Aff})) : S c= S1 } ; ::_thesis: ( ( conv (@ S) meets Int Aff implies card X = 2 ) & ( conv (@ S) misses Int Aff implies card X = 1 ) )
thus ( ( conv (@ S) meets Int Aff implies card X = 2 ) & ( conv (@ S) misses Int Aff implies card X = 1 ) ) by A1, A15, A16, A17, Th44; ::_thesis: verum
end;
for k being Nat holds S1[k] from NAT_1:sch_2(A5, A14);
hence ( not X = { S1 where S1 is Simplex of n, BCS (k,(Complex_of {Aff})) : S c= S1 } or ( ( conv (@ S) meets Int Aff implies card X = 2 ) & ( conv (@ S) misses Int Aff implies card X = 1 ) ) ) ; ::_thesis: verum
end;
begin
theorem Th46: :: SIMPLEX1:46
for k being Nat
for V being RealLinearSpace
for Aff being finite affinely-independent Subset of V
for F being Function of (Vertices (BCS (k,(Complex_of {Aff})))),Aff st ( for v being Vertex of (BCS (k,(Complex_of {Aff})))
for B being Subset of V st B c= Aff & v in conv B holds
F . v in B ) holds
ex n being Nat st card { S where S is Simplex of (card Aff) - 1, BCS (k,(Complex_of {Aff})) : F .: S = Aff } = (2 * n) + 1
proof
let k be Nat; ::_thesis: for V being RealLinearSpace
for Aff being finite affinely-independent Subset of V
for F being Function of (Vertices (BCS (k,(Complex_of {Aff})))),Aff st ( for v being Vertex of (BCS (k,(Complex_of {Aff})))
for B being Subset of V st B c= Aff & v in conv B holds
F . v in B ) holds
ex n being Nat st card { S where S is Simplex of (card Aff) - 1, BCS (k,(Complex_of {Aff})) : F .: S = Aff } = (2 * n) + 1
let V be RealLinearSpace; ::_thesis: for Aff being finite affinely-independent Subset of V
for F being Function of (Vertices (BCS (k,(Complex_of {Aff})))),Aff st ( for v being Vertex of (BCS (k,(Complex_of {Aff})))
for B being Subset of V st B c= Aff & v in conv B holds
F . v in B ) holds
ex n being Nat st card { S where S is Simplex of (card Aff) - 1, BCS (k,(Complex_of {Aff})) : F .: S = Aff } = (2 * n) + 1
let Aff be finite affinely-independent Subset of V; ::_thesis: for F being Function of (Vertices (BCS (k,(Complex_of {Aff})))),Aff st ( for v being Vertex of (BCS (k,(Complex_of {Aff})))
for B being Subset of V st B c= Aff & v in conv B holds
F . v in B ) holds
ex n being Nat st card { S where S is Simplex of (card Aff) - 1, BCS (k,(Complex_of {Aff})) : F .: S = Aff } = (2 * n) + 1
reconsider O = 1 as ext-real number ;
reconsider Z = 0 as ext-real number ;
defpred S1[ Nat] means for A being finite affinely-independent Subset of V st card A = $1 holds
for F being Function of (Vertices (BCS (k,(Complex_of {A})))),A st ( for v being Vertex of (BCS (k,(Complex_of {A})))
for B being Subset of V st B c= A & v in conv B holds
F . v in B ) holds
ex n being Nat st card { S where S is Simplex of (card A) - 1, BCS (k,(Complex_of {A})) : F .: S = A } = (2 * n) + 1;
A1: for m being Nat st S1[m] holds
S1[m + 1]
proof
let m be Nat; ::_thesis: ( S1[m] implies S1[m + 1] )
assume A2: S1[m] ; ::_thesis: S1[m + 1]
let A be finite affinely-independent Subset of V; ::_thesis: ( card A = m + 1 implies for F being Function of (Vertices (BCS (k,(Complex_of {A})))),A st ( for v being Vertex of (BCS (k,(Complex_of {A})))
for B being Subset of V st B c= A & v in conv B holds
F . v in B ) holds
ex n being Nat st card { S where S is Simplex of (card A) - 1, BCS (k,(Complex_of {A})) : F .: S = A } = (2 * n) + 1 )
assume A3: card A = m + 1 ; ::_thesis: for F being Function of (Vertices (BCS (k,(Complex_of {A})))),A st ( for v being Vertex of (BCS (k,(Complex_of {A})))
for B being Subset of V st B c= A & v in conv B holds
F . v in B ) holds
ex n being Nat st card { S where S is Simplex of (card A) - 1, BCS (k,(Complex_of {A})) : F .: S = A } = (2 * n) + 1
not A is empty by A3;
then consider a being set such that
A4: a in A by XBOOLE_0:def_1;
reconsider a = a as Element of V by A4;
A5: card (A \ {a}) = m by A3, A4, STIRL2_1:55;
reconsider Aa = A \ {a} as finite affinely-independent Subset of V by RLAFFIN1:43, XBOOLE_1:36;
set CAa = Complex_of {Aa};
the topology of (Complex_of {Aa}) = bool Aa by SIMPLEX0:4;
then A6: Vertices (Complex_of {Aa}) = union (bool Aa) by SIMPLEX0:16
.= Aa by ZFMISC_1:81 ;
A7: ( [#] (Complex_of {Aa}) = [#] V & |.(Complex_of {Aa}).| c= [#] V ) ;
then A8: Vertices (Complex_of {Aa}) c= Vertices (BCS (k,(Complex_of {Aa}))) by Th24;
set CA = Complex_of {A};
let F be Function of (Vertices (BCS (k,(Complex_of {A})))),A; ::_thesis: ( ( for v being Vertex of (BCS (k,(Complex_of {A})))
for B being Subset of V st B c= A & v in conv B holds
F . v in B ) implies ex n being Nat st card { S where S is Simplex of (card A) - 1, BCS (k,(Complex_of {A})) : F .: S = A } = (2 * n) + 1 )
assume A9: for v being Vertex of (BCS (k,(Complex_of {A})))
for B being Subset of V st B c= A & v in conv B holds
F . v in B ; ::_thesis: ex n being Nat st card { S where S is Simplex of (card A) - 1, BCS (k,(Complex_of {A})) : F .: S = A } = (2 * n) + 1
set XX = { S where S is Simplex of (card A) - 1, BCS (k,(Complex_of {A})) : F .: S = A } ;
A10: { S where S is Simplex of (card A) - 1, BCS (k,(Complex_of {A})) : F .: S = A } c= the topology of (BCS (k,(Complex_of {A})))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { S where S is Simplex of (card A) - 1, BCS (k,(Complex_of {A})) : F .: S = A } or x in the topology of (BCS (k,(Complex_of {A}))) )
assume x in { S where S is Simplex of (card A) - 1, BCS (k,(Complex_of {A})) : F .: S = A } ; ::_thesis: x in the topology of (BCS (k,(Complex_of {A})))
then ex S being Simplex of (card A) - 1, BCS (k,(Complex_of {A})) st
( S = x & A = F .: S ) ;
hence x in the topology of (BCS (k,(Complex_of {A}))) by PRE_TOPC:def_2; ::_thesis: verum
end;
then reconsider XX = { S where S is Simplex of (card A) - 1, BCS (k,(Complex_of {A})) : F .: S = A } as Subset-Family of (BCS (k,(Complex_of {A}))) by XBOOLE_1:1;
reconsider XX = XX as simplex-like Subset-Family of (BCS (k,(Complex_of {A}))) by A10, SIMPLEX0:14;
A11: ( [#] (Complex_of {A}) = [#] V & |.(Complex_of {A}).| c= [#] V ) ;
A12: A \ {a} c= A by XBOOLE_1:36;
for x being set st x in {Aa} holds
ex y being set st
( y in {A} & x c= y )
proof
let x be set ; ::_thesis: ( x in {Aa} implies ex y being set st
( y in {A} & x c= y ) )
assume A13: x in {Aa} ; ::_thesis: ex y being set st
( y in {A} & x c= y )
take A ; ::_thesis: ( A in {A} & x c= A )
thus ( A in {A} & x c= A ) by A12, A13, TARSKI:def_1; ::_thesis: verum
end;
then {Aa} is_finer_than {A} by SETFAM_1:def_2;
then Complex_of {Aa} is SubSimplicialComplex of Complex_of {A} by SIMPLEX0:30;
then A14: BCS (k,(Complex_of {Aa})) is SubSimplicialComplex of BCS (k,(Complex_of {A})) by A11, A7, Th23;
then A15: Vertices (BCS (k,(Complex_of {Aa}))) c= Vertices (BCS (k,(Complex_of {A}))) by SIMPLEX0:31;
A16: the topology of (Complex_of {A}) = bool A by SIMPLEX0:4;
then A17: Vertices (Complex_of {A}) = union (bool A) by SIMPLEX0:16
.= A by ZFMISC_1:81 ;
A18: dom F = Vertices (BCS (k,(Complex_of {A}))) by A4, FUNCT_2:def_1;
percases ( m = 0 or m > 0 ) ;
supposeA19: m = 0 ; ::_thesis: ex n being Nat st card { S where S is Simplex of (card A) - 1, BCS (k,(Complex_of {A})) : F .: S = A } = (2 * n) + 1
A20: O - 1 = 0 by XXREAL_3:7;
then A21: degree (Complex_of {A}) = 0 by A3, A19, SIMPLEX0:26;
( k = 0 or k > 0 ) ;
then A22: BCS (k,(Complex_of {A})) = Complex_of {A} by A11, A21, Th16, Th22;
then A23: dom F = Vertices (Complex_of {A}) by A4, FUNCT_2:def_1;
take 0 ; ::_thesis: card { S where S is Simplex of (card A) - 1, BCS (k,(Complex_of {A})) : F .: S = A } = (2 * 0) + 1
A in bool A by ZFMISC_1:def_1;
then reconsider A1 = A as Simplex of (Complex_of {A}) by A16, PRE_TOPC:def_2;
A24: A1 is Simplex of 0 , Complex_of {A} by A3, A19, A20, SIMPLEX0:48;
ex x being set st A = {x} by A3, A19, CARD_2:42;
then A25: A = {a} by A4, TARSKI:def_1;
then conv A = A by RLAFFIN1:1;
then F . a in A by A4, A9, A17, A22;
then A26: F . a = a by A25, TARSKI:def_1;
A27: XX c= {A}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in XX or x in {A} )
assume x in XX ; ::_thesis: x in {A}
then consider S being Simplex of 0 , Complex_of {A} such that
A28: x = S and
F .: S = A by A3, A19, A22;
A29: S in the topology of (Complex_of {A}) by PRE_TOPC:def_2;
card S = Z + 1 by A21, SIMPLEX0:def_18
.= 1 by XXREAL_3:4 ;
then S = A by A3, A16, A19, A29, CARD_FIN:1;
hence x in {A} by A28, TARSKI:def_1; ::_thesis: verum
end;
F .: A = Im (F,a) by A25, RELAT_1:def_16
.= A by A4, A17, A23, A25, A26, FUNCT_1:59 ;
then A in XX by A3, A19, A24, A22;
then XX = {A} by A27, ZFMISC_1:33;
hence card { S where S is Simplex of (card A) - 1, BCS (k,(Complex_of {A})) : F .: S = A } = (2 * 0) + 1 by CARD_1:30; ::_thesis: verum
end;
supposeA30: m > 0 ; ::_thesis: ex n being Nat st card { S where S is Simplex of (card A) - 1, BCS (k,(Complex_of {A})) : F .: S = A } = (2 * n) + 1
defpred S2[ set , set ] means $1 c= $2;
set XXA = { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } ;
reconsider m1 = m - 1 as ext-real number ;
reconsider M = m as ext-real number ;
reconsider cA = card A as ext-real number ;
set YA = { S where S is Simplex of m, BCS (k,(Complex_of {A})) : Aa = F .: S } ;
A31: { S where S is Simplex of m, BCS (k,(Complex_of {A})) : Aa = F .: S } c= the topology of (BCS (k,(Complex_of {A})))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { S where S is Simplex of m, BCS (k,(Complex_of {A})) : Aa = F .: S } or x in the topology of (BCS (k,(Complex_of {A}))) )
assume x in { S where S is Simplex of m, BCS (k,(Complex_of {A})) : Aa = F .: S } ; ::_thesis: x in the topology of (BCS (k,(Complex_of {A})))
then ex S being Simplex of m, BCS (k,(Complex_of {A})) st
( S = x & Aa = F .: S ) ;
hence x in the topology of (BCS (k,(Complex_of {A}))) by PRE_TOPC:def_2; ::_thesis: verum
end;
then reconsider YA = { S where S is Simplex of m, BCS (k,(Complex_of {A})) : Aa = F .: S } as Subset-Family of (BCS (k,(Complex_of {A}))) by XBOOLE_1:1;
reconsider YA = YA as simplex-like Subset-Family of (BCS (k,(Complex_of {A}))) by A31, SIMPLEX0:14;
defpred S3[ set , set ] means $2 c= $1;
set Xm1 = { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : Aa = F .: S } ;
set Xm = { S where S is Simplex of m, BCS (k,(Complex_of {A})) : verum } ;
consider R1 being Relation such that
A32: for x, y being set holds
( [x,y] in R1 iff ( x in { S where S is Simplex of m, BCS (k,(Complex_of {A})) : verum } & y in { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : Aa = F .: S } & S3[x,y] ) ) from RELAT_1:sch_1();
set DY = (dom R1) \ YA;
A33: (dom R1) \ YA c= XX
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (dom R1) \ YA or x in XX )
assume A34: x in (dom R1) \ YA ; ::_thesis: x in XX
then consider y being set such that
A35: [x,y] in R1 by XTUPLE_0:def_12;
x in { S where S is Simplex of m, BCS (k,(Complex_of {A})) : verum } by A32, A35;
then consider S being Simplex of m, BCS (k,(Complex_of {A})) such that
A36: x = S and
verum ;
not x in YA by A34, XBOOLE_0:def_5;
then A37: F .: S <> Aa by A36;
y in { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : Aa = F .: S } by A32, A35;
then A38: ex W being Simplex of m - 1, BCS (k,(Complex_of {A})) st
( y = W & Aa = F .: W ) ;
y c= x by A32, A35;
then Aa c= F .: S by A36, A38, RELAT_1:123;
then Aa c< F .: S by A37, XBOOLE_0:def_8;
then m < card (F .: S) by A5, CARD_2:48;
then A39: m + 1 <= card (F .: S) by NAT_1:13;
card (F .: S) <= m + 1 by A3, NAT_1:43;
then F .: S = A by A3, A39, CARD_FIN:1, XXREAL_0:1;
hence x in XX by A3, A36; ::_thesis: verum
end;
set RDY = R1 | ((dom R1) \ YA);
A40: (R1 | ((dom R1) \ YA)) | ((dom (R1 | ((dom R1) \ YA))) \ ((dom R1) \ YA)) = {}
proof
assume (R1 | ((dom R1) \ YA)) | ((dom (R1 | ((dom R1) \ YA))) \ ((dom R1) \ YA)) <> {} ; ::_thesis: contradiction
then consider xy being set such that
A41: xy in (R1 | ((dom R1) \ YA)) | ((dom (R1 | ((dom R1) \ YA))) \ ((dom R1) \ YA)) by XBOOLE_0:def_1;
consider x, y being set such that
A42: xy = [x,y] by A41, RELAT_1:def_1;
A43: x in (dom (R1 | ((dom R1) \ YA))) \ ((dom R1) \ YA) by A41, A42, RELAT_1:def_11;
then ( dom (R1 | ((dom R1) \ YA)) c= (dom R1) \ YA & x in dom (R1 | ((dom R1) \ YA)) ) by RELAT_1:58;
hence contradiction by A43, XBOOLE_0:def_5; ::_thesis: verum
end;
A44: 2 *` (card YA) = (card 2) *` (card (card YA)) by Lm1
.= card (2 * (card YA)) by CARD_2:39 ;
A45: 2 * (card YA) in NAT by ORDINAL1:def_12;
cA - 1 = (m + 1) + (- 1) by A3, XXREAL_3:def_2;
then A46: degree (Complex_of {A}) = m by SIMPLEX0:26;
consider R being Relation such that
A47: for x, y being set holds
( [x,y] in R iff ( x in { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : Aa = F .: S } & y in { S where S is Simplex of m, BCS (k,(Complex_of {A})) : verum } & S2[x,y] ) ) from RELAT_1:sch_1();
A48: card R = card R1
proof
deffunc H1( set ) -> set = [($1 `2),($1 `1)];
A49: for x being set st x in R holds
H1(x) in R1
proof
let z be set ; ::_thesis: ( z in R implies H1(z) in R1 )
assume A50: z in R ; ::_thesis: H1(z) in R1
then ex x, y being set st z = [x,y] by RELAT_1:def_1;
then A51: z = [(z `1),(z `2)] by MCART_1:8;
then A52: z `2 in { S where S is Simplex of m, BCS (k,(Complex_of {A})) : verum } by A47, A50;
( S2[z `1 ,z `2 ] & z `1 in { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : Aa = F .: S } ) by A47, A50, A51;
hence H1(z) in R1 by A32, A52; ::_thesis: verum
end;
consider f being Function of R,R1 such that
A53: for x being set st x in R holds
f . x = H1(x) from FUNCT_2:sch_2(A49);
percases ( R1 is empty or not R1 is empty ) ;
supposeA54: R1 is empty ; ::_thesis: card R = card R1
R is empty
proof
assume not R is empty ; ::_thesis: contradiction
then ex x being set st x in R by XBOOLE_0:def_1;
hence contradiction by A49, A54; ::_thesis: verum
end;
hence card R = card R1 by A54; ::_thesis: verum
end;
suppose not R1 is empty ; ::_thesis: card R = card R1
then A55: dom f = R by FUNCT_2:def_1;
R1 c= rng f
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in R1 or z in rng f )
assume A56: z in R1 ; ::_thesis: z in rng f
then ex x, y being set st z = [x,y] by RELAT_1:def_1;
then A57: z = [(z `1),(z `2)] by MCART_1:8;
then A58: z `2 in { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : Aa = F .: S } by A32, A56;
( S3[z `1 ,z `2 ] & z `1 in { S where S is Simplex of m, BCS (k,(Complex_of {A})) : verum } ) by A32, A56, A57;
then A59: [(z `2),(z `1)] in R by A47, A58;
H1([(z `2),(z `1)]) = z by A57;
then z = f . [(z `2),(z `1)] by A53, A59;
hence z in rng f by A55, A59, FUNCT_1:def_3; ::_thesis: verum
end;
then A60: rng f = R1 by XBOOLE_0:def_10;
now__::_thesis:_for_x1,_x2_being_set_st_x1_in_R_&_x2_in_R_&_f_._x1_=_f_._x2_holds_
x1_=_x2
let x1, x2 be set ; ::_thesis: ( x1 in R & x2 in R & f . x1 = f . x2 implies x1 = x2 )
assume that
A61: x1 in R and
A62: x2 in R and
A63: f . x1 = f . x2 ; ::_thesis: x1 = x2
( f . x1 = H1(x1) & f . x2 = H1(x2) ) by A53, A61, A62;
then A64: ( x1 `2 = x2 `2 & x1 `1 = x2 `1 ) by A63, XTUPLE_0:1;
A65: ex x, y being set st x2 = [x,y] by A62, RELAT_1:def_1;
ex x, y being set st x1 = [x,y] by A61, RELAT_1:def_1;
hence x1 = [(x2 `1),(x2 `2)] by A64, MCART_1:8
.= x2 by A65, MCART_1:8 ;
::_thesis: verum
end;
then f is one-to-one by A55, FUNCT_1:def_4;
then R,R1 are_equipotent by A55, A60, WELLORD2:def_4;
hence card R = card R1 by CARD_1:5; ::_thesis: verum
end;
end;
end;
A66: ( |.(BCS (k,(Complex_of {Aa}))).| = |.(Complex_of {Aa}).| & |.(Complex_of {Aa}).| = conv Aa ) by Th8, Th10;
set DX = (dom R) \ { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } ;
A67: (dom R) \ { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } c= the topology of (BCS (k,(Complex_of {A})))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (dom R) \ { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } or x in the topology of (BCS (k,(Complex_of {A}))) )
assume x in (dom R) \ { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } ; ::_thesis: x in the topology of (BCS (k,(Complex_of {A})))
then ex y being set st [x,y] in R by XTUPLE_0:def_12;
then x in { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : Aa = F .: S } by A47;
then ex S being Simplex of m - 1, BCS (k,(Complex_of {A})) st
( S = x & Aa = F .: S ) ;
hence x in the topology of (BCS (k,(Complex_of {A}))) by PRE_TOPC:def_2; ::_thesis: verum
end;
set RDX = R | ((dom R) \ { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } );
reconsider DX = (dom R) \ { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } as Subset-Family of (BCS (k,(Complex_of {A}))) by A67, XBOOLE_1:1;
reconsider DX = DX as simplex-like Subset-Family of (BCS (k,(Complex_of {A}))) by A67, SIMPLEX0:14;
A68: (R | ((dom R) \ { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } )) | ((dom (R | ((dom R) \ { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } ))) \ DX) = {}
proof
assume (R | ((dom R) \ { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } )) | ((dom (R | ((dom R) \ { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } ))) \ DX) <> {} ; ::_thesis: contradiction
then consider xy being set such that
A69: xy in (R | ((dom R) \ { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } )) | ((dom (R | ((dom R) \ { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } ))) \ DX) by XBOOLE_0:def_1;
consider x, y being set such that
A70: xy = [x,y] by A69, RELAT_1:def_1;
A71: x in (dom (R | ((dom R) \ { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } ))) \ DX by A69, A70, RELAT_1:def_11;
then ( dom (R | ((dom R) \ { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } )) c= DX & x in dom (R | ((dom R) \ { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } )) ) by RELAT_1:58;
hence contradiction by A71, XBOOLE_0:def_5; ::_thesis: verum
end;
A72: m1 + 1 = (m - 1) + 1 by XXREAL_3:def_2
.= m ;
set FA = F | (Vertices (BCS (k,(Complex_of {Aa}))));
A73: dom (F | (Vertices (BCS (k,(Complex_of {Aa}))))) = Vertices (BCS (k,(Complex_of {Aa}))) by A18, A14, RELAT_1:62, SIMPLEX0:31;
A74: not Vertices (BCS (k,(Complex_of {Aa}))) is empty by A5, A6, A8, A30;
A75: for v being Vertex of (BCS (k,(Complex_of {Aa})))
for B being Subset of V st B c= Aa & v in conv B holds
(F | (Vertices (BCS (k,(Complex_of {Aa}))))) . v in B
proof
let v be Vertex of (BCS (k,(Complex_of {Aa}))); ::_thesis: for B being Subset of V st B c= Aa & v in conv B holds
(F | (Vertices (BCS (k,(Complex_of {Aa}))))) . v in B
let B be Subset of V; ::_thesis: ( B c= Aa & v in conv B implies (F | (Vertices (BCS (k,(Complex_of {Aa}))))) . v in B )
assume A76: ( B c= Aa & v in conv B ) ; ::_thesis: (F | (Vertices (BCS (k,(Complex_of {Aa}))))) . v in B
v in Vertices (BCS (k,(Complex_of {Aa}))) by A74;
then F . v in B by A9, A12, A15, A76, XBOOLE_1:1;
hence (F | (Vertices (BCS (k,(Complex_of {Aa}))))) . v in B by A73, A74, FUNCT_1:47; ::_thesis: verum
end;
rng (F | (Vertices (BCS (k,(Complex_of {Aa}))))) c= Aa
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (F | (Vertices (BCS (k,(Complex_of {Aa}))))) or y in Aa )
assume y in rng (F | (Vertices (BCS (k,(Complex_of {Aa}))))) ; ::_thesis: y in Aa
then consider x being set such that
A77: x in dom (F | (Vertices (BCS (k,(Complex_of {Aa}))))) and
A78: (F | (Vertices (BCS (k,(Complex_of {Aa}))))) . x = y by FUNCT_1:def_3;
reconsider v = x as Element of (BCS (k,(Complex_of {Aa}))) by A73, A77;
v is vertex-like by A73, A77, SIMPLEX0:def_4;
then consider S being Subset of (BCS (k,(Complex_of {Aa}))) such that
A79: S is simplex-like and
A80: v in S by SIMPLEX0:def_3;
A81: conv (@ S) c= |.(BCS (k,(Complex_of {Aa}))).| by A79, Th5;
S c= conv (@ S) by RLAFFIN1:2;
then A82: v in conv (@ S) by A80;
x in Vertices (BCS (k,(Complex_of {Aa}))) by A18, A14, A77, RELAT_1:62, SIMPLEX0:31;
hence y in Aa by A66, A75, A78, A81, A82; ::_thesis: verum
end;
then reconsider FA = F | (Vertices (BCS (k,(Complex_of {Aa})))) as Function of (Vertices (BCS (k,(Complex_of {Aa})))),Aa by A73, FUNCT_2:2;
set XXa = { S where S is Simplex of m - 1, BCS (k,(Complex_of {Aa})) : FA .: S = Aa } ;
consider n being Nat such that
A83: card { S where S is Simplex of m - 1, BCS (k,(Complex_of {Aa})) : FA .: S = Aa } = (2 * n) + 1 by A2, A5, A75;
A84: ( m - 1 <= m - 0 & - 1 <= m + (- 1) ) by XREAL_1:10, XREAL_1:31;
A85: for x being set st x in { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } holds
card (Im (R,x)) = 1
proof
let x be set ; ::_thesis: ( x in { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } implies card (Im (R,x)) = 1 )
assume x in { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } ; ::_thesis: card (Im (R,x)) = 1
then consider S being Simplex of m - 1, BCS (k,(Complex_of {A})) such that
A86: x = S and
A87: F .: S = Aa and
A88: conv (@ S) misses Int A ;
set XX = { S1 where S1 is Simplex of m, BCS (k,(Complex_of {A})) : S c= S1 } ;
A89: R .: {S} c= { S1 where S1 is Simplex of m, BCS (k,(Complex_of {A})) : S c= S1 }
proof
let w be set ; :: according to TARSKI:def_3 ::_thesis: ( not w in R .: {S} or w in { S1 where S1 is Simplex of m, BCS (k,(Complex_of {A})) : S c= S1 } )
assume w in R .: {S} ; ::_thesis: w in { S1 where S1 is Simplex of m, BCS (k,(Complex_of {A})) : S c= S1 }
then consider s being set such that
A90: [s,w] in R and
A91: s in {S} by RELAT_1:def_13;
w in { S where S is Simplex of m, BCS (k,(Complex_of {A})) : verum } by A47, A90;
then A92: ex W being Simplex of m, BCS (k,(Complex_of {A})) st w = W ;
( s = S & s c= w ) by A47, A90, A91, TARSKI:def_1;
hence w in { S1 where S1 is Simplex of m, BCS (k,(Complex_of {A})) : S c= S1 } by A92; ::_thesis: verum
end;
{ S1 where S1 is Simplex of m, BCS (k,(Complex_of {A})) : S c= S1 } c= R .: {S}
proof
let w be set ; :: according to TARSKI:def_3 ::_thesis: ( not w in { S1 where S1 is Simplex of m, BCS (k,(Complex_of {A})) : S c= S1 } or w in R .: {S} )
assume w in { S1 where S1 is Simplex of m, BCS (k,(Complex_of {A})) : S c= S1 } ; ::_thesis: w in R .: {S}
then consider W being Simplex of m, BCS (k,(Complex_of {A})) such that
A93: w = W and
A94: S c= W ;
( W in { S where S is Simplex of m, BCS (k,(Complex_of {A})) : verum } & S in { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : Aa = F .: S } ) by A87;
then ( S in {S} & [S,W] in R ) by A47, A94, TARSKI:def_1;
hence w in R .: {S} by A93, RELAT_1:def_13; ::_thesis: verum
end;
then A95: R .: {S} = { S1 where S1 is Simplex of m, BCS (k,(Complex_of {A})) : S c= S1 } by A89, XBOOLE_0:def_10;
card { S1 where S1 is Simplex of m, BCS (k,(Complex_of {A})) : S c= S1 } = 1 by A3, A88, Th45;
hence card (Im (R,x)) = 1 by A86, A95, RELAT_1:def_16; ::_thesis: verum
end;
A96: degree (Complex_of {A}) = degree (BCS (k,(Complex_of {A}))) by A11, Th32;
A97: M + 1 = m + 1 by XXREAL_3:def_2;
A98: for x being set st x in YA holds
card (Im (R1,x)) = 2
proof
let x be set ; ::_thesis: ( x in YA implies card (Im (R1,x)) = 2 )
assume x in YA ; ::_thesis: card (Im (R1,x)) = 2
then consider S being Simplex of m, BCS (k,(Complex_of {A})) such that
A99: x = S and
A100: Aa = F .: S ;
set FS = F | S;
A101: rng (F | S) = Aa by A100, RELAT_1:115;
A102: not Aa is empty by A5, A30;
A103: S in {x} by A99, TARSKI:def_1;
A104: dom (F | S) = S by A18, RELAT_1:62, SIMPLEX0:17;
A105: card S = m + 1 by A96, A46, A97, SIMPLEX0:def_18;
reconsider FS = F | S as Function of S,Aa by A101, A104, FUNCT_2:1;
FS is onto by A101, FUNCT_2:def_3;
then consider b being set such that
A106: b in Aa and
A107: card (FS " {b}) = 2 and
A108: for x being set st x in Aa & x <> b holds
card (FS " {x}) = 1 by A5, A102, A105, Th2;
consider a1, a2 being set such that
A109: a1 <> a2 and
A110: FS " {b} = {a1,a2} by A107, CARD_2:60;
reconsider S1 = S \ {a1}, S2 = S \ {a2} as Simplex of (BCS (k,(Complex_of {A}))) ;
A111: a1 in {a1,a2} by TARSKI:def_2;
then A112: a1 in S2 by A109, A110, ZFMISC_1:56;
A113: card S1 = m by A105, A110, A111, STIRL2_1:55;
A114: a2 in {a1,a2} by TARSKI:def_2;
then A115: card S2 = m by A105, A110, STIRL2_1:55;
then reconsider S1 = S1, S2 = S2 as Simplex of m - 1, BCS (k,(Complex_of {A})) by A96, A84, A72, A46, A113, SIMPLEX0:def_18;
A116: {a1} c= S by A110, A111, ZFMISC_1:31;
A117: FS . a2 = F . a2 by A104, A110, A114, FUNCT_1:47;
A118: {a2} c= S by A110, A114, ZFMISC_1:31;
A119: R1 .: {x} c= {S1,S2}
proof
let Y be set ; :: according to TARSKI:def_3 ::_thesis: ( not Y in R1 .: {x} or Y in {S1,S2} )
assume Y in R1 .: {x} ; ::_thesis: Y in {S1,S2}
then consider X being set such that
A120: [X,Y] in R1 and
A121: X in {x} by RELAT_1:def_13;
Y in { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : Aa = F .: S } by A32, A120;
then consider W being Simplex of m - 1, BCS (k,(Complex_of {A})) such that
A122: Y = W and
A123: Aa = F .: W ;
X = x by A121, TARSKI:def_1;
then W c= S by A32, A99, A120, A122;
then A124: Aa = FS .: W by A123, RELAT_1:129;
then consider w being set such that
A125: w in dom FS and
A126: w in W and
A127: FS . w = b by A106, FUNCT_1:def_6;
A128: {w} c= W by A126, ZFMISC_1:31;
A129: S \ {a1,a2} c= W
proof
let s be set ; :: according to TARSKI:def_3 ::_thesis: ( not s in S \ {a1,a2} or s in W )
assume A130: s in S \ {a1,a2} ; ::_thesis: s in W
then A131: s in dom FS by A104, XBOOLE_0:def_5;
then A132: FS . s in Aa by A101, FUNCT_1:def_3;
then consider w being set such that
A133: w in dom FS and
A134: w in W and
A135: FS . w = FS . s by A124, FUNCT_1:def_6;
not s in FS " {b} by A110, A130, XBOOLE_0:def_5;
then not FS . s in {b} by A131, FUNCT_1:def_7;
then FS . s <> b by TARSKI:def_1;
then card (FS " {(FS . s)}) = 1 by A108, A132;
then consider z being set such that
A136: FS " {(FS . s)} = {z} by CARD_2:42;
A137: FS . s in {(FS . s)} by TARSKI:def_1;
then A138: s in FS " {(FS . s)} by A131, FUNCT_1:def_7;
w in FS " {(FS . s)} by A133, A135, A137, FUNCT_1:def_7;
then w = z by A136, TARSKI:def_1;
hence s in W by A134, A136, A138, TARSKI:def_1; ::_thesis: verum
end;
b in {b} by TARSKI:def_1;
then A139: w in FS " {b} by A125, A127, FUNCT_1:def_7;
A140: card W = m by A96, A84, A72, A46, SIMPLEX0:def_18;
A141: S /\ {a1} = {a1} by A116, XBOOLE_1:28;
A142: S /\ {a2} = {a2} by A118, XBOOLE_1:28;
percases ( w = a1 or w = a2 ) by A110, A139, TARSKI:def_2;
suppose w = a1 ; ::_thesis: Y in {S1,S2}
then (S \ {a1,a2}) \/ {w} = S \ ({a1,a2} \ {a1}) by A141, XBOOLE_1:52
.= S2 by A109, ZFMISC_1:17 ;
then S2 = W by A115, A128, A129, A140, CARD_FIN:1, XBOOLE_1:8;
hence Y in {S1,S2} by A122, TARSKI:def_2; ::_thesis: verum
end;
suppose w = a2 ; ::_thesis: Y in {S1,S2}
then (S \ {a1,a2}) \/ {w} = S \ ({a1,a2} \ {a2}) by A142, XBOOLE_1:52
.= S1 by A109, ZFMISC_1:17 ;
then S1 = W by A113, A128, A129, A140, CARD_FIN:1, XBOOLE_1:8;
hence Y in {S1,S2} by A122, TARSKI:def_2; ::_thesis: verum
end;
end;
end;
A143: S c= dom F by A18, SIMPLEX0:17;
A144: FS . a1 = F . a1 by A104, A110, A111, FUNCT_1:47;
A145: FS . a1 in {b} by A110, A111, FUNCT_1:def_7;
then A146: FS . a1 = b by TARSKI:def_1;
A147: FS . a2 in {b} by A110, A114, FUNCT_1:def_7;
then A148: FS . a2 = b by TARSKI:def_1;
A149: ( a2 in S & a2 in S1 ) by A109, A110, A114, ZFMISC_1:56;
A150: Aa c= F .: S1
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in Aa or z in F .: S1 )
assume A151: z in Aa ; ::_thesis: z in F .: S1
percases ( z = b or z <> b ) ;
supposeA152: z = b ; ::_thesis: z in F .: S1
FS . a2 in F .: S1 by A143, A117, A149, FUNCT_1:def_6;
hence z in F .: S1 by A147, A152, TARSKI:def_1; ::_thesis: verum
end;
supposeA153: z <> b ; ::_thesis: z in F .: S1
consider c being set such that
A154: c in dom F and
A155: c in S and
A156: z = F . c by A100, A151, FUNCT_1:def_6;
c in S1 by A144, A146, A153, A155, A156, ZFMISC_1:56;
hence z in F .: S1 by A154, A156, FUNCT_1:def_6; ::_thesis: verum
end;
end;
end;
A157: S in { S where S is Simplex of m, BCS (k,(Complex_of {A})) : verum } ;
A158: ( a1 in S & a1 in S2 ) by A109, A110, A111, ZFMISC_1:56;
A159: Aa c= F .: S2
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in Aa or z in F .: S2 )
assume A160: z in Aa ; ::_thesis: z in F .: S2
percases ( z = b or z <> b ) ;
supposeA161: z = b ; ::_thesis: z in F .: S2
FS . a1 in F .: S2 by A144, A143, A158, FUNCT_1:def_6;
hence z in F .: S2 by A145, A161, TARSKI:def_1; ::_thesis: verum
end;
supposeA162: z <> b ; ::_thesis: z in F .: S2
consider c being set such that
A163: c in dom F and
A164: c in S and
A165: z = F . c by A100, A160, FUNCT_1:def_6;
c in S2 by A117, A148, A162, A164, A165, ZFMISC_1:56;
hence z in F .: S2 by A163, A165, FUNCT_1:def_6; ::_thesis: verum
end;
end;
end;
F .: S1 c= Aa by A100, RELAT_1:123, XBOOLE_1:36;
then Aa = F .: S1 by A150, XBOOLE_0:def_10;
then ( S \ {a1} c= S & S1 in { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : Aa = F .: S } ) by XBOOLE_1:36;
then [S,S1] in R1 by A32, A157;
then A166: S1 in R1 .: {x} by A103, RELAT_1:def_13;
F .: S2 c= Aa by A100, RELAT_1:123, XBOOLE_1:36;
then Aa = F .: S2 by A159, XBOOLE_0:def_10;
then ( S \ {a2} c= S & S2 in { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : Aa = F .: S } ) by XBOOLE_1:36;
then [S,S2] in R1 by A32, A157;
then S2 in R1 .: {x} by A103, RELAT_1:def_13;
then {S1,S2} c= R1 .: {x} by A166, ZFMISC_1:32;
then A167: R1 .: {x} = {S1,S2} by A119, XBOOLE_0:def_10;
S1 <> S2 by A112, ZFMISC_1:56;
then card (R1 .: {x}) = 2 by A167, CARD_2:57;
hence card (Im (R1,x)) = 2 by RELAT_1:def_16; ::_thesis: verum
end;
A168: M - 1 = m + (- 1) by XXREAL_3:def_2;
XX c= (dom R1) \ YA
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in XX or x in (dom R1) \ YA )
assume x in XX ; ::_thesis: x in (dom R1) \ YA
then consider S being Simplex of m, BCS (k,(Complex_of {A})) such that
A169: x = S and
A170: F .: S = A by A3;
set FS = F | S;
A171: rng (F | S) = A by A170, RELAT_1:115;
A172: card A = card S by A3, A96, A46, A97, SIMPLEX0:def_18;
A173: dom (F | S) = S by A18, RELAT_1:62, SIMPLEX0:17;
then reconsider FS = F | S as Function of S,A by A171, FUNCT_2:1;
consider s being set such that
A174: ( s in dom FS & FS . s = a ) by A4, A171, FUNCT_1:def_3;
set Ss = S \ {s};
FS is onto by A171, FUNCT_2:def_3;
then A175: FS is one-to-one by A172, STIRL2_1:60;
then A176: FS .: (S \ {s}) = (FS .: S) \ (FS .: {s}) by FUNCT_1:64
.= A \ (FS .: {s}) by A171, A173, RELAT_1:113
.= A \ (Im (FS,s)) by RELAT_1:def_16
.= Aa by A174, FUNCT_1:59 ;
S \ {s},FS .: (S \ {s}) are_equipotent by A173, A175, CARD_1:33, XBOOLE_1:36;
then A177: card (S \ {s}) = m by A5, A176, CARD_1:5;
reconsider Ss = S \ {s} as Simplex of (BCS (k,(Complex_of {A}))) ;
reconsider Ss = Ss as Simplex of m - 1, BCS (k,(Complex_of {A})) by A168, A177, SIMPLEX0:48;
FS .: Ss = F .: Ss by RELAT_1:129, XBOOLE_1:36;
then A178: Ss in { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : Aa = F .: S } by A176;
( Ss c= S & S in { S where S is Simplex of m, BCS (k,(Complex_of {A})) : verum } ) by XBOOLE_1:36;
then [S,Ss] in R1 by A32, A178;
then A179: S in dom R1 by XTUPLE_0:def_12;
for W being Simplex of m, BCS (k,(Complex_of {A})) st S = W holds
Aa <> F .: W by A4, A170, ZFMISC_1:56;
then not S in YA ;
hence x in (dom R1) \ YA by A169, A179, XBOOLE_0:def_5; ::_thesis: verum
end;
then A180: (dom R1) \ YA = XX by A33, XBOOLE_0:def_10;
for x being set st x in (dom R1) \ YA holds
card (Im ((R1 | ((dom R1) \ YA)),x)) = 1
proof
let x be set ; ::_thesis: ( x in (dom R1) \ YA implies card (Im ((R1 | ((dom R1) \ YA)),x)) = 1 )
assume A181: x in (dom R1) \ YA ; ::_thesis: card (Im ((R1 | ((dom R1) \ YA)),x)) = 1
then consider y being set such that
A182: [x,y] in R1 by XTUPLE_0:def_12;
A183: ex W being Simplex of m, BCS (k,(Complex_of {A})) st
( x = W & F .: W = A ) by A3, A180, A181;
x in { S where S is Simplex of m, BCS (k,(Complex_of {A})) : verum } by A32, A182;
then consider S being Simplex of m, BCS (k,(Complex_of {A})) such that
A184: x = S and
verum ;
y in { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : Aa = F .: S } by A32, A182;
then consider W being Simplex of m - 1, BCS (k,(Complex_of {A})) such that
A185: y = W and
A186: Aa = F .: W ;
A187: card S = m + 1 by A96, A46, A97, SIMPLEX0:def_18;
A188: (R1 | ((dom R1) \ YA)) .: {x} c= {y}
proof
let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in (R1 | ((dom R1) \ YA)) .: {x} or u in {y} )
set FS = F | S;
assume u in (R1 | ((dom R1) \ YA)) .: {x} ; ::_thesis: u in {y}
then consider s being set such that
A189: [s,u] in R1 | ((dom R1) \ YA) and
A190: s in {x} by RELAT_1:def_13;
A191: [s,u] in R1 by A189, RELAT_1:def_11;
then u in { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : Aa = F .: S } by A32;
then consider U being Simplex of m - 1, BCS (k,(Complex_of {A})) such that
A192: u = U and
A193: Aa = F .: U ;
A194: dom (F | S) = S by A18, RELAT_1:62, SIMPLEX0:17;
A195: rng (F | S) = A by A183, A184, RELAT_1:115;
then reconsider FS = F | S as Function of S,A by A194, FUNCT_2:1;
A196: W c= S by A32, A182, A184, A185;
then A197: FS .: W = F .: W by RELAT_1:129;
s = S by A184, A190, TARSKI:def_1;
then A198: U c= S by A32, A191, A192;
then A199: FS .: U = F .: U by RELAT_1:129;
FS is onto by A195, FUNCT_2:def_3;
then A200: FS is one-to-one by A3, A187, STIRL2_1:60;
then A201: U c= W by A186, A193, A194, A197, A198, A199, FUNCT_1:87;
W c= U by A186, A193, A194, A196, A197, A199, A200, FUNCT_1:87;
then u = y by A185, A192, A201, XBOOLE_0:def_10;
hence u in {y} by TARSKI:def_1; ::_thesis: verum
end;
( x in {x} & [x,y] in R1 | ((dom R1) \ YA) ) by A181, A182, RELAT_1:def_11, TARSKI:def_1;
then y in (R1 | ((dom R1) \ YA)) .: {x} by RELAT_1:def_13;
then (R1 | ((dom R1) \ YA)) .: {x} = {y} by A188, ZFMISC_1:33;
then Im ((R1 | ((dom R1) \ YA)),x) = {y} by RELAT_1:def_16;
hence card (Im ((R1 | ((dom R1) \ YA)),x)) = 1 by CARD_1:30; ::_thesis: verum
end;
then card (R1 | ((dom R1) \ YA)) = (card {}) +` (1 *` (card ((dom R1) \ YA))) by A40, Th1
.= 1 *` (card ((dom R1) \ YA)) by CARD_2:18
.= card ((dom R1) \ YA) by CARD_2:21 ;
then A202: card R1 = (card (card XX)) +` (card (2 * (card YA))) by A44, A98, A180, Th1
.= card ((card XX) + (2 * (card YA))) by A45, CARD_2:38
.= (card XX) + (2 * (card YA)) by Lm1 ;
A203: ( |.(BCS (k,(Complex_of {A}))).| = |.(Complex_of {A}).| & |.(Complex_of {A}).| = conv A ) by Th8, Th10;
A204: { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } c= { S where S is Simplex of m - 1, BCS (k,(Complex_of {Aa})) : FA .: S = Aa }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } or x in { S where S is Simplex of m - 1, BCS (k,(Complex_of {Aa})) : FA .: S = Aa } )
assume x in { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } ; ::_thesis: x in { S where S is Simplex of m - 1, BCS (k,(Complex_of {Aa})) : FA .: S = Aa }
then consider S being Simplex of m - 1, BCS (k,(Complex_of {A})) such that
A205: x = S and
A206: F .: S = Aa and
A207: conv (@ S) misses Int A ;
conv (@ S) c= conv A by A203, Th5;
then consider B being Subset of V such that
A208: B c< A and
A209: conv (@ S) c= conv B by A4, A207, RLAFFIN2:23;
A210: B c= A by A208, XBOOLE_0:def_8;
then reconsider B = B as finite Subset of V ;
card B < m + 1 by A3, A208, CARD_2:48;
then A211: card B <= m by NAT_1:13;
A212: Aa c= B
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in Aa or y in B )
assume y in Aa ; ::_thesis: y in B
then consider v being set such that
A213: v in dom F and
A214: v in S and
A215: F . v = y by A206, FUNCT_1:def_6;
S c= conv (@ S) by RLAFFIN1:2;
then v in conv (@ S) by A214;
hence y in B by A9, A209, A210, A213, A215; ::_thesis: verum
end;
then card Aa <= card B by NAT_1:43;
then A216: Aa = B by A5, A211, A212, CARD_FIN:1, XXREAL_0:1;
A217: the topology of (BCS (k,(Complex_of {Aa}))) c= the topology of (BCS (k,(Complex_of {A}))) by A14, SIMPLEX0:def_13;
A218: card S = m by A96, A84, A72, A46, SIMPLEX0:def_18;
then not S is empty by A30;
then A219: (center_of_mass V) . S in Int (@ S) by RLAFFIN2:20;
Int (@ S) c= conv (@ S) by RLAFFIN2:5;
then (center_of_mass V) . S in conv (@ S) by A219;
then consider w being Subset of (BCS (k,(Complex_of {Aa}))) such that
A220: w is simplex-like and
A221: (center_of_mass V) . S in conv (@ w) by A66, A209, A216, Def3;
w in the topology of (BCS (k,(Complex_of {Aa}))) by A220, PRE_TOPC:def_2;
then w in the topology of (BCS (k,(Complex_of {A}))) by A217;
then reconsider W = w as Simplex of (BCS (k,(Complex_of {A}))) by PRE_TOPC:def_2;
Int (@ S) meets conv (@ W) by A219, A221, XBOOLE_0:3;
then A222: S c= w by Th26;
then reconsider s = S as Subset of (BCS (k,(Complex_of {Aa}))) by XBOOLE_1:1;
reconsider s = s as Simplex of (BCS (k,(Complex_of {Aa}))) by A220, A222, MATROID0:1;
A223: FA .: s = Aa by A206, RELAT_1:129, SIMPLEX0:17;
s is Simplex of m - 1, BCS (k,(Complex_of {Aa})) by A168, A218, SIMPLEX0:48;
hence x in { S where S is Simplex of m - 1, BCS (k,(Complex_of {Aa})) : FA .: S = Aa } by A205, A223; ::_thesis: verum
end;
A224: degree (Complex_of {Aa}) = m - 1 by A5, A168, SIMPLEX0:26;
{ S where S is Simplex of m - 1, BCS (k,(Complex_of {Aa})) : FA .: S = Aa } c= { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) }
proof
A <> Aa by A3, A5;
then A225: Aa c< A by A12, XBOOLE_0:def_8;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { S where S is Simplex of m - 1, BCS (k,(Complex_of {Aa})) : FA .: S = Aa } or x in { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } )
assume x in { S where S is Simplex of m - 1, BCS (k,(Complex_of {Aa})) : FA .: S = Aa } ; ::_thesis: x in { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) }
then consider S being Simplex of m - 1, BCS (k,(Complex_of {Aa})) such that
A226: x = S and
A227: FA .: S = Aa ;
m - 1 <= degree (BCS (k,(Complex_of {Aa}))) by A7, A224, Th32;
then reconsider S1 = x as Simplex of m - 1, BCS (k,(Complex_of {A})) by A14, A226, SIMPLEX0:49;
A228: FA .: S = F .: S by RELAT_1:129, SIMPLEX0:17;
conv (@ S) c= conv Aa by A66, Th5;
then conv (@ S1) misses Int A by A225, A226, RLAFFIN2:7, XBOOLE_1:63;
hence x in { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } by A226, A227, A228; ::_thesis: verum
end;
then A229: { S where S is Simplex of m - 1, BCS (k,(Complex_of {Aa})) : FA .: S = Aa } = { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } by A204, XBOOLE_0:def_10;
A230: ( (2 * n) + 1 in NAT & 2 * (card DX) in NAT ) by ORDINAL1:def_12;
for x being set st x in DX holds
card (Im ((R | ((dom R) \ { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } )),x)) = 2
proof
let x be set ; ::_thesis: ( x in DX implies card (Im ((R | ((dom R) \ { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } )),x)) = 2 )
assume A231: x in DX ; ::_thesis: card (Im ((R | ((dom R) \ { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } )),x)) = 2
then ex y being set st [x,y] in R by XTUPLE_0:def_12;
then A232: x in { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : Aa = F .: S } by A47;
then consider S being Simplex of m - 1, BCS (k,(Complex_of {A})) such that
A233: x = S and
A234: Aa = F .: S ;
set XX = { S1 where S1 is Simplex of m, BCS (k,(Complex_of {A})) : S c= S1 } ;
not x in { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } by A231, XBOOLE_0:def_5;
then conv (@ S) meets Int A by A233, A234;
then A235: card { S1 where S1 is Simplex of m, BCS (k,(Complex_of {A})) : S c= S1 } = 2 by A3, Th45;
A236: (R | ((dom R) \ { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } )) .: {S} c= { S1 where S1 is Simplex of m, BCS (k,(Complex_of {A})) : S c= S1 }
proof
let w be set ; :: according to TARSKI:def_3 ::_thesis: ( not w in (R | ((dom R) \ { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } )) .: {S} or w in { S1 where S1 is Simplex of m, BCS (k,(Complex_of {A})) : S c= S1 } )
assume w in (R | ((dom R) \ { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } )) .: {S} ; ::_thesis: w in { S1 where S1 is Simplex of m, BCS (k,(Complex_of {A})) : S c= S1 }
then consider s being set such that
A237: [s,w] in R | ((dom R) \ { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } ) and
A238: s in {S} by RELAT_1:def_13;
A239: [s,w] in R by A237, RELAT_1:def_11;
then w in { S where S is Simplex of m, BCS (k,(Complex_of {A})) : verum } by A47;
then A240: ex W being Simplex of m, BCS (k,(Complex_of {A})) st w = W ;
s = S by A238, TARSKI:def_1;
then S c= w by A47, A239;
hence w in { S1 where S1 is Simplex of m, BCS (k,(Complex_of {A})) : S c= S1 } by A240; ::_thesis: verum
end;
{ S1 where S1 is Simplex of m, BCS (k,(Complex_of {A})) : S c= S1 } c= (R | ((dom R) \ { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } )) .: {S}
proof
let w be set ; :: according to TARSKI:def_3 ::_thesis: ( not w in { S1 where S1 is Simplex of m, BCS (k,(Complex_of {A})) : S c= S1 } or w in (R | ((dom R) \ { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } )) .: {S} )
assume w in { S1 where S1 is Simplex of m, BCS (k,(Complex_of {A})) : S c= S1 } ; ::_thesis: w in (R | ((dom R) \ { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } )) .: {S}
then consider W being Simplex of m, BCS (k,(Complex_of {A})) such that
A241: w = W and
A242: S c= W ;
W in { S where S is Simplex of m, BCS (k,(Complex_of {A})) : verum } ;
then [S,W] in R by A47, A232, A233, A242;
then A243: [S,W] in R | ((dom R) \ { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } ) by A231, A233, RELAT_1:def_11;
S in {S} by TARSKI:def_1;
hence w in (R | ((dom R) \ { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } )) .: {S} by A241, A243, RELAT_1:def_13; ::_thesis: verum
end;
then { S1 where S1 is Simplex of m, BCS (k,(Complex_of {A})) : S c= S1 } = (R | ((dom R) \ { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } )) .: {S} by A236, XBOOLE_0:def_10;
hence card (Im ((R | ((dom R) \ { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } )),x)) = 2 by A233, A235, RELAT_1:def_16; ::_thesis: verum
end;
then card (R | ((dom R) \ { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } )) = (card ((R | ((dom R) \ { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } )) | ((dom (R | ((dom R) \ { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } ))) \ DX))) +` (2 *` (card DX)) by Th1
.= 0 +` (2 *` (card DX)) by A68
.= 2 *` (card DX) by CARD_2:18 ;
then A244: card R = (2 *` (card DX)) +` (1 *` (card { S where S is Simplex of m - 1, BCS (k,(Complex_of {A})) : ( F .: S = Aa & conv (@ S) misses Int A ) } )) by A85, Th1
.= (2 *` (card DX)) +` ((2 * n) + 1) by A83, A229, CARD_2:21
.= ((card 2) *` (card (card DX))) +` ((2 * n) + 1) by Lm1
.= (card (2 * (card DX))) +` ((2 * n) + 1) by CARD_2:39
.= (card (2 * (card DX))) +` (card ((2 * n) + 1)) by Lm1
.= card ((2 * (card DX)) + ((2 * n) + 1)) by A230, CARD_2:38
.= (2 * (card DX)) + ((2 * n) + 1) by Lm1 ;
then card XX = (2 * (((card DX) + n) - (card YA))) + 1 by A48, A202;
then 2 * (((card DX) + n) - (card YA)) >= - 1 by INT_1:7;
then ((card DX) + n) - (card YA) >= (- 1) / 2 by XREAL_1:79;
then ((card DX) + n) - (card YA) > - 1 by XXREAL_0:2;
then ((card DX) + n) - (card YA) >= 0 by INT_1:8;
then reconsider cnc = ((card DX) + n) - (card YA) as Element of NAT by INT_1:3;
take cnc ; ::_thesis: card { S where S is Simplex of (card A) - 1, BCS (k,(Complex_of {A})) : F .: S = A } = (2 * cnc) + 1
thus card { S where S is Simplex of (card A) - 1, BCS (k,(Complex_of {A})) : F .: S = A } = (2 * cnc) + 1 by A48, A202, A244; ::_thesis: verum
end;
end;
end;
A245: S1[ 0 ]
proof
let A be finite affinely-independent Subset of V; ::_thesis: ( card A = 0 implies for F being Function of (Vertices (BCS (k,(Complex_of {A})))),A st ( for v being Vertex of (BCS (k,(Complex_of {A})))
for B being Subset of V st B c= A & v in conv B holds
F . v in B ) holds
ex n being Nat st card { S where S is Simplex of (card A) - 1, BCS (k,(Complex_of {A})) : F .: S = A } = (2 * n) + 1 )
assume A246: card A = 0 ; ::_thesis: for F being Function of (Vertices (BCS (k,(Complex_of {A})))),A st ( for v being Vertex of (BCS (k,(Complex_of {A})))
for B being Subset of V st B c= A & v in conv B holds
F . v in B ) holds
ex n being Nat st card { S where S is Simplex of (card A) - 1, BCS (k,(Complex_of {A})) : F .: S = A } = (2 * n) + 1
A247: A = {} by A246;
set C = Complex_of {A};
A248: ( |.(Complex_of {A}).| c= [#] V & [#] (Complex_of {A}) = [#] V ) ;
let F be Function of (Vertices (BCS (k,(Complex_of {A})))),A; ::_thesis: ( ( for v being Vertex of (BCS (k,(Complex_of {A})))
for B being Subset of V st B c= A & v in conv B holds
F . v in B ) implies ex n being Nat st card { S where S is Simplex of (card A) - 1, BCS (k,(Complex_of {A})) : F .: S = A } = (2 * n) + 1 )
assume for v being Vertex of (BCS (k,(Complex_of {A})))
for B being Subset of V st B c= A & v in conv B holds
F . v in B ; ::_thesis: ex n being Nat st card { S where S is Simplex of (card A) - 1, BCS (k,(Complex_of {A})) : F .: S = A } = (2 * n) + 1
set X = { S where S is Simplex of (card A) - 1, BCS (k,(Complex_of {A})) : F .: S = A } ;
take 0 ; ::_thesis: card { S where S is Simplex of (card A) - 1, BCS (k,(Complex_of {A})) : F .: S = A } = (2 * 0) + 1
A249: ( k = 0 or k > 0 ) ;
A250: Z - 1 = - 1 by XXREAL_3:4;
then degree (Complex_of {A}) = - 1 by A246, SIMPLEX0:26;
then A251: Complex_of {A} = BCS (k,(Complex_of {A})) by A248, A249, Th16, Th22;
A252: the topology of (Complex_of {A}) = bool A by SIMPLEX0:4;
A253: { S where S is Simplex of (card A) - 1, BCS (k,(Complex_of {A})) : F .: S = A } c= {A}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { S where S is Simplex of (card A) - 1, BCS (k,(Complex_of {A})) : F .: S = A } or x in {A} )
assume A254: x in { S where S is Simplex of (card A) - 1, BCS (k,(Complex_of {A})) : F .: S = A } ; ::_thesis: x in {A}
consider S being Simplex of (card A) - 1, BCS (k,(Complex_of {A})) such that
A255: S = x and
F .: S = A by A254;
S in the topology of (Complex_of {A}) by A251, PRE_TOPC:def_2;
then S is empty by A247, A252;
hence x in {A} by A247, A255, TARSKI:def_1; ::_thesis: verum
end;
A in bool A by ZFMISC_1:def_1;
then reconsider A1 = A as Simplex of (Complex_of {A}) by A252, PRE_TOPC:def_2;
A256: F .: A1 = A by A247;
A1 is Simplex of - 1, Complex_of {A} by A246, A250, SIMPLEX0:48;
then A in { S where S is Simplex of (card A) - 1, BCS (k,(Complex_of {A})) : F .: S = A } by A246, A251, A256;
then { S where S is Simplex of (card A) - 1, BCS (k,(Complex_of {A})) : F .: S = A } = {A} by A253, ZFMISC_1:33;
hence card { S where S is Simplex of (card A) - 1, BCS (k,(Complex_of {A})) : F .: S = A } = (2 * 0) + 1 by CARD_1:30; ::_thesis: verum
end;
for k being Nat holds S1[k] from NAT_1:sch_2(A245, A1);
hence for F being Function of (Vertices (BCS (k,(Complex_of {Aff})))),Aff st ( for v being Vertex of (BCS (k,(Complex_of {Aff})))
for B being Subset of V st B c= Aff & v in conv B holds
F . v in B ) holds
ex n being Nat st card { S where S is Simplex of (card Aff) - 1, BCS (k,(Complex_of {Aff})) : F .: S = Aff } = (2 * n) + 1 ; ::_thesis: verum
end;
theorem :: SIMPLEX1:47
for k being Nat
for V being RealLinearSpace
for Aff being finite affinely-independent Subset of V
for F being Function of (Vertices (BCS (k,(Complex_of {Aff})))),Aff st ( for v being Vertex of (BCS (k,(Complex_of {Aff})))
for B being Subset of V st B c= Aff & v in conv B holds
F . v in B ) holds
ex S being Simplex of (card Aff) - 1, BCS (k,(Complex_of {Aff})) st F .: S = Aff
proof
let k be Nat; ::_thesis: for V being RealLinearSpace
for Aff being finite affinely-independent Subset of V
for F being Function of (Vertices (BCS (k,(Complex_of {Aff})))),Aff st ( for v being Vertex of (BCS (k,(Complex_of {Aff})))
for B being Subset of V st B c= Aff & v in conv B holds
F . v in B ) holds
ex S being Simplex of (card Aff) - 1, BCS (k,(Complex_of {Aff})) st F .: S = Aff
let V be RealLinearSpace; ::_thesis: for Aff being finite affinely-independent Subset of V
for F being Function of (Vertices (BCS (k,(Complex_of {Aff})))),Aff st ( for v being Vertex of (BCS (k,(Complex_of {Aff})))
for B being Subset of V st B c= Aff & v in conv B holds
F . v in B ) holds
ex S being Simplex of (card Aff) - 1, BCS (k,(Complex_of {Aff})) st F .: S = Aff
let Aff be finite affinely-independent Subset of V; ::_thesis: for F being Function of (Vertices (BCS (k,(Complex_of {Aff})))),Aff st ( for v being Vertex of (BCS (k,(Complex_of {Aff})))
for B being Subset of V st B c= Aff & v in conv B holds
F . v in B ) holds
ex S being Simplex of (card Aff) - 1, BCS (k,(Complex_of {Aff})) st F .: S = Aff
let F be Function of (Vertices (BCS (k,(Complex_of {Aff})))),Aff; ::_thesis: ( ( for v being Vertex of (BCS (k,(Complex_of {Aff})))
for B being Subset of V st B c= Aff & v in conv B holds
F . v in B ) implies ex S being Simplex of (card Aff) - 1, BCS (k,(Complex_of {Aff})) st F .: S = Aff )
set XX = { S where S is Simplex of (card Aff) - 1, BCS (k,(Complex_of {Aff})) : F .: S = Aff } ;
assume for v being Vertex of (BCS (k,(Complex_of {Aff})))
for B being Subset of V st B c= Aff & v in conv B holds
F . v in B ; ::_thesis: ex S being Simplex of (card Aff) - 1, BCS (k,(Complex_of {Aff})) st F .: S = Aff
then ex n being Nat st card { S where S is Simplex of (card Aff) - 1, BCS (k,(Complex_of {Aff})) : F .: S = Aff } = (2 * n) + 1 by Th46;
then not { S where S is Simplex of (card Aff) - 1, BCS (k,(Complex_of {Aff})) : F .: S = Aff } is empty ;
then consider x being set such that
A1: x in { S where S is Simplex of (card Aff) - 1, BCS (k,(Complex_of {Aff})) : F .: S = Aff } by XBOOLE_0:def_1;
ex S being Simplex of (card Aff) - 1, BCS (k,(Complex_of {Aff})) st
( x = S & F .: S = Aff ) by A1;
hence ex S being Simplex of (card Aff) - 1, BCS (k,(Complex_of {Aff})) st F .: S = Aff ; ::_thesis: verum
end;