Volume 15, 2003

University of Bialystok

Copyright (c) 2003 Association of Mizar Users

### The abstract of the Mizar article:

### Some Properties for Convex Combinations

**by****Noboru Endou,****Yasumasa Suzuki, and****Yasunari Shidama**- Received April 3, 2003
- MML identifier: CONVEX2

- [ Mizar article, MML identifier index ]

environ vocabulary RLVECT_1, FUNCT_1, PROB_2, ARYTM_1, RELAT_1, BHSP_1, FINSEQ_1, BOOLE, JORDAN1, SETFAM_1, CONNSP_3, ARYTM_3, RLSUB_1, CONVEX1, FINSEQ_4, SEQ_1, CARD_1, RLVECT_2, JORDAN3, TARSKI, FUNCT_2, RFINSEQ, FINSET_1, CONVEX2; notation TARSKI, XBOOLE_0, SUBSET_1, NUMBERS, XCMPLX_0, XREAL_0, REAL_1, FINSEQ_1, RELAT_1, FUNCT_1, FINSET_1, CARD_1, NAT_1, SETFAM_1, STRUCT_0, FUNCT_2, RLVECT_1, RLSUB_1, RLSUB_2, FINSEQ_4, RLVECT_2, RVSUM_1, BHSP_1, RUSUB_4, CONVEX1, TOPREAL1, JORDAN3, RFINSEQ; constructors REAL_1, SEQ_1, RUSUB_5, FINSEQ_4, CONVEX1, JORDAN3, NAT_1, MONOID_0, WELLORD2, TOPREAL1, MATRIX_2, RFINSEQ, RLSUB_2, MEMBERED; clusters RELSET_1, FINSEQ_1, ARYTM_3, STRUCT_0, RLVECT_1, SEQ_1, RUSUB_4, CONVEX1, BINARITH, XREAL_0, MEMBERED; requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM; begin :: Convex Combination on Convex Family theorem :: CONVEX2:1 for V being non empty RLSStruct, M,N being convex Subset of V holds M /\ N is convex; theorem :: CONVEX2:2 for V being RealUnitarySpace-like (non empty UNITSTR), M being Subset of V, F being FinSequence of the carrier of V, B being FinSequence of REAL st M = {u where u is VECTOR of V : for i being Nat st i in (dom F /\ dom B) holds ex v being VECTOR of V st v = F.i & u .|. v <= B.i} holds M is convex; theorem :: CONVEX2:3 for V being RealUnitarySpace-like (non empty UNITSTR), M being Subset of V, F being FinSequence of the carrier of V, B being FinSequence of REAL st M = {u where u is VECTOR of V : for i being Nat st i in (dom F /\ dom B) holds ex v being VECTOR of V st v = F.i & u .|. v < B.i} holds M is convex; theorem :: CONVEX2:4 for V being RealUnitarySpace-like (non empty UNITSTR), M being Subset of V, F being FinSequence of the carrier of V, B being FinSequence of REAL st M = {u where u is VECTOR of V : for i being Nat st i in (dom F /\ dom B) holds ex v being VECTOR of V st v = F.i & u .|. v >= B.i} holds M is convex; theorem :: CONVEX2:5 for V being RealUnitarySpace-like (non empty UNITSTR), M being Subset of V, F being FinSequence of the carrier of V, B being FinSequence of REAL st M = {u where u is VECTOR of V : for i being Nat st i in (dom F /\ dom B) holds ex v being VECTOR of V st v = F.i & u .|. v > B.i} holds M is convex; theorem :: CONVEX2:6 for V being RealLinearSpace, M being Subset of V holds (for N being Subset of V, L being Linear_Combination of N st L is convex & N c= M holds Sum(L) in M) iff M is convex; definition let V be RealLinearSpace, M be Subset of V; func LinComb(M) -> set means :: CONVEX2:def 1 for L being set holds L in it iff L is Linear_Combination of M; end; definition let V be RealLinearSpace; cluster convex Linear_Combination of V; end; definition let V be RealLinearSpace; mode Convex_Combination of V is convex Linear_Combination of V; end; definition let V be RealLinearSpace, M be non empty Subset of V; cluster convex Linear_Combination of M; end; definition let V be RealLinearSpace, M be non empty Subset of V; mode Convex_Combination of M is convex Linear_Combination of M; end; theorem :: CONVEX2:7 for V being RealLinearSpace, M be Subset of V holds Convex-Family M <> {}; theorem :: CONVEX2:8 for V being RealLinearSpace, M being Subset of V holds M c= conv(M); theorem :: CONVEX2:9 for V being RealLinearSpace, L1,L2 being Convex_Combination of V, r being Real st 0 < r & r < 1 holds r*L1 + (1-r)*L2 is Convex_Combination of V; theorem :: CONVEX2:10 for V being RealLinearSpace, M being non empty Subset of V, L1,L2 being Convex_Combination of M, r being Real st 0 < r & r < 1 holds r*L1 + (1-r)*L2 is Convex_Combination of M; theorem :: CONVEX2:11 for V being RealLinearSpace holds ex L being Linear_Combination of V st L is convex; theorem :: CONVEX2:12 for V being RealLinearSpace, M being non empty Subset of V holds ex L being Linear_Combination of M st L is convex; begin :: Miscellaneous theorem :: CONVEX2:13 for V being RealLinearSpace, W1,W2 being Subspace of V holds Up(W1+W2) = Up(W1) + Up(W2); theorem :: CONVEX2:14 for V being RealLinearSpace, W1,W2 being Subspace of V holds Up(W1 /\ W2) = Up(W1) /\ Up(W2); theorem :: CONVEX2:15 for V being RealLinearSpace, L1, L2 being Convex_Combination of V, a,b being Real st a * b > 0 holds Carrier(a*L1 + b*L2) = Carrier(a * L1) \/ Carrier(b * L2); theorem :: CONVEX2:16 for F,G being Function st F,G are_fiberwise_equipotent for x1,x2 being set st x1 in dom F & x2 in dom F & x1 <> x2 holds ex z1,z2 being set st z1 in dom G & z2 in dom G & z1 <> z2 & F.x1 = G.z1 & F.x2 = G.z2; theorem :: CONVEX2:17 for V being RealLinearSpace, L being Linear_Combination of V, A being Subset of V st A c= Carrier(L) holds ex L1 being Linear_Combination of V st Carrier(L1) = A;

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