fact stringlengths 6 39.6k | type stringclasses 2
values | library stringlengths 5 8 | imports listlengths 0 52 | filename stringlengths 9 12 | symbolic_name stringlengths 7 40 | docstring stringlengths 6 76 |
|---|---|---|---|---|---|---|
u in v + W iff ex v1 st v1 in W & u = v + v1 proof thus u in v + W implies ex v1 st v1 in W & u = v + v1 proof assume u in v + W; then ex v1 st u = v + v1 & v1 in W; hence thesis; end; given v1 such that A1: v1 in W & u = v + v1; thus thesis by A1; end; | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th72 | $\mathbb Z$-modules |
u in v + W iff ex v1 st v1 in W & u = v - v1 proof thus u in v + W implies ex v1 st v1 in W & u = v - v1 proof assume u in v + W; then consider v1 such that A1: u = v + v1 and A2: v1 in W; take x = - v1; thus x in W by A2,Th38; thus thesis by A1,RLVECT_1:17; end; given v1 such that A3: v1 in W and A4: u = v - v1; - v1 ... | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th73 | $\mathbb Z$-modules |
(ex v st v1 in v + W & v2 in v + W) iff v1 - v2 in W proof thus (ex v st v1 in v + W & v2 in v + W) implies v1 - v2 in W proof given v such that A1: v1 in v + W and A2: v2 in v + W; consider u2 such that A3: u2 in W and A4: v2 = v + u2 by A2,Th72; consider u1 such that A5: u1 in W and A6: v1 = v + u1 by A1,Th72; v1 - v... | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th74 | $\mathbb Z$-modules |
v + W = u + W implies ex v1 st v1 in W & v + v1 = u proof assume v + W = u + W; then v in u + W by Th58; then consider u1 such that A1: v = u + u1 and A2: u1 in W; take v1 = u - v; 0.V = (u + u1) - v by A1,RLVECT_1:15 .= u1 + (u - v) by RLVECT_1:def 3; then v1 = - u1 by RLVECT_1:def 10; hence v1 in W by A2,Th38; thus v... | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th75 | $\mathbb Z$-modules |
v + W = u + W implies ex v1 st v1 in W & v - v1 = u proof assume v + W = u + W; then u in v + W by Th58; then consider u1 such that A1: u = v + u1 and A2: u1 in W; take v1 = v - u; 0.V = (v + u1) - u by A1,RLVECT_1:15 .= u1 + (v - u) by RLVECT_1:def 3; then v1 = - u1 by RLVECT_1:def 10; hence v1 in W by A2,Th38; thus v... | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th76 | $\mathbb Z$-modules |
for W1, W2 being strict Submodule of V st v + W1 = v + W2 holds W1 = W2 proof let W1, W2 be strict Submodule of V; assume A1: v + W1 = v + W2; the carrier of W1 = the carrier of W2 proof A2: the carrier of W1 c= the carrier of V by Def9; thus the carrier of W1 c= the carrier of W2 proof let x be set; assume A3: x in th... | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th77 | $\mathbb Z$-modules |
for W1, W2 being strict Submodule of V st v + W1 = u + W2 holds W1 = W2 proof let W1, W2 be strict Submodule of V; assume A1: v + W1 = u + W2; set V2 = the carrier of W2; set V1 = the carrier of W1; assume A2: W1 <> W2; A3: now set x = the Element of V1 \ V2; assume V1 \ V2 <> {}; then x in V1 by XBOOLE_0:def 5; then A... | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th78 | $\mathbb Z$-modules |
V1 is Coset of (0).V implies ex v st V1 = {v} proof assume V1 is Coset of (0).V; then consider v such that A1: V1 = v + (0).V by Def13; take v; thus thesis by A1,Th60; end; | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th82 | $\mathbb Z$-modules |
the carrier of W is Coset of W proof the carrier of W = 0.V + W by Lm4; hence thesis by Def13; end; | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th83 | $\mathbb Z$-modules |
u in C iff C = u + W proof thus u in C implies C = u + W proof assume A1: u in C; ex v st C = v + W by Def13; hence thesis by A1,Th67; end; thus thesis by Th58; end; | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th87 | $\mathbb Z$-modules |
u in C & v in C implies ex v1 st v1 in W & u - v1 = v proof assume u in C & v in C; then C = u + W & C = v + W by Th87; hence thesis by Th76; end; | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th89 | $\mathbb Z$-modules |
x in W1 + W2 iff ex v1, v2 st v1 in W1 & v2 in W2 & x = v1 + v2 proof thus x in W1 + W2 implies ex v1, v2 st v1 in W1 & v2 in W2 & x = v1 + v2 proof assume x in W1 + W2; then x in the carrier of W1 + W2 by STRUCT_0:def 5; then x in {v + u : v in W1 & u in W2} by Def14; then consider v1, v2 such that A1: x = v1 + v2 & v... | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th92 | $\mathbb Z$-modules |
x in W1 /\ W2 iff x in W1 & x in W2 proof x in W1 /\ W2 iff x in the carrier of W1 /\ W2 by STRUCT_0:def 5; then x in W1 /\ W2 iff x in (the carrier of W1) /\ (the carrier of W2) by Def15; then x in W1 /\ W2 iff x in the carrier of W1 & x in the carrier of W2 by XBOOLE_0:def 4; hence thesis by STRUCT_0:def 5; end; Lm6:... | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th94 | $\mathbb Z$-modules |
W1 + (W2 + W3) = (W1 + W2) + W3 proof set A = {v + u : v in W1 & u in W2}; set B = {v + u : v in W2 & u in W3}; set C = {v + u : v in W1 + W2 & u in W3}; set D = {v + u : v in W1 & u in W2 + W3}; A1: the carrier of W1 + (W2 + W3) = D by Def14; A2: C c= D proof let x be set; assume x in C; then consider v,u such that A3... | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th96 | $\mathbb Z$-modules |
W1 is Submodule of W1 + W2 proof the carrier of W1 c= the carrier of W1 + W2 by Lm6; hence W1 is Submodule of W1 + W2 by Th43; end; | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th97 | $\mathbb Z$-modules |
for W2 being strict Submodule of V holds W1 is Submodule of W2 iff W1 + W2 = W2 proof let W2 be strict Submodule of V; thus W1 is Submodule of W2 implies W1 + W2 = W2 proof assume W1 is Submodule of W2; then the carrier of W1 c= the carrier of W2 by Def9; hence thesis by Lm7; end; thus thesis by Th97; end; | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th98 | $\mathbb Z$-modules |
for W being strict Submodule of V holds (0).V + W = W proof let W be strict Submodule of V; (0).V is Submodule of W by Th54; then the carrier of (0).V c= the carrier of W by Def9; hence thesis by Lm7; end; | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th99 | $\mathbb Z$-modules |
(Omega).V + W = the Z_ModuleStruct of V proof the carrier of W c= the carrier of V by Def9; hence thesis by Lm7; end; | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th101 | $\mathbb Z$-modules |
W1 /\ (W2 /\ W3) = (W1 /\ W2) /\ W3 proof set V1 = the carrier of W1; set V2 = the carrier of W2; set V3 = the carrier of W3; the carrier of W1 /\ (W2 /\ W3) = V1 /\ (the carrier of W2 /\ W3) by Def15 .= V1 /\ (V2 /\ V3) by Def15 .= (V1 /\ V2) /\ V3 by XBOOLE_1:16 .= (the carrier of W1 /\ W2) /\ V3 by Def15; hence thes... | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th104 | $\mathbb Z$-modules |
W1 /\ W2 is Submodule of W1 proof the carrier of W1 /\ W2 c= the carrier of W1 by Lm8; hence W1 /\ W2 is Submodule of W1 by Th43; end; | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th105 | $\mathbb Z$-modules |
for W1 being strict Submodule of V holds W1 is Submodule of W2 iff W1 /\ W2 = W1 proof let W1 be strict Submodule of V; thus W1 is Submodule of W2 implies W1 /\ W2 = W1 proof assume W1 is Submodule of W2; then A1: the carrier of W1 c= the carrier of W2 by Def9; the carrier of W1 /\ W2 = (the carrier of W1) /\ (the carr... | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th106 | $\mathbb Z$-modules |
(0).V /\ W = (0).V proof 0.V in W by Th33; then 0.V in the carrier of W by STRUCT_0:def 5; then {0.V} c= the carrier of W by ZFMISC_1:31; then A1: {0.V} /\ (the carrier of W) = {0.V} by XBOOLE_1:28; the carrier of (0).V /\ W = (the carrier of (0).V) /\ (the carrier of W) by Def15 .= {0.V} /\ (the carrier of W) by Def10... | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th107 | $\mathbb Z$-modules |
for W being strict Submodule of V holds (Omega).V /\ W = W proof let W be strict Submodule of V; the carrier of (Omega). V /\ W = (the carrier of V) /\ (the carrier of W) & the carrier of W c= the carrier of V by Def15,Def9; hence thesis by Th45,XBOOLE_1:28; end; | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th109 | $\mathbb Z$-modules |
W1 is strict Submodule of W3 implies W1 + (W2 /\ W3) = (W1 + W2) /\ W3 proof assume A1: W1 is strict Submodule of W3; thus (W1 + W2) /\ W3 = (W1 /\ W3) + (W3 /\ W2) by A1,Lm13,Th45 .= W1 + (W2 /\ W3) by A1,Th106; end; | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th118 | $\mathbb Z$-modules |
for V being Z_Module, W being with_Linear_Compl Submodule of V, L being Linear_Compl of W holds V is_the_direct_sum_of L,W & V is_the_direct_sum_of W,L proof let V be Z_Module, W be with_Linear_Compl Submodule of V, L be Linear_Compl of W; thus V is_the_direct_sum_of L,W by Def19; hence thesis by Lm18; end; | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th124 | $\mathbb Z$-modules |
for V being Z_Module holds V is_the_direct_sum_of (0).V, (Omega).V & V is_the_direct_sum_of (Omega).V,(0).V proof let V be Z_Module; A1: (0).V + (Omega).V = the Z_ModuleStruct of V & (0).V = (0).V /\ (Omega). V by Th99,Th107; hence V is_the_direct_sum_of (0).V,(Omega).V by Def17; thus thesis by A1,Def17; end; | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th129 | $\mathbb Z$-modules |
C1 meets C2 implies C1 /\ C2 is Coset of W1 /\ W2 proof set v = the Element of C1 /\ C2; set C = C1 /\ C2; assume A1: C1 /\ C2 <> {}; then reconsider v as Element of V by TARSKI:def 3; v in C2 by A1,XBOOLE_0:def 4; then A2: C2 = v + W2 by Th87; v in C1 by A1,XBOOLE_0:def 4; then A3: C1 = v + W1 by Th87; C is Coset of W... | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th131 | $\mathbb Z$-modules |
for V being Z_Module, W1, W2 being Submodule of V holds V is_the_direct_sum_of W1,W2 iff for C1 being Coset of W1, C2 being Coset of W2 ex v being VECTOR of V st C1 /\ C2 = {v} proof let V be Z_Module, W1, W2 be Submodule of V; set VW1 = the carrier of W1; set VW2 = the carrier of W2; 0.V in W2 by Th33; then A1: 0.V in... | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th132 | $\mathbb Z$-modules |
V is_the_direct_sum_of W1,W2 & v1 + v2 = u1 + u2 & v1 in W1 & u1 in W1 & v2 in W2 & u2 in W2 implies v1 = u1 & v2 = u2 proof reconsider C2 = v1 + W2 as Coset of W2 by Def13; reconsider C1 = the carrier of W1 as Coset of W1 by Th83; A1: v1 in C2 by Th58; assume V is_the_direct_sum_of W1,W2; then consider u being VECTOR ... | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th134 | $\mathbb Z$-modules |
V is_the_direct_sum_of W1,W2 implies (v |-- (W1,W2))`1 = (v |-- (W2,W1))`2 proof assume A1: V is_the_direct_sum_of W1,W2; then A2: (v |-- (W1,W2))`2 in W2 by Def20; A3: V is_the_direct_sum_of W2,W1 by A1,Lm18; then A4: v = (v |-- (W2,W1))`2 + (v |-- (W2,W1))`1 & (v |-- (W2,W1))`1 in W2 by Def20; A5: (v |-- (W2,W1))`2 i... | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th136 | $\mathbb Z$-modules |
V is_the_direct_sum_of W1,W2 implies (v |-- (W1,W2))`2 = (v |-- (W2,W1))`1 proof assume A1: V is_the_direct_sum_of W1,W2; then A2: (v |-- (W1,W2))`2 in W2 by Def20; A3: V is_the_direct_sum_of W2,W1 by A1,Lm18; then A4: v = (v |-- (W2,W1))`2 + (v |-- (W2,W1))`1 & (v |-- (W2,W1))`1 in W2 by Def20; A5: (v |-- (W2,W1))`2 i... | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th137 | $\mathbb Z$-modules |
LattStr (# Submodules(V), SubJoin(V), SubMeet(V) #) is Lattice proof set S = LattStr (# Submodules(V), SubJoin(V), SubMeet(V) #); A1: for A, B being Element of S holds A "/\" B = B "/\" A proof let A, B be Element of S; reconsider W1 = A, W2 = B as Submodule of V by Def16; thus A "/\" B = W1 /\ W2 by Def22 .= B "/\" A ... | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th143 | $\mathbb Z$-modules |
for V being Z_Module holds LattStr (# Submodules(V), SubJoin(V), SubMeet(V) #) is lower-bounded proof let V be Z_Module; set S = LattStr (# Submodules(V), SubJoin(V), SubMeet(V) #); ex C being Element of S st for A being Element of S holds C "/\" A = C & A "/\" C = C proof reconsider C = (0).V as Element of S by Def16;... | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th144 | $\mathbb Z$-modules |
for V being Z_Module holds LattStr (# Submodules(V), SubJoin(V), SubMeet(V) #) is upper-bounded proof let V be Z_Module; set S = LattStr (# Submodules(V), SubJoin(V), SubMeet(V) #); ex C being Element of S st for A being Element of S holds C "\/" A = C & A "\/" C = C proof reconsider C = (Omega).V as Element of S by De... | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th145 | $\mathbb Z$-modules |
for R being non empty addLoopStr, a being Element of R, i be Element of INT st i=0 holds (Int-mult-left(R)).(i,a) = 0.R proof let R be non empty addLoopStr, a be Element of R, i be Element of INT; assume i=0; hence (Int-mult-left(R)).(i,a) = 0 * a by Def23 .=0.R by BINOM:12; end; | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th152 | $\mathbb Z$-modules |
for R being add-associative right_zeroed right_complementable non empty addLoopStr, i be Element of NAT holds (Nat-mult-left(R)).(i,0.R) = 0.R proof let R being add-associative right_zeroed right_complementable non empty addLoopStr, i be Element of NAT; defpred P[Element of NAT] means (Nat-mult-left(R)).($1,0.R) = 0.R;... | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th153 | $\mathbb Z$-modules |
for R being add-associative right_zeroed right_complementable non empty addLoopStr, i be Element of INT holds (Int-mult-left(R)).(i,0.R) = 0.R proof let R being add-associative right_zeroed right_complementable non empty addLoopStr, i be Element of INT; per cases; suppose 0 <= i; then reconsider i1=i as Element of NAT ... | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th154 | $\mathbb Z$-modules |
for R being right_zeroed non empty addLoopStr, a being Element of R, i be Element of INT st i = 1 holds (Int-mult-left(R)).(i,a) = a proof let R be right_zeroed non empty addLoopStr, a be Element of R, i be Element of INT; assume i=1; hence (Int-mult-left(R)).(i,a) = 1 * a by Def23 .= a by BINOM:13; end; | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th155 | $\mathbb Z$-modules |
for R being Abelian right_zeroed add-associative right_complementable non empty addLoopStr, a being Element of R, i, j, k being Element of NAT st i <= j & k = j-i holds (Nat-mult-left(R)).(k,a) = (Nat-mult-left(R)).(j,a) - (Nat-mult-left(R)).(i,a) proof let R be Abelian right_zeroed add-associative right_complementable... | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th156 | $\mathbb Z$-modules |
for R being Abelian right_zeroed add-associative right_complementable non empty addLoopStr, a being Element of R, i being Element of NAT holds -(Nat-mult-left(R)).(i,a) = (Nat-mult-left(R)).(i,-a) proof let R be Abelian right_zeroed add-associative right_complementable non empty addLoopStr, a being Element of R, i bein... | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th157 | $\mathbb Z$-modules |
for R being Abelian right_zeroed add-associative right_complementable non empty addLoopStr, a being Element of R, i, j being Element of INT st i in NAT & not j in NAT holds (Int-mult-left(R)).(i+j,a) = (Int-mult-left(R)).(i,a) + (Int-mult-left(R)).(j,a) proof let R be Abelian right_zeroed add-associative right_compleme... | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th158 | $\mathbb Z$-modules |
for R being Abelian right_zeroed add-associative right_complementable non empty addLoopStr, a being Element of R, i, j being Element of INT holds (Int-mult-left(R)).(i+j,a) = (Int-mult-left(R)).(i,a) + (Int-mult-left(R)).(j,a) proof let R be Abelian right_zeroed add-associative right_complementable non empty addLoopStr... | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th159 | $\mathbb Z$-modules |
for R being Abelian right_zeroed add-associative right_complementable non empty addLoopStr, a, b being Element of R, i being Element of NAT holds (Nat-mult-left(R)).(i,a+b) = (Nat-mult-left(R)).(i,a) + (Nat-mult-left(R)).(i,b) proof let R be Abelian right_zeroed add-associative right_complementable non empty addLoopStr... | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th160 | $\mathbb Z$-modules |
for R being Abelian right_zeroed add-associative right_complementable non empty addLoopStr, a, b being Element of R, i being Element of INT holds (Int-mult-left(R)).(i,a+b) = (Int-mult-left(R)).(i,a) + (Int-mult-left(R)).(i,b) proof let R be Abelian right_zeroed add-associative right_complementable non empty addLoopStr... | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th161 | $\mathbb Z$-modules |
for R being Abelian right_zeroed add-associative right_complementable non empty addLoopStr, a being Element of R, i, j being Element of NAT holds (Nat-mult-left(R)).(i*j,a) = (Nat-mult-left(R)).(i,(Nat-mult-left(R)).(j,a)) proof let R be Abelian right_zeroed add-associative right_complementable non empty addLoopStr, a ... | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th162 | $\mathbb Z$-modules |
for R being Abelian right_zeroed add-associative right_complementable non empty addLoopStr, a being Element of R, i, j being Element of INT holds (Int-mult-left(R)).(i*j,a) = (Int-mult-left(R)).(i,(Int-mult-left(R)).(j,a)) proof let R be Abelian right_zeroed add-associative right_complementable non empty addLoopStr, a ... | theorem | zmodul01 | [
"vocabularies NUMBERS, ALGSTR_0, STRUCT_0, SUBSET_1, BINOP_1, FUNCT_1,",
"ZFMISC_1, XBOOLE_0, ORDINAL1, RELAT_1, ARYTM_3, PARTFUN1, SUPINF_2,",
"FUNCT_5, MCART_1, ARYTM_1, CARD_1, FINSEQ_1, CARD_3, TARSKI, XXREAL_0,",
"RLVECT_1, REALSET1, RLSUB_1, ZMODUL01, INT_1, FINSEQ_4, LATTICES,",
"EQREL_1, PBOOLE, RLS... | zmodul01.miz | zmodul01:Th163 | $\mathbb Z$-modules |
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