mapdindp0 - Mathbox for Norm Megill

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Theorem mapdindp0 39660

Description: Vector independence lemma. (Contributed by NM, 29-Apr-2015.)

**Hypotheses**
Ref	Expression
mapdindp1.v	⊢ 𝑉 = (Base‘𝑊)
mapdindp1.p	⊢ + = (+_g‘𝑊)
mapdindp1.o	⊢ 0 = (0_g‘𝑊)
mapdindp1.n	⊢ 𝑁 = (LSpan‘𝑊)
mapdindp1.w	⊢ (𝜑 → 𝑊 ∈ LVec)
mapdindp1.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
mapdindp1.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
mapdindp1.z	⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
mapdindp1.W	⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))
mapdindp1.e	⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))
mapdindp1.ne	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
mapdindp1.f	⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))
mapdindp1.yz	⊢ (𝜑 → (𝑌 + 𝑍) ≠ 0 )

**Assertion**
Ref	Expression
mapdindp0	⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) = (𝑁‘{𝑌}))

**Proof of Theorem mapdindp0**
Step	Hyp	Ref	Expression
1		eqid 2738	. . . 4 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
2		mapdindp1.n	. . . 4 ⊢ 𝑁 = (LSpan‘𝑊)
3		mapdindp1.w	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ LVec)
4		lveclmod 20283	. . . . 5 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
5	3, 4	syl 17	. . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod)
6		mapdindp1.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
7	6	eldifad 3895	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
8		mapdindp1.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑊)
9	8, 1, 2	lspsncl 20154	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
10	5, 7, 9	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
11		mapdindp1.e	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))
12	11, 10	eqeltrrd 2840	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
13		eqid 2738	. . . . . 6 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊)
14	1, 13	lsmcl 20260	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊) ∧ (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊)) → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})) ∈ (LSubSp‘𝑊))
15	5, 10, 12, 14	syl3anc 1369	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})) ∈ (LSubSp‘𝑊))
16	1	lsssssubg 20135	. . . . . . 7 ⊢ (𝑊 ∈ LMod → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊))
17	5, 16	syl 17	. . . . . 6 ⊢ (𝜑 → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊))
18	17, 10	sseldd 3918	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊))
19	11, 18	eqeltrrd 2840	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (SubGrp‘𝑊))
20	8, 2	lspsnid 20170	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → 𝑌 ∈ (𝑁‘{𝑌}))
21	5, 7, 20	syl2anc 583	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑌}))
22		mapdindp1.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
23	22	eldifad 3895	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
24	8, 2	lspsnid 20170	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑍 ∈ 𝑉) → 𝑍 ∈ (𝑁‘{𝑍}))
25	5, 23, 24	syl2anc 583	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ (𝑁‘{𝑍}))
26		mapdindp1.p	. . . . . 6 ⊢ + = (+_g‘𝑊)
27	26, 13	lsmelvali 19170	. . . . 5 ⊢ ((((𝑁‘{𝑌}) ∈ (SubGrp‘𝑊) ∧ (𝑁‘{𝑍}) ∈ (SubGrp‘𝑊)) ∧ (𝑌 ∈ (𝑁‘{𝑌}) ∧ 𝑍 ∈ (𝑁‘{𝑍}))) → (𝑌 + 𝑍) ∈ ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})))
28	18, 19, 21, 25, 27	syl22anc 835	. . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})))
29	1, 2, 5, 15, 28	lspsnel5a 20173	. . 3 ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) ⊆ ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})))
30	11	oveq2d 7271	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑌})) = ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})))
31	13	lsmidm 19183	. . . . 5 ⊢ ((𝑁‘{𝑌}) ∈ (SubGrp‘𝑊) → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑌})) = (𝑁‘{𝑌}))
32	18, 31	syl 17	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑌})) = (𝑁‘{𝑌}))
33	30, 32	eqtr3d 2780	. . 3 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})) = (𝑁‘{𝑌}))
34	29, 33	sseqtrd 3957	. 2 ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) ⊆ (𝑁‘{𝑌}))
35		mapdindp1.o	. . 3 ⊢ 0 = (0_g‘𝑊)
36	8, 26	lmodvacl 20052	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) → (𝑌 + 𝑍) ∈ 𝑉)
37	5, 7, 23, 36	syl3anc 1369	. . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ 𝑉)
38		mapdindp1.yz	. . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) ≠ 0 )
39		eldifsn 4717	. . . 4 ⊢ ((𝑌 + 𝑍) ∈ (𝑉 ∖ { 0 }) ↔ ((𝑌 + 𝑍) ∈ 𝑉 ∧ (𝑌 + 𝑍) ≠ 0 ))
40	37, 38, 39	sylanbrc 582	. . 3 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ (𝑉 ∖ { 0 }))
41	8, 35, 2, 3, 40, 7	lspsncmp 20293	. 2 ⊢ (𝜑 → ((𝑁‘{(𝑌 + 𝑍)}) ⊆ (𝑁‘{𝑌}) ↔ (𝑁‘{(𝑌 + 𝑍)}) = (𝑁‘{𝑌})))
42	34, 41	mpbid 231	1 ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) = (𝑁‘{𝑌}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∖ cdif 3880 ⊆ wss 3883 {csn 4558 {cpr 4560 ‘cfv 6418 (class class class)co 7255 Basecbs 16840 +_gcplusg 16888 0_gc0g 17067 SubGrpcsubg 18664 LSSumclsm 19154 LModclmod 20038 LSubSpclss 20108 LSpanclspn 20148 LVecclvec 20279

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-tpos 8013 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-0g 17069 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-grp 18495 df-minusg 18496 df-sbg 18497 df-subg 18667 df-cntz 18838 df-lsm 19156 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-drng 19908 df-lmod 20040 df-lss 20109 df-lsp 20149 df-lvec 20280

This theorem is referenced by: mapdindp1 39661 mapdindp2 39662

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