mapdindp0 - Mathbox for Norm Megill

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Theorem mapdindp0 39470

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 2737	. . . 4 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
2		mapdindp1.n	. . . 4 ⊢ 𝑁 = (LSpan‘𝑊)
3		mapdindp1.w	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ LVec)
4		lveclmod 20143	. . . . 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 3878	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
8		mapdindp1.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑊)
9	8, 1, 2	lspsncl 20014	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
10	5, 7, 9	syl2anc 587	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
11		mapdindp1.e	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))
12	11, 10	eqeltrrd 2839	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
13		eqid 2737	. . . . . 6 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊)
14	1, 13	lsmcl 20120	. . . . 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 1373	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})) ∈ (LSubSp‘𝑊))
16	1	lsssssubg 19995	. . . . . . 7 ⊢ (𝑊 ∈ LMod → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊))
17	5, 16	syl 17	. . . . . 6 ⊢ (𝜑 → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊))
18	17, 10	sseldd 3902	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊))
19	11, 18	eqeltrrd 2839	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (SubGrp‘𝑊))
20	8, 2	lspsnid 20030	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → 𝑌 ∈ (𝑁‘{𝑌}))
21	5, 7, 20	syl2anc 587	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑌}))
22		mapdindp1.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
23	22	eldifad 3878	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
24	8, 2	lspsnid 20030	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑍 ∈ 𝑉) → 𝑍 ∈ (𝑁‘{𝑍}))
25	5, 23, 24	syl2anc 587	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ (𝑁‘{𝑍}))
26		mapdindp1.p	. . . . . 6 ⊢ + = (+_g‘𝑊)
27	26, 13	lsmelvali 19039	. . . . 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 839	. . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})))
29	1, 2, 5, 15, 28	lspsnel5a 20033	. . 3 ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) ⊆ ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})))
30	11	oveq2d 7229	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑌})) = ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})))
31	13	lsmidm 19052	. . . . 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 2779	. . 3 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{𝑍})) = (𝑁‘{𝑌}))
34	29, 33	sseqtrd 3941	. 2 ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) ⊆ (𝑁‘{𝑌}))
35		mapdindp1.o	. . 3 ⊢ 0 = (0_g‘𝑊)
36	8, 26	lmodvacl 19913	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) → (𝑌 + 𝑍) ∈ 𝑉)
37	5, 7, 23, 36	syl3anc 1373	. . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ 𝑉)
38		mapdindp1.yz	. . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) ≠ 0 )
39		eldifsn 4700	. . . 4 ⊢ ((𝑌 + 𝑍) ∈ (𝑉 ∖ { 0 }) ↔ ((𝑌 + 𝑍) ∈ 𝑉 ∧ (𝑌 + 𝑍) ≠ 0 ))
40	37, 38, 39	sylanbrc 586	. . 3 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ (𝑉 ∖ { 0 }))
41	8, 35, 2, 3, 40, 7	lspsncmp 20153	. 2 ⊢ (𝜑 → ((𝑁‘{(𝑌 + 𝑍)}) ⊆ (𝑁‘{𝑌}) ↔ (𝑁‘{(𝑌 + 𝑍)}) = (𝑁‘{𝑌})))
42	34, 41	mpbid 235	1 ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) = (𝑁‘{𝑌}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 ∖ cdif 3863 ⊆ wss 3866 {csn 4541 {cpr 4543 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 +_gcplusg 16802 0_gc0g 16944 SubGrpcsubg 18537 LSSumclsm 19023 LModclmod 19899 LSubSpclss 19968 LSpanclspn 20008 LVecclvec 20139

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-tpos 7968 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-0g 16946 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-grp 18368 df-minusg 18369 df-sbg 18370 df-subg 18540 df-cntz 18711 df-lsm 19025 df-cmn 19172 df-abl 19173 df-mgp 19505 df-ur 19517 df-ring 19564 df-oppr 19641 df-dvdsr 19659 df-unit 19660 df-invr 19690 df-drng 19769 df-lmod 19901 df-lss 19969 df-lsp 20009 df-lvec 20140

This theorem is referenced by: mapdindp1 39471 mapdindp2 39472

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