mapdindp2 - Mathbox for Norm Megill

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Theorem mapdindp2 41745

Description: Vector independence lemma. (Contributed by NM, 1-May-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}))

**Assertion**
Ref	Expression
mapdindp2	⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, (𝑌 + 𝑍)}))

**Proof of Theorem mapdindp2**
Step	Hyp	Ref	Expression
1		preq2 4715	. . . . . 6 ⊢ ((𝑌 + 𝑍) = 0 → {𝑋, (𝑌 + 𝑍)} = {𝑋, 0 })
2	1	fveq2d 6885	. . . . 5 ⊢ ((𝑌 + 𝑍) = 0 → (𝑁‘{𝑋, (𝑌 + 𝑍)}) = (𝑁‘{𝑋, 0 }))
3		mapdindp1.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑊)
4		mapdindp1.o	. . . . . 6 ⊢ 0 = (0_g‘𝑊)
5		mapdindp1.n	. . . . . 6 ⊢ 𝑁 = (LSpan‘𝑊)
6		mapdindp1.w	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LVec)
7		lveclmod 21069	. . . . . . 7 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
8	6, 7	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod)
9		mapdindp1.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
10	9	eldifad 3943	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
11	3, 4, 5, 8, 10	lsppr0 21055	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋, 0 }) = (𝑁‘{𝑋}))
12	2, 11	sylan9eqr 2793	. . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → (𝑁‘{𝑋, (𝑌 + 𝑍)}) = (𝑁‘{𝑋}))
13		mapdindp1.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
14	13	eldifad 3943	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
15		prssi 4802	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {𝑋, 𝑌} ⊆ 𝑉)
16	10, 14, 15	syl2anc 584	. . . . . 6 ⊢ (𝜑 → {𝑋, 𝑌} ⊆ 𝑉)
17		snsspr1 4795	. . . . . . 7 ⊢ {𝑋} ⊆ {𝑋, 𝑌}
18	17	a1i 11	. . . . . 6 ⊢ (𝜑 → {𝑋} ⊆ {𝑋, 𝑌})
19	3, 5	lspss 20946	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ {𝑋, 𝑌} ⊆ 𝑉 ∧ {𝑋} ⊆ {𝑋, 𝑌}) → (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑋, 𝑌}))
20	8, 16, 18, 19	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑋, 𝑌}))
21	20	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑋, 𝑌}))
22	12, 21	eqsstrd 3998	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → (𝑁‘{𝑋, (𝑌 + 𝑍)}) ⊆ (𝑁‘{𝑋, 𝑌}))
23		mapdindp1.f	. . . 4 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))
24	23	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))
25	22, 24	ssneldd 3966	. 2 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → ¬ 𝑤 ∈ (𝑁‘{𝑋, (𝑌 + 𝑍)}))
26	23	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))
27		mapdindp1.p	. . . . . 6 ⊢ + = (+_g‘𝑊)
28	6	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑊 ∈ LVec)
29	9	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑋 ∈ (𝑉 ∖ { 0 }))
30	13	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑌 ∈ (𝑉 ∖ { 0 }))
31		mapdindp1.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
32	31	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑍 ∈ (𝑉 ∖ { 0 }))
33		mapdindp1.W	. . . . . . 7 ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))
34	33	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑤 ∈ (𝑉 ∖ { 0 }))
35		mapdindp1.e	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))
36	35	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))
37		mapdindp1.ne	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
38	37	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
39		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑌 + 𝑍) ≠ 0 )
40	3, 27, 4, 5, 28, 29, 30, 32, 34, 36, 38, 26, 39	mapdindp0 41743	. . . . 5 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{(𝑌 + 𝑍)}) = (𝑁‘{𝑌}))
41	40	oveq2d 7426	. . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{(𝑌 + 𝑍)})) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})))
42		eqid 2736	. . . . . 6 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊)
43	31	eldifad 3943	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
44	3, 27	lmodvacl 20837	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) → (𝑌 + 𝑍) ∈ 𝑉)
45	8, 14, 43, 44	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ 𝑉)
46	3, 5, 42, 8, 10, 45	lsmpr 21052	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋, (𝑌 + 𝑍)}) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{(𝑌 + 𝑍)})))
47	46	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{𝑋, (𝑌 + 𝑍)}) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{(𝑌 + 𝑍)})))
48	3, 5, 42, 8, 10, 14	lsmpr 21052	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})))
49	48	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})))
50	41, 47, 49	3eqtr4d 2781	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{𝑋, (𝑌 + 𝑍)}) = (𝑁‘{𝑋, 𝑌}))
51	26, 50	neleqtrrd 2858	. 2 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → ¬ 𝑤 ∈ (𝑁‘{𝑋, (𝑌 + 𝑍)}))
52	25, 51	pm2.61dane 3020	1 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, (𝑌 + 𝑍)}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2933 ∖ cdif 3928 ⊆ wss 3931 {csn 4606 {cpr 4608 ‘cfv 6536 (class class class)co 7410 Basecbs 17233 +_gcplusg 17276 0_gc0g 17458 LSSumclsm 19620 LModclmod 20822 LSpanclspn 20933 LVecclvec 21065

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-nn 12246 df-2 12308 df-3 12309 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-0g 17460 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-submnd 18767 df-grp 18924 df-minusg 18925 df-sbg 18926 df-subg 19111 df-cntz 19305 df-lsm 19622 df-cmn 19768 df-abl 19769 df-mgp 20106 df-rng 20118 df-ur 20147 df-ring 20200 df-oppr 20302 df-dvdsr 20322 df-unit 20323 df-invr 20353 df-drng 20696 df-lmod 20824 df-lss 20894 df-lsp 20934 df-lvec 21066

This theorem is referenced by: mapdh6dN 41763 hdmap1l6d 41837

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