mapdindp2 - Mathbox for Norm Megill

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Theorem mapdindp2 42097

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 4693	. . . . . 6 ⊢ ((𝑌 + 𝑍) = 0 → {𝑋, (𝑌 + 𝑍)} = {𝑋, 0 })
2	1	fveq2d 6846	. . . . 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 21070	. . . . . . 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 3915	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
11	3, 4, 5, 8, 10	lsppr0 21056	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋, 0 }) = (𝑁‘{𝑋}))
12	2, 11	sylan9eqr 2794	. . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → (𝑁‘{𝑋, (𝑌 + 𝑍)}) = (𝑁‘{𝑋}))
13		mapdindp1.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
14	13	eldifad 3915	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
15		prssi 4779	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {𝑋, 𝑌} ⊆ 𝑉)
16	10, 14, 15	syl2anc 585	. . . . . 6 ⊢ (𝜑 → {𝑋, 𝑌} ⊆ 𝑉)
17		snsspr1 4772	. . . . . . 7 ⊢ {𝑋} ⊆ {𝑋, 𝑌}
18	17	a1i 11	. . . . . 6 ⊢ (𝜑 → {𝑋} ⊆ {𝑋, 𝑌})
19	3, 5	lspss 20947	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ {𝑋, 𝑌} ⊆ 𝑉 ∧ {𝑋} ⊆ {𝑋, 𝑌}) → (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑋, 𝑌}))
20	8, 16, 18, 19	syl3anc 1374	. . . . 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 3970	. . 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 3938	. 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 42095	. . . . 5 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{(𝑌 + 𝑍)}) = (𝑁‘{𝑌}))
41	40	oveq2d 7384	. . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{(𝑌 + 𝑍)})) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})))
42		eqid 2737	. . . . . 6 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊)
43	31	eldifad 3915	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
44	3, 27	lmodvacl 20838	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) → (𝑌 + 𝑍) ∈ 𝑉)
45	8, 14, 43, 44	syl3anc 1374	. . . . . 6 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ 𝑉)
46	3, 5, 42, 8, 10, 45	lsmpr 21053	. . . . 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 21053	. . . . 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 2782	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{𝑋, (𝑌 + 𝑍)}) = (𝑁‘{𝑋, 𝑌}))
51	26, 50	neleqtrrd 2860	. 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 1542 ∈ wcel 2114 ≠ wne 2933 ∖ cdif 3900 ⊆ wss 3903 {csn 4582 {cpr 4584 ‘cfv 6500 (class class class)co 7368 Basecbs 17148 +_gcplusg 17189 0_gc0g 17371 LSSumclsm 19575 LModclmod 20823 LSpanclspn 20934 LVecclvec 21066

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-tpos 8178 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-0g 17373 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-grp 18878 df-minusg 18879 df-sbg 18880 df-subg 19065 df-cntz 19258 df-lsm 19577 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-ring 20182 df-oppr 20285 df-dvdsr 20305 df-unit 20306 df-invr 20336 df-drng 20676 df-lmod 20825 df-lss 20895 df-lsp 20935 df-lvec 21067

This theorem is referenced by: mapdh6dN 42115 hdmap1l6d 42189

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