mapdindp4 - Mathbox for Norm Megill

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Theorem mapdindp4 39019

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}))

**Assertion**
Ref	Expression
mapdindp4	⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, (𝑤 + 𝑌)}))

**Proof of Theorem mapdindp4**
Step	Hyp	Ref	Expression
1		mapdindp1.v	. . 3 ⊢ 𝑉 = (Base‘𝑊)
2		mapdindp1.o	. . 3 ⊢ 0 = (0_g‘𝑊)
3		mapdindp1.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑊)
4		mapdindp1.w	. . 3 ⊢ (𝜑 → 𝑊 ∈ LVec)
5		mapdindp1.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
6		lveclmod 19871	. . . . 5 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
7	4, 6	syl 17	. . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod)
8		mapdindp1.W	. . . . 5 ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))
9	8	eldifad 3893	. . . 4 ⊢ (𝜑 → 𝑤 ∈ 𝑉)
10		mapdindp1.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
11	10	eldifad 3893	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
12		mapdindp1.p	. . . . 5 ⊢ + = (+_g‘𝑊)
13	1, 12	lmodvacl 19641	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑤 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑤 + 𝑌) ∈ 𝑉)
14	7, 9, 11, 13	syl3anc 1368	. . 3 ⊢ (𝜑 → (𝑤 + 𝑌) ∈ 𝑉)
15		mapdindp1.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
16	15	eldifad 3893	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
17		mapdindp1.e	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))
18		mapdindp1.f	. . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))
19	1, 3, 4, 9, 16, 11, 18	lspindpi 19897	. . . . . . . 8 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌})))
20	19	simprd 499	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌}))
21	20	necomd 3042	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤}))
22	1, 12, 2, 3, 4, 11, 8, 21	lspindp3 19901	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{(𝑌 + 𝑤)}))
23	1, 12	lmodcom 19673	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑤 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑤 + 𝑌) = (𝑌 + 𝑤))
24	7, 9, 11, 23	syl3anc 1368	. . . . . . 7 ⊢ (𝜑 → (𝑤 + 𝑌) = (𝑌 + 𝑤))
25	24	sneqd 4537	. . . . . 6 ⊢ (𝜑 → {(𝑤 + 𝑌)} = {(𝑌 + 𝑤)})
26	25	fveq2d 6649	. . . . 5 ⊢ (𝜑 → (𝑁‘{(𝑤 + 𝑌)}) = (𝑁‘{(𝑌 + 𝑤)}))
27	22, 26	neeqtrrd 3061	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{(𝑤 + 𝑌)}))
28	17, 27	eqnetrrd 3055	. . 3 ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{(𝑤 + 𝑌)}))
29		mapdindp1.ne	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
30	1, 2, 3, 4, 15, 11, 9, 29, 18	lspindp1 19898	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑌}) ∧ ¬ 𝑋 ∈ (𝑁‘{𝑤, 𝑌})))
31	30	simprd 499	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑤, 𝑌}))
32		eqid 2798	. . . . . 6 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊)
33	5	eldifad 3893	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
34	1, 3, 32, 7, 33, 14	lsmpr 19854	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑍, (𝑤 + 𝑌)}) = ((𝑁‘{𝑍})(LSSum‘𝑊)(𝑁‘{(𝑤 + 𝑌)})))
35	1, 12	lmodcom 19673	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) → (𝑌 + 𝑤) = (𝑤 + 𝑌))
36	7, 11, 9, 35	syl3anc 1368	. . . . . . . . 9 ⊢ (𝜑 → (𝑌 + 𝑤) = (𝑤 + 𝑌))
37	36	preq2d 4636	. . . . . . . 8 ⊢ (𝜑 → {𝑌, (𝑌 + 𝑤)} = {𝑌, (𝑤 + 𝑌)})
38	37	fveq2d 6649	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌, (𝑌 + 𝑤)}) = (𝑁‘{𝑌, (𝑤 + 𝑌)}))
39	1, 12, 3, 7, 11, 9	lspprabs 19860	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌, (𝑌 + 𝑤)}) = (𝑁‘{𝑌, 𝑤}))
40	1, 3, 32, 7, 11, 14	lsmpr 19854	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌, (𝑤 + 𝑌)}) = ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{(𝑤 + 𝑌)})))
41	38, 39, 40	3eqtr3rd 2842	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{(𝑤 + 𝑌)})) = (𝑁‘{𝑌, 𝑤}))
42	17	oveq1d 7150	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{(𝑤 + 𝑌)})) = ((𝑁‘{𝑍})(LSSum‘𝑊)(𝑁‘{(𝑤 + 𝑌)})))
43		prcom 4628	. . . . . . . 8 ⊢ {𝑌, 𝑤} = {𝑤, 𝑌}
44	43	fveq2i 6648	. . . . . . 7 ⊢ (𝑁‘{𝑌, 𝑤}) = (𝑁‘{𝑤, 𝑌})
45	44	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑤}) = (𝑁‘{𝑤, 𝑌}))
46	41, 42, 45	3eqtr3d 2841	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑍})(LSSum‘𝑊)(𝑁‘{(𝑤 + 𝑌)})) = (𝑁‘{𝑤, 𝑌}))
47	34, 46	eqtrd 2833	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑍, (𝑤 + 𝑌)}) = (𝑁‘{𝑤, 𝑌}))
48	31, 47	neleqtrrd 2912	. . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑍, (𝑤 + 𝑌)}))
49	1, 2, 3, 4, 5, 14, 16, 28, 48	lspindp1 19898	. 2 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{(𝑤 + 𝑌)}) ∧ ¬ 𝑍 ∈ (𝑁‘{𝑋, (𝑤 + 𝑌)})))
50	49	simprd 499	1 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, (𝑤 + 𝑌)}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∖ cdif 3878 {csn 4525 {cpr 4527 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +_gcplusg 16557 0_gc0g 16705 LSSumclsm 18751 LModclmod 19627 LSpanclspn 19736 LVecclvec 19867

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-cntz 18439 df-lsm 18753 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-drng 19497 df-lmod 19629 df-lss 19697 df-lsp 19737 df-lvec 19868

This theorem is referenced by: mapdh6eN 39036 hdmap1l6e 39110

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