mapdindp4 - Mathbox for Norm Megill

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Theorem mapdindp4 41742

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 21064	. . . . 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 3938	. . . 4 ⊢ (𝜑 → 𝑤 ∈ 𝑉)
10		mapdindp1.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
11	10	eldifad 3938	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
12		mapdindp1.p	. . . . 5 ⊢ + = (+_g‘𝑊)
13	1, 12	lmodvacl 20832	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑤 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑤 + 𝑌) ∈ 𝑉)
14	7, 9, 11, 13	syl3anc 1373	. . 3 ⊢ (𝜑 → (𝑤 + 𝑌) ∈ 𝑉)
15		mapdindp1.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
16	15	eldifad 3938	. . 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 21093	. . . . . . . 8 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌})))
20	19	simprd 495	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌}))
21	20	necomd 2987	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤}))
22	1, 12, 2, 3, 4, 11, 8, 21	lspindp3 21097	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{(𝑌 + 𝑤)}))
23	1, 12	lmodcom 20865	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑤 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑤 + 𝑌) = (𝑌 + 𝑤))
24	7, 9, 11, 23	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → (𝑤 + 𝑌) = (𝑌 + 𝑤))
25	24	sneqd 4613	. . . . . 6 ⊢ (𝜑 → {(𝑤 + 𝑌)} = {(𝑌 + 𝑤)})
26	25	fveq2d 6880	. . . . 5 ⊢ (𝜑 → (𝑁‘{(𝑤 + 𝑌)}) = (𝑁‘{(𝑌 + 𝑤)}))
27	22, 26	neeqtrrd 3006	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{(𝑤 + 𝑌)}))
28	17, 27	eqnetrrd 3000	. . 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 21094	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑌}) ∧ ¬ 𝑋 ∈ (𝑁‘{𝑤, 𝑌})))
31	30	simprd 495	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑤, 𝑌}))
32		eqid 2735	. . . . . 6 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊)
33	5	eldifad 3938	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
34	1, 3, 32, 7, 33, 14	lsmpr 21047	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑍, (𝑤 + 𝑌)}) = ((𝑁‘{𝑍})(LSSum‘𝑊)(𝑁‘{(𝑤 + 𝑌)})))
35	1, 12	lmodcom 20865	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) → (𝑌 + 𝑤) = (𝑤 + 𝑌))
36	7, 11, 9, 35	syl3anc 1373	. . . . . . . . 9 ⊢ (𝜑 → (𝑌 + 𝑤) = (𝑤 + 𝑌))
37	36	preq2d 4716	. . . . . . . 8 ⊢ (𝜑 → {𝑌, (𝑌 + 𝑤)} = {𝑌, (𝑤 + 𝑌)})
38	37	fveq2d 6880	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌, (𝑌 + 𝑤)}) = (𝑁‘{𝑌, (𝑤 + 𝑌)}))
39	1, 12, 3, 7, 11, 9	lspprabs 21053	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌, (𝑌 + 𝑤)}) = (𝑁‘{𝑌, 𝑤}))
40	1, 3, 32, 7, 11, 14	lsmpr 21047	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌, (𝑤 + 𝑌)}) = ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{(𝑤 + 𝑌)})))
41	38, 39, 40	3eqtr3rd 2779	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{(𝑤 + 𝑌)})) = (𝑁‘{𝑌, 𝑤}))
42	17	oveq1d 7420	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{(𝑤 + 𝑌)})) = ((𝑁‘{𝑍})(LSSum‘𝑊)(𝑁‘{(𝑤 + 𝑌)})))
43		prcom 4708	. . . . . . . 8 ⊢ {𝑌, 𝑤} = {𝑤, 𝑌}
44	43	fveq2i 6879	. . . . . . 7 ⊢ (𝑁‘{𝑌, 𝑤}) = (𝑁‘{𝑤, 𝑌})
45	44	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑤}) = (𝑁‘{𝑤, 𝑌}))
46	41, 42, 45	3eqtr3d 2778	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑍})(LSSum‘𝑊)(𝑁‘{(𝑤 + 𝑌)})) = (𝑁‘{𝑤, 𝑌}))
47	34, 46	eqtrd 2770	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑍, (𝑤 + 𝑌)}) = (𝑁‘{𝑤, 𝑌}))
48	31, 47	neleqtrrd 2857	. . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑍, (𝑤 + 𝑌)}))
49	1, 2, 3, 4, 5, 14, 16, 28, 48	lspindp1 21094	. 2 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{(𝑤 + 𝑌)}) ∧ ¬ 𝑍 ∈ (𝑁‘{𝑋, (𝑤 + 𝑌)})))
50	49	simprd 495	1 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, (𝑤 + 𝑌)}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 ∖ cdif 3923 {csn 4601 {cpr 4603 ‘cfv 6531 (class class class)co 7405 Basecbs 17228 +_gcplusg 17271 0_gc0g 17453 LSSumclsm 19615 LModclmod 20817 LSpanclspn 20928 LVecclvec 21060

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8719 df-en 8960 df-dom 8961 df-sdom 8962 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-nn 12241 df-2 12303 df-3 12304 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-0g 17455 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-submnd 18762 df-grp 18919 df-minusg 18920 df-sbg 18921 df-subg 19106 df-cntz 19300 df-lsm 19617 df-cmn 19763 df-abl 19764 df-mgp 20101 df-rng 20113 df-ur 20142 df-ring 20195 df-oppr 20297 df-dvdsr 20317 df-unit 20318 df-invr 20348 df-drng 20691 df-lmod 20819 df-lss 20889 df-lsp 20929 df-lvec 21061

This theorem is referenced by: mapdh6eN 41759 hdmap1l6e 41833

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