mapdindp4 - Mathbox for Norm Megill

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Theorem mapdindp4 41832

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 21040	. . . . 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 3909	. . . 4 ⊢ (𝜑 → 𝑤 ∈ 𝑉)
10		mapdindp1.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
11	10	eldifad 3909	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
12		mapdindp1.p	. . . . 5 ⊢ + = (+_g‘𝑊)
13	1, 12	lmodvacl 20808	. . . 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 3909	. . 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 21069	. . . . . . . 8 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌})))
20	19	simprd 495	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌}))
21	20	necomd 2983	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤}))
22	1, 12, 2, 3, 4, 11, 8, 21	lspindp3 21073	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{(𝑌 + 𝑤)}))
23	1, 12	lmodcom 20841	. . . . . . . 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 4585	. . . . . 6 ⊢ (𝜑 → {(𝑤 + 𝑌)} = {(𝑌 + 𝑤)})
26	25	fveq2d 6826	. . . . 5 ⊢ (𝜑 → (𝑁‘{(𝑤 + 𝑌)}) = (𝑁‘{(𝑌 + 𝑤)}))
27	22, 26	neeqtrrd 3002	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{(𝑤 + 𝑌)}))
28	17, 27	eqnetrrd 2996	. . 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 21070	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑌}) ∧ ¬ 𝑋 ∈ (𝑁‘{𝑤, 𝑌})))
31	30	simprd 495	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑤, 𝑌}))
32		eqid 2731	. . . . . 6 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊)
33	5	eldifad 3909	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
34	1, 3, 32, 7, 33, 14	lsmpr 21023	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑍, (𝑤 + 𝑌)}) = ((𝑁‘{𝑍})(LSSum‘𝑊)(𝑁‘{(𝑤 + 𝑌)})))
35	1, 12	lmodcom 20841	. . . . . . . . . 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 4690	. . . . . . . 8 ⊢ (𝜑 → {𝑌, (𝑌 + 𝑤)} = {𝑌, (𝑤 + 𝑌)})
38	37	fveq2d 6826	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌, (𝑌 + 𝑤)}) = (𝑁‘{𝑌, (𝑤 + 𝑌)}))
39	1, 12, 3, 7, 11, 9	lspprabs 21029	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌, (𝑌 + 𝑤)}) = (𝑁‘{𝑌, 𝑤}))
40	1, 3, 32, 7, 11, 14	lsmpr 21023	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌, (𝑤 + 𝑌)}) = ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{(𝑤 + 𝑌)})))
41	38, 39, 40	3eqtr3rd 2775	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{(𝑤 + 𝑌)})) = (𝑁‘{𝑌, 𝑤}))
42	17	oveq1d 7361	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{(𝑤 + 𝑌)})) = ((𝑁‘{𝑍})(LSSum‘𝑊)(𝑁‘{(𝑤 + 𝑌)})))
43		prcom 4682	. . . . . . . 8 ⊢ {𝑌, 𝑤} = {𝑤, 𝑌}
44	43	fveq2i 6825	. . . . . . 7 ⊢ (𝑁‘{𝑌, 𝑤}) = (𝑁‘{𝑤, 𝑌})
45	44	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑤}) = (𝑁‘{𝑤, 𝑌}))
46	41, 42, 45	3eqtr3d 2774	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑍})(LSSum‘𝑊)(𝑁‘{(𝑤 + 𝑌)})) = (𝑁‘{𝑤, 𝑌}))
47	34, 46	eqtrd 2766	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑍, (𝑤 + 𝑌)}) = (𝑁‘{𝑤, 𝑌}))
48	31, 47	neleqtrrd 2854	. . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑍, (𝑤 + 𝑌)}))
49	1, 2, 3, 4, 5, 14, 16, 28, 48	lspindp1 21070	. 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 1541 ∈ wcel 2111 ≠ wne 2928 ∖ cdif 3894 {csn 4573 {cpr 4575 ‘cfv 6481 (class class class)co 7346 Basecbs 17120 +_gcplusg 17161 0_gc0g 17343 LSSumclsm 19546 LModclmod 20793 LSpanclspn 20904 LVecclvec 21036

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-tpos 8156 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-2 12188 df-3 12189 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-0g 17345 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-grp 18849 df-minusg 18850 df-sbg 18851 df-subg 19036 df-cntz 19229 df-lsm 19548 df-cmn 19694 df-abl 19695 df-mgp 20059 df-rng 20071 df-ur 20100 df-ring 20153 df-oppr 20255 df-dvdsr 20275 df-unit 20276 df-invr 20306 df-drng 20646 df-lmod 20795 df-lss 20865 df-lsp 20905 df-lvec 21037

This theorem is referenced by: mapdh6eN 41849 hdmap1l6e 41923

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