mapdindp2 - Mathbox for Norm Megill

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Theorem mapdindp2 37010

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 4269	. . . . . 6 ⊢ ((𝑌 + 𝑍) = 0 → {𝑋, (𝑌 + 𝑍)} = {𝑋, 0 })
2	1	fveq2d 6195	. . . . 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 19106	. . . . . . 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 3586	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
11	3, 4, 5, 8, 10	lsppr0 19092	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋, 0 }) = (𝑁‘{𝑋}))
12	2, 11	sylan9eqr 2678	. . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → (𝑁‘{𝑋, (𝑌 + 𝑍)}) = (𝑁‘{𝑋}))
13		mapdindp1.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
14	13	eldifad 3586	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
15		prssi 4353	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {𝑋, 𝑌} ⊆ 𝑉)
16	10, 14, 15	syl2anc 693	. . . . . 6 ⊢ (𝜑 → {𝑋, 𝑌} ⊆ 𝑉)
17		snsspr1 4345	. . . . . . 7 ⊢ {𝑋} ⊆ {𝑋, 𝑌}
18	17	a1i 11	. . . . . 6 ⊢ (𝜑 → {𝑋} ⊆ {𝑋, 𝑌})
19	3, 5	lspss 18984	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ {𝑋, 𝑌} ⊆ 𝑉 ∧ {𝑋} ⊆ {𝑋, 𝑌}) → (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑋, 𝑌}))
20	8, 16, 18, 19	syl3anc 1326	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑋, 𝑌}))
21	20	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑋, 𝑌}))
22	12, 21	eqsstrd 3639	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → (𝑁‘{𝑋, (𝑌 + 𝑍)}) ⊆ (𝑁‘{𝑋, 𝑌}))
23		mapdindp1.f	. . . 4 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))
24	23	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))
25	22, 24	ssneldd 3606	. 2 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → ¬ 𝑤 ∈ (𝑁‘{𝑋, (𝑌 + 𝑍)}))
26	23	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))
27		mapdindp1.p	. . . . . 6 ⊢ + = (+_g‘𝑊)
28	6	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑊 ∈ LVec)
29	9	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑋 ∈ (𝑉 ∖ { 0 }))
30	13	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑌 ∈ (𝑉 ∖ { 0 }))
31		mapdindp1.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
32	31	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑍 ∈ (𝑉 ∖ { 0 }))
33		mapdindp1.W	. . . . . . 7 ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))
34	33	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑤 ∈ (𝑉 ∖ { 0 }))
35		mapdindp1.e	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))
36	35	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))
37		mapdindp1.ne	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
38	37	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
39		simpr 477	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑌 + 𝑍) ≠ 0 )
40	3, 27, 4, 5, 28, 29, 30, 32, 34, 36, 38, 26, 39	mapdindp0 37008	. . . . 5 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{(𝑌 + 𝑍)}) = (𝑁‘{𝑌}))
41	40	oveq2d 6666	. . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{(𝑌 + 𝑍)})) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})))
42		eqid 2622	. . . . . 6 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊)
43	31	eldifad 3586	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
44	3, 27	lmodvacl 18877	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) → (𝑌 + 𝑍) ∈ 𝑉)
45	8, 14, 43, 44	syl3anc 1326	. . . . . 6 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ 𝑉)
46	3, 5, 42, 8, 10, 45	lsmpr 19089	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋, (𝑌 + 𝑍)}) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{(𝑌 + 𝑍)})))
47	46	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{𝑋, (𝑌 + 𝑍)}) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{(𝑌 + 𝑍)})))
48	3, 5, 42, 8, 10, 14	lsmpr 19089	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})))
49	48	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})))
50	41, 47, 49	3eqtr4d 2666	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{𝑋, (𝑌 + 𝑍)}) = (𝑁‘{𝑋, 𝑌}))
51	26, 50	neleqtrrd 2723	. 2 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → ¬ 𝑤 ∈ (𝑁‘{𝑋, (𝑌 + 𝑍)}))
52	25, 51	pm2.61dane 2881	1 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, (𝑌 + 𝑍)}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∖ cdif 3571 ⊆ wss 3574 {csn 4177 {cpr 4179 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +_gcplusg 15941 0_gc0g 16100 LSSumclsm 18049 LModclmod 18863 LSpanclspn 18971 LVecclvec 19102

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-cntz 17750 df-lsm 18051 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-drng 18749 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lvec 19103

This theorem is referenced by: mapdh6dN 37028 hdmap1l6d 37103

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