mapdindp2 - Mathbox for Norm Megill

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Theorem mapdindp2 38872

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 4670	. . . . . 6 ⊢ ((𝑌 + 𝑍) = 0 → {𝑋, (𝑌 + 𝑍)} = {𝑋, 0 })
2	1	fveq2d 6674	. . . . 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 19878	. . . . . . 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 3948	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
11	3, 4, 5, 8, 10	lsppr0 19864	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋, 0 }) = (𝑁‘{𝑋}))
12	2, 11	sylan9eqr 2878	. . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → (𝑁‘{𝑋, (𝑌 + 𝑍)}) = (𝑁‘{𝑋}))
13		mapdindp1.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
14	13	eldifad 3948	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
15		prssi 4754	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {𝑋, 𝑌} ⊆ 𝑉)
16	10, 14, 15	syl2anc 586	. . . . . 6 ⊢ (𝜑 → {𝑋, 𝑌} ⊆ 𝑉)
17		snsspr1 4747	. . . . . . 7 ⊢ {𝑋} ⊆ {𝑋, 𝑌}
18	17	a1i 11	. . . . . 6 ⊢ (𝜑 → {𝑋} ⊆ {𝑋, 𝑌})
19	3, 5	lspss 19756	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ {𝑋, 𝑌} ⊆ 𝑉 ∧ {𝑋} ⊆ {𝑋, 𝑌}) → (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑋, 𝑌}))
20	8, 16, 18, 19	syl3anc 1367	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑋, 𝑌}))
21	20	adantr 483	. . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑋, 𝑌}))
22	12, 21	eqsstrd 4005	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → (𝑁‘{𝑋, (𝑌 + 𝑍)}) ⊆ (𝑁‘{𝑋, 𝑌}))
23		mapdindp1.f	. . . 4 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))
24	23	adantr 483	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))
25	22, 24	ssneldd 3970	. 2 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → ¬ 𝑤 ∈ (𝑁‘{𝑋, (𝑌 + 𝑍)}))
26	23	adantr 483	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))
27		mapdindp1.p	. . . . . 6 ⊢ + = (+_g‘𝑊)
28	6	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑊 ∈ LVec)
29	9	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑋 ∈ (𝑉 ∖ { 0 }))
30	13	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑌 ∈ (𝑉 ∖ { 0 }))
31		mapdindp1.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
32	31	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑍 ∈ (𝑉 ∖ { 0 }))
33		mapdindp1.W	. . . . . . 7 ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))
34	33	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑤 ∈ (𝑉 ∖ { 0 }))
35		mapdindp1.e	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))
36	35	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))
37		mapdindp1.ne	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
38	37	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
39		simpr 487	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑌 + 𝑍) ≠ 0 )
40	3, 27, 4, 5, 28, 29, 30, 32, 34, 36, 38, 26, 39	mapdindp0 38870	. . . . 5 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{(𝑌 + 𝑍)}) = (𝑁‘{𝑌}))
41	40	oveq2d 7172	. . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{(𝑌 + 𝑍)})) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})))
42		eqid 2821	. . . . . 6 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊)
43	31	eldifad 3948	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
44	3, 27	lmodvacl 19648	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) → (𝑌 + 𝑍) ∈ 𝑉)
45	8, 14, 43, 44	syl3anc 1367	. . . . . 6 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ 𝑉)
46	3, 5, 42, 8, 10, 45	lsmpr 19861	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋, (𝑌 + 𝑍)}) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{(𝑌 + 𝑍)})))
47	46	adantr 483	. . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{𝑋, (𝑌 + 𝑍)}) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{(𝑌 + 𝑍)})))
48	3, 5, 42, 8, 10, 14	lsmpr 19861	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})))
49	48	adantr 483	. . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})))
50	41, 47, 49	3eqtr4d 2866	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{𝑋, (𝑌 + 𝑍)}) = (𝑁‘{𝑋, 𝑌}))
51	26, 50	neleqtrrd 2935	. 2 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → ¬ 𝑤 ∈ (𝑁‘{𝑋, (𝑌 + 𝑍)}))
52	25, 51	pm2.61dane 3104	1 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, (𝑌 + 𝑍)}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∖ cdif 3933 ⊆ wss 3936 {csn 4567 {cpr 4569 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +_gcplusg 16565 0_gc0g 16713 LSSumclsm 18759 LModclmod 19634 LSpanclspn 19743 LVecclvec 19874

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-cntz 18447 df-lsm 18761 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-drng 19504 df-lmod 19636 df-lss 19704 df-lsp 19744 df-lvec 19875

This theorem is referenced by: mapdh6dN 38890 hdmap1l6d 38964

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