mapdpglem11 - Mathbox for Norm Megill

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Theorem mapdpglem11 37830

Description: Lemma for mapdpg 37854. (Contributed by NM, 20-Mar-2015.)

**Hypotheses**
Ref	Expression
mapdpglem.h	⊢ 𝐻 = (LHyp‘𝐾)
mapdpglem.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
mapdpglem.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
mapdpglem.v	⊢ 𝑉 = (Base‘𝑈)
mapdpglem.s	⊢ − = (-_g‘𝑈)
mapdpglem.n	⊢ 𝑁 = (LSpan‘𝑈)
mapdpglem.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
mapdpglem.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
mapdpglem.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
mapdpglem.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
mapdpglem1.p	⊢ ⊕ = (LSSum‘𝐶)
mapdpglem2.j	⊢ 𝐽 = (LSpan‘𝐶)
mapdpglem3.f	⊢ 𝐹 = (Base‘𝐶)
mapdpglem3.te	⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌}))))
mapdpglem3.a	⊢ 𝐴 = (Scalar‘𝑈)
mapdpglem3.b	⊢ 𝐵 = (Base‘𝐴)
mapdpglem3.t	⊢ · = ( ·_𝑠 ‘𝐶)
mapdpglem3.r	⊢ 𝑅 = (-_g‘𝐶)
mapdpglem3.g	⊢ (𝜑 → 𝐺 ∈ 𝐹)
mapdpglem3.e	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))
mapdpglem4.q	⊢ 𝑄 = (0_g‘𝑈)
mapdpglem.ne	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
mapdpglem4.jt	⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡}))
mapdpglem4.z	⊢ 0 = (0_g‘𝐴)
mapdpglem4.g4	⊢ (𝜑 → 𝑔 ∈ 𝐵)
mapdpglem4.z4	⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))
mapdpglem4.t4	⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧))
mapdpglem4.xn	⊢ (𝜑 → 𝑋 ≠ 𝑄)

**Assertion**
Ref	Expression
mapdpglem11	⊢ (𝜑 → 𝑔 ≠ 0 )

Distinct variable groups: 𝑡, − 𝑡,𝐶 𝑡,𝐽 𝑡,𝑀 𝑡,𝑁 𝑡,𝑋 𝑡,𝑌 𝐵,𝑔 𝑧,𝑔,𝐶 𝑔,𝐹 𝑔,𝐺,𝑧 𝑔,𝐽,𝑧 𝑔,𝑀,𝑧 𝑔,𝑁,𝑧 𝑅,𝑔,𝑧 · ,𝑔,𝑧 𝑔,𝑌,𝑧,𝑡 Allowed substitution hints: 𝜑(𝑧,𝑡,𝑔) 𝐴(𝑧,𝑡,𝑔) 𝐵(𝑧,𝑡) ⊕ (𝑧,𝑡,𝑔) 𝑄(𝑧,𝑡,𝑔) 𝑅(𝑡) · (𝑡) 𝑈(𝑧,𝑡,𝑔) 𝐹(𝑧,𝑡) 𝐺(𝑡) 𝐻(𝑧,𝑡,𝑔) 𝐾(𝑧,𝑡,𝑔) − (𝑧,𝑔) 𝑉(𝑧,𝑡,𝑔) 𝑊(𝑧,𝑡,𝑔) 𝑋(𝑧,𝑔) 0 (𝑧,𝑡,𝑔)

**Proof of Theorem mapdpglem11**
Step	Hyp	Ref	Expression
1		mapdpglem.ne	. 2 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
2		mapdpglem.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
3		mapdpglem.m	. . . . 5 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
4		mapdpglem.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
5		mapdpglem.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑈)
6		mapdpglem.s	. . . . 5 ⊢ − = (-_g‘𝑈)
7		mapdpglem.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑈)
8		mapdpglem.c	. . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
9		mapdpglem.k	. . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
10	9	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 0 ) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
11		mapdpglem.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
12	11	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 0 ) → 𝑋 ∈ 𝑉)
13		mapdpglem.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
14	13	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 0 ) → 𝑌 ∈ 𝑉)
15		mapdpglem1.p	. . . . 5 ⊢ ⊕ = (LSSum‘𝐶)
16		mapdpglem2.j	. . . . 5 ⊢ 𝐽 = (LSpan‘𝐶)
17		mapdpglem3.f	. . . . 5 ⊢ 𝐹 = (Base‘𝐶)
18		mapdpglem3.te	. . . . . 6 ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌}))))
19	18	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 0 ) → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌}))))
20		mapdpglem3.a	. . . . 5 ⊢ 𝐴 = (Scalar‘𝑈)
21		mapdpglem3.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐴)
22		mapdpglem3.t	. . . . 5 ⊢ · = ( ·_𝑠 ‘𝐶)
23		mapdpglem3.r	. . . . 5 ⊢ 𝑅 = (-_g‘𝐶)
24		mapdpglem3.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
25	24	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 0 ) → 𝐺 ∈ 𝐹)
26		mapdpglem3.e	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))
27	26	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 0 ) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))
28		mapdpglem4.q	. . . . 5 ⊢ 𝑄 = (0_g‘𝑈)
29	1	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 0 ) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
30		mapdpglem4.jt	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡}))
31	30	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 0 ) → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡}))
32		mapdpglem4.z	. . . . 5 ⊢ 0 = (0_g‘𝐴)
33		mapdpglem4.g4	. . . . . 6 ⊢ (𝜑 → 𝑔 ∈ 𝐵)
34	33	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 0 ) → 𝑔 ∈ 𝐵)
35		mapdpglem4.z4	. . . . . 6 ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))
36	35	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 0 ) → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))
37		mapdpglem4.t4	. . . . . 6 ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧))
38	37	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 0 ) → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧))
39		mapdpglem4.xn	. . . . . 6 ⊢ (𝜑 → 𝑋 ≠ 𝑄)
40	39	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 0 ) → 𝑋 ≠ 𝑄)
41		simpr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 0 ) → 𝑔 = 0 )
42	2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 16, 17, 19, 20, 21, 22, 23, 25, 27, 28, 29, 31, 32, 34, 36, 38, 40, 41	mapdpglem10 37829	. . . 4 ⊢ ((𝜑 ∧ 𝑔 = 0 ) → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))
43	42	ex 403	. . 3 ⊢ (𝜑 → (𝑔 = 0 → (𝑁‘{𝑋}) = (𝑁‘{𝑌})))
44	43	necon3d 2989	. 2 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) → 𝑔 ≠ 0 ))
45	1, 44	mpd 15	1 ⊢ (𝜑 → 𝑔 ≠ 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ≠ wne 2968 {csn 4397 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 Scalarcsca 16341 ·_𝑠 cvsca 16342 0_gc0g 16486 -_gcsg 17811 LSSumclsm 18433 LSpanclspn 19366 HLchlt 35498 LHypclh 36132 DVecHcdvh 37226 LCDualclcd 37734 mapdcmpd 37772

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-riotaBAD 35101

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-tpos 7634 df-undef 7681 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-0g 16488 df-mre 16632 df-mrc 16633 df-acs 16635 df-proset 17314 df-poset 17332 df-plt 17344 df-lub 17360 df-glb 17361 df-join 17362 df-meet 17363 df-p0 17425 df-p1 17426 df-lat 17432 df-clat 17494 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-grp 17812 df-minusg 17813 df-sbg 17814 df-subg 17975 df-cntz 18133 df-oppg 18159 df-lsm 18435 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-oppr 19010 df-dvdsr 19028 df-unit 19029 df-invr 19059 df-dvr 19070 df-drng 19141 df-lmod 19257 df-lss 19325 df-lsp 19367 df-lvec 19498 df-lsatoms 35124 df-lshyp 35125 df-lcv 35167 df-lfl 35206 df-lkr 35234 df-ldual 35272 df-oposet 35324 df-ol 35326 df-oml 35327 df-covers 35414 df-ats 35415 df-atl 35446 df-cvlat 35470 df-hlat 35499 df-llines 35646 df-lplanes 35647 df-lvols 35648 df-lines 35649 df-psubsp 35651 df-pmap 35652 df-padd 35944 df-lhyp 36136 df-laut 36137 df-ldil 36252 df-ltrn 36253 df-trl 36307 df-tgrp 36891 df-tendo 36903 df-edring 36905 df-dveca 37151 df-disoa 37177 df-dvech 37227 df-dib 37287 df-dic 37321 df-dih 37377 df-doch 37496 df-djh 37543 df-lcdual 37735 df-mapd 37773

This theorem is referenced by: mapdpglem17N 37836 mapdpglem18 37837 mapdpglem19 37838 mapdpglem21 37840 mapdpglem22 37841

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