Theorem hdmap1eq4N 37962

Description: Convert mapdheq4 37888 to use HDMap1 function. (Contributed by NM, 17-May-2015.) (New usage is discouraged.)

Hypotheses

Ref	Expression
hdmap1eq2.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmap1eq2.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmap1eq2.v	⊢ 𝑉 = (Base‘𝑈)
hdmap1eq2.o	⊢ 0 = (0g‘𝑈)
hdmap1eq2.n	⊢ 𝑁 = (LSpan‘𝑈)
hdmap1eq2.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmap1eq2.d	⊢ 𝐷 = (Base‘𝐶)
hdmap1eq2.l	⊢ 𝐿 = (LSpan‘𝐶)
hdmap1eq2.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
hdmap1eq2.i	⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)
hdmap1eq2.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmap1eq2.f	⊢ (𝜑 → 𝐹 ∈ 𝐷)
hdmap1eq2.mn	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))
hdmap1eq4.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
hdmap1eq4.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
hdmap1eq4.z	⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
hdmap1eq4.ne	⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))
hdmap1eq4.xn	⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
hdmap1eq4.eg	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)
hdmap1eq4.ee	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐵)

Assertion

Ref	Expression
hdmap1eq4N	⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑍⟩) = 𝐵)

Proof of Theorem hdmap1eq4N

Dummy variables 𝑥 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hdmap1eq2.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmap1eq2.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		hdmap1eq2.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
4		eqid 2778	. . 3 ⊢ (-g‘𝑈) = (-g‘𝑈)
5		hdmap1eq2.o	. . 3 ⊢ 0 = (0g‘𝑈)
6		hdmap1eq2.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
7		hdmap1eq2.c	. . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
8		hdmap1eq2.d	. . 3 ⊢ 𝐷 = (Base‘𝐶)
9		eqid 2778	. . 3 ⊢ (-g‘𝐶) = (-g‘𝐶)
10		eqid 2778	. . 3 ⊢ (0g‘𝐶) = (0g‘𝐶)
11		hdmap1eq2.l	. . 3 ⊢ 𝐿 = (LSpan‘𝐶)
12		hdmap1eq2.m	. . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
13		hdmap1eq2.i	. . 3 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)
14		hdmap1eq2.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
15		hdmap1eq4.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
16		hdmap1eq4.eg	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)
17		hdmap1eq2.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
18		hdmap1eq2.mn	. . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))
19	1, 2, 14	dvhlvec 37265	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LVec)
20		hdmap1eq4.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
21	20	eldifad 3804	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
22	15	eldifad 3804	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
23		hdmap1eq4.z	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
24	23	eldifad 3804	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
25		hdmap1eq4.xn	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
26	3, 6, 19, 21, 22, 24, 25	lspindpi 19528	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})))
27	26	simpld 490	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
28	1, 2, 3, 5, 6, 7, 8, 11, 12, 13, 14, 17, 18, 27, 20, 22	hdmap1cl 37960	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷)
29	16, 28	eqeltrrd 2860	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐷)
30		eqid 2778	. . 3 ⊢ (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , (0g‘𝐶), (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐿‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥))(-g‘𝑈)(2nd ‘𝑥))})) = (𝐿‘{((2nd ‘(1st ‘𝑥))(-g‘𝐶)ℎ)}))))) = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , (0g‘𝐶), (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐿‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥))(-g‘𝑈)(2nd ‘𝑥))})) = (𝐿‘{((2nd ‘(1st ‘𝑥))(-g‘𝐶)ℎ)})))))
31	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 29, 24, 30	hdmap1valc 37959	. 2 ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑍⟩) = ((𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , (0g‘𝐶), (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐿‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥))(-g‘𝑈)(2nd ‘𝑥))})) = (𝐿‘{((2nd ‘(1st ‘𝑥))(-g‘𝐶)ℎ)})))))‘⟨𝑌, 𝐺, 𝑍⟩))
32		hdmap1eq4.ne	. . 3 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))
33	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 20, 17, 22, 30	hdmap1valc 37959	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = ((𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , (0g‘𝐶), (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐿‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥))(-g‘𝑈)(2nd ‘𝑥))})) = (𝐿‘{((2nd ‘(1st ‘𝑥))(-g‘𝐶)ℎ)})))))‘⟨𝑋, 𝐹, 𝑌⟩))
34	33, 16	eqtr3d 2816	. . 3 ⊢ (𝜑 → ((𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , (0g‘𝐶), (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐿‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥))(-g‘𝑈)(2nd ‘𝑥))})) = (𝐿‘{((2nd ‘(1st ‘𝑥))(-g‘𝐶)ℎ)})))))‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)
35	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 20, 17, 24, 30	hdmap1valc 37959	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = ((𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , (0g‘𝐶), (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐿‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥))(-g‘𝑈)(2nd ‘𝑥))})) = (𝐿‘{((2nd ‘(1st ‘𝑥))(-g‘𝐶)ℎ)})))))‘⟨𝑋, 𝐹, 𝑍⟩))
36		hdmap1eq4.ee	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐵)
37	35, 36	eqtr3d 2816	. . 3 ⊢ (𝜑 → ((𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , (0g‘𝐶), (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐿‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥))(-g‘𝑈)(2nd ‘𝑥))})) = (𝐿‘{((2nd ‘(1st ‘𝑥))(-g‘𝐶)ℎ)})))))‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐵)
38	10, 30, 1, 12, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 17, 18, 20, 15, 23, 25, 32, 34, 37	mapdheq4 37888	. 2 ⊢ (𝜑 → ((𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , (0g‘𝐶), (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐿‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥))(-g‘𝑈)(2nd ‘𝑥))})) = (𝐿‘{((2nd ‘(1st ‘𝑥))(-g‘𝐶)ℎ)})))))‘⟨𝑌, 𝐺, 𝑍⟩) = 𝐵)
39	31, 38	eqtrd 2814	1 ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑍⟩) = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107   ≠ wne 2969  Vcvv 3398   ∖ cdif 3789  ifcif 4307  {csn 4398  {cpr 4400  ⟨cotp 4406   ↦ cmpt 4965  ‘cfv 6135  ℩crio 6882  (class class class)co 6922  1^st c1st 7443  2^nd c2nd 7444  Basecbs 16255  0_gc0g 16486  -_gcsg 17811  LSpanclspn 19366  HLchlt 35506  LHypclh 36140  DVecHcdvh 37234  LCDualclcd 37742  mapdcmpd 37780  HDMap1chdma1 37947

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 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  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 35109

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 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-ot 4407  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 35132  df-lshyp 35133  df-lcv 35175  df-lfl 35214  df-lkr 35242  df-ldual 35280  df-oposet 35332  df-ol 35334  df-oml 35335  df-covers 35422  df-ats 35423  df-atl 35454  df-cvlat 35478  df-hlat 35507  df-llines 35654  df-lplanes 35655  df-lvols 35656  df-lines 35657  df-psubsp 35659  df-pmap 35660  df-padd 35952  df-lhyp 36144  df-laut 36145  df-ldil 36260  df-ltrn 36261  df-trl 36315  df-tgrp 36899  df-tendo 36911  df-edring 36913  df-dveca 37159  df-disoa 37185  df-dvech 37235  df-dib 37295  df-dic 37329  df-dih 37385  df-doch 37504  df-djh 37551  df-lcdual 37743  df-mapd 37781  df-hdmap1 37949

This theorem is referenced by:  hdmapval3lemN  37993