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Theorem hdmap1eq4N 39557
Description: Convert mapdheq4 39483 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 = (0g𝑈)
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 ∧ 𝑊𝐻))
hdmap1eq2.f (𝜑𝐹𝐷)
hdmap1eq2.mn (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))
hdmap1eq4.x (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))
hdmap1eq4.y (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))
hdmap1eq4.z (𝜑𝑍 ∈ (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 are mutually distinct and distinct from all other variables.
StepHypRef 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 2737 . . 3 (-g𝑈) = (-g𝑈)
5 hdmap1eq2.o . . 3 0 = (0g𝑈)
6 hdmap1eq2.n . . 3 𝑁 = (LSpan‘𝑈)
7 hdmap1eq2.c . . 3 𝐶 = ((LCDual‘𝐾)‘𝑊)
8 hdmap1eq2.d . . 3 𝐷 = (Base‘𝐶)
9 eqid 2737 . . 3 (-g𝐶) = (-g𝐶)
10 eqid 2737 . . 3 (0g𝐶) = (0g𝐶)
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 ∧ 𝑊𝐻))
15 hdmap1eq4.y . . 3 (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))
16 hdmap1eq4.eg . . . 4 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)
17 hdmap1eq2.f . . . . 5 (𝜑𝐹𝐷)
18 hdmap1eq2.mn . . . . 5 (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))
191, 2, 14dvhlvec 38860 . . . . . . 7 (𝜑𝑈 ∈ LVec)
20 hdmap1eq4.x . . . . . . . 8 (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))
2120eldifad 3878 . . . . . . 7 (𝜑𝑋𝑉)
2215eldifad 3878 . . . . . . 7 (𝜑𝑌𝑉)
23 hdmap1eq4.z . . . . . . . 8 (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))
2423eldifad 3878 . . . . . . 7 (𝜑𝑍𝑉)
25 hdmap1eq4.xn . . . . . . 7 (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
263, 6, 19, 21, 22, 24, 25lspindpi 20169 . . . . . 6 (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})))
2726simpld 498 . . . . 5 (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
281, 2, 3, 5, 6, 7, 8, 11, 12, 13, 14, 17, 18, 27, 20, 22hdmap1cl 39555 . . . 4 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷)
2916, 28eqeltrrd 2839 . . 3 (𝜑𝐺𝐷)
30 eqid 2737 . . 3 (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , (0g𝐶), (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐿‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥))(-g𝑈)(2nd𝑥))})) = (𝐿‘{((2nd ‘(1st𝑥))(-g𝐶))}))))) = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , (0g𝐶), (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐿‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥))(-g𝑈)(2nd𝑥))})) = (𝐿‘{((2nd ‘(1st𝑥))(-g𝐶))})))))
311, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 29, 24, 30hdmap1valc 39554 . 2 (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑍⟩) = ((𝑥 ∈ V ↦ if((2nd𝑥) = 0 , (0g𝐶), (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐿‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥))(-g𝑈)(2nd𝑥))})) = (𝐿‘{((2nd ‘(1st𝑥))(-g𝐶))})))))‘⟨𝑌, 𝐺, 𝑍⟩))
32 hdmap1eq4.ne . . 3 (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))
331, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 20, 17, 22, 30hdmap1valc 39554 . . . 4 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = ((𝑥 ∈ V ↦ if((2nd𝑥) = 0 , (0g𝐶), (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐿‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥))(-g𝑈)(2nd𝑥))})) = (𝐿‘{((2nd ‘(1st𝑥))(-g𝐶))})))))‘⟨𝑋, 𝐹, 𝑌⟩))
3433, 16eqtr3d 2779 . . 3 (𝜑 → ((𝑥 ∈ V ↦ if((2nd𝑥) = 0 , (0g𝐶), (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐿‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥))(-g𝑈)(2nd𝑥))})) = (𝐿‘{((2nd ‘(1st𝑥))(-g𝐶))})))))‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)
351, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 20, 17, 24, 30hdmap1valc 39554 . . . 4 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = ((𝑥 ∈ V ↦ if((2nd𝑥) = 0 , (0g𝐶), (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐿‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥))(-g𝑈)(2nd𝑥))})) = (𝐿‘{((2nd ‘(1st𝑥))(-g𝐶))})))))‘⟨𝑋, 𝐹, 𝑍⟩))
36 hdmap1eq4.ee . . . 4 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐵)
3735, 36eqtr3d 2779 . . 3 (𝜑 → ((𝑥 ∈ V ↦ if((2nd𝑥) = 0 , (0g𝐶), (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐿‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥))(-g𝑈)(2nd𝑥))})) = (𝐿‘{((2nd ‘(1st𝑥))(-g𝐶))})))))‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐵)
3810, 30, 1, 12, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 17, 18, 20, 15, 23, 25, 32, 34, 37mapdheq4 39483 . 2 (𝜑 → ((𝑥 ∈ V ↦ if((2nd𝑥) = 0 , (0g𝐶), (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐿‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥))(-g𝑈)(2nd𝑥))})) = (𝐿‘{((2nd ‘(1st𝑥))(-g𝐶))})))))‘⟨𝑌, 𝐺, 𝑍⟩) = 𝐵)
3931, 38eqtrd 2777 1 (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑍⟩) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1543  wcel 2110  wne 2940  Vcvv 3408  cdif 3863  ifcif 4439  {csn 4541  {cpr 4543  cotp 4549  cmpt 5135  cfv 6380  crio 7169  (class class class)co 7213  1st c1st 7759  2nd c2nd 7760  Basecbs 16760  0gc0g 16944  -gcsg 18367  LSpanclspn 20008  HLchlt 37101  LHypclh 37735  DVecHcdvh 38829  LCDualclcd 39337  mapdcmpd 39375  HDMap1chdma1 39542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-riotaBAD 36704
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-ot 4550  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-of 7469  df-om 7645  df-1st 7761  df-2nd 7762  df-tpos 7968  df-undef 8015  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-er 8391  df-map 8510  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-2 11893  df-3 11894  df-4 11895  df-5 11896  df-6 11897  df-n0 12091  df-z 12177  df-uz 12439  df-fz 13096  df-struct 16700  df-sets 16717  df-slot 16735  df-ndx 16745  df-base 16761  df-ress 16785  df-plusg 16815  df-mulr 16816  df-sca 16818  df-vsca 16819  df-0g 16946  df-mre 17089  df-mrc 17090  df-acs 17092  df-proset 17802  df-poset 17820  df-plt 17836  df-lub 17852  df-glb 17853  df-join 17854  df-meet 17855  df-p0 17931  df-p1 17932  df-lat 17938  df-clat 18005  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-submnd 18219  df-grp 18368  df-minusg 18369  df-sbg 18370  df-subg 18540  df-cntz 18711  df-oppg 18738  df-lsm 19025  df-cmn 19172  df-abl 19173  df-mgp 19505  df-ur 19517  df-ring 19564  df-oppr 19641  df-dvdsr 19659  df-unit 19660  df-invr 19690  df-dvr 19701  df-drng 19769  df-lmod 19901  df-lss 19969  df-lsp 20009  df-lvec 20140  df-lsatoms 36727  df-lshyp 36728  df-lcv 36770  df-lfl 36809  df-lkr 36837  df-ldual 36875  df-oposet 36927  df-ol 36929  df-oml 36930  df-covers 37017  df-ats 37018  df-atl 37049  df-cvlat 37073  df-hlat 37102  df-llines 37249  df-lplanes 37250  df-lvols 37251  df-lines 37252  df-psubsp 37254  df-pmap 37255  df-padd 37547  df-lhyp 37739  df-laut 37740  df-ldil 37855  df-ltrn 37856  df-trl 37910  df-tgrp 38494  df-tendo 38506  df-edring 38508  df-dveca 38754  df-disoa 38780  df-dvech 38830  df-dib 38890  df-dic 38924  df-dih 38980  df-doch 39099  df-djh 39146  df-lcdual 39338  df-mapd 39376  df-hdmap1 39544
This theorem is referenced by:  hdmapval3lemN  39588
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