Theorem hdmap1eq4N 42431

Description: Convert mapdheq4 42357 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 2763	. . 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 2763	. . 3 ⊢ (-g‘𝐶) = (-g‘𝐶)
10		eqid 2763	. . 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 41734	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LVec)
20		hdmap1eq4.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
21	20	eldifad 3917	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
22	15	eldifad 3917	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
23		hdmap1eq4.z	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
24	23	eldifad 3917	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
25		hdmap1eq4.xn	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
26	3, 6, 19, 21, 22, 24, 25	lspindpi 21203	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})))
27	26	simpld 498	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
28	1, 2, 3, 5, 6, 7, 8, 11, 12, 13, 14, 17, 18, 27, 20, 22	hdmap1cl 42429	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷)
29	16, 28	eqeltrrd 2864	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐷)
30		eqid 2763	. . 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 42428	. 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 42428	. . . 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 2800	. . 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 42428	. . . 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 2800	. . 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 42357	. 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 2798	1 ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑍⟩) = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1561   ∈ wcel 2143   ≠ wne 2958  Vcvv 3455   ∖ cdif 3902  ifcif 4481  {csn 4583  {cpr 4585  ⟨cotp 4591   ↦ cmpt 5182  ‘cfv 6522  ℩crio 7353  (class class class)co 7397  1^st c1st 7969  2^nd c2nd 7970  Basecbs 17246  0_gc0g 17469  -_gcsg 18978  LSpanclspn 21039  HLchlt 39975  LHypclh 40609  DVecHcdvh 41703  LCDualclcd 42211  mapdcmpd 42249  HDMap1chdma1 42416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719  ax-cnex 11130  ax-resscn 11131  ax-1cn 11132  ax-icn 11133  ax-addcl 11134  ax-addrcl 11135  ax-mulcl 11136  ax-mulrcl 11137  ax-mulcom 11138  ax-addass 11139  ax-mulass 11140  ax-distr 11141  ax-i2m1 11142  ax-1ne0 11143  ax-1rid 11144  ax-rnegex 11145  ax-rrecex 11146  ax-cnre 11147  ax-pre-lttri 11148  ax-pre-lttrn 11149  ax-pre-ltadd 11150  ax-pre-mulgt0 11151  ax-riotaBAD 39578

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-ot 4592  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6289  df-ord 6350  df-on 6351  df-lim 6352  df-suc 6353  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-riota 7354  df-ov 7400  df-oprab 7401  df-mpo 7402  df-of 7661  df-om 7848  df-1st 7971  df-2nd 7972  df-tpos 8207  df-undef 8254  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8382  df-1o 8438  df-2o 8439  df-er 8679  df-map 8811  df-en 8929  df-dom 8930  df-sdom 8931  df-fin 8932  df-pnf 11219  df-mnf 11220  df-xr 11221  df-ltxr 11222  df-le 11223  df-sub 11417  df-neg 11418  df-nn 12212  df-2 12281  df-3 12282  df-4 12283  df-5 12284  df-6 12285  df-n0 12483  df-z 12570  df-uz 12841  df-fz 13514  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17247  df-ress 17268  df-plusg 17300  df-mulr 17301  df-sca 17303  df-vsca 17304  df-0g 17471  df-mre 17615  df-mrc 17616  df-acs 17618  df-proset 18327  df-poset 18346  df-plt 18361  df-lub 18377  df-glb 18378  df-join 18379  df-meet 18380  df-p0 18456  df-p1 18457  df-lat 18465  df-clat 18532  df-mgm 18675  df-sgrp 18754  df-mnd 18770  df-submnd 18819  df-grp 18979  df-minusg 18980  df-sbg 18981  df-subg 19166  df-cntz 19358  df-oppg 19387  df-lsm 19677  df-cmn 19823  df-abl 19824  df-mgp 20188  df-rng 20200  df-ur 20233  df-ring 20286  df-oppr 20387  df-dvdsr 20407  df-unit 20408  df-invr 20438  df-dvr 20451  df-nzr 20564  df-rlreg 20745  df-domn 20746  df-drng 20782  df-lmod 20930  df-lss 21000  df-lsp 21040  df-lvec 21171  df-lsatoms 39601  df-lshyp 39602  df-lcv 39644  df-lfl 39683  df-lkr 39711  df-ldual 39749  df-oposet 39801  df-ol 39803  df-oml 39804  df-covers 39891  df-ats 39892  df-atl 39923  df-cvlat 39947  df-hlat 39976  df-llines 40123  df-lplanes 40124  df-lvols 40125  df-lines 40126  df-psubsp 40128  df-pmap 40129  df-padd 40421  df-lhyp 40613  df-laut 40614  df-ldil 40729  df-ltrn 40730  df-trl 40784  df-tgrp 41368  df-tendo 41380  df-edring 41382  df-dveca 41628  df-disoa 41654  df-dvech 41704  df-dib 41764  df-dic 41798  df-dih 41854  df-doch 41973  df-djh 42020  df-lcdual 42212  df-mapd 42250  df-hdmap1 42418

This theorem is referenced by:  hdmapval3lemN  42462