Theorem hdmap1eq4N 40482

Description: Convert mapdheq4 40408 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 2731	. . 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 2731	. . 3 ⊢ (-g‘𝐶) = (-g‘𝐶)
10		eqid 2731	. . 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 39785	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LVec)
20		hdmap1eq4.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
21	20	eldifad 3956	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
22	15	eldifad 3956	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
23		hdmap1eq4.z	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
24	23	eldifad 3956	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
25		hdmap1eq4.xn	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
26	3, 6, 19, 21, 22, 24, 25	lspindpi 20694	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})))
27	26	simpld 495	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
28	1, 2, 3, 5, 6, 7, 8, 11, 12, 13, 14, 17, 18, 27, 20, 22	hdmap1cl 40480	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷)
29	16, 28	eqeltrrd 2833	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐷)
30		eqid 2731	. . 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 40479	. 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 40479	. . . 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 2773	. . 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 40479	. . . 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 2773	. . 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 40408	. 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 2771	1 ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑍⟩) = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2939  Vcvv 3473   ∖ cdif 3941  ifcif 4522  {csn 4622  {cpr 4624  ⟨cotp 4630   ↦ cmpt 5224  ‘cfv 6532  ℩crio 7348  (class class class)co 7393  1^st c1st 7955  2^nd c2nd 7956  Basecbs 17126  0_gc0g 17367  -_gcsg 18796  LSpanclspn 20531  HLchlt 38025  LHypclh 38660  DVecHcdvh 39754  LCDualclcd 40262  mapdcmpd 40300  HDMap1chdma1 40467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-cnex 11148  ax-resscn 11149  ax-1cn 11150  ax-icn 11151  ax-addcl 11152  ax-addrcl 11153  ax-mulcl 11154  ax-mulrcl 11155  ax-mulcom 11156  ax-addass 11157  ax-mulass 11158  ax-distr 11159  ax-i2m1 11160  ax-1ne0 11161  ax-1rid 11162  ax-rnegex 11163  ax-rrecex 11164  ax-cnre 11165  ax-pre-lttri 11166  ax-pre-lttrn 11167  ax-pre-ltadd 11168  ax-pre-mulgt0 11169  ax-riotaBAD 37628

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-tp 4627  df-op 4629  df-ot 4631  df-uni 4902  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-of 7653  df-om 7839  df-1st 7957  df-2nd 7958  df-tpos 8193  df-undef 8240  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-1o 8448  df-er 8686  df-map 8805  df-en 8923  df-dom 8924  df-sdom 8925  df-fin 8926  df-pnf 11232  df-mnf 11233  df-xr 11234  df-ltxr 11235  df-le 11236  df-sub 11428  df-neg 11429  df-nn 12195  df-2 12257  df-3 12258  df-4 12259  df-5 12260  df-6 12261  df-n0 12455  df-z 12541  df-uz 12805  df-fz 13467  df-struct 17062  df-sets 17079  df-slot 17097  df-ndx 17109  df-base 17127  df-ress 17156  df-plusg 17192  df-mulr 17193  df-sca 17195  df-vsca 17196  df-0g 17369  df-mre 17512  df-mrc 17513  df-acs 17515  df-proset 18230  df-poset 18248  df-plt 18265  df-lub 18281  df-glb 18282  df-join 18283  df-meet 18284  df-p0 18360  df-p1 18361  df-lat 18367  df-clat 18434  df-mgm 18543  df-sgrp 18592  df-mnd 18603  df-submnd 18648  df-grp 18797  df-minusg 18798  df-sbg 18799  df-subg 18975  df-cntz 19147  df-oppg 19174  df-lsm 19468  df-cmn 19614  df-abl 19615  df-mgp 19947  df-ur 19964  df-ring 20016  df-oppr 20102  df-dvdsr 20123  df-unit 20124  df-invr 20154  df-dvr 20165  df-drng 20267  df-lmod 20422  df-lss 20492  df-lsp 20532  df-lvec 20663  df-lsatoms 37651  df-lshyp 37652  df-lcv 37694  df-lfl 37733  df-lkr 37761  df-ldual 37799  df-oposet 37851  df-ol 37853  df-oml 37854  df-covers 37941  df-ats 37942  df-atl 37973  df-cvlat 37997  df-hlat 38026  df-llines 38174  df-lplanes 38175  df-lvols 38176  df-lines 38177  df-psubsp 38179  df-pmap 38180  df-padd 38472  df-lhyp 38664  df-laut 38665  df-ldil 38780  df-ltrn 38781  df-trl 38835  df-tgrp 39419  df-tendo 39431  df-edring 39433  df-dveca 39679  df-disoa 39705  df-dvech 39755  df-dib 39815  df-dic 39849  df-dih 39905  df-doch 40024  df-djh 40071  df-lcdual 40263  df-mapd 40301  df-hdmap1 40469

This theorem is referenced by:  hdmapval3lemN  40513