Theorem hdmapinvlem3 32406

Description: Line 30 in [Baer] p. 110, f(sw + u, tw - v) = 0. (Contributed by NM, 12-Jun-2015.)

Hypotheses

Ref	Expression
hdmapinvlem3.h	|- H = ( LHyp ` K )
hdmapinvlem3.e	|- E = <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` W ) ) >.
hdmapinvlem3.o	|- O = ( ( ocH ` K ) ` W )
hdmapinvlem3.u	|- U = ( ( DVecH ` K ) ` W )
hdmapinvlem3.v	|- V = ( Base ` U )
hdmapinvlem3.p	|- .+ = ( +g ` U )
hdmapinvlem3.m	|- .- = ( -g ` U )
hdmapinvlem3.q	|- .x. = ( .s ` U )
hdmapinvlem3.r	|- R = (Scalar ` U )
hdmapinvlem3.b	|- B = ( Base ` R )
hdmapinvlem3.t	|- .X. = ( .r ` R )
hdmapinvlem3.z	|- .0. = ( 0g ` R )
hdmapinvlem3.s	|- S = ( (HDMap ` K ) ` W )
hdmapinvlem3.g	|- G = ( (HGMap ` K ) ` W )
hdmapinvlem3.k	|- ( ph -> ( K e. HL /\ W e. H ) )
hdmapinvlem3.c	|- ( ph -> C e. ( O ` { E } ) )
hdmapinvlem3.d	|- ( ph -> D e. ( O ` { E } ) )
hdmapinvlem3.i	|- ( ph -> I e. B )
hdmapinvlem3.j	|- ( ph -> J e. B )
hdmapinvlem3.ij	|- ( ph -> ( I .X. ( G ` J ) ) = ( ( S ` D ) ` C ) )

Assertion

Ref	Expression
hdmapinvlem3	|- ( ph -> ( ( S ` ( ( J .x. E ) .- D ) ) ` ( ( I .x. E ) .+ C ) ) = .0. )

Proof of Theorem hdmapinvlem3

Step	Hyp	Ref	Expression
1		hdmapinvlem3.h	. . . 4 |- H = ( LHyp ` K )
2		hdmapinvlem3.u	. . . 4 |- U = ( ( DVecH ` K ) ` W )
3		hdmapinvlem3.v	. . . 4 |- V = ( Base ` U )
4		hdmapinvlem3.m	. . . 4 |- .- = ( -g ` U )
5		eqid 2404	. . . 4 |- ( (LCDual ` K ) ` W ) = ( (LCDual ` K ) ` W )
6		eqid 2404	. . . 4 |- ( -g ` ( (LCDual ` K ) ` W ) ) = ( -g ` ( (LCDual ` K ) ` W ) )
7		hdmapinvlem3.s	. . . 4 |- S = ( (HDMap ` K ) ` W )
8		hdmapinvlem3.k	. . . 4 |- ( ph -> ( K e. HL /\ W e. H ) )
9	1, 2, 8	dvhlmod 31593	. . . . 5 |- ( ph -> U e. LMod )
10		hdmapinvlem3.j	. . . . 5 |- ( ph -> J e. B )
11		eqid 2404	. . . . . . 7 |- ( Base ` K ) = ( Base ` K )
12		eqid 2404	. . . . . . 7 |- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
13		eqid 2404	. . . . . . 7 |- ( 0g ` U ) = ( 0g ` U )
14		hdmapinvlem3.e	. . . . . . 7 |- E = <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` W ) ) >.
15	1, 11, 12, 2, 3, 13, 14, 8	dvheveccl 31595	. . . . . 6 |- ( ph -> E e. ( V \ { ( 0g ` U ) } ) )
16	15	eldifad 3292	. . . . 5 |- ( ph -> E e. V )
17		hdmapinvlem3.r	. . . . . 6 |- R = (Scalar ` U )
18		hdmapinvlem3.q	. . . . . 6 |- .x. = ( .s ` U )
19		hdmapinvlem3.b	. . . . . 6 |- B = ( Base ` R )
20	3, 17, 18, 19	lmodvscl 15922	. . . . 5 |- ( ( U e. LMod /\ J e. B /\ E e. V ) -> ( J .x. E ) e. V )
21	9, 10, 16, 20	syl3anc 1184	. . . 4 |- ( ph -> ( J .x. E ) e. V )
22	16	snssd 3903	. . . . . 6 |- ( ph -> { E } C_ V )
23		hdmapinvlem3.o	. . . . . . 7 |- O = ( ( ocH ` K ) ` W )
24	1, 2, 3, 23	dochssv 31838	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ { E } C_ V ) -> ( O ` { E } ) C_ V )
25	8, 22, 24	syl2anc 643	. . . . 5 |- ( ph -> ( O ` { E } ) C_ V )
26		hdmapinvlem3.d	. . . . 5 |- ( ph -> D e. ( O ` { E } ) )
27	25, 26	sseldd 3309	. . . 4 |- ( ph -> D e. V )
28	1, 2, 3, 4, 5, 6, 7, 8, 21, 27	hdmapsub 32333	. . 3 |- ( ph -> ( S ` ( ( J .x. E ) .- D ) ) = ( ( S ` ( J .x. E ) ) ( -g ` ( (LCDual ` K ) ` W ) ) ( S ` D ) ) )
29	28	fveq1d 5689	. 2 |- ( ph -> ( ( S ` ( ( J .x. E ) .- D ) ) ` ( ( I .x. E ) .+ C ) ) = ( ( ( S ` ( J .x. E ) ) ( -g ` ( (LCDual ` K ) ` W ) ) ( S ` D ) ) ` ( ( I .x. E ) .+ C ) ) )
30		eqid 2404	. . . 4 |- ( -g ` R ) = ( -g ` R )
31		eqid 2404	. . . 4 |- ( Base ` ( (LCDual ` K ) ` W ) ) = ( Base ` ( (LCDual ` K ) ` W ) )
32	1, 2, 3, 5, 31, 7, 8, 21	hdmapcl 32316	. . . 4 |- ( ph -> ( S ` ( J .x. E ) ) e. ( Base ` ( (LCDual ` K ) ` W ) ) )
33	1, 2, 3, 5, 31, 7, 8, 27	hdmapcl 32316	. . . 4 |- ( ph -> ( S ` D ) e. ( Base ` ( (LCDual ` K ) ` W ) ) )
34		hdmapinvlem3.i	. . . . . 6 |- ( ph -> I e. B )
35	3, 17, 18, 19	lmodvscl 15922	. . . . . 6 |- ( ( U e. LMod /\ I e. B /\ E e. V ) -> ( I .x. E ) e. V )
36	9, 34, 16, 35	syl3anc 1184	. . . . 5 |- ( ph -> ( I .x. E ) e. V )
37		hdmapinvlem3.c	. . . . . 6 |- ( ph -> C e. ( O ` { E } ) )
38	25, 37	sseldd 3309	. . . . 5 |- ( ph -> C e. V )
39		hdmapinvlem3.p	. . . . . 6 |- .+ = ( +g ` U )
40	3, 39	lmodvacl 15919	. . . . 5 |- ( ( U e. LMod /\ ( I .x. E ) e. V /\ C e. V ) -> ( ( I .x. E ) .+ C ) e. V )
41	9, 36, 38, 40	syl3anc 1184	. . . 4 |- ( ph -> ( ( I .x. E ) .+ C ) e. V )
42	1, 2, 3, 17, 30, 5, 31, 6, 8, 32, 33, 41	lcdvsubval 32101	. . 3 |- ( ph -> ( ( ( S ` ( J .x. E ) ) ( -g ` ( (LCDual ` K ) ` W ) ) ( S ` D ) ) ` ( ( I .x. E ) .+ C ) ) = ( ( ( S ` ( J .x. E ) ) ` ( ( I .x. E ) .+ C ) ) ( -g ` R ) ( ( S ` D ) ` ( ( I .x. E ) .+ C ) ) ) )
43		eqid 2404	. . . . . 6 |- ( +g ` R ) = ( +g ` R )
44	1, 2, 3, 39, 17, 43, 7, 8, 36, 38, 21	hdmaplna1 32393	. . . . 5 |- ( ph -> ( ( S ` ( J .x. E ) ) ` ( ( I .x. E ) .+ C ) ) = ( ( ( S ` ( J .x. E ) ) ` ( I .x. E ) ) ( +g ` R ) ( ( S ` ( J .x. E ) ) ` C ) ) )
45		hdmapinvlem3.t	. . . . . . . 8 |- .X. = ( .r ` R )
46		hdmapinvlem3.g	. . . . . . . 8 |- G = ( (HGMap ` K ) ` W )
47	1, 2, 3, 18, 17, 19, 45, 7, 46, 8, 36, 16, 10	hdmapglnm2 32397	. . . . . . 7 |- ( ph -> ( ( S ` ( J .x. E ) ) ` ( I .x. E ) ) = ( ( ( S ` E ) ` ( I .x. E ) ) .X. ( G ` J ) ) )
48	1, 2, 3, 18, 17, 19, 45, 7, 8, 16, 16, 34	hdmaplnm1 32395	. . . . . . . . 9 |- ( ph -> ( ( S ` E ) ` ( I .x. E ) ) = ( I .X. ( ( S ` E ) ` E ) ) )
49		eqid 2404	. . . . . . . . . . 11 |- ( (HVMap ` K ) ` W ) = ( (HVMap ` K ) ` W )
50		eqid 2404	. . . . . . . . . . 11 |- ( 1r ` R ) = ( 1r ` R )
51	1, 14, 49, 7, 8, 2, 17, 50	hdmapevec2 32322	. . . . . . . . . 10 |- ( ph -> ( ( S ` E ) ` E ) = ( 1r ` R ) )
52	51	oveq2d 6056	. . . . . . . . 9 |- ( ph -> ( I .X. ( ( S ` E ) ` E ) ) = ( I .X. ( 1r ` R ) ) )
53	17	lmodrng 15913	. . . . . . . . . . 11 |- ( U e. LMod -> R e. Ring )
54	9, 53	syl 16	. . . . . . . . . 10 |- ( ph -> R e. Ring )
55	19, 45, 50	rngridm 15643	. . . . . . . . . 10 |- ( ( R e. Ring /\ I e. B ) -> ( I .X. ( 1r ` R ) ) = I )
56	54, 34, 55	syl2anc 643	. . . . . . . . 9 |- ( ph -> ( I .X. ( 1r ` R ) ) = I )
57	48, 52, 56	3eqtrd 2440	. . . . . . . 8 |- ( ph -> ( ( S ` E ) ` ( I .x. E ) ) = I )
58	57	oveq1d 6055	. . . . . . 7 |- ( ph -> ( ( ( S ` E ) ` ( I .x. E ) ) .X. ( G ` J ) ) = ( I .X. ( G ` J ) ) )
59	47, 58	eqtrd 2436	. . . . . 6 |- ( ph -> ( ( S ` ( J .x. E ) ) ` ( I .x. E ) ) = ( I .X. ( G ` J ) ) )
60	1, 2, 3, 18, 17, 19, 45, 7, 46, 8, 38, 16, 10	hdmapglnm2 32397	. . . . . . 7 |- ( ph -> ( ( S ` ( J .x. E ) ) ` C ) = ( ( ( S ` E ) ` C ) .X. ( G ` J ) ) )
61		hdmapinvlem3.z	. . . . . . . . 9 |- .0. = ( 0g ` R )
62	1, 14, 23, 2, 3, 17, 19, 45, 61, 7, 8, 37	hdmapinvlem1 32404	. . . . . . . 8 |- ( ph -> ( ( S ` E ) ` C ) = .0. )
63	62	oveq1d 6055	. . . . . . 7 |- ( ph -> ( ( ( S ` E ) ` C ) .X. ( G ` J ) ) = ( .0. .X. ( G ` J ) ) )
64	1, 2, 17, 19, 46, 8, 10	hgmapcl 32375	. . . . . . . 8 |- ( ph -> ( G ` J ) e. B )
65	19, 45, 61	rnglz 15655	. . . . . . . 8 |- ( ( R e. Ring /\ ( G ` J ) e. B ) -> ( .0. .X. ( G ` J ) ) = .0. )
66	54, 64, 65	syl2anc 643	. . . . . . 7 |- ( ph -> ( .0. .X. ( G ` J ) ) = .0. )
67	60, 63, 66	3eqtrd 2440	. . . . . 6 |- ( ph -> ( ( S ` ( J .x. E ) ) ` C ) = .0. )
68	59, 67	oveq12d 6058	. . . . 5 |- ( ph -> ( ( ( S ` ( J .x. E ) ) ` ( I .x. E ) ) ( +g ` R ) ( ( S ` ( J .x. E ) ) ` C ) ) = ( ( I .X. ( G ` J ) ) ( +g ` R ) .0. ) )
69		rnggrp 15624	. . . . . . 7 |- ( R e. Ring -> R e. Grp )
70	54, 69	syl 16	. . . . . 6 |- ( ph -> R e. Grp )
71	17, 19, 45	lmodmcl 15917	. . . . . . 7 |- ( ( U e. LMod /\ I e. B /\ ( G ` J ) e. B ) -> ( I .X. ( G ` J ) ) e. B )
72	9, 34, 64, 71	syl3anc 1184	. . . . . 6 |- ( ph -> ( I .X. ( G ` J ) ) e. B )
73	19, 43, 61	grprid 14791	. . . . . 6 |- ( ( R e. Grp /\ ( I .X. ( G ` J ) ) e. B ) -> ( ( I .X. ( G ` J ) ) ( +g ` R ) .0. ) = ( I .X. ( G ` J ) ) )
74	70, 72, 73	syl2anc 643	. . . . 5 |- ( ph -> ( ( I .X. ( G ` J ) ) ( +g ` R ) .0. ) = ( I .X. ( G ` J ) ) )
75	44, 68, 74	3eqtrd 2440	. . . 4 |- ( ph -> ( ( S ` ( J .x. E ) ) ` ( ( I .x. E ) .+ C ) ) = ( I .X. ( G ` J ) ) )
76	1, 2, 3, 39, 17, 43, 7, 8, 36, 38, 27	hdmaplna1 32393	. . . . 5 |- ( ph -> ( ( S ` D ) ` ( ( I .x. E ) .+ C ) ) = ( ( ( S ` D ) ` ( I .x. E ) ) ( +g ` R ) ( ( S ` D ) ` C ) ) )
77	1, 2, 3, 18, 17, 19, 45, 7, 8, 16, 27, 34	hdmaplnm1 32395	. . . . . . 7 |- ( ph -> ( ( S ` D ) ` ( I .x. E ) ) = ( I .X. ( ( S ` D ) ` E ) ) )
78	1, 14, 23, 2, 3, 17, 19, 45, 61, 7, 8, 26	hdmapinvlem2 32405	. . . . . . . 8 |- ( ph -> ( ( S ` D ) ` E ) = .0. )
79	78	oveq2d 6056	. . . . . . 7 |- ( ph -> ( I .X. ( ( S ` D ) ` E ) ) = ( I .X. .0. ) )
80	19, 45, 61	rngrz 15656	. . . . . . . 8 |- ( ( R e. Ring /\ I e. B ) -> ( I .X. .0. ) = .0. )
81	54, 34, 80	syl2anc 643	. . . . . . 7 |- ( ph -> ( I .X. .0. ) = .0. )
82	77, 79, 81	3eqtrrd 2441	. . . . . 6 |- ( ph -> .0. = ( ( S ` D ) ` ( I .x. E ) ) )
83		hdmapinvlem3.ij	. . . . . 6 |- ( ph -> ( I .X. ( G ` J ) ) = ( ( S ` D ) ` C ) )
84	82, 83	oveq12d 6058	. . . . 5 |- ( ph -> ( .0. ( +g ` R ) ( I .X. ( G ` J ) ) ) = ( ( ( S ` D ) ` ( I .x. E ) ) ( +g ` R ) ( ( S ` D ) ` C ) ) )
85	19, 43, 61	grplid 14790	. . . . . 6 |- ( ( R e. Grp /\ ( I .X. ( G ` J ) ) e. B ) -> ( .0. ( +g ` R ) ( I .X. ( G ` J ) ) ) = ( I .X. ( G ` J ) ) )
86	70, 72, 85	syl2anc 643	. . . . 5 |- ( ph -> ( .0. ( +g ` R ) ( I .X. ( G ` J ) ) ) = ( I .X. ( G ` J ) ) )
87	76, 84, 86	3eqtr2d 2442	. . . 4 |- ( ph -> ( ( S ` D ) ` ( ( I .x. E ) .+ C ) ) = ( I .X. ( G ` J ) ) )
88	75, 87	oveq12d 6058	. . 3 |- ( ph -> ( ( ( S ` ( J .x. E ) ) ` ( ( I .x. E ) .+ C ) ) ( -g ` R ) ( ( S ` D ) ` ( ( I .x. E ) .+ C ) ) ) = ( ( I .X. ( G ` J ) ) ( -g ` R ) ( I .X. ( G ` J ) ) ) )
89	42, 88	eqtrd 2436	. 2 |- ( ph -> ( ( ( S ` ( J .x. E ) ) ( -g ` ( (LCDual ` K ) ` W ) ) ( S ` D ) ) ` ( ( I .x. E ) .+ C ) ) = ( ( I .X. ( G ` J ) ) ( -g ` R ) ( I .X. ( G ` J ) ) ) )
90	19, 61, 30	grpsubid 14828	. . 3 |- ( ( R e. Grp /\ ( I .X. ( G ` J ) ) e. B ) -> ( ( I .X. ( G ` J ) ) ( -g ` R ) ( I .X. ( G ` J ) ) ) = .0. )
91	70, 72, 90	syl2anc 643	. 2 |- ( ph -> ( ( I .X. ( G ` J ) ) ( -g ` R ) ( I .X. ( G ` J ) ) ) = .0. )
92	29, 89, 91	3eqtrd 2440	1 |- ( ph -> ( ( S ` ( ( J .x. E ) .- D ) ) ` ( ( I .x. E ) .+ C ) ) = .0. )

Syntax hints:   -> wi 4   /\ wa 359   = wceq 1649   e. wcel 1721   C_ wss 3280   {csn 3774   <.cop 3777   _I cid 4453   |` cres 4839   ` cfv 5413  (class class class)co 6040   Basecbs 13424   +g cplusg 13484   .rcmulr 13485  Scalarcsca 13487   .scvsca 13488   0gc0g 13678   Grpcgrp 14640   -gcsg 14643   Ringcrg 15615   1rcur 15617   LModclmod 15905   HLchlt 29833   LHypclh 30466   LTrncltrn 30583   DVecHcdvh 31561   ocHcoch 31830  LCDualclcd 32069  HVMapchvm 32239  HDMapchdma 32276  HGMapchg 32369

This theorem is referenced by:  hdmapinvlem4  32407

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-fal 1326  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-ot 3784  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-tpos 6438  df-undef 6502  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-0g 13682  df-mre 13766  df-mrc 13767  df-acs 13769  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-mnd 14645  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-subg 14896  df-cntz 15071  df-oppg 15097  df-lsm 15225  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-dvr 15743  df-drng 15792  df-lmod 15907  df-lss 15964  df-lsp 16003  df-lvec 16130  df-lsatoms 29459  df-lshyp 29460  df-lcv 29502  df-lfl 29541  df-lkr 29569  df-ldual 29607  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lvols 29982  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641  df-tgrp 31225  df-tendo 31237  df-edring 31239  df-dveca 31485  df-disoa 31512  df-dvech 31562  df-dib 31622  df-dic 31656  df-dih 31712  df-doch 31831  df-djh 31878  df-lcdual 32070  df-mapd 32108  df-hvmap 32240  df-hdmap1 32277  df-hdmap 32278  df-hgmap 32370