Theorem hdmapinvlem3 36738

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 2467	. . . 4 |- ( (LCDual ` K ) ` W ) = ( (LCDual ` K ) ` W )
6		eqid 2467	. . . 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 35925	. . . . 5 |- ( ph -> U e. LMod )
10		hdmapinvlem3.j	. . . . 5 |- ( ph -> J e. B )
11		eqid 2467	. . . . . . 7 |- ( Base ` K ) = ( Base ` K )
12		eqid 2467	. . . . . . 7 |- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
13		eqid 2467	. . . . . . 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 35927	. . . . . 6 |- ( ph -> E e. ( V \ { ( 0g ` U ) } ) )
16	15	eldifad 3488	. . . . 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 17329	. . . . 5 |- ( ( U e. LMod /\ J e. B /\ E e. V ) -> ( J .x. E ) e. V )
21	9, 10, 16, 20	syl3anc 1228	. . . 4 |- ( ph -> ( J .x. E ) e. V )
22	16	snssd 4172	. . . . . 6 |- ( ph -> { E } C_ V )
23		hdmapinvlem3.o	. . . . . . 7 |- O = ( ( ocH ` K ) ` W )
24	1, 2, 3, 23	dochssv 36170	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ { E } C_ V ) -> ( O ` { E } ) C_ V )
25	8, 22, 24	syl2anc 661	. . . . 5 |- ( ph -> ( O ` { E } ) C_ V )
26		hdmapinvlem3.d	. . . . 5 |- ( ph -> D e. ( O ` { E } ) )
27	25, 26	sseldd 3505	. . . 4 |- ( ph -> D e. V )
28	1, 2, 3, 4, 5, 6, 7, 8, 21, 27	hdmapsub 36665	. . 3 |- ( ph -> ( S ` ( ( J .x. E ) .- D ) ) = ( ( S ` ( J .x. E ) ) ( -g ` ( (LCDual ` K ) ` W ) ) ( S ` D ) ) )
29	28	fveq1d 5868	. 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 2467	. . . 4 |- ( -g ` R ) = ( -g ` R )
31		eqid 2467	. . . 4 |- ( Base ` ( (LCDual ` K ) ` W ) ) = ( Base ` ( (LCDual ` K ) ` W ) )
32	1, 2, 3, 5, 31, 7, 8, 21	hdmapcl 36648	. . . 4 |- ( ph -> ( S ` ( J .x. E ) ) e. ( Base ` ( (LCDual ` K ) ` W ) ) )
33	1, 2, 3, 5, 31, 7, 8, 27	hdmapcl 36648	. . . 4 |- ( ph -> ( S ` D ) e. ( Base ` ( (LCDual ` K ) ` W ) ) )
34		hdmapinvlem3.i	. . . . . 6 |- ( ph -> I e. B )
35	3, 17, 18, 19	lmodvscl 17329	. . . . . 6 |- ( ( U e. LMod /\ I e. B /\ E e. V ) -> ( I .x. E ) e. V )
36	9, 34, 16, 35	syl3anc 1228	. . . . 5 |- ( ph -> ( I .x. E ) e. V )
37		hdmapinvlem3.c	. . . . . 6 |- ( ph -> C e. ( O ` { E } ) )
38	25, 37	sseldd 3505	. . . . 5 |- ( ph -> C e. V )
39		hdmapinvlem3.p	. . . . . 6 |- .+ = ( +g ` U )
40	3, 39	lmodvacl 17326	. . . . 5 |- ( ( U e. LMod /\ ( I .x. E ) e. V /\ C e. V ) -> ( ( I .x. E ) .+ C ) e. V )
41	9, 36, 38, 40	syl3anc 1228	. . . 4 |- ( ph -> ( ( I .x. E ) .+ C ) e. V )
42	1, 2, 3, 17, 30, 5, 31, 6, 8, 32, 33, 41	lcdvsubval 36433	. . 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 2467	. . . . . 6 |- ( +g ` R ) = ( +g ` R )
44	1, 2, 3, 39, 17, 43, 7, 8, 36, 38, 21	hdmaplna1 36725	. . . . 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 36729	. . . . . . 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 36727	. . . . . . . . 9 |- ( ph -> ( ( S ` E ) ` ( I .x. E ) ) = ( I .X. ( ( S ` E ) ` E ) ) )
49		eqid 2467	. . . . . . . . . . 11 |- ( (HVMap ` K ) ` W ) = ( (HVMap ` K ) ` W )
50		eqid 2467	. . . . . . . . . . 11 |- ( 1r ` R ) = ( 1r ` R )
51	1, 14, 49, 7, 8, 2, 17, 50	hdmapevec2 36654	. . . . . . . . . 10 |- ( ph -> ( ( S ` E ) ` E ) = ( 1r ` R ) )
52	51	oveq2d 6300	. . . . . . . . 9 |- ( ph -> ( I .X. ( ( S ` E ) ` E ) ) = ( I .X. ( 1r ` R ) ) )
53	17	lmodrng 17320	. . . . . . . . . . 11 |- ( U e. LMod -> R e. Ring )
54	9, 53	syl 16	. . . . . . . . . 10 |- ( ph -> R e. Ring )
55	19, 45, 50	rngridm 17024	. . . . . . . . . 10 |- ( ( R e. Ring /\ I e. B ) -> ( I .X. ( 1r ` R ) ) = I )
56	54, 34, 55	syl2anc 661	. . . . . . . . 9 |- ( ph -> ( I .X. ( 1r ` R ) ) = I )
57	48, 52, 56	3eqtrd 2512	. . . . . . . 8 |- ( ph -> ( ( S ` E ) ` ( I .x. E ) ) = I )
58	57	oveq1d 6299	. . . . . . 7 |- ( ph -> ( ( ( S ` E ) ` ( I .x. E ) ) .X. ( G ` J ) ) = ( I .X. ( G ` J ) ) )
59	47, 58	eqtrd 2508	. . . . . 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 36729	. . . . . . 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 36736	. . . . . . . 8 |- ( ph -> ( ( S ` E ) ` C ) = .0. )
63	62	oveq1d 6299	. . . . . . 7 |- ( ph -> ( ( ( S ` E ) ` C ) .X. ( G ` J ) ) = ( .0. .X. ( G ` J ) ) )
64	1, 2, 17, 19, 46, 8, 10	hgmapcl 36707	. . . . . . . 8 |- ( ph -> ( G ` J ) e. B )
65	19, 45, 61	rnglz 17036	. . . . . . . 8 |- ( ( R e. Ring /\ ( G ` J ) e. B ) -> ( .0. .X. ( G ` J ) ) = .0. )
66	54, 64, 65	syl2anc 661	. . . . . . 7 |- ( ph -> ( .0. .X. ( G ` J ) ) = .0. )
67	60, 63, 66	3eqtrd 2512	. . . . . 6 |- ( ph -> ( ( S ` ( J .x. E ) ) ` C ) = .0. )
68	59, 67	oveq12d 6302	. . . . 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 17005	. . . . . . 7 |- ( R e. Ring -> R e. Grp )
70	54, 69	syl 16	. . . . . 6 |- ( ph -> R e. Grp )
71	17, 19, 45	lmodmcl 17324	. . . . . . 7 |- ( ( U e. LMod /\ I e. B /\ ( G ` J ) e. B ) -> ( I .X. ( G ` J ) ) e. B )
72	9, 34, 64, 71	syl3anc 1228	. . . . . 6 |- ( ph -> ( I .X. ( G ` J ) ) e. B )
73	19, 43, 61	grprid 15891	. . . . . 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 661	. . . . 5 |- ( ph -> ( ( I .X. ( G ` J ) ) ( +g ` R ) .0. ) = ( I .X. ( G ` J ) ) )
75	44, 68, 74	3eqtrd 2512	. . . 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 36725	. . . . 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 36727	. . . . . . 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 36737	. . . . . . . 8 |- ( ph -> ( ( S ` D ) ` E ) = .0. )
79	78	oveq2d 6300	. . . . . . 7 |- ( ph -> ( I .X. ( ( S ` D ) ` E ) ) = ( I .X. .0. ) )
80	19, 45, 61	rngrz 17037	. . . . . . . 8 |- ( ( R e. Ring /\ I e. B ) -> ( I .X. .0. ) = .0. )
81	54, 34, 80	syl2anc 661	. . . . . . 7 |- ( ph -> ( I .X. .0. ) = .0. )
82	77, 79, 81	3eqtrrd 2513	. . . . . 6 |- ( ph -> .0. = ( ( S ` D ) ` ( I .x. E ) ) )
83		hdmapinvlem3.ij	. . . . . 6 |- ( ph -> ( I .X. ( G ` J ) ) = ( ( S ` D ) ` C ) )
84	82, 83	oveq12d 6302	. . . . 5 |- ( ph -> ( .0. ( +g ` R ) ( I .X. ( G ` J ) ) ) = ( ( ( S ` D ) ` ( I .x. E ) ) ( +g ` R ) ( ( S ` D ) ` C ) ) )
85	19, 43, 61	grplid 15890	. . . . . 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 661	. . . . 5 |- ( ph -> ( .0. ( +g ` R ) ( I .X. ( G ` J ) ) ) = ( I .X. ( G ` J ) ) )
87	76, 84, 86	3eqtr2d 2514	. . . 4 |- ( ph -> ( ( S ` D ) ` ( ( I .x. E ) .+ C ) ) = ( I .X. ( G ` J ) ) )
88	75, 87	oveq12d 6302	. . 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 2508	. 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 15932	. . 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 661	. 2 |- ( ph -> ( ( I .X. ( G ` J ) ) ( -g ` R ) ( I .X. ( G ` J ) ) ) = .0. )
92	29, 89, 91	3eqtrd 2512	1 |- ( ph -> ( ( S ` ( ( J .x. E ) .- D ) ) ` ( ( I .x. E ) .+ C ) ) = .0. )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1379   e. wcel 1767   C_ wss 3476   {csn 4027   <.cop 4033   _I cid 4790   |` cres 5001   ` cfv 5588  (class class class)co 6284   Basecbs 14490   +g cplusg 14555   .rcmulr 14556  Scalarcsca 14558   .scvsca 14559   0gc0g 14695   Grpcgrp 15727   -gcsg 15730   1rcur 16955   Ringcrg 17000   LModclmod 17312   HLchlt 34165   LHypclh 34798   LTrncltrn 34915   DVecHcdvh 35893   ocHcoch 36162  LCDualclcd 36401  HVMapchvm 36571  HDMapchdma 36608  HGMapchg 36701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-riotaBAD 33774

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-ot 4036  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-tpos 6955  df-undef 7002  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-sca 14571  df-vsca 14572  df-0g 14697  df-mre 14841  df-mrc 14842  df-acs 14844  df-poset 15433  df-plt 15445  df-lub 15461  df-glb 15462  df-join 15463  df-meet 15464  df-p0 15526  df-p1 15527  df-lat 15533  df-clat 15595  df-mnd 15732  df-submnd 15787  df-grp 15867  df-minusg 15868  df-sbg 15869  df-subg 16003  df-cntz 16160  df-oppg 16186  df-lsm 16462  df-cmn 16606  df-abl 16607  df-mgp 16944  df-ur 16956  df-rng 17002  df-oppr 17073  df-dvdsr 17091  df-unit 17092  df-invr 17122  df-dvr 17133  df-drng 17198  df-lmod 17314  df-lss 17379  df-lsp 17418  df-lvec 17549  df-lsatoms 33791  df-lshyp 33792  df-lcv 33834  df-lfl 33873  df-lkr 33901  df-ldual 33939  df-oposet 33991  df-ol 33993  df-oml 33994  df-covers 34081  df-ats 34082  df-atl 34113  df-cvlat 34137  df-hlat 34166  df-llines 34312  df-lplanes 34313  df-lvols 34314  df-lines 34315  df-psubsp 34317  df-pmap 34318  df-padd 34610  df-lhyp 34802  df-laut 34803  df-ldil 34918  df-ltrn 34919  df-trl 34973  df-tgrp 35557  df-tendo 35569  df-edring 35571  df-dveca 35817  df-disoa 35844  df-dvech 35894  df-dib 35954  df-dic 35988  df-dih 36044  df-doch 36163  df-djh 36210  df-lcdual 36402  df-mapd 36440  df-hvmap 36572  df-hdmap1 36609  df-hdmap 36610  df-hgmap 36702

This theorem is referenced by:  hdmapinvlem4  36739