Theorem hdmapinvlem3 35200

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 2429	. . . 4 |- ( (LCDual ` K ) ` W ) = ( (LCDual ` K ) ` W )
6		eqid 2429	. . . 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 34387	. . . . 5 |- ( ph -> U e. LMod )
10		hdmapinvlem3.j	. . . . 5 |- ( ph -> J e. B )
11		eqid 2429	. . . . . . 7 |- ( Base ` K ) = ( Base ` K )
12		eqid 2429	. . . . . . 7 |- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
13		eqid 2429	. . . . . . 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 34389	. . . . . 6 |- ( ph -> E e. ( V \ { ( 0g ` U ) } ) )
16	15	eldifad 3454	. . . . 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 18043	. . . . 5 |- ( ( U e. LMod /\ J e. B /\ E e. V ) -> ( J .x. E ) e. V )
21	9, 10, 16, 20	syl3anc 1264	. . . 4 |- ( ph -> ( J .x. E ) e. V )
22	16	snssd 4148	. . . . . 6 |- ( ph -> { E } C_ V )
23		hdmapinvlem3.o	. . . . . . 7 |- O = ( ( ocH ` K ) ` W )
24	1, 2, 3, 23	dochssv 34632	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ { E } C_ V ) -> ( O ` { E } ) C_ V )
25	8, 22, 24	syl2anc 665	. . . . 5 |- ( ph -> ( O ` { E } ) C_ V )
26		hdmapinvlem3.d	. . . . 5 |- ( ph -> D e. ( O ` { E } ) )
27	25, 26	sseldd 3471	. . . 4 |- ( ph -> D e. V )
28	1, 2, 3, 4, 5, 6, 7, 8, 21, 27	hdmapsub 35127	. . 3 |- ( ph -> ( S ` ( ( J .x. E ) .- D ) ) = ( ( S ` ( J .x. E ) ) ( -g ` ( (LCDual ` K ) ` W ) ) ( S ` D ) ) )
29	28	fveq1d 5883	. 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 2429	. . . 4 |- ( -g ` R ) = ( -g ` R )
31		eqid 2429	. . . 4 |- ( Base ` ( (LCDual ` K ) ` W ) ) = ( Base ` ( (LCDual ` K ) ` W ) )
32	1, 2, 3, 5, 31, 7, 8, 21	hdmapcl 35110	. . . 4 |- ( ph -> ( S ` ( J .x. E ) ) e. ( Base ` ( (LCDual ` K ) ` W ) ) )
33	1, 2, 3, 5, 31, 7, 8, 27	hdmapcl 35110	. . . 4 |- ( ph -> ( S ` D ) e. ( Base ` ( (LCDual ` K ) ` W ) ) )
34		hdmapinvlem3.i	. . . . . 6 |- ( ph -> I e. B )
35	3, 17, 18, 19	lmodvscl 18043	. . . . . 6 |- ( ( U e. LMod /\ I e. B /\ E e. V ) -> ( I .x. E ) e. V )
36	9, 34, 16, 35	syl3anc 1264	. . . . 5 |- ( ph -> ( I .x. E ) e. V )
37		hdmapinvlem3.c	. . . . . 6 |- ( ph -> C e. ( O ` { E } ) )
38	25, 37	sseldd 3471	. . . . 5 |- ( ph -> C e. V )
39		hdmapinvlem3.p	. . . . . 6 |- .+ = ( +g ` U )
40	3, 39	lmodvacl 18040	. . . . 5 |- ( ( U e. LMod /\ ( I .x. E ) e. V /\ C e. V ) -> ( ( I .x. E ) .+ C ) e. V )
41	9, 36, 38, 40	syl3anc 1264	. . . 4 |- ( ph -> ( ( I .x. E ) .+ C ) e. V )
42	1, 2, 3, 17, 30, 5, 31, 6, 8, 32, 33, 41	lcdvsubval 34895	. . 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 2429	. . . . . 6 |- ( +g ` R ) = ( +g ` R )
44	1, 2, 3, 39, 17, 43, 7, 8, 36, 38, 21	hdmaplna1 35187	. . . . 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 35191	. . . . . . 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 35189	. . . . . . . . 9 |- ( ph -> ( ( S ` E ) ` ( I .x. E ) ) = ( I .X. ( ( S ` E ) ` E ) ) )
49		eqid 2429	. . . . . . . . . . 11 |- ( (HVMap ` K ) ` W ) = ( (HVMap ` K ) ` W )
50		eqid 2429	. . . . . . . . . . 11 |- ( 1r ` R ) = ( 1r ` R )
51	1, 14, 49, 7, 8, 2, 17, 50	hdmapevec2 35116	. . . . . . . . . 10 |- ( ph -> ( ( S ` E ) ` E ) = ( 1r ` R ) )
52	51	oveq2d 6321	. . . . . . . . 9 |- ( ph -> ( I .X. ( ( S ` E ) ` E ) ) = ( I .X. ( 1r ` R ) ) )
53	17	lmodring 18034	. . . . . . . . . . 11 |- ( U e. LMod -> R e. Ring )
54	9, 53	syl 17	. . . . . . . . . 10 |- ( ph -> R e. Ring )
55	19, 45, 50	ringridm 17740	. . . . . . . . . 10 |- ( ( R e. Ring /\ I e. B ) -> ( I .X. ( 1r ` R ) ) = I )
56	54, 34, 55	syl2anc 665	. . . . . . . . 9 |- ( ph -> ( I .X. ( 1r ` R ) ) = I )
57	48, 52, 56	3eqtrd 2474	. . . . . . . 8 |- ( ph -> ( ( S ` E ) ` ( I .x. E ) ) = I )
58	57	oveq1d 6320	. . . . . . 7 |- ( ph -> ( ( ( S ` E ) ` ( I .x. E ) ) .X. ( G ` J ) ) = ( I .X. ( G ` J ) ) )
59	47, 58	eqtrd 2470	. . . . . 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 35191	. . . . . . 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 35198	. . . . . . . 8 |- ( ph -> ( ( S ` E ) ` C ) = .0. )
63	62	oveq1d 6320	. . . . . . 7 |- ( ph -> ( ( ( S ` E ) ` C ) .X. ( G ` J ) ) = ( .0. .X. ( G ` J ) ) )
64	1, 2, 17, 19, 46, 8, 10	hgmapcl 35169	. . . . . . . 8 |- ( ph -> ( G ` J ) e. B )
65	19, 45, 61	ringlz 17752	. . . . . . . 8 |- ( ( R e. Ring /\ ( G ` J ) e. B ) -> ( .0. .X. ( G ` J ) ) = .0. )
66	54, 64, 65	syl2anc 665	. . . . . . 7 |- ( ph -> ( .0. .X. ( G ` J ) ) = .0. )
67	60, 63, 66	3eqtrd 2474	. . . . . 6 |- ( ph -> ( ( S ` ( J .x. E ) ) ` C ) = .0. )
68	59, 67	oveq12d 6323	. . . . 5 |- ( ph -> ( ( ( S ` ( J .x. E ) ) ` ( I .x. E ) ) ( +g ` R ) ( ( S ` ( J .x. E ) ) ` C ) ) = ( ( I .X. ( G ` J ) ) ( +g ` R ) .0. ) )
69		ringgrp 17720	. . . . . . 7 |- ( R e. Ring -> R e. Grp )
70	54, 69	syl 17	. . . . . 6 |- ( ph -> R e. Grp )
71	17, 19, 45	lmodmcl 18038	. . . . . . 7 |- ( ( U e. LMod /\ I e. B /\ ( G ` J ) e. B ) -> ( I .X. ( G ` J ) ) e. B )
72	9, 34, 64, 71	syl3anc 1264	. . . . . 6 |- ( ph -> ( I .X. ( G ` J ) ) e. B )
73	19, 43, 61	grprid 16648	. . . . . 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 665	. . . . 5 |- ( ph -> ( ( I .X. ( G ` J ) ) ( +g ` R ) .0. ) = ( I .X. ( G ` J ) ) )
75	44, 68, 74	3eqtrd 2474	. . . 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 35187	. . . . 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 35189	. . . . . . 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 35199	. . . . . . . 8 |- ( ph -> ( ( S ` D ) ` E ) = .0. )
79	78	oveq2d 6321	. . . . . . 7 |- ( ph -> ( I .X. ( ( S ` D ) ` E ) ) = ( I .X. .0. ) )
80	19, 45, 61	ringrz 17753	. . . . . . . 8 |- ( ( R e. Ring /\ I e. B ) -> ( I .X. .0. ) = .0. )
81	54, 34, 80	syl2anc 665	. . . . . . 7 |- ( ph -> ( I .X. .0. ) = .0. )
82	77, 79, 81	3eqtrrd 2475	. . . . . 6 |- ( ph -> .0. = ( ( S ` D ) ` ( I .x. E ) ) )
83		hdmapinvlem3.ij	. . . . . 6 |- ( ph -> ( I .X. ( G ` J ) ) = ( ( S ` D ) ` C ) )
84	82, 83	oveq12d 6323	. . . . 5 |- ( ph -> ( .0. ( +g ` R ) ( I .X. ( G ` J ) ) ) = ( ( ( S ` D ) ` ( I .x. E ) ) ( +g ` R ) ( ( S ` D ) ` C ) ) )
85	19, 43, 61	grplid 16647	. . . . . 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 665	. . . . 5 |- ( ph -> ( .0. ( +g ` R ) ( I .X. ( G ` J ) ) ) = ( I .X. ( G ` J ) ) )
87	76, 84, 86	3eqtr2d 2476	. . . 4 |- ( ph -> ( ( S ` D ) ` ( ( I .x. E ) .+ C ) ) = ( I .X. ( G ` J ) ) )
88	75, 87	oveq12d 6323	. . 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 2470	. 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 16689	. . 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 665	. 2 |- ( ph -> ( ( I .X. ( G ` J ) ) ( -g ` R ) ( I .X. ( G ` J ) ) ) = .0. )
92	29, 89, 91	3eqtrd 2474	1 |- ( ph -> ( ( S ` ( ( J .x. E ) .- D ) ) ` ( ( I .x. E ) .+ C ) ) = .0. )

Syntax hints:   -> wi 4   /\ wa 370   = wceq 1437   e. wcel 1870   C_ wss 3442   {csn 4002   <.cop 4008   _I cid 4764   |` cres 4856   ` cfv 5601  (class class class)co 6305   Basecbs 15084   +g cplusg 15152   .rcmulr 15153  Scalarcsca 15155   .scvsca 15156   0gc0g 15297   Grpcgrp 16620   -gcsg 16622   1rcur 17670   Ringcrg 17715   LModclmod 18026   HLchlt 32625   LHypclh 33258   LTrncltrn 33375   DVecHcdvh 34355   ocHcoch 34624  LCDualclcd 34863  HVMapchvm 35033  HDMapchdma 35070  HGMapchg 35163

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-riotaBAD 32234

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-ot 4011  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-tpos 6981  df-undef 7028  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-sca 15168  df-vsca 15169  df-0g 15299  df-mre 15443  df-mrc 15444  df-acs 15446  df-preset 16124  df-poset 16142  df-plt 16155  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-p0 16236  df-p1 16237  df-lat 16243  df-clat 16305  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-grp 16624  df-minusg 16625  df-sbg 16626  df-subg 16765  df-cntz 16922  df-oppg 16948  df-lsm 17223  df-cmn 17367  df-abl 17368  df-mgp 17659  df-ur 17671  df-ring 17717  df-oppr 17786  df-dvdsr 17804  df-unit 17805  df-invr 17835  df-dvr 17846  df-drng 17912  df-lmod 18028  df-lss 18091  df-lsp 18130  df-lvec 18261  df-lsatoms 32251  df-lshyp 32252  df-lcv 32294  df-lfl 32333  df-lkr 32361  df-ldual 32399  df-oposet 32451  df-ol 32453  df-oml 32454  df-covers 32541  df-ats 32542  df-atl 32573  df-cvlat 32597  df-hlat 32626  df-llines 32772  df-lplanes 32773  df-lvols 32774  df-lines 32775  df-psubsp 32777  df-pmap 32778  df-padd 33070  df-lhyp 33262  df-laut 33263  df-ldil 33378  df-ltrn 33379  df-trl 33434  df-tgrp 34019  df-tendo 34031  df-edring 34033  df-dveca 34279  df-disoa 34306  df-dvech 34356  df-dib 34416  df-dic 34450  df-dih 34506  df-doch 34625  df-djh 34672  df-lcdual 34864  df-mapd 34902  df-hvmap 35034  df-hdmap1 35071  df-hdmap 35072  df-hgmap 35164

This theorem is referenced by:  hdmapinvlem4  35201