Theorem hdmapinvlem4 37213

Description: Part 1.1 of Proposition 1 of [Baer] p. 110. We use C, D, I, and J for Baer's u, v, s, and t. Our unit vector E has the required properties for his w by hdmapevec2 37128. Our ( ( S ` D ) ` C ) means his f(u,v) (note argument reversal). (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
hdmapinvlem4	|- ( ph -> ( J .X. ( G ` I ) ) = ( ( S ` C ) ` D ) )

Proof of Theorem hdmapinvlem4

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		hdmapinvlem3.r	. . . 4 |- R = (Scalar ` U )
6		eqid 2622	. . . 4 |- ( -g ` R ) = ( -g ` R )
7		hdmapinvlem3.s	. . . 4 |- S = ( (HDMap ` K ) ` W )
8		hdmapinvlem3.k	. . . 4 |- ( ph -> ( K e. HL /\ W e. H ) )
9	1, 2, 8	dvhlmod 36399	. . . . 5 |- ( ph -> U e. LMod )
10		hdmapinvlem3.j	. . . . 5 |- ( ph -> J e. B )
11		eqid 2622	. . . . . . 7 |- ( Base ` K ) = ( Base ` K )
12		eqid 2622	. . . . . . 7 |- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
13		eqid 2622	. . . . . . 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 36401	. . . . . 6 |- ( ph -> E e. ( V \ { ( 0g ` U ) } ) )
16	15	eldifad 3586	. . . . 5 |- ( ph -> E e. V )
17		hdmapinvlem3.q	. . . . . 6 |- .x. = ( .s ` U )
18		hdmapinvlem3.b	. . . . . 6 |- B = ( Base ` R )
19	3, 5, 17, 18	lmodvscl 18880	. . . . 5 |- ( ( U e. LMod /\ J e. B /\ E e. V ) -> ( J .x. E ) e. V )
20	9, 10, 16, 19	syl3anc 1326	. . . 4 |- ( ph -> ( J .x. E ) e. V )
21	16	snssd 4340	. . . . . 6 |- ( ph -> { E } C_ V )
22		hdmapinvlem3.o	. . . . . . 7 |- O = ( ( ocH ` K ) ` W )
23	1, 2, 3, 22	dochssv 36644	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ { E } C_ V ) -> ( O ` { E } ) C_ V )
24	8, 21, 23	syl2anc 693	. . . . 5 |- ( ph -> ( O ` { E } ) C_ V )
25		hdmapinvlem3.d	. . . . 5 |- ( ph -> D e. ( O ` { E } ) )
26	24, 25	sseldd 3604	. . . 4 |- ( ph -> D e. V )
27		hdmapinvlem3.i	. . . . . 6 |- ( ph -> I e. B )
28	3, 5, 17, 18	lmodvscl 18880	. . . . . 6 |- ( ( U e. LMod /\ I e. B /\ E e. V ) -> ( I .x. E ) e. V )
29	9, 27, 16, 28	syl3anc 1326	. . . . 5 |- ( ph -> ( I .x. E ) e. V )
30		hdmapinvlem3.c	. . . . . 6 |- ( ph -> C e. ( O ` { E } ) )
31	24, 30	sseldd 3604	. . . . 5 |- ( ph -> C e. V )
32		hdmapinvlem3.p	. . . . . 6 |- .+ = ( +g ` U )
33	3, 32	lmodvacl 18877	. . . . 5 |- ( ( U e. LMod /\ ( I .x. E ) e. V /\ C e. V ) -> ( ( I .x. E ) .+ C ) e. V )
34	9, 29, 31, 33	syl3anc 1326	. . . 4 |- ( ph -> ( ( I .x. E ) .+ C ) e. V )
35	1, 2, 3, 4, 5, 6, 7, 8, 20, 26, 34	hdmaplns1 37200	. . 3 |- ( ph -> ( ( S ` ( ( I .x. E ) .+ C ) ) ` ( ( J .x. E ) .- D ) ) = ( ( ( S ` ( ( I .x. E ) .+ C ) ) ` ( J .x. E ) ) ( -g ` R ) ( ( S ` ( ( I .x. E ) .+ C ) ) ` D ) ) )
36		hdmapinvlem3.t	. . . . 5 |- .X. = ( .r ` R )
37		hdmapinvlem3.z	. . . . 5 |- .0. = ( 0g ` R )
38		hdmapinvlem3.g	. . . . 5 |- G = ( (HGMap ` K ) ` W )
39		hdmapinvlem3.ij	. . . . 5 |- ( ph -> ( I .X. ( G ` J ) ) = ( ( S ` D ) ` C ) )
40	1, 14, 22, 2, 3, 32, 4, 17, 5, 18, 36, 37, 7, 38, 8, 30, 25, 27, 10, 39	hdmapinvlem3 37212	. . . 4 |- ( ph -> ( ( S ` ( ( J .x. E ) .- D ) ) ` ( ( I .x. E ) .+ C ) ) = .0. )
41	3, 4	lmodvsubcl 18908	. . . . . 6 |- ( ( U e. LMod /\ ( J .x. E ) e. V /\ D e. V ) -> ( ( J .x. E ) .- D ) e. V )
42	9, 20, 26, 41	syl3anc 1326	. . . . 5 |- ( ph -> ( ( J .x. E ) .- D ) e. V )
43	1, 2, 3, 5, 37, 7, 8, 42, 34	hdmapip0com 37209	. . . 4 |- ( ph -> ( ( ( S ` ( ( J .x. E ) .- D ) ) ` ( ( I .x. E ) .+ C ) ) = .0. <-> ( ( S ` ( ( I .x. E ) .+ C ) ) ` ( ( J .x. E ) .- D ) ) = .0. ) )
44	40, 43	mpbid 222	. . 3 |- ( ph -> ( ( S ` ( ( I .x. E ) .+ C ) ) ` ( ( J .x. E ) .- D ) ) = .0. )
45	1, 2, 3, 17, 5, 18, 36, 7, 8, 16, 34, 10	hdmaplnm1 37201	. . . . 5 |- ( ph -> ( ( S ` ( ( I .x. E ) .+ C ) ) ` ( J .x. E ) ) = ( J .X. ( ( S ` ( ( I .x. E ) .+ C ) ) ` E ) ) )
46		eqid 2622	. . . . . . . 8 |- ( +g ` R ) = ( +g ` R )
47	1, 2, 3, 32, 5, 46, 7, 8, 16, 29, 31	hdmaplna2 37202	. . . . . . 7 |- ( ph -> ( ( S ` ( ( I .x. E ) .+ C ) ) ` E ) = ( ( ( S ` ( I .x. E ) ) ` E ) ( +g ` R ) ( ( S ` C ) ` E ) ) )
48	1, 14, 22, 2, 3, 5, 18, 36, 37, 7, 8, 30	hdmapinvlem2 37211	. . . . . . . 8 |- ( ph -> ( ( S ` C ) ` E ) = .0. )
49	48	oveq2d 6666	. . . . . . 7 |- ( ph -> ( ( ( S ` ( I .x. E ) ) ` E ) ( +g ` R ) ( ( S ` C ) ` E ) ) = ( ( ( S ` ( I .x. E ) ) ` E ) ( +g ` R ) .0. ) )
50	5	lmodring 18871	. . . . . . . . . . 11 |- ( U e. LMod -> R e. Ring )
51	9, 50	syl 17	. . . . . . . . . 10 |- ( ph -> R e. Ring )
52		ringgrp 18552	. . . . . . . . . 10 |- ( R e. Ring -> R e. Grp )
53	51, 52	syl 17	. . . . . . . . 9 |- ( ph -> R e. Grp )
54	1, 2, 3, 5, 18, 7, 8, 16, 29	hdmapipcl 37197	. . . . . . . . 9 |- ( ph -> ( ( S ` ( I .x. E ) ) ` E ) e. B )
55	18, 46, 37	grprid 17453	. . . . . . . . 9 |- ( ( R e. Grp /\ ( ( S ` ( I .x. E ) ) ` E ) e. B ) -> ( ( ( S ` ( I .x. E ) ) ` E ) ( +g ` R ) .0. ) = ( ( S ` ( I .x. E ) ) ` E ) )
56	53, 54, 55	syl2anc 693	. . . . . . . 8 |- ( ph -> ( ( ( S ` ( I .x. E ) ) ` E ) ( +g ` R ) .0. ) = ( ( S ` ( I .x. E ) ) ` E ) )
57	1, 2, 3, 17, 5, 18, 36, 7, 38, 8, 16, 16, 27	hdmapglnm2 37203	. . . . . . . 8 |- ( ph -> ( ( S ` ( I .x. E ) ) ` E ) = ( ( ( S ` E ) ` E ) .X. ( G ` I ) ) )
58		eqid 2622	. . . . . . . . . . 11 |- ( (HVMap ` K ) ` W ) = ( (HVMap ` K ) ` W )
59		eqid 2622	. . . . . . . . . . 11 |- ( 1r ` R ) = ( 1r ` R )
60	1, 14, 58, 7, 8, 2, 5, 59	hdmapevec2 37128	. . . . . . . . . 10 |- ( ph -> ( ( S ` E ) ` E ) = ( 1r ` R ) )
61	60	oveq1d 6665	. . . . . . . . 9 |- ( ph -> ( ( ( S ` E ) ` E ) .X. ( G ` I ) ) = ( ( 1r ` R ) .X. ( G ` I ) ) )
62	1, 2, 5, 18, 38, 8, 27	hgmapcl 37181	. . . . . . . . . 10 |- ( ph -> ( G ` I ) e. B )
63	18, 36, 59	ringlidm 18571	. . . . . . . . . 10 |- ( ( R e. Ring /\ ( G ` I ) e. B ) -> ( ( 1r ` R ) .X. ( G ` I ) ) = ( G ` I ) )
64	51, 62, 63	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( ( 1r ` R ) .X. ( G ` I ) ) = ( G ` I ) )
65	61, 64	eqtrd 2656	. . . . . . . 8 |- ( ph -> ( ( ( S ` E ) ` E ) .X. ( G ` I ) ) = ( G ` I ) )
66	56, 57, 65	3eqtrd 2660	. . . . . . 7 |- ( ph -> ( ( ( S ` ( I .x. E ) ) ` E ) ( +g ` R ) .0. ) = ( G ` I ) )
67	47, 49, 66	3eqtrd 2660	. . . . . 6 |- ( ph -> ( ( S ` ( ( I .x. E ) .+ C ) ) ` E ) = ( G ` I ) )
68	67	oveq2d 6666	. . . . 5 |- ( ph -> ( J .X. ( ( S ` ( ( I .x. E ) .+ C ) ) ` E ) ) = ( J .X. ( G ` I ) ) )
69	45, 68	eqtrd 2656	. . . 4 |- ( ph -> ( ( S ` ( ( I .x. E ) .+ C ) ) ` ( J .x. E ) ) = ( J .X. ( G ` I ) ) )
70	1, 2, 3, 32, 5, 46, 7, 8, 26, 29, 31	hdmaplna2 37202	. . . . 5 |- ( ph -> ( ( S ` ( ( I .x. E ) .+ C ) ) ` D ) = ( ( ( S ` ( I .x. E ) ) ` D ) ( +g ` R ) ( ( S ` C ) ` D ) ) )
71	1, 2, 3, 17, 5, 18, 36, 7, 38, 8, 26, 16, 27	hdmapglnm2 37203	. . . . . . 7 |- ( ph -> ( ( S ` ( I .x. E ) ) ` D ) = ( ( ( S ` E ) ` D ) .X. ( G ` I ) ) )
72	1, 14, 22, 2, 3, 5, 18, 36, 37, 7, 8, 25	hdmapinvlem1 37210	. . . . . . . 8 |- ( ph -> ( ( S ` E ) ` D ) = .0. )
73	72	oveq1d 6665	. . . . . . 7 |- ( ph -> ( ( ( S ` E ) ` D ) .X. ( G ` I ) ) = ( .0. .X. ( G ` I ) ) )
74	18, 36, 37	ringlz 18587	. . . . . . . 8 |- ( ( R e. Ring /\ ( G ` I ) e. B ) -> ( .0. .X. ( G ` I ) ) = .0. )
75	51, 62, 74	syl2anc 693	. . . . . . 7 |- ( ph -> ( .0. .X. ( G ` I ) ) = .0. )
76	71, 73, 75	3eqtrd 2660	. . . . . 6 |- ( ph -> ( ( S ` ( I .x. E ) ) ` D ) = .0. )
77	76	oveq1d 6665	. . . . 5 |- ( ph -> ( ( ( S ` ( I .x. E ) ) ` D ) ( +g ` R ) ( ( S ` C ) ` D ) ) = ( .0. ( +g ` R ) ( ( S ` C ) ` D ) ) )
78	1, 2, 3, 5, 18, 7, 8, 26, 31	hdmapipcl 37197	. . . . . 6 |- ( ph -> ( ( S ` C ) ` D ) e. B )
79	18, 46, 37	grplid 17452	. . . . . 6 |- ( ( R e. Grp /\ ( ( S ` C ) ` D ) e. B ) -> ( .0. ( +g ` R ) ( ( S ` C ) ` D ) ) = ( ( S ` C ) ` D ) )
80	53, 78, 79	syl2anc 693	. . . . 5 |- ( ph -> ( .0. ( +g ` R ) ( ( S ` C ) ` D ) ) = ( ( S ` C ) ` D ) )
81	70, 77, 80	3eqtrd 2660	. . . 4 |- ( ph -> ( ( S ` ( ( I .x. E ) .+ C ) ) ` D ) = ( ( S ` C ) ` D ) )
82	69, 81	oveq12d 6668	. . 3 |- ( ph -> ( ( ( S ` ( ( I .x. E ) .+ C ) ) ` ( J .x. E ) ) ( -g ` R ) ( ( S ` ( ( I .x. E ) .+ C ) ) ` D ) ) = ( ( J .X. ( G ` I ) ) ( -g ` R ) ( ( S ` C ) ` D ) ) )
83	35, 44, 82	3eqtr3rd 2665	. 2 |- ( ph -> ( ( J .X. ( G ` I ) ) ( -g ` R ) ( ( S ` C ) ` D ) ) = .0. )
84	5, 18, 36	lmodmcl 18875	. . . 4 |- ( ( U e. LMod /\ J e. B /\ ( G ` I ) e. B ) -> ( J .X. ( G ` I ) ) e. B )
85	9, 10, 62, 84	syl3anc 1326	. . 3 |- ( ph -> ( J .X. ( G ` I ) ) e. B )
86	18, 37, 6	grpsubeq0 17501	. . 3 |- ( ( R e. Grp /\ ( J .X. ( G ` I ) ) e. B /\ ( ( S ` C ) ` D ) e. B ) -> ( ( ( J .X. ( G ` I ) ) ( -g ` R ) ( ( S ` C ) ` D ) ) = .0. <-> ( J .X. ( G ` I ) ) = ( ( S ` C ) ` D ) ) )
87	53, 85, 78, 86	syl3anc 1326	. 2 |- ( ph -> ( ( ( J .X. ( G ` I ) ) ( -g ` R ) ( ( S ` C ) ` D ) ) = .0. <-> ( J .X. ( G ` I ) ) = ( ( S ` C ) ` D ) ) )
88	83, 87	mpbid 222	1 |- ( ph -> ( J .X. ( G ` I ) ) = ( ( S ` C ) ` D ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   {csn 4177   <.cop 4183   _I cid 5023   |` cres 5116   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   Grpcgrp 17422   -gcsg 17424   1rcur 18501   Ringcrg 18547   LModclmod 18863   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   DVecHcdvh 36367   ocHcoch 36636  HVMapchvm 37045  HDMapchdma 37082  HGMapchg 37175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-riotaBAD 34239

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-ot 4186  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-undef 7399  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-0g 16102  df-mre 16246  df-mrc 16247  df-acs 16249  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-cntz 17750  df-oppg 17776  df-lsm 18051  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-drng 18749  df-lmod 18865  df-lss 18933  df-lsp 18972  df-lvec 19103  df-lsatoms 34263  df-lshyp 34264  df-lcv 34306  df-lfl 34345  df-lkr 34373  df-ldual 34411  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-llines 34784  df-lplanes 34785  df-lvols 34786  df-lines 34787  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446  df-tgrp 36031  df-tendo 36043  df-edring 36045  df-dveca 36291  df-disoa 36318  df-dvech 36368  df-dib 36428  df-dic 36462  df-dih 36518  df-doch 36637  df-djh 36684  df-lcdual 36876  df-mapd 36914  df-hvmap 37046  df-hdmap1 37083  df-hdmap 37084  df-hgmap 37176

This theorem is referenced by:  hdmapglem5  37214  hgmapvvlem1  37215