Theorem hdmap1neglem1N 37117

Description: Lemma for hdmapneg 37138. TODO: Not used; delete. (Contributed by NM, 23-May-2015.) (New usage is discouraged.)

Hypotheses

Ref	Expression
hdmap1neglem1.h	|- H = ( LHyp ` K )
hdmap1neglem1.u	|- U = ( ( DVecH ` K ) ` W )
hdmap1neglem1.v	|- V = ( Base ` U )
hdmap1neglem1.r	|- R = ( invg ` U )
hdmap1neglem1.o	|- .0. = ( 0g ` U )
hdmap1neglem1.n	|- N = ( LSpan ` U )
hdmap1neglem1.c	|- C = ( (LCDual ` K ) ` W )
hdmap1neglem1.d	|- D = ( Base ` C )
hdmap1neglem1.s	|- S = ( invg ` C )
hdmap1neglem1.l	|- L = ( LSpan ` C )
hdmap1neglem1.m	|- M = ( (mapd ` K ) ` W )
hdmap1neglem1.i	|- I = ( (HDMap1 ` K ) ` W )
hdmap1neglem1.k	|- ( ph -> ( K e. HL /\ W e. H ) )
hdmap1neglem1.f	|- ( ph -> F e. D )
hdmap1neglem1.mn	|- ( ph -> ( M ` ( N ` { X } ) ) = ( L ` { F } ) )
hdmap1neglem1.ne	|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
hdmap1neglem1.x	|- ( ph -> X e. ( V \ { .0. } ) )
hdmap1neglem1.y	|- ( ph -> Y e. ( V \ { .0. } ) )
hdmap1neglem1.e	|- ( ph -> ( I ` <. X , F , Y >. ) = G )

Assertion

Ref	Expression
hdmap1neglem1N	|- ( ph -> ( I ` <. ( R ` X ) , ( S ` F ) , ( R ` Y ) >. ) = ( S ` G ) )

Proof of Theorem hdmap1neglem1N

Step	Hyp	Ref	Expression
1		hdmap1neglem1.e	. . . . 5 |- ( ph -> ( I ` <. X , F , Y >. ) = G )
2		hdmap1neglem1.h	. . . . . 6 |- H = ( LHyp ` K )
3		hdmap1neglem1.u	. . . . . 6 |- U = ( ( DVecH ` K ) ` W )
4		hdmap1neglem1.v	. . . . . 6 |- V = ( Base ` U )
5		eqid 2622	. . . . . 6 |- ( -g ` U ) = ( -g ` U )
6		hdmap1neglem1.o	. . . . . 6 |- .0. = ( 0g ` U )
7		hdmap1neglem1.n	. . . . . 6 |- N = ( LSpan ` U )
8		hdmap1neglem1.c	. . . . . 6 |- C = ( (LCDual ` K ) ` W )
9		hdmap1neglem1.d	. . . . . 6 |- D = ( Base ` C )
10		eqid 2622	. . . . . 6 |- ( -g ` C ) = ( -g ` C )
11		hdmap1neglem1.l	. . . . . 6 |- L = ( LSpan ` C )
12		hdmap1neglem1.m	. . . . . 6 |- M = ( (mapd ` K ) ` W )
13		hdmap1neglem1.i	. . . . . 6 |- I = ( (HDMap1 ` K ) ` W )
14		hdmap1neglem1.k	. . . . . 6 |- ( ph -> ( K e. HL /\ W e. H ) )
15		hdmap1neglem1.x	. . . . . 6 |- ( ph -> X e. ( V \ { .0. } ) )
16		hdmap1neglem1.f	. . . . . 6 |- ( ph -> F e. D )
17		hdmap1neglem1.y	. . . . . 6 |- ( ph -> Y e. ( V \ { .0. } ) )
18		hdmap1neglem1.mn	. . . . . . . 8 |- ( ph -> ( M ` ( N ` { X } ) ) = ( L ` { F } ) )
19		hdmap1neglem1.ne	. . . . . . . 8 |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
20	17	eldifad 3586	. . . . . . . 8 |- ( ph -> Y e. V )
21	2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 18, 19, 15, 20	hdmap1cl 37094	. . . . . . 7 |- ( ph -> ( I ` <. X , F , Y >. ) e. D )
22	1, 21	eqeltrrd 2702	. . . . . 6 |- ( ph -> G e. D )
23	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 22, 19, 18	hdmap1eq 37091	. . . . 5 |- ( ph -> ( ( I ` <. X , F , Y >. ) = G <-> ( ( M ` ( N ` { Y } ) ) = ( L ` { G } ) /\ ( M ` ( N ` { ( X ( -g ` U ) Y ) } ) ) = ( L ` { ( F ( -g ` C ) G ) } ) ) ) )
24	1, 23	mpbid 222	. . . 4 |- ( ph -> ( ( M ` ( N ` { Y } ) ) = ( L ` { G } ) /\ ( M ` ( N ` { ( X ( -g ` U ) Y ) } ) ) = ( L ` { ( F ( -g ` C ) G ) } ) ) )
25	24	simpld 475	. . 3 |- ( ph -> ( M ` ( N ` { Y } ) ) = ( L ` { G } ) )
26	2, 3, 14	dvhlmod 36399	. . . . 5 |- ( ph -> U e. LMod )
27		hdmap1neglem1.r	. . . . . 6 |- R = ( invg ` U )
28	4, 27, 7	lspsnneg 19006	. . . . 5 |- ( ( U e. LMod /\ Y e. V ) -> ( N ` { ( R ` Y ) } ) = ( N ` { Y } ) )
29	26, 20, 28	syl2anc 693	. . . 4 |- ( ph -> ( N ` { ( R ` Y ) } ) = ( N ` { Y } ) )
30	29	fveq2d 6195	. . 3 |- ( ph -> ( M ` ( N ` { ( R ` Y ) } ) ) = ( M ` ( N ` { Y } ) ) )
31	2, 8, 14	lcdlmod 36881	. . . 4 |- ( ph -> C e. LMod )
32		hdmap1neglem1.s	. . . . 5 |- S = ( invg ` C )
33	9, 32, 11	lspsnneg 19006	. . . 4 |- ( ( C e. LMod /\ G e. D ) -> ( L ` { ( S ` G ) } ) = ( L ` { G } ) )
34	31, 22, 33	syl2anc 693	. . 3 |- ( ph -> ( L ` { ( S ` G ) } ) = ( L ` { G } ) )
35	25, 30, 34	3eqtr4d 2666	. 2 |- ( ph -> ( M ` ( N ` { ( R ` Y ) } ) ) = ( L ` { ( S ` G ) } ) )
36	24	simprd 479	. . 3 |- ( ph -> ( M ` ( N ` { ( X ( -g ` U ) Y ) } ) ) = ( L ` { ( F ( -g ` C ) G ) } ) )
37		lmodabl 18910	. . . . . . . . 9 |- ( U e. LMod -> U e. Abel )
38	26, 37	syl 17	. . . . . . . 8 |- ( ph -> U e. Abel )
39	15	eldifad 3586	. . . . . . . 8 |- ( ph -> X e. V )
40	4, 5, 27, 38, 39, 20	ablsub2inv 18216	. . . . . . 7 |- ( ph -> ( ( R ` X ) ( -g ` U ) ( R ` Y ) ) = ( Y ( -g ` U ) X ) )
41	40	sneqd 4189	. . . . . 6 |- ( ph -> { ( ( R ` X ) ( -g ` U ) ( R ` Y ) ) } = { ( Y ( -g ` U ) X ) } )
42	41	fveq2d 6195	. . . . 5 |- ( ph -> ( N ` { ( ( R ` X ) ( -g ` U ) ( R ` Y ) ) } ) = ( N ` { ( Y ( -g ` U ) X ) } ) )
43	4, 5, 7, 26, 20, 39	lspsnsub 19007	. . . . 5 |- ( ph -> ( N ` { ( Y ( -g ` U ) X ) } ) = ( N ` { ( X ( -g ` U ) Y ) } ) )
44	42, 43	eqtrd 2656	. . . 4 |- ( ph -> ( N ` { ( ( R ` X ) ( -g ` U ) ( R ` Y ) ) } ) = ( N ` { ( X ( -g ` U ) Y ) } ) )
45	44	fveq2d 6195	. . 3 |- ( ph -> ( M ` ( N ` { ( ( R ` X ) ( -g ` U ) ( R ` Y ) ) } ) ) = ( M ` ( N ` { ( X ( -g ` U ) Y ) } ) ) )
46		lmodabl 18910	. . . . . . . 8 |- ( C e. LMod -> C e. Abel )
47	31, 46	syl 17	. . . . . . 7 |- ( ph -> C e. Abel )
48	9, 10, 32, 47, 16, 22	ablsub2inv 18216	. . . . . 6 |- ( ph -> ( ( S ` F ) ( -g ` C ) ( S ` G ) ) = ( G ( -g ` C ) F ) )
49	48	sneqd 4189	. . . . 5 |- ( ph -> { ( ( S ` F ) ( -g ` C ) ( S ` G ) ) } = { ( G ( -g ` C ) F ) } )
50	49	fveq2d 6195	. . . 4 |- ( ph -> ( L ` { ( ( S ` F ) ( -g ` C ) ( S ` G ) ) } ) = ( L ` { ( G ( -g ` C ) F ) } ) )
51	9, 10, 11, 31, 22, 16	lspsnsub 19007	. . . 4 |- ( ph -> ( L ` { ( G ( -g ` C ) F ) } ) = ( L ` { ( F ( -g ` C ) G ) } ) )
52	50, 51	eqtrd 2656	. . 3 |- ( ph -> ( L ` { ( ( S ` F ) ( -g ` C ) ( S ` G ) ) } ) = ( L ` { ( F ( -g ` C ) G ) } ) )
53	36, 45, 52	3eqtr4d 2666	. 2 |- ( ph -> ( M ` ( N ` { ( ( R ` X ) ( -g ` U ) ( R ` Y ) ) } ) ) = ( L ` { ( ( S ` F ) ( -g ` C ) ( S ` G ) ) } ) )
54		lmodgrp 18870	. . . . 5 |- ( U e. LMod -> U e. Grp )
55	26, 54	syl 17	. . . 4 |- ( ph -> U e. Grp )
56	4, 6, 27	grpinvnzcl 17487	. . . 4 |- ( ( U e. Grp /\ X e. ( V \ { .0. } ) ) -> ( R ` X ) e. ( V \ { .0. } ) )
57	55, 15, 56	syl2anc 693	. . 3 |- ( ph -> ( R ` X ) e. ( V \ { .0. } ) )
58	9, 32	lmodvnegcl 18904	. . . 4 |- ( ( C e. LMod /\ F e. D ) -> ( S ` F ) e. D )
59	31, 16, 58	syl2anc 693	. . 3 |- ( ph -> ( S ` F ) e. D )
60	4, 6, 27	grpinvnzcl 17487	. . . 4 |- ( ( U e. Grp /\ Y e. ( V \ { .0. } ) ) -> ( R ` Y ) e. ( V \ { .0. } ) )
61	55, 17, 60	syl2anc 693	. . 3 |- ( ph -> ( R ` Y ) e. ( V \ { .0. } ) )
62	9, 32	lmodvnegcl 18904	. . . 4 |- ( ( C e. LMod /\ G e. D ) -> ( S ` G ) e. D )
63	31, 22, 62	syl2anc 693	. . 3 |- ( ph -> ( S ` G ) e. D )
64	4, 27, 7	lspsnneg 19006	. . . . 5 |- ( ( U e. LMod /\ X e. V ) -> ( N ` { ( R ` X ) } ) = ( N ` { X } ) )
65	26, 39, 64	syl2anc 693	. . . 4 |- ( ph -> ( N ` { ( R ` X ) } ) = ( N ` { X } ) )
66	19, 65, 29	3netr4d 2871	. . 3 |- ( ph -> ( N ` { ( R ` X ) } ) =/= ( N ` { ( R ` Y ) } ) )
67	65	fveq2d 6195	. . . 4 |- ( ph -> ( M ` ( N ` { ( R ` X ) } ) ) = ( M ` ( N ` { X } ) ) )
68	9, 32, 11	lspsnneg 19006	. . . . 5 |- ( ( C e. LMod /\ F e. D ) -> ( L ` { ( S ` F ) } ) = ( L ` { F } ) )
69	31, 16, 68	syl2anc 693	. . . 4 |- ( ph -> ( L ` { ( S ` F ) } ) = ( L ` { F } ) )
70	18, 67, 69	3eqtr4d 2666	. . 3 |- ( ph -> ( M ` ( N ` { ( R ` X ) } ) ) = ( L ` { ( S ` F ) } ) )
71	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 57, 59, 61, 63, 66, 70	hdmap1eq 37091	. 2 |- ( ph -> ( ( I ` <. ( R ` X ) , ( S ` F ) , ( R ` Y ) >. ) = ( S ` G ) <-> ( ( M ` ( N ` { ( R ` Y ) } ) ) = ( L ` { ( S ` G ) } ) /\ ( M ` ( N ` { ( ( R ` X ) ( -g ` U ) ( R ` Y ) ) } ) ) = ( L ` { ( ( S ` F ) ( -g ` C ) ( S ` G ) ) } ) ) ) )
72	35, 53, 71	mpbir2and 957	1 |- ( ph -> ( I ` <. ( R ` X ) , ( S ` F ) , ( R ` Y ) >. ) = ( S ` G ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   {csn 4177   <.cotp 4185   ` cfv 5888  (class class class)co 6650   Basecbs 15857   0gc0g 16100   Grpcgrp 17422   invgcminusg 17423   -gcsg 17424   Abelcabl 18194   LModclmod 18863   LSpanclspn 18971   HLchlt 34637   LHypclh 35270   DVecHcdvh 36367  LCDualclcd 36875  mapdcmpd 36913  HDMap1chdma1 37081

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-hdmap1 37083

This theorem is referenced by: (None)