Theorem vcm 27431

Description: Minus 1 times a vector is the underlying group's inverse element. Equation 2 of [Kreyszig] p. 51. (Contributed by NM, 25-Nov-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
vcm.1	|- G = ( 1st ` W )
vcm.2	|- S = ( 2nd ` W )
vcm.3	|- X = ran G
vcm.4	|- M = ( inv ` G )

Assertion

Ref	Expression
vcm	|- ( ( W e. CVecOLD /\ A e. X ) -> ( -u 1 S A ) = ( M ` A ) )

Proof of Theorem vcm

Step	Hyp	Ref	Expression
1		vcm.1	. . . . 5 |- G = ( 1st ` W )
2	1	vcgrp 27425	. . . 4 |- ( W e. CVecOLD -> G e. GrpOp )
3	2	adantr 481	. . 3 |- ( ( W e. CVecOLD /\ A e. X ) -> G e. GrpOp )
4		neg1cn 11124	. . . 4 |- -u 1 e. CC
5		vcm.2	. . . . 5 |- S = ( 2nd ` W )
6		vcm.3	. . . . 5 |- X = ran G
7	1, 5, 6	vccl 27418	. . . 4 |- ( ( W e. CVecOLD /\ -u 1 e. CC /\ A e. X ) -> ( -u 1 S A ) e. X )
8	4, 7	mp3an2 1412	. . 3 |- ( ( W e. CVecOLD /\ A e. X ) -> ( -u 1 S A ) e. X )
9		eqid 2622	. . . 4 |- (GId ` G ) = (GId ` G )
10	6, 9	grporid 27371	. . 3 |- ( ( G e. GrpOp /\ ( -u 1 S A ) e. X ) -> ( ( -u 1 S A ) G (GId ` G ) ) = ( -u 1 S A ) )
11	3, 8, 10	syl2anc 693	. 2 |- ( ( W e. CVecOLD /\ A e. X ) -> ( ( -u 1 S A ) G (GId ` G ) ) = ( -u 1 S A ) )
12		simpr 477	. . . . . 6 |- ( ( W e. CVecOLD /\ A e. X ) -> A e. X )
13		vcm.4	. . . . . . . 8 |- M = ( inv ` G )
14	6, 13	grpoinvcl 27378	. . . . . . 7 |- ( ( G e. GrpOp /\ A e. X ) -> ( M ` A ) e. X )
15	2, 14	sylan 488	. . . . . 6 |- ( ( W e. CVecOLD /\ A e. X ) -> ( M ` A ) e. X )
16	6	grpoass 27357	. . . . . 6 |- ( ( G e. GrpOp /\ ( ( -u 1 S A ) e. X /\ A e. X /\ ( M ` A ) e. X ) ) -> ( ( ( -u 1 S A ) G A ) G ( M ` A ) ) = ( ( -u 1 S A ) G ( A G ( M ` A ) ) ) )
17	3, 8, 12, 15, 16	syl13anc 1328	. . . . 5 |- ( ( W e. CVecOLD /\ A e. X ) -> ( ( ( -u 1 S A ) G A ) G ( M ` A ) ) = ( ( -u 1 S A ) G ( A G ( M ` A ) ) ) )
18	1, 5, 6	vcidOLD 27419	. . . . . . . 8 |- ( ( W e. CVecOLD /\ A e. X ) -> ( 1 S A ) = A )
19	18	oveq2d 6666	. . . . . . 7 |- ( ( W e. CVecOLD /\ A e. X ) -> ( ( -u 1 S A ) G ( 1 S A ) ) = ( ( -u 1 S A ) G A ) )
20		ax-1cn 9994	. . . . . . . . . 10 |- 1 e. CC
21		1pneg1e0 11129	. . . . . . . . . 10 |- ( 1 + -u 1 ) = 0
22	20, 4, 21	addcomli 10228	. . . . . . . . 9 |- ( -u 1 + 1 ) = 0
23	22	oveq1i 6660	. . . . . . . 8 |- ( ( -u 1 + 1 ) S A ) = ( 0 S A )
24	1, 5, 6	vcdir 27421	. . . . . . . . . 10 |- ( ( W e. CVecOLD /\ ( -u 1 e. CC /\ 1 e. CC /\ A e. X ) ) -> ( ( -u 1 + 1 ) S A ) = ( ( -u 1 S A ) G ( 1 S A ) ) )
25	4, 24	mp3anr1 1421	. . . . . . . . 9 |- ( ( W e. CVecOLD /\ ( 1 e. CC /\ A e. X ) ) -> ( ( -u 1 + 1 ) S A ) = ( ( -u 1 S A ) G ( 1 S A ) ) )
26	20, 25	mpanr1 719	. . . . . . . 8 |- ( ( W e. CVecOLD /\ A e. X ) -> ( ( -u 1 + 1 ) S A ) = ( ( -u 1 S A ) G ( 1 S A ) ) )
27	1, 5, 6, 9	vc0 27429	. . . . . . . 8 |- ( ( W e. CVecOLD /\ A e. X ) -> ( 0 S A ) = (GId ` G ) )
28	23, 26, 27	3eqtr3a 2680	. . . . . . 7 |- ( ( W e. CVecOLD /\ A e. X ) -> ( ( -u 1 S A ) G ( 1 S A ) ) = (GId ` G ) )
29	19, 28	eqtr3d 2658	. . . . . 6 |- ( ( W e. CVecOLD /\ A e. X ) -> ( ( -u 1 S A ) G A ) = (GId ` G ) )
30	29	oveq1d 6665	. . . . 5 |- ( ( W e. CVecOLD /\ A e. X ) -> ( ( ( -u 1 S A ) G A ) G ( M ` A ) ) = ( (GId ` G ) G ( M ` A ) ) )
31	17, 30	eqtr3d 2658	. . . 4 |- ( ( W e. CVecOLD /\ A e. X ) -> ( ( -u 1 S A ) G ( A G ( M ` A ) ) ) = ( (GId ` G ) G ( M ` A ) ) )
32	6, 9, 13	grporinv 27381	. . . . . 6 |- ( ( G e. GrpOp /\ A e. X ) -> ( A G ( M ` A ) ) = (GId ` G ) )
33	2, 32	sylan 488	. . . . 5 |- ( ( W e. CVecOLD /\ A e. X ) -> ( A G ( M ` A ) ) = (GId ` G ) )
34	33	oveq2d 6666	. . . 4 |- ( ( W e. CVecOLD /\ A e. X ) -> ( ( -u 1 S A ) G ( A G ( M ` A ) ) ) = ( ( -u 1 S A ) G (GId ` G ) ) )
35	31, 34	eqtr3d 2658	. . 3 |- ( ( W e. CVecOLD /\ A e. X ) -> ( (GId ` G ) G ( M ` A ) ) = ( ( -u 1 S A ) G (GId ` G ) ) )
36	6, 9	grpolid 27370	. . . 4 |- ( ( G e. GrpOp /\ ( M ` A ) e. X ) -> ( (GId ` G ) G ( M ` A ) ) = ( M ` A ) )
37	3, 15, 36	syl2anc 693	. . 3 |- ( ( W e. CVecOLD /\ A e. X ) -> ( (GId ` G ) G ( M ` A ) ) = ( M ` A ) )
38	35, 37	eqtr3d 2658	. 2 |- ( ( W e. CVecOLD /\ A e. X ) -> ( ( -u 1 S A ) G (GId ` G ) ) = ( M ` A ) )
39	11, 38	eqtr3d 2658	1 |- ( ( W e. CVecOLD /\ A e. X ) -> ( -u 1 S A ) = ( M ` A ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ran crn 5115   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   -ucneg 10267   GrpOpcgr 27343  GIdcgi 27344   invcgn 27345   CVecOLDcvc 27413

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-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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-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-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-ltxr 10079  df-sub 10268  df-neg 10269  df-grpo 27347  df-gid 27348  df-ginv 27349  df-ablo 27399  df-vc 27414

This theorem is referenced by:  nvinv  27494