Theorem vcm 30635

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 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) = (𝑀‘𝐴))

Proof of Theorem vcm

Step	Hyp	Ref	Expression
1		vcm.1	. . . . 5 ⊢ 𝐺 = (1st ‘𝑊)
2	1	vcgrp 30629	. . . 4 ⊢ (𝑊 ∈ CVecOLD → 𝐺 ∈ GrpOp)
3	2	adantr 480	. . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → 𝐺 ∈ GrpOp)
4		neg1cn 12133	. . . 4 ⊢ -1 ∈ ℂ
5		vcm.2	. . . . 5 ⊢ 𝑆 = (2nd ‘𝑊)
6		vcm.3	. . . . 5 ⊢ 𝑋 = ran 𝐺
7	1, 5, 6	vccl 30622	. . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ -1 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) ∈ 𝑋)
8	4, 7	mp3an2 1452	. . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) ∈ 𝑋)
9		eqid 2735	. . . 4 ⊢ (GId‘𝐺) = (GId‘𝐺)
10	6, 9	grporid 30576	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (-1𝑆𝐴) ∈ 𝑋) → ((-1𝑆𝐴)𝐺(GId‘𝐺)) = (-1𝑆𝐴))
11	3, 8, 10	syl2anc 585	. 2 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺(GId‘𝐺)) = (-1𝑆𝐴))
12		simpr 484	. . . . . 6 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
13		vcm.4	. . . . . . . 8 ⊢ 𝑀 = (inv‘𝐺)
14	6, 13	grpoinvcl 30583	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑀‘𝐴) ∈ 𝑋)
15	2, 14	sylan 581	. . . . . 6 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (𝑀‘𝐴) ∈ 𝑋)
16	6	grpoass 30562	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ ((-1𝑆𝐴) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (𝑀‘𝐴) ∈ 𝑋)) → (((-1𝑆𝐴)𝐺𝐴)𝐺(𝑀‘𝐴)) = ((-1𝑆𝐴)𝐺(𝐴𝐺(𝑀‘𝐴))))
17	3, 8, 12, 15, 16	syl13anc 1375	. . . . 5 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (((-1𝑆𝐴)𝐺𝐴)𝐺(𝑀‘𝐴)) = ((-1𝑆𝐴)𝐺(𝐴𝐺(𝑀‘𝐴))))
18	1, 5, 6	vcidOLD 30623	. . . . . . . 8 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴)
19	18	oveq2d 7372	. . . . . . 7 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺(1𝑆𝐴)) = ((-1𝑆𝐴)𝐺𝐴))
20		ax-1cn 11085	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
21		1pneg1e0 12284	. . . . . . . . . 10 ⊢ (1 + -1) = 0
22	20, 4, 21	addcomli 11327	. . . . . . . . 9 ⊢ (-1 + 1) = 0
23	22	oveq1i 7366	. . . . . . . 8 ⊢ ((-1 + 1)𝑆𝐴) = (0𝑆𝐴)
24	1, 5, 6	vcdir 30625	. . . . . . . . . 10 ⊢ ((𝑊 ∈ CVecOLD ∧ (-1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → ((-1 + 1)𝑆𝐴) = ((-1𝑆𝐴)𝐺(1𝑆𝐴)))
25	4, 24	mp3anr1 1461	. . . . . . . . 9 ⊢ ((𝑊 ∈ CVecOLD ∧ (1 ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → ((-1 + 1)𝑆𝐴) = ((-1𝑆𝐴)𝐺(1𝑆𝐴)))
26	20, 25	mpanr1 704	. . . . . . . 8 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1 + 1)𝑆𝐴) = ((-1𝑆𝐴)𝐺(1𝑆𝐴)))
27	1, 5, 6, 9	vc0 30633	. . . . . . . 8 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = (GId‘𝐺))
28	23, 26, 27	3eqtr3a 2794	. . . . . . 7 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺(1𝑆𝐴)) = (GId‘𝐺))
29	19, 28	eqtr3d 2772	. . . . . 6 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺𝐴) = (GId‘𝐺))
30	29	oveq1d 7371	. . . . 5 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (((-1𝑆𝐴)𝐺𝐴)𝐺(𝑀‘𝐴)) = ((GId‘𝐺)𝐺(𝑀‘𝐴)))
31	17, 30	eqtr3d 2772	. . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺(𝐴𝐺(𝑀‘𝐴))) = ((GId‘𝐺)𝐺(𝑀‘𝐴)))
32	6, 9, 13	grporinv 30586	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(𝑀‘𝐴)) = (GId‘𝐺))
33	2, 32	sylan 581	. . . . 5 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(𝑀‘𝐴)) = (GId‘𝐺))
34	33	oveq2d 7372	. . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺(𝐴𝐺(𝑀‘𝐴))) = ((-1𝑆𝐴)𝐺(GId‘𝐺)))
35	31, 34	eqtr3d 2772	. . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((GId‘𝐺)𝐺(𝑀‘𝐴)) = ((-1𝑆𝐴)𝐺(GId‘𝐺)))
36	6, 9	grpolid 30575	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑀‘𝐴) ∈ 𝑋) → ((GId‘𝐺)𝐺(𝑀‘𝐴)) = (𝑀‘𝐴))
37	3, 15, 36	syl2anc 585	. . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((GId‘𝐺)𝐺(𝑀‘𝐴)) = (𝑀‘𝐴))
38	35, 37	eqtr3d 2772	. 2 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺(GId‘𝐺)) = (𝑀‘𝐴))
39	11, 38	eqtr3d 2772	1 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) = (𝑀‘𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ran crn 5621  ‘cfv 6487  (class class class)co 7356  1^st c1st 7929  2^nd c2nd 7930  ℂcc 11025  0cc0 11027  1c1 11028   + caddc 11030  -cneg 11367  GrpOpcgr 30548  GIdcgi 30549  invcgn 30550  CVec_OLDcvc 30617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678  ax-resscn 11084  ax-1cn 11085  ax-icn 11086  ax-addcl 11087  ax-addrcl 11088  ax-mulcl 11089  ax-mulrcl 11090  ax-mulcom 11091  ax-addass 11092  ax-mulass 11093  ax-distr 11094  ax-i2m1 11095  ax-1ne0 11096  ax-1rid 11097  ax-rnegex 11098  ax-rrecex 11099  ax-cnre 11100  ax-pre-lttri 11101  ax-pre-lttrn 11102  ax-pre-ltadd 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3060  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-po 5528  df-so 5529  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-er 8632  df-en 8883  df-dom 8884  df-sdom 8885  df-pnf 11170  df-mnf 11171  df-ltxr 11173  df-sub 11368  df-neg 11369  df-grpo 30552  df-gid 30553  df-ginv 30554  df-ablo 30604  df-vc 30618

This theorem is referenced by:  nvinv  30698