Theorem vcm 30546

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 30540	. . . 4 ⊢ (𝑊 ∈ CVecOLD → 𝐺 ∈ GrpOp)
3	2	adantr 480	. . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → 𝐺 ∈ GrpOp)
4		neg1cn 12102	. . . 4 ⊢ -1 ∈ ℂ
5		vcm.2	. . . . 5 ⊢ 𝑆 = (2nd ‘𝑊)
6		vcm.3	. . . . 5 ⊢ 𝑋 = ran 𝐺
7	1, 5, 6	vccl 30533	. . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ -1 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) ∈ 𝑋)
8	4, 7	mp3an2 1451	. . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) ∈ 𝑋)
9		eqid 2730	. . . 4 ⊢ (GId‘𝐺) = (GId‘𝐺)
10	6, 9	grporid 30487	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (-1𝑆𝐴) ∈ 𝑋) → ((-1𝑆𝐴)𝐺(GId‘𝐺)) = (-1𝑆𝐴))
11	3, 8, 10	syl2anc 584	. 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 30494	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑀‘𝐴) ∈ 𝑋)
15	2, 14	sylan 580	. . . . . 6 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (𝑀‘𝐴) ∈ 𝑋)
16	6	grpoass 30473	. . . . . 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 1374	. . . . 5 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (((-1𝑆𝐴)𝐺𝐴)𝐺(𝑀‘𝐴)) = ((-1𝑆𝐴)𝐺(𝐴𝐺(𝑀‘𝐴))))
18	1, 5, 6	vcidOLD 30534	. . . . . . . 8 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴)
19	18	oveq2d 7357	. . . . . . 7 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺(1𝑆𝐴)) = ((-1𝑆𝐴)𝐺𝐴))
20		ax-1cn 11056	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
21		1pneg1e0 12231	. . . . . . . . . 10 ⊢ (1 + -1) = 0
22	20, 4, 21	addcomli 11297	. . . . . . . . 9 ⊢ (-1 + 1) = 0
23	22	oveq1i 7351	. . . . . . . 8 ⊢ ((-1 + 1)𝑆𝐴) = (0𝑆𝐴)
24	1, 5, 6	vcdir 30536	. . . . . . . . . 10 ⊢ ((𝑊 ∈ CVecOLD ∧ (-1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → ((-1 + 1)𝑆𝐴) = ((-1𝑆𝐴)𝐺(1𝑆𝐴)))
25	4, 24	mp3anr1 1460	. . . . . . . . 9 ⊢ ((𝑊 ∈ CVecOLD ∧ (1 ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → ((-1 + 1)𝑆𝐴) = ((-1𝑆𝐴)𝐺(1𝑆𝐴)))
26	20, 25	mpanr1 703	. . . . . . . 8 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1 + 1)𝑆𝐴) = ((-1𝑆𝐴)𝐺(1𝑆𝐴)))
27	1, 5, 6, 9	vc0 30544	. . . . . . . 8 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = (GId‘𝐺))
28	23, 26, 27	3eqtr3a 2789	. . . . . . 7 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺(1𝑆𝐴)) = (GId‘𝐺))
29	19, 28	eqtr3d 2767	. . . . . 6 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺𝐴) = (GId‘𝐺))
30	29	oveq1d 7356	. . . . 5 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (((-1𝑆𝐴)𝐺𝐴)𝐺(𝑀‘𝐴)) = ((GId‘𝐺)𝐺(𝑀‘𝐴)))
31	17, 30	eqtr3d 2767	. . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺(𝐴𝐺(𝑀‘𝐴))) = ((GId‘𝐺)𝐺(𝑀‘𝐴)))
32	6, 9, 13	grporinv 30497	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(𝑀‘𝐴)) = (GId‘𝐺))
33	2, 32	sylan 580	. . . . 5 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(𝑀‘𝐴)) = (GId‘𝐺))
34	33	oveq2d 7357	. . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺(𝐴𝐺(𝑀‘𝐴))) = ((-1𝑆𝐴)𝐺(GId‘𝐺)))
35	31, 34	eqtr3d 2767	. . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((GId‘𝐺)𝐺(𝑀‘𝐴)) = ((-1𝑆𝐴)𝐺(GId‘𝐺)))
36	6, 9	grpolid 30486	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑀‘𝐴) ∈ 𝑋) → ((GId‘𝐺)𝐺(𝑀‘𝐴)) = (𝑀‘𝐴))
37	3, 15, 36	syl2anc 584	. . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((GId‘𝐺)𝐺(𝑀‘𝐴)) = (𝑀‘𝐴))
38	35, 37	eqtr3d 2767	. 2 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺(GId‘𝐺)) = (𝑀‘𝐴))
39	11, 38	eqtr3d 2767	1 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) = (𝑀‘𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2110  ran crn 5615  ‘cfv 6477  (class class class)co 7341  1^st c1st 7914  2^nd c2nd 7915  ℂcc 10996  0cc0 10998  1c1 10999   + caddc 11001  -cneg 11337  GrpOpcgr 30459  GIdcgi 30460  invcgn 30461  CVec_OLDcvc 30528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663  ax-resscn 11055  ax-1cn 11056  ax-icn 11057  ax-addcl 11058  ax-addrcl 11059  ax-mulcl 11060  ax-mulrcl 11061  ax-mulcom 11062  ax-addass 11063  ax-mulass 11064  ax-distr 11065  ax-i2m1 11066  ax-1ne0 11067  ax-1rid 11068  ax-rnegex 11069  ax-rrecex 11070  ax-cnre 11071  ax-pre-lttri 11072  ax-pre-lttrn 11073  ax-pre-ltadd 11074

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-po 5522  df-so 5523  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-er 8617  df-en 8865  df-dom 8866  df-sdom 8867  df-pnf 11140  df-mnf 11141  df-ltxr 11143  df-sub 11338  df-neg 11339  df-grpo 30463  df-gid 30464  df-ginv 30465  df-ablo 30515  df-vc 30529

This theorem is referenced by:  nvinv  30609