Theorem vcm 30647

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 30641	. . . 4 ⊢ (𝑊 ∈ CVecOLD → 𝐺 ∈ GrpOp)
3	2	adantr 480	. . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → 𝐺 ∈ GrpOp)
4		neg1cn 12144	. . . 4 ⊢ -1 ∈ ℂ
5		vcm.2	. . . . 5 ⊢ 𝑆 = (2nd ‘𝑊)
6		vcm.3	. . . . 5 ⊢ 𝑋 = ran 𝐺
7	1, 5, 6	vccl 30634	. . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ -1 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) ∈ 𝑋)
8	4, 7	mp3an2 1452	. . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) ∈ 𝑋)
9		eqid 2737	. . . 4 ⊢ (GId‘𝐺) = (GId‘𝐺)
10	6, 9	grporid 30588	. . 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 30595	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑀‘𝐴) ∈ 𝑋)
15	2, 14	sylan 581	. . . . . 6 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (𝑀‘𝐴) ∈ 𝑋)
16	6	grpoass 30574	. . . . . 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 30635	. . . . . . . 8 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴)
19	18	oveq2d 7383	. . . . . . 7 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺(1𝑆𝐴)) = ((-1𝑆𝐴)𝐺𝐴))
20		ax-1cn 11096	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
21		1pneg1e0 12295	. . . . . . . . . 10 ⊢ (1 + -1) = 0
22	20, 4, 21	addcomli 11338	. . . . . . . . 9 ⊢ (-1 + 1) = 0
23	22	oveq1i 7377	. . . . . . . 8 ⊢ ((-1 + 1)𝑆𝐴) = (0𝑆𝐴)
24	1, 5, 6	vcdir 30637	. . . . . . . . . 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 30645	. . . . . . . 8 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = (GId‘𝐺))
28	23, 26, 27	3eqtr3a 2796	. . . . . . 7 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺(1𝑆𝐴)) = (GId‘𝐺))
29	19, 28	eqtr3d 2774	. . . . . 6 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺𝐴) = (GId‘𝐺))
30	29	oveq1d 7382	. . . . 5 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (((-1𝑆𝐴)𝐺𝐴)𝐺(𝑀‘𝐴)) = ((GId‘𝐺)𝐺(𝑀‘𝐴)))
31	17, 30	eqtr3d 2774	. . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺(𝐴𝐺(𝑀‘𝐴))) = ((GId‘𝐺)𝐺(𝑀‘𝐴)))
32	6, 9, 13	grporinv 30598	. . . . . 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 7383	. . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺(𝐴𝐺(𝑀‘𝐴))) = ((-1𝑆𝐴)𝐺(GId‘𝐺)))
35	31, 34	eqtr3d 2774	. . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((GId‘𝐺)𝐺(𝑀‘𝐴)) = ((-1𝑆𝐴)𝐺(GId‘𝐺)))
36	6, 9	grpolid 30587	. . . 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 2774	. 2 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺(GId‘𝐺)) = (𝑀‘𝐴))
39	11, 38	eqtr3d 2774	1 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) = (𝑀‘𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ran crn 5632  ‘cfv 6499  (class class class)co 7367  1^st c1st 7940  2^nd c2nd 7941  ℂcc 11036  0cc0 11038  1c1 11039   + caddc 11041  -cneg 11378  GrpOpcgr 30560  GIdcgi 30561  invcgn 30562  CVec_OLDcvc 30629

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 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376  ax-un 7689  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5526  df-po 5539  df-so 5540  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-ltxr 11184  df-sub 11379  df-neg 11380  df-grpo 30564  df-gid 30565  df-ginv 30566  df-ablo 30616  df-vc 30630

This theorem is referenced by:  nvinv  30710