Theorem vcm 31043

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 31037	. . . 4 ⊢ (𝑊 ∈ CVecOLD → 𝐺 ∈ GrpOp)
3	2	adantr 486	. . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → 𝐺 ∈ GrpOp)
4		neg1cn 12230	. . . 4 ⊢ -1 ∈ ℂ
5		vcm.2	. . . . 5 ⊢ 𝑆 = (2nd ‘𝑊)
6		vcm.3	. . . . 5 ⊢ 𝑋 = ran 𝐺
7	1, 5, 6	vccl 31030	. . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ -1 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) ∈ 𝑋)
8	4, 7	mp3an2 1478	. . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) ∈ 𝑋)
9		eqid 2762	. . . 4 ⊢ (GId‘𝐺) = (GId‘𝐺)
10	6, 9	grporid 30984	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (-1𝑆𝐴) ∈ 𝑋) → ((-1𝑆𝐴)𝐺(GId‘𝐺)) = (-1𝑆𝐴))
11	3, 8, 10	syl2anc 596	. 2 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺(GId‘𝐺)) = (-1𝑆𝐴))
12		simpr 490	. . . . . 6 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
13		vcm.4	. . . . . . . 8 ⊢ 𝑀 = (inv‘𝐺)
14	6, 13	grpoinvcl 30991	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑀‘𝐴) ∈ 𝑋)
15	2, 14	sylan 592	. . . . . 6 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (𝑀‘𝐴) ∈ 𝑋)
16	6	grpoass 30970	. . . . . 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 1399	. . . . 5 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (((-1𝑆𝐴)𝐺𝐴)𝐺(𝑀‘𝐴)) = ((-1𝑆𝐴)𝐺(𝐴𝐺(𝑀‘𝐴))))
18	1, 5, 6	vcidOLD 31031	. . . . . . . 8 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴)
19	18	oveq2d 7432	. . . . . . 7 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺(1𝑆𝐴)) = ((-1𝑆𝐴)𝐺𝐴))
20		ax-1cn 11185	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
21		1pneg1e0 12385	. . . . . . . . . 10 ⊢ (1 + -1) = 0
22	20, 4, 21	addcomli 11429	. . . . . . . . 9 ⊢ (-1 + 1) = 0
23	22	oveq1i 7426	. . . . . . . 8 ⊢ ((-1 + 1)𝑆𝐴) = (0𝑆𝐴)
24	1, 5, 6	vcdir 31033	. . . . . . . . . 10 ⊢ ((𝑊 ∈ CVecOLD ∧ (-1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → ((-1 + 1)𝑆𝐴) = ((-1𝑆𝐴)𝐺(1𝑆𝐴)))
25	4, 24	mp3anr1 1487	. . . . . . . . 9 ⊢ ((𝑊 ∈ CVecOLD ∧ (1 ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → ((-1 + 1)𝑆𝐴) = ((-1𝑆𝐴)𝐺(1𝑆𝐴)))
26	20, 25	mpanr1 716	. . . . . . . 8 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1 + 1)𝑆𝐴) = ((-1𝑆𝐴)𝐺(1𝑆𝐴)))
27	1, 5, 6, 9	vc0 31041	. . . . . . . 8 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = (GId‘𝐺))
28	23, 26, 27	3eqtr3a 2821	. . . . . . 7 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺(1𝑆𝐴)) = (GId‘𝐺))
29	19, 28	eqtr3d 2799	. . . . . 6 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺𝐴) = (GId‘𝐺))
30	29	oveq1d 7431	. . . . 5 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (((-1𝑆𝐴)𝐺𝐴)𝐺(𝑀‘𝐴)) = ((GId‘𝐺)𝐺(𝑀‘𝐴)))
31	17, 30	eqtr3d 2799	. . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺(𝐴𝐺(𝑀‘𝐴))) = ((GId‘𝐺)𝐺(𝑀‘𝐴)))
32	6, 9, 13	grporinv 30994	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(𝑀‘𝐴)) = (GId‘𝐺))
33	2, 32	sylan 592	. . . . 5 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(𝑀‘𝐴)) = (GId‘𝐺))
34	33	oveq2d 7432	. . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺(𝐴𝐺(𝑀‘𝐴))) = ((-1𝑆𝐴)𝐺(GId‘𝐺)))
35	31, 34	eqtr3d 2799	. . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((GId‘𝐺)𝐺(𝑀‘𝐴)) = ((-1𝑆𝐴)𝐺(GId‘𝐺)))
36	6, 9	grpolid 30983	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑀‘𝐴) ∈ 𝑋) → ((GId‘𝐺)𝐺(𝑀‘𝐴)) = (𝑀‘𝐴))
37	3, 15, 36	syl2anc 596	. . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((GId‘𝐺)𝐺(𝑀‘𝐴)) = (𝑀‘𝐴))
38	35, 37	eqtr3d 2799	. 2 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺(GId‘𝐺)) = (𝑀‘𝐴))
39	11, 38	eqtr3d 2799	1 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) = (𝑀‘𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ran crn 5660  ‘cfv 6537  (class class class)co 7416  1^st c1st 7987  2^nd c2nd 7988  ℂcc 11125  0cc0 11127  1c1 11128   + caddc 11130  -cneg 11469  GrpOpcgr 30956  GIdcgi 30957  invcgn 30958  CVec_OLDcvc 31025

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-po 5567  df-so 5568  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7989  df-2nd 7990  df-er 8699  df-en 8956  df-dom 8957  df-sdom 8958  df-pnf 11272  df-mnf 11273  df-ltxr 11275  df-sub 11470  df-neg 11471  df-grpo 30960  df-gid 30961  df-ginv 30962  df-ablo 31012  df-vc 31026

This theorem is used by:  nvinv  31106