Theorem vcm 31111

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 31105	. . . 4 ⊢ (𝑊 ∈ CVecOLD → 𝐺 ∈ GrpOp)
3	2	adantr 486	. . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → 𝐺 ∈ GrpOp)
4		neg1cn 12274	. . . 4 ⊢ -1 ∈ ℂ
5		vcm.2	. . . . 5 ⊢ 𝑆 = (2nd ‘𝑊)
6		vcm.3	. . . . 5 ⊢ 𝑋 = ran 𝐺
7	1, 5, 6	vccl 31098	. . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ -1 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) ∈ 𝑋)
8	4, 7	mp3an2 1478	. . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) ∈ 𝑋)
9		eqid 2760	. . . 4 ⊢ (GId‘𝐺) = (GId‘𝐺)
10	6, 9	grporid 31052	. . 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 31059	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑀‘𝐴) ∈ 𝑋)
15	2, 14	sylan 592	. . . . . 6 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (𝑀‘𝐴) ∈ 𝑋)
16	6	grpoass 31038	. . . . . 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 31099	. . . . . . . 8 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴)
19	18	oveq2d 7424	. . . . . . 7 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺(1𝑆𝐴)) = ((-1𝑆𝐴)𝐺𝐴))
20		ax-1cn 11229	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
21		1pneg1e0 12429	. . . . . . . . . 10 ⊢ (1 + -1) = 0
22	20, 4, 21	addcomli 11473	. . . . . . . . 9 ⊢ (-1 + 1) = 0
23	22	oveq1i 7418	. . . . . . . 8 ⊢ ((-1 + 1)𝑆𝐴) = (0𝑆𝐴)
24	1, 5, 6	vcdir 31101	. . . . . . . . . 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 31109	. . . . . . . 8 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = (GId‘𝐺))
28	23, 26, 27	3eqtr3a 2819	. . . . . . 7 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺(1𝑆𝐴)) = (GId‘𝐺))
29	19, 28	eqtr3d 2797	. . . . . 6 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺𝐴) = (GId‘𝐺))
30	29	oveq1d 7423	. . . . 5 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (((-1𝑆𝐴)𝐺𝐴)𝐺(𝑀‘𝐴)) = ((GId‘𝐺)𝐺(𝑀‘𝐴)))
31	17, 30	eqtr3d 2797	. . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺(𝐴𝐺(𝑀‘𝐴))) = ((GId‘𝐺)𝐺(𝑀‘𝐴)))
32	6, 9, 13	grporinv 31062	. . . . . 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 7424	. . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺(𝐴𝐺(𝑀‘𝐴))) = ((-1𝑆𝐴)𝐺(GId‘𝐺)))
35	31, 34	eqtr3d 2797	. . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((GId‘𝐺)𝐺(𝑀‘𝐴)) = ((-1𝑆𝐴)𝐺(GId‘𝐺)))
36	6, 9	grpolid 31051	. . . 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 2797	. 2 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺(GId‘𝐺)) = (𝑀‘𝐴))
39	11, 38	eqtr3d 2797	1 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) = (𝑀‘𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ran crn 5648  ‘cfv 6527  (class class class)co 7408  1^st c1st 7982  2^nd c2nd 7983  ℂcc 11169  0cc0 11171  1c1 11172   + caddc 11174  -cneg 11513  GrpOpcgr 31024  GIdcgi 31025  invcgn 31026  CVec_OLDcvc 31093

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 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-resscn 11228  ax-1cn 11229  ax-icn 11230  ax-addcl 11231  ax-addrcl 11232  ax-mulcl 11233  ax-mulrcl 11234  ax-mulcom 11235  ax-addass 11236  ax-mulass 11237  ax-distr 11238  ax-i2m1 11239  ax-1ne0 11240  ax-1rid 11241  ax-rnegex 11242  ax-rrecex 11243  ax-cnre 11244  ax-pre-lttri 11245  ax-pre-lttrn 11246  ax-pre-ltadd 11247

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-po 5555  df-so 5556  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7984  df-2nd 7985  df-er 8695  df-en 8952  df-dom 8953  df-sdom 8954  df-pnf 11316  df-mnf 11317  df-ltxr 11319  df-sub 11514  df-neg 11515  df-grpo 31028  df-gid 31029  df-ginv 31030  df-ablo 31080  df-vc 31094

This theorem is used by:  nvinv  31174