Theorem vcm 8186

Description: Minus 1 times a vector is the underlying group's inverse element. Equation 2 of [Kreyszig] p. 51.

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 ∈ CVec ⋀ A ∈ X) → (-1SA) = (M ‘A))

Proof of Theorem vcm

Step	Hyp	Ref	Expression
1		vcm.3	. . . . 5 ⊢ X = ran G
2	1	grpass 8044	. . . 4 ⊢ ((G ∈ Grp ⋀ ((-1SA) ∈ X ⋀ A ∈ X ⋀ (M ‘A) ∈ X)) → (((-1SA)GA)G(M ‘A)) = ((-1SA)G(AG(M ‘A))))
3		vcm.1	. . . . . 6 ⊢ G = (1st ‘W)
4	3	vcgrp 8173	. . . . 5 ⊢ (W ∈ CVec → G ∈ Grp)
5	4	adantr 391	. . . 4 ⊢ ((W ∈ CVec ⋀ A ∈ X) → G ∈ Grp)
6		ax1cn 5281	. . . . . . 7 ⊢ 1 ∈ ℂ
7	6	negcl 5381	. . . . . 6 ⊢ -1 ∈ ℂ
8		vcm.2	. . . . . . 7 ⊢ S = (2nd ‘W)
9	3, 8, 1	vccl 8165	. . . . . 6 ⊢ ((W ∈ CVec ⋀ -1 ∈ ℂ ⋀ A ∈ X) → (-1SA) ∈ X)
10	7, 9	mp3an2 906	. . . . 5 ⊢ ((W ∈ CVec ⋀ A ∈ X) → (-1SA) ∈ X)
11		pm3.27 323	. . . . 5 ⊢ ((W ∈ CVec ⋀ A ∈ X) → A ∈ X)
12		vcm.4	. . . . . . 7 ⊢ M = (inv ‘G)
13	1, 12	grpinvcl 8064	. . . . . 6 ⊢ ((G ∈ Grp ⋀ A ∈ X) → (M ‘A) ∈ X)
14	13, 4	sylan 450	. . . . 5 ⊢ ((W ∈ CVec ⋀ A ∈ X) → (M ‘A) ∈ X)
15	10, 11, 14	3jca 821	. . . 4 ⊢ ((W ∈ CVec ⋀ A ∈ X) → ((-1SA) ∈ X ⋀ A ∈ X ⋀ (M ‘A) ∈ X))
16	2, 5, 15	sylanc 473	. . 3 ⊢ ((W ∈ CVec ⋀ A ∈ X) → (((-1SA)GA)G(M ‘A)) = ((-1SA)G(AG(M ‘A))))
17	3, 8, 1	vcid 8166	. . . . . 6 ⊢ ((W ∈ CVec ⋀ A ∈ X) → (1SA) = A)
18	17	opreq2d 3982	. . . . 5 ⊢ ((W ∈ CVec ⋀ A ∈ X) → ((-1SA)G(1SA)) = ((-1SA)GA))
19	7, 6	addcom 5334	. . . . . . . . 9 ⊢ (-1 + 1) = (1 + -1)
20	6	negid 5392	. . . . . . . . 9 ⊢ (1 + -1) = 0
21	19, 20	eqtr 1498	. . . . . . . 8 ⊢ (-1 + 1) = 0
22	21	opreq1i 3977	. . . . . . 7 ⊢ ((-1 + 1)SA) = (0SA)
23	22	a1i 8	. . . . . 6 ⊢ ((W ∈ CVec ⋀ A ∈ X) → ((-1 + 1)SA) = (0SA))
24	3, 8, 1	vcdir 8168	. . . . . . . 8 ⊢ ((W ∈ CVec ⋀ (-1 ∈ ℂ ⋀ 1 ∈ ℂ ⋀ A ∈ X)) → ((-1 + 1)SA) = ((-1SA)G(1SA)))
25	7, 24	mp3anr1 915	. . . . . . 7 ⊢ ((W ∈ CVec ⋀ (1 ∈ ℂ ⋀ A ∈ X)) → ((-1 + 1)SA) = ((-1SA)G(1SA)))
26	6, 25	mpanr1 711	. . . . . 6 ⊢ ((W ∈ CVec ⋀ A ∈ X) → ((-1 + 1)SA) = ((-1SA)G(1SA)))
27		eqid 1478	. . . . . . 7 ⊢ (Id ‘G) = (Id ‘G)
28	3, 8, 1, 27	vc0 8184	. . . . . 6 ⊢ ((W ∈ CVec ⋀ A ∈ X) → (0SA) = (Id ‘G))
29	23, 26, 28	3eqtr3d 1518	. . . . 5 ⊢ ((W ∈ CVec ⋀ A ∈ X) → ((-1SA)G(1SA)) = (Id ‘G))
30	18, 29	eqtr3d 1512	. . . 4 ⊢ ((W ∈ CVec ⋀ A ∈ X) → ((-1SA)GA) = (Id ‘G))
31	30	opreq1d 3981	. . 3 ⊢ ((W ∈ CVec ⋀ A ∈ X) → (((-1SA)GA)G(M ‘A)) = ((Id ‘G)G(M ‘A)))
32	1, 27, 12	grprinv 8067	. . . . 5 ⊢ ((G ∈ Grp ⋀ A ∈ X) → (AG(M ‘A)) = (Id ‘G))
33	32, 4	sylan 450	. . . 4 ⊢ ((W ∈ CVec ⋀ A ∈ X) → (AG(M ‘A)) = (Id ‘G))
34	33	opreq2d 3982	. . 3 ⊢ ((W ∈ CVec ⋀ A ∈ X) → ((-1SA)G(AG(M ‘A))) = ((-1SA)G(Id ‘G)))
35	16, 31, 34	3eqtr3d 1518	. 2 ⊢ ((W ∈ CVec ⋀ A ∈ X) → ((Id ‘G)G(M ‘A)) = ((-1SA)G(Id ‘G)))
36	1, 27	grplid 8057	. . 3 ⊢ ((G ∈ Grp ⋀ (M ‘A) ∈ X) → ((Id ‘G)G(M ‘A)) = (M ‘A))
37	36, 5, 14	sylanc 473	. 2 ⊢ ((W ∈ CVec ⋀ A ∈ X) → ((Id ‘G)G(M ‘A)) = (M ‘A))
38	1, 27	grprid 8058	. . 3 ⊢ ((G ∈ Grp ⋀ (-1SA) ∈ X) → ((-1SA)G(Id ‘G)) = (-1SA))
39	38, 5, 10	sylanc 473	. 2 ⊢ ((W ∈ CVec ⋀ A ∈ X) → ((-1SA)G(Id ‘G)) = (-1SA))
40	35, 37, 39	3eqtr3rd 1519	1 ⊢ ((W ∈ CVec ⋀ A ∈ X) → (-1SA) = (M ‘A))

Syntax hints:   → wi 3   ⋀ wa 223   ⋀ w3a 777   = wceq 958   ∈ wcel 960  ran crn 3177   ‘cfv 3188  (class class class)co 3969  1^st c1st 4083  2^nd c2nd 4084  ℂcc 5244  0cc0 5246  1c1 5247   + caddc 5249  -cneg 5305  Grpcgr 8030  Idcgi 8031  invcgn 8032  CVeccvc 8160

This theorem is referenced by:  vcrinv 8187  vclinv 8188  nvinv 8256

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fo 3202  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-sub 5368  df-neg 5370  df-grp 8034  df-gid 8035  df-ginv 8036  df-abl 8096  df-vc 8161

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