Theorem grplactcnv 18922

Description: The left group action of element 𝐴 of group 𝐺 maps the underlying set 𝑋 of 𝐺 one-to-one onto itself. (Contributed by Paul Chapman, 18-Mar-2008.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)

Hypotheses

Ref	Expression
grplact.1	⊢ 𝐹 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎)))
grplact.2	⊢ 𝑋 = (Base‘𝐺)
grplact.3	⊢ + = (+g‘𝐺)
grplactcnv.4	⊢ 𝐼 = (invg‘𝐺)

Assertion

Ref	Expression
grplactcnv	⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝐹‘𝐴):𝑋–1-1-onto→𝑋 ∧ ◡(𝐹‘𝐴) = (𝐹‘(𝐼‘𝐴))))

Distinct variable groups:   𝑔,𝑎,𝐴   𝐺,𝑎,𝑔   𝐼,𝑎,𝑔   + ,𝑎,𝑔   𝑋,𝑎,𝑔

Allowed substitution hints:   𝐹(𝑔,𝑎)

Proof of Theorem grplactcnv

Dummy variable 𝑏 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2732	. . 3 ⊢ (𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎)) = (𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎))
2		grplact.2	. . . . 5 ⊢ 𝑋 = (Base‘𝐺)
3		grplact.3	. . . . 5 ⊢ + = (+g‘𝐺)
4	2, 3	grpcl 18823	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋) → (𝐴 + 𝑎) ∈ 𝑋)
5	4	3expa 1118	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑎 ∈ 𝑋) → (𝐴 + 𝑎) ∈ 𝑋)
6		grplactcnv.4	. . . . 5 ⊢ 𝐼 = (invg‘𝐺)
7	2, 6	grpinvcl 18868	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐼‘𝐴) ∈ 𝑋)
8	2, 3	grpcl 18823	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝐼‘𝐴) ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → ((𝐼‘𝐴) + 𝑏) ∈ 𝑋)
9	8	3expa 1118	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ (𝐼‘𝐴) ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → ((𝐼‘𝐴) + 𝑏) ∈ 𝑋)
10	7, 9	syldanl 602	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → ((𝐼‘𝐴) + 𝑏) ∈ 𝑋)
11		eqcom 2739	. . . . 5 ⊢ (𝑎 = ((𝐼‘𝐴) + 𝑏) ↔ ((𝐼‘𝐴) + 𝑏) = 𝑎)
12		eqid 2732	. . . . . . . . . 10 ⊢ (0g‘𝐺) = (0g‘𝐺)
13	2, 3, 12, 6	grplinv 18870	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝐼‘𝐴) + 𝐴) = (0g‘𝐺))
14	13	adantr 481	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝐼‘𝐴) + 𝐴) = (0g‘𝐺))
15	14	oveq1d 7420	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (((𝐼‘𝐴) + 𝐴) + 𝑎) = ((0g‘𝐺) + 𝑎))
16		simpll 765	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝐺 ∈ Grp)
17	7	adantr 481	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝐼‘𝐴) ∈ 𝑋)
18		simplr 767	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
19		simprl 769	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑎 ∈ 𝑋)
20	2, 3	grpass 18824	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ ((𝐼‘𝐴) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋)) → (((𝐼‘𝐴) + 𝐴) + 𝑎) = ((𝐼‘𝐴) + (𝐴 + 𝑎)))
21	16, 17, 18, 19, 20	syl13anc 1372	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (((𝐼‘𝐴) + 𝐴) + 𝑎) = ((𝐼‘𝐴) + (𝐴 + 𝑎)))
22	2, 3, 12	grplid 18848	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑎 ∈ 𝑋) → ((0g‘𝐺) + 𝑎) = 𝑎)
23	22	ad2ant2r 745	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((0g‘𝐺) + 𝑎) = 𝑎)
24	15, 21, 23	3eqtr3rd 2781	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑎 = ((𝐼‘𝐴) + (𝐴 + 𝑎)))
25	24	eqeq2d 2743	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (((𝐼‘𝐴) + 𝑏) = 𝑎 ↔ ((𝐼‘𝐴) + 𝑏) = ((𝐼‘𝐴) + (𝐴 + 𝑎))))
26	11, 25	bitrid 282	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎 = ((𝐼‘𝐴) + 𝑏) ↔ ((𝐼‘𝐴) + 𝑏) = ((𝐼‘𝐴) + (𝐴 + 𝑎))))
27		simprr 771	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑏 ∈ 𝑋)
28	5	adantrr 715	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝐴 + 𝑎) ∈ 𝑋)
29	2, 3	grplcan 18881	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑏 ∈ 𝑋 ∧ (𝐴 + 𝑎) ∈ 𝑋 ∧ (𝐼‘𝐴) ∈ 𝑋)) → (((𝐼‘𝐴) + 𝑏) = ((𝐼‘𝐴) + (𝐴 + 𝑎)) ↔ 𝑏 = (𝐴 + 𝑎)))
30	16, 27, 28, 17, 29	syl13anc 1372	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (((𝐼‘𝐴) + 𝑏) = ((𝐼‘𝐴) + (𝐴 + 𝑎)) ↔ 𝑏 = (𝐴 + 𝑎)))
31	26, 30	bitrd 278	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎 = ((𝐼‘𝐴) + 𝑏) ↔ 𝑏 = (𝐴 + 𝑎)))
32	1, 5, 10, 31	f1ocnv2d 7655	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎)):𝑋–1-1-onto→𝑋 ∧ ◡(𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎)) = (𝑏 ∈ 𝑋 ↦ ((𝐼‘𝐴) + 𝑏))))
33		grplact.1	. . . . . 6 ⊢ 𝐹 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎)))
34	33, 2	grplactfval 18920	. . . . 5 ⊢ (𝐴 ∈ 𝑋 → (𝐹‘𝐴) = (𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎)))
35	34	adantl 482	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) = (𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎)))
36	35	f1oeq1d 6825	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝐹‘𝐴):𝑋–1-1-onto→𝑋 ↔ (𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎)):𝑋–1-1-onto→𝑋))
37	35	cnveqd 5873	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ◡(𝐹‘𝐴) = ◡(𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎)))
38	33, 2	grplactfval 18920	. . . . . 6 ⊢ ((𝐼‘𝐴) ∈ 𝑋 → (𝐹‘(𝐼‘𝐴)) = (𝑎 ∈ 𝑋 ↦ ((𝐼‘𝐴) + 𝑎)))
39		oveq2 7413	. . . . . . 7 ⊢ (𝑎 = 𝑏 → ((𝐼‘𝐴) + 𝑎) = ((𝐼‘𝐴) + 𝑏))
40	39	cbvmptv 5260	. . . . . 6 ⊢ (𝑎 ∈ 𝑋 ↦ ((𝐼‘𝐴) + 𝑎)) = (𝑏 ∈ 𝑋 ↦ ((𝐼‘𝐴) + 𝑏))
41	38, 40	eqtrdi 2788	. . . . 5 ⊢ ((𝐼‘𝐴) ∈ 𝑋 → (𝐹‘(𝐼‘𝐴)) = (𝑏 ∈ 𝑋 ↦ ((𝐼‘𝐴) + 𝑏)))
42	7, 41	syl 17	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐹‘(𝐼‘𝐴)) = (𝑏 ∈ 𝑋 ↦ ((𝐼‘𝐴) + 𝑏)))
43	37, 42	eqeq12d 2748	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (◡(𝐹‘𝐴) = (𝐹‘(𝐼‘𝐴)) ↔ ◡(𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎)) = (𝑏 ∈ 𝑋 ↦ ((𝐼‘𝐴) + 𝑏))))
44	36, 43	anbi12d 631	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (((𝐹‘𝐴):𝑋–1-1-onto→𝑋 ∧ ◡(𝐹‘𝐴) = (𝐹‘(𝐼‘𝐴))) ↔ ((𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎)):𝑋–1-1-onto→𝑋 ∧ ◡(𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎)) = (𝑏 ∈ 𝑋 ↦ ((𝐼‘𝐴) + 𝑏)))))
45	32, 44	mpbird 256	1 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝐹‘𝐴):𝑋–1-1-onto→𝑋 ∧ ◡(𝐹‘𝐴) = (𝐹‘(𝐼‘𝐴))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ↦ cmpt 5230  ^◡ccnv 5674  –1-1-onto→wf1o 6539  ‘cfv 6540  (class class class)co 7405  Basecbs 17140  +_gcplusg 17193  0_gc0g 17381  Grpcgrp 18815  inv_gcminusg 18816

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-0g 17383  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-grp 18818  df-minusg 18819

This theorem is referenced by:  grplactf1o  18923  eqglact  19053  tgplacthmeo  23598  tgpconncompeqg  23607