Theorem grplactcnv 17518

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 2622	. . 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 17430	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋) → (𝐴 + 𝑎) ∈ 𝑋)
5	4	3expa 1265	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑎 ∈ 𝑋) → (𝐴 + 𝑎) ∈ 𝑋)
6		simpl 473	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → 𝐺 ∈ Grp)
7		grplactcnv.4	. . . . . 6 ⊢ 𝐼 = (invg‘𝐺)
8	2, 7	grpinvcl 17467	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐼‘𝐴) ∈ 𝑋)
9	6, 8	jca 554	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐺 ∈ Grp ∧ (𝐼‘𝐴) ∈ 𝑋))
10	2, 3	grpcl 17430	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝐼‘𝐴) ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → ((𝐼‘𝐴) + 𝑏) ∈ 𝑋)
11	10	3expa 1265	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ (𝐼‘𝐴) ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → ((𝐼‘𝐴) + 𝑏) ∈ 𝑋)
12	9, 11	sylan 488	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → ((𝐼‘𝐴) + 𝑏) ∈ 𝑋)
13		eqcom 2629	. . . . 5 ⊢ (𝑎 = ((𝐼‘𝐴) + 𝑏) ↔ ((𝐼‘𝐴) + 𝑏) = 𝑎)
14		eqid 2622	. . . . . . . . . 10 ⊢ (0g‘𝐺) = (0g‘𝐺)
15	2, 3, 14, 7	grplinv 17468	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝐼‘𝐴) + 𝐴) = (0g‘𝐺))
16	15	adantr 481	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝐼‘𝐴) + 𝐴) = (0g‘𝐺))
17	16	oveq1d 6665	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (((𝐼‘𝐴) + 𝐴) + 𝑎) = ((0g‘𝐺) + 𝑎))
18		simpll 790	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝐺 ∈ Grp)
19	8	adantr 481	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝐼‘𝐴) ∈ 𝑋)
20		simplr 792	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
21		simprl 794	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑎 ∈ 𝑋)
22	2, 3	grpass 17431	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ ((𝐼‘𝐴) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋)) → (((𝐼‘𝐴) + 𝐴) + 𝑎) = ((𝐼‘𝐴) + (𝐴 + 𝑎)))
23	18, 19, 20, 21, 22	syl13anc 1328	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (((𝐼‘𝐴) + 𝐴) + 𝑎) = ((𝐼‘𝐴) + (𝐴 + 𝑎)))
24	2, 3, 14	grplid 17452	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑎 ∈ 𝑋) → ((0g‘𝐺) + 𝑎) = 𝑎)
25	24	ad2ant2r 783	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((0g‘𝐺) + 𝑎) = 𝑎)
26	17, 23, 25	3eqtr3rd 2665	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑎 = ((𝐼‘𝐴) + (𝐴 + 𝑎)))
27	26	eqeq2d 2632	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (((𝐼‘𝐴) + 𝑏) = 𝑎 ↔ ((𝐼‘𝐴) + 𝑏) = ((𝐼‘𝐴) + (𝐴 + 𝑎))))
28	13, 27	syl5bb 272	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎 = ((𝐼‘𝐴) + 𝑏) ↔ ((𝐼‘𝐴) + 𝑏) = ((𝐼‘𝐴) + (𝐴 + 𝑎))))
29		simprr 796	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑏 ∈ 𝑋)
30	5	adantrr 753	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝐴 + 𝑎) ∈ 𝑋)
31	2, 3	grplcan 17477	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑏 ∈ 𝑋 ∧ (𝐴 + 𝑎) ∈ 𝑋 ∧ (𝐼‘𝐴) ∈ 𝑋)) → (((𝐼‘𝐴) + 𝑏) = ((𝐼‘𝐴) + (𝐴 + 𝑎)) ↔ 𝑏 = (𝐴 + 𝑎)))
32	18, 29, 30, 19, 31	syl13anc 1328	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (((𝐼‘𝐴) + 𝑏) = ((𝐼‘𝐴) + (𝐴 + 𝑎)) ↔ 𝑏 = (𝐴 + 𝑎)))
33	28, 32	bitrd 268	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎 = ((𝐼‘𝐴) + 𝑏) ↔ 𝑏 = (𝐴 + 𝑎)))
34	1, 5, 12, 33	f1ocnv2d 6886	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎)):𝑋–1-1-onto→𝑋 ∧ ◡(𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎)) = (𝑏 ∈ 𝑋 ↦ ((𝐼‘𝐴) + 𝑏))))
35		grplact.1	. . . . . 6 ⊢ 𝐹 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎)))
36	35, 2	grplactfval 17516	. . . . 5 ⊢ (𝐴 ∈ 𝑋 → (𝐹‘𝐴) = (𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎)))
37	36	adantl 482	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) = (𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎)))
38		f1oeq1 6127	. . . 4 ⊢ ((𝐹‘𝐴) = (𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎)) → ((𝐹‘𝐴):𝑋–1-1-onto→𝑋 ↔ (𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎)):𝑋–1-1-onto→𝑋))
39	37, 38	syl 17	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝐹‘𝐴):𝑋–1-1-onto→𝑋 ↔ (𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎)):𝑋–1-1-onto→𝑋))
40	37	cnveqd 5298	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ◡(𝐹‘𝐴) = ◡(𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎)))
41	35, 2	grplactfval 17516	. . . . . 6 ⊢ ((𝐼‘𝐴) ∈ 𝑋 → (𝐹‘(𝐼‘𝐴)) = (𝑎 ∈ 𝑋 ↦ ((𝐼‘𝐴) + 𝑎)))
42		oveq2 6658	. . . . . . 7 ⊢ (𝑎 = 𝑏 → ((𝐼‘𝐴) + 𝑎) = ((𝐼‘𝐴) + 𝑏))
43	42	cbvmptv 4750	. . . . . 6 ⊢ (𝑎 ∈ 𝑋 ↦ ((𝐼‘𝐴) + 𝑎)) = (𝑏 ∈ 𝑋 ↦ ((𝐼‘𝐴) + 𝑏))
44	41, 43	syl6eq 2672	. . . . 5 ⊢ ((𝐼‘𝐴) ∈ 𝑋 → (𝐹‘(𝐼‘𝐴)) = (𝑏 ∈ 𝑋 ↦ ((𝐼‘𝐴) + 𝑏)))
45	8, 44	syl 17	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐹‘(𝐼‘𝐴)) = (𝑏 ∈ 𝑋 ↦ ((𝐼‘𝐴) + 𝑏)))
46	40, 45	eqeq12d 2637	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (◡(𝐹‘𝐴) = (𝐹‘(𝐼‘𝐴)) ↔ ◡(𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎)) = (𝑏 ∈ 𝑋 ↦ ((𝐼‘𝐴) + 𝑏))))
47	39, 46	anbi12d 747	. 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)))))
48	34, 47	mpbird 247	1 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝐹‘𝐴):𝑋–1-1-onto→𝑋 ∧ ◡(𝐹‘𝐴) = (𝐹‘(𝐼‘𝐴))))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ↦ cmpt 4729  ^◡ccnv 5113  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  +_gcplusg 15941  0_gc0g 16100  Grpcgrp 17422  inv_gcminusg 17423

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426

This theorem is referenced by:  grplactf1o  17519  eqglact  17645  tgplacthmeo  21907  tgpconncompeqg  21915