Theorem grplactcnv 13635

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 2229	. . 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 13541	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋) → (𝐴 + 𝑎) ∈ 𝑋)
5	4	3expa 1227	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑎 ∈ 𝑋) → (𝐴 + 𝑎) ∈ 𝑋)
6		grplactcnv.4	. . . . 5 ⊢ 𝐼 = (invg‘𝐺)
7	2, 6	grpinvcl 13581	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐼‘𝐴) ∈ 𝑋)
8	2, 3	grpcl 13541	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝐼‘𝐴) ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → ((𝐼‘𝐴) + 𝑏) ∈ 𝑋)
9	8	3expa 1227	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ (𝐼‘𝐴) ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → ((𝐼‘𝐴) + 𝑏) ∈ 𝑋)
10	7, 9	syldanl 449	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → ((𝐼‘𝐴) + 𝑏) ∈ 𝑋)
11		eqcom 2231	. . . . 5 ⊢ (𝑎 = ((𝐼‘𝐴) + 𝑏) ↔ ((𝐼‘𝐴) + 𝑏) = 𝑎)
12		eqid 2229	. . . . . . . . . 10 ⊢ (0g‘𝐺) = (0g‘𝐺)
13	2, 3, 12, 6	grplinv 13583	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝐼‘𝐴) + 𝐴) = (0g‘𝐺))
14	13	adantr 276	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝐼‘𝐴) + 𝐴) = (0g‘𝐺))
15	14	oveq1d 6016	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (((𝐼‘𝐴) + 𝐴) + 𝑎) = ((0g‘𝐺) + 𝑎))
16		simpll 527	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝐺 ∈ Grp)
17	7	adantr 276	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝐼‘𝐴) ∈ 𝑋)
18		simplr 528	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
19		simprl 529	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑎 ∈ 𝑋)
20	2, 3	grpass 13542	. . . . . . . 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 1273	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (((𝐼‘𝐴) + 𝐴) + 𝑎) = ((𝐼‘𝐴) + (𝐴 + 𝑎)))
22	2, 3, 12	grplid 13564	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑎 ∈ 𝑋) → ((0g‘𝐺) + 𝑎) = 𝑎)
23	22	ad2ant2r 509	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((0g‘𝐺) + 𝑎) = 𝑎)
24	15, 21, 23	3eqtr3rd 2271	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑎 = ((𝐼‘𝐴) + (𝐴 + 𝑎)))
25	24	eqeq2d 2241	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (((𝐼‘𝐴) + 𝑏) = 𝑎 ↔ ((𝐼‘𝐴) + 𝑏) = ((𝐼‘𝐴) + (𝐴 + 𝑎))))
26	11, 25	bitrid 192	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎 = ((𝐼‘𝐴) + 𝑏) ↔ ((𝐼‘𝐴) + 𝑏) = ((𝐼‘𝐴) + (𝐴 + 𝑎))))
27		simprr 531	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑏 ∈ 𝑋)
28	5	adantrr 479	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝐴 + 𝑎) ∈ 𝑋)
29	2, 3	grplcan 13595	. . . . 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 1273	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (((𝐼‘𝐴) + 𝑏) = ((𝐼‘𝐴) + (𝐴 + 𝑎)) ↔ 𝑏 = (𝐴 + 𝑎)))
31	26, 30	bitrd 188	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎 = ((𝐼‘𝐴) + 𝑏) ↔ 𝑏 = (𝐴 + 𝑎)))
32	1, 5, 10, 31	f1ocnv2d 6210	. 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 13634	. . . . 5 ⊢ (𝐴 ∈ 𝑋 → (𝐹‘𝐴) = (𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎)))
35	34	adantl 277	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) = (𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎)))
36	35	f1oeq1d 5567	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝐹‘𝐴):𝑋–1-1-onto→𝑋 ↔ (𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎)):𝑋–1-1-onto→𝑋))
37	35	cnveqd 4898	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ◡(𝐹‘𝐴) = ◡(𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎)))
38	33, 2	grplactfval 13634	. . . . . 6 ⊢ ((𝐼‘𝐴) ∈ 𝑋 → (𝐹‘(𝐼‘𝐴)) = (𝑎 ∈ 𝑋 ↦ ((𝐼‘𝐴) + 𝑎)))
39		oveq2 6009	. . . . . . 7 ⊢ (𝑎 = 𝑏 → ((𝐼‘𝐴) + 𝑎) = ((𝐼‘𝐴) + 𝑏))
40	39	cbvmptv 4180	. . . . . 6 ⊢ (𝑎 ∈ 𝑋 ↦ ((𝐼‘𝐴) + 𝑎)) = (𝑏 ∈ 𝑋 ↦ ((𝐼‘𝐴) + 𝑏))
41	38, 40	eqtrdi 2278	. . . . 5 ⊢ ((𝐼‘𝐴) ∈ 𝑋 → (𝐹‘(𝐼‘𝐴)) = (𝑏 ∈ 𝑋 ↦ ((𝐼‘𝐴) + 𝑏)))
42	7, 41	syl 14	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐹‘(𝐼‘𝐴)) = (𝑏 ∈ 𝑋 ↦ ((𝐼‘𝐴) + 𝑏)))
43	37, 42	eqeq12d 2244	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (◡(𝐹‘𝐴) = (𝐹‘(𝐼‘𝐴)) ↔ ◡(𝑎 ∈ 𝑋 ↦ (𝐴 + 𝑎)) = (𝑏 ∈ 𝑋 ↦ ((𝐼‘𝐴) + 𝑏))))
44	36, 43	anbi12d 473	. 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 167	1 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝐹‘𝐴):𝑋–1-1-onto→𝑋 ∧ ◡(𝐹‘𝐴) = (𝐹‘(𝐼‘𝐴))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200   ↦ cmpt 4145  ^◡ccnv 4718  –1-1-onto→wf1o 5317  ‘cfv 5318  (class class class)co 6001  Basecbs 13032  +_gcplusg 13110  0_gc0g 13289  Grpcgrp 13533  inv_gcminusg 13534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-cnex 8090  ax-resscn 8091  ax-1re 8093  ax-addrcl 8096

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-inn 9111  df-2 9169  df-ndx 13035  df-slot 13036  df-base 13038  df-plusg 13123  df-0g 13291  df-mgm 13389  df-sgrp 13435  df-mnd 13450  df-grp 13536  df-minusg 13537

This theorem is referenced by:  grplactf1o  13636  eqglact  13762