Theorem grplactcnv 13304

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 e. X |-> ( a e. X |-> ( g .+ a ) ) )
grplact.2	|- X = ( Base ` G )
grplact.3	|- .+ = ( +g ` G )
grplactcnv.4	|- I = ( invg ` G )

Assertion

Ref	Expression
grplactcnv	|- ( ( G e. Grp /\ A e. 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 2196	. . 3 |- ( a e. X |-> ( A .+ a ) ) = ( a e. X |-> ( A .+ a ) )
2		grplact.2	. . . . 5 |- X = ( Base ` G )
3		grplact.3	. . . . 5 |- .+ = ( +g ` G )
4	2, 3	grpcl 13210	. . . 4 |- ( ( G e. Grp /\ A e. X /\ a e. X ) -> ( A .+ a ) e. X )
5	4	3expa 1205	. . 3 |- ( ( ( G e. Grp /\ A e. X ) /\ a e. X ) -> ( A .+ a ) e. X )
6		grplactcnv.4	. . . . 5 |- I = ( invg ` G )
7	2, 6	grpinvcl 13250	. . . 4 |- ( ( G e. Grp /\ A e. X ) -> ( I ` A ) e. X )
8	2, 3	grpcl 13210	. . . . 5 |- ( ( G e. Grp /\ ( I ` A ) e. X /\ b e. X ) -> ( ( I ` A ) .+ b ) e. X )
9	8	3expa 1205	. . . 4 |- ( ( ( G e. Grp /\ ( I ` A ) e. X ) /\ b e. X ) -> ( ( I ` A ) .+ b ) e. X )
10	7, 9	syldanl 449	. . 3 |- ( ( ( G e. Grp /\ A e. X ) /\ b e. X ) -> ( ( I ` A ) .+ b ) e. X )
11		eqcom 2198	. . . . 5 |- ( a = ( ( I ` A ) .+ b ) <-> ( ( I ` A ) .+ b ) = a )
12		eqid 2196	. . . . . . . . . 10 |- ( 0g ` G ) = ( 0g ` G )
13	2, 3, 12, 6	grplinv 13252	. . . . . . . . 9 |- ( ( G e. Grp /\ A e. X ) -> ( ( I ` A ) .+ A ) = ( 0g ` G ) )
14	13	adantr 276	. . . . . . . 8 |- ( ( ( G e. Grp /\ A e. X ) /\ ( a e. X /\ b e. X ) ) -> ( ( I ` A ) .+ A ) = ( 0g ` G ) )
15	14	oveq1d 5940	. . . . . . 7 |- ( ( ( G e. Grp /\ A e. X ) /\ ( a e. X /\ b e. X ) ) -> ( ( ( I ` A ) .+ A ) .+ a ) = ( ( 0g ` G ) .+ a ) )
16		simpll 527	. . . . . . . 8 |- ( ( ( G e. Grp /\ A e. X ) /\ ( a e. X /\ b e. X ) ) -> G e. Grp )
17	7	adantr 276	. . . . . . . 8 |- ( ( ( G e. Grp /\ A e. X ) /\ ( a e. X /\ b e. X ) ) -> ( I ` A ) e. X )
18		simplr 528	. . . . . . . 8 |- ( ( ( G e. Grp /\ A e. X ) /\ ( a e. X /\ b e. X ) ) -> A e. X )
19		simprl 529	. . . . . . . 8 |- ( ( ( G e. Grp /\ A e. X ) /\ ( a e. X /\ b e. X ) ) -> a e. X )
20	2, 3	grpass 13211	. . . . . . . 8 |- ( ( G e. Grp /\ ( ( I ` A ) e. X /\ A e. X /\ a e. X ) ) -> ( ( ( I ` A ) .+ A ) .+ a ) = ( ( I ` A ) .+ ( A .+ a ) ) )
21	16, 17, 18, 19, 20	syl13anc 1251	. . . . . . 7 |- ( ( ( G e. Grp /\ A e. X ) /\ ( a e. X /\ b e. X ) ) -> ( ( ( I ` A ) .+ A ) .+ a ) = ( ( I ` A ) .+ ( A .+ a ) ) )
22	2, 3, 12	grplid 13233	. . . . . . . 8 |- ( ( G e. Grp /\ a e. X ) -> ( ( 0g ` G ) .+ a ) = a )
23	22	ad2ant2r 509	. . . . . . 7 |- ( ( ( G e. Grp /\ A e. X ) /\ ( a e. X /\ b e. X ) ) -> ( ( 0g ` G ) .+ a ) = a )
24	15, 21, 23	3eqtr3rd 2238	. . . . . 6 |- ( ( ( G e. Grp /\ A e. X ) /\ ( a e. X /\ b e. X ) ) -> a = ( ( I ` A ) .+ ( A .+ a ) ) )
25	24	eqeq2d 2208	. . . . 5 |- ( ( ( G e. Grp /\ A e. X ) /\ ( a e. X /\ b e. X ) ) -> ( ( ( I ` A ) .+ b ) = a <-> ( ( I ` A ) .+ b ) = ( ( I ` A ) .+ ( A .+ a ) ) ) )
26	11, 25	bitrid 192	. . . 4 |- ( ( ( G e. Grp /\ A e. X ) /\ ( a e. X /\ b e. X ) ) -> ( a = ( ( I ` A ) .+ b ) <-> ( ( I ` A ) .+ b ) = ( ( I ` A ) .+ ( A .+ a ) ) ) )
27		simprr 531	. . . . 5 |- ( ( ( G e. Grp /\ A e. X ) /\ ( a e. X /\ b e. X ) ) -> b e. X )
28	5	adantrr 479	. . . . 5 |- ( ( ( G e. Grp /\ A e. X ) /\ ( a e. X /\ b e. X ) ) -> ( A .+ a ) e. X )
29	2, 3	grplcan 13264	. . . . 5 |- ( ( G e. Grp /\ ( b e. X /\ ( A .+ a ) e. X /\ ( I ` A ) e. X ) ) -> ( ( ( I ` A ) .+ b ) = ( ( I ` A ) .+ ( A .+ a ) ) <-> b = ( A .+ a ) ) )
30	16, 27, 28, 17, 29	syl13anc 1251	. . . 4 |- ( ( ( G e. Grp /\ A e. X ) /\ ( a e. X /\ b e. X ) ) -> ( ( ( I ` A ) .+ b ) = ( ( I ` A ) .+ ( A .+ a ) ) <-> b = ( A .+ a ) ) )
31	26, 30	bitrd 188	. . 3 |- ( ( ( G e. Grp /\ A e. X ) /\ ( a e. X /\ b e. X ) ) -> ( a = ( ( I ` A ) .+ b ) <-> b = ( A .+ a ) ) )
32	1, 5, 10, 31	f1ocnv2d 6131	. 2 |- ( ( G e. Grp /\ A e. X ) -> ( ( a e. X |-> ( A .+ a ) ) : X -1-1-onto-> X /\ `' ( a e. X |-> ( A .+ a ) ) = ( b e. X |-> ( ( I ` A ) .+ b ) ) ) )
33		grplact.1	. . . . . 6 |- F = ( g e. X |-> ( a e. X |-> ( g .+ a ) ) )
34	33, 2	grplactfval 13303	. . . . 5 |- ( A e. X -> ( F ` A ) = ( a e. X |-> ( A .+ a ) ) )
35	34	adantl 277	. . . 4 |- ( ( G e. Grp /\ A e. X ) -> ( F ` A ) = ( a e. X |-> ( A .+ a ) ) )
36	35	f1oeq1d 5502	. . 3 |- ( ( G e. Grp /\ A e. X ) -> ( ( F ` A ) : X -1-1-onto-> X <-> ( a e. X |-> ( A .+ a ) ) : X -1-1-onto-> X ) )
37	35	cnveqd 4843	. . . 4 |- ( ( G e. Grp /\ A e. X ) -> `' ( F ` A ) = `' ( a e. X |-> ( A .+ a ) ) )
38	33, 2	grplactfval 13303	. . . . . 6 |- ( ( I ` A ) e. X -> ( F ` ( I ` A ) ) = ( a e. X |-> ( ( I ` A ) .+ a ) ) )
39		oveq2 5933	. . . . . . 7 |- ( a = b -> ( ( I ` A ) .+ a ) = ( ( I ` A ) .+ b ) )
40	39	cbvmptv 4130	. . . . . 6 |- ( a e. X |-> ( ( I ` A ) .+ a ) ) = ( b e. X |-> ( ( I ` A ) .+ b ) )
41	38, 40	eqtrdi 2245	. . . . 5 |- ( ( I ` A ) e. X -> ( F ` ( I ` A ) ) = ( b e. X |-> ( ( I ` A ) .+ b ) ) )
42	7, 41	syl 14	. . . 4 |- ( ( G e. Grp /\ A e. X ) -> ( F ` ( I ` A ) ) = ( b e. X |-> ( ( I ` A ) .+ b ) ) )
43	37, 42	eqeq12d 2211	. . 3 |- ( ( G e. Grp /\ A e. X ) -> ( `' ( F ` A ) = ( F ` ( I ` A ) ) <-> `' ( a e. X |-> ( A .+ a ) ) = ( b e. X |-> ( ( I ` A ) .+ b ) ) ) )
44	36, 43	anbi12d 473	. 2 |- ( ( G e. Grp /\ A e. X ) -> ( ( ( F ` A ) : X -1-1-onto-> X /\ `' ( F ` A ) = ( F ` ( I ` A ) ) ) <-> ( ( a e. X |-> ( A .+ a ) ) : X -1-1-onto-> X /\ `' ( a e. X |-> ( A .+ a ) ) = ( b e. X |-> ( ( I ` A ) .+ b ) ) ) ) )
45	32, 44	mpbird 167	1 |- ( ( G e. Grp /\ A e. X ) -> ( ( F ` A ) : X -1-1-onto-> X /\ `' ( F ` A ) = ( F ` ( I ` A ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   |-> cmpt 4095   `'ccnv 4663   -1-1-onto->wf1o 5258   ` cfv 5259  (class class class)co 5925   Basecbs 12703   +g cplusg 12780   0gc0g 12958   Grpcgrp 13202   invgcminusg 13203

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-inn 9008  df-2 9066  df-ndx 12706  df-slot 12707  df-base 12709  df-plusg 12793  df-0g 12960  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-grp 13205  df-minusg 13206

This theorem is referenced by:  grplactf1o  13305  eqglact  13431