Theorem grplactcnv 12977

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 2177	. . 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 12890	. . . 4 |- ( ( G e. Grp /\ A e. X /\ a e. X ) -> ( A .+ a ) e. X )
5	4	3expa 1203	. . 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 12926	. . . 4 |- ( ( G e. Grp /\ A e. X ) -> ( I ` A ) e. X )
8	2, 3	grpcl 12890	. . . . 5 |- ( ( G e. Grp /\ ( I ` A ) e. X /\ b e. X ) -> ( ( I ` A ) .+ b ) e. X )
9	8	3expa 1203	. . . 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 2179	. . . . 5 |- ( a = ( ( I ` A ) .+ b ) <-> ( ( I ` A ) .+ b ) = a )
12		eqid 2177	. . . . . . . . . 10 |- ( 0g ` G ) = ( 0g ` G )
13	2, 3, 12, 6	grplinv 12927	. . . . . . . . 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 5892	. . . . . . 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 12891	. . . . . . . 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 1240	. . . . . . 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 12911	. . . . . . . 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 2219	. . . . . 6 |- ( ( ( G e. Grp /\ A e. X ) /\ ( a e. X /\ b e. X ) ) -> a = ( ( I ` A ) .+ ( A .+ a ) ) )
25	24	eqeq2d 2189	. . . . 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 12937	. . . . 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 1240	. . . 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 6077	. 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 12976	. . . . 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 5458	. . 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 4805	. . . 4 |- ( ( G e. Grp /\ A e. X ) -> `' ( F ` A ) = `' ( a e. X |-> ( A .+ a ) ) )
38	33, 2	grplactfval 12976	. . . . . 6 |- ( ( I ` A ) e. X -> ( F ` ( I ` A ) ) = ( a e. X |-> ( ( I ` A ) .+ a ) ) )
39		oveq2 5885	. . . . . . 7 |- ( a = b -> ( ( I ` A ) .+ a ) = ( ( I ` A ) .+ b ) )
40	39	cbvmptv 4101	. . . . . 6 |- ( a e. X |-> ( ( I ` A ) .+ a ) ) = ( b e. X |-> ( ( I ` A ) .+ b ) )
41	38, 40	eqtrdi 2226	. . . . 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 2192	. . 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 1353   e. wcel 2148   |-> cmpt 4066   `'ccnv 4627   -1-1-onto->wf1o 5217   ` cfv 5218  (class class class)co 5877   Basecbs 12464   +g cplusg 12538   0gc0g 12710   Grpcgrp 12882   invgcminusg 12883

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-inn 8922  df-2 8980  df-ndx 12467  df-slot 12468  df-base 12470  df-plusg 12551  df-0g 12712  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-grp 12885  df-minusg 12886

This theorem is referenced by:  grplactf1o  12978  eqglact  13089