Theorem grplactcnv 13857

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 2234	. . 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 13763	. . . 4 |- ( ( G e. Grp /\ A e. X /\ a e. X ) -> ( A .+ a ) e. X )
5	4	3expa 1230	. . 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 13803	. . . 4 |- ( ( G e. Grp /\ A e. X ) -> ( I ` A ) e. X )
8	2, 3	grpcl 13763	. . . . 5 |- ( ( G e. Grp /\ ( I ` A ) e. X /\ b e. X ) -> ( ( I ` A ) .+ b ) e. X )
9	8	3expa 1230	. . . 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 2236	. . . . 5 |- ( a = ( ( I ` A ) .+ b ) <-> ( ( I ` A ) .+ b ) = a )
12		eqid 2234	. . . . . . . . . 10 |- ( 0g ` G ) = ( 0g ` G )
13	2, 3, 12, 6	grplinv 13805	. . . . . . . . 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 6073	. . . . . . 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 529	. . . . . . . 8 |- ( ( ( G e. Grp /\ A e. X ) /\ ( a e. X /\ b e. X ) ) -> A e. X )
19		simprl 531	. . . . . . . 8 |- ( ( ( G e. Grp /\ A e. X ) /\ ( a e. X /\ b e. X ) ) -> a e. X )
20	2, 3	grpass 13764	. . . . . . . 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 1276	. . . . . . 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 13786	. . . . . . . 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 2276	. . . . . 6 |- ( ( ( G e. Grp /\ A e. X ) /\ ( a e. X /\ b e. X ) ) -> a = ( ( I ` A ) .+ ( A .+ a ) ) )
25	24	eqeq2d 2246	. . . . 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 533	. . . . 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 13817	. . . . 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 1276	. . . 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 6267	. 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 13856	. . . . 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 5614	. . 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 4936	. . . 4 |- ( ( G e. Grp /\ A e. X ) -> `' ( F ` A ) = `' ( a e. X |-> ( A .+ a ) ) )
38	33, 2	grplactfval 13856	. . . . . 6 |- ( ( I ` A ) e. X -> ( F ` ( I ` A ) ) = ( a e. X |-> ( ( I ` A ) .+ a ) ) )
39		oveq2 6066	. . . . . . 7 |- ( a = b -> ( ( I ` A ) .+ a ) = ( ( I ` A ) .+ b ) )
40	39	cbvmptv 4211	. . . . . 6 |- ( a e. X |-> ( ( I ` A ) .+ a ) ) = ( b e. X |-> ( ( I ` A ) .+ b ) )
41	38, 40	eqtrdi 2283	. . . . 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 2249	. . 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 1398   e. wcel 2205   |-> cmpt 4176   `'ccnv 4753   -1-1-onto->wf1o 5356   ` cfv 5357  (class class class)co 6058   Basecbs 13296   +g cplusg 13374   0gc0g 13553   Grpcgrp 13755   invgcminusg 13756

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-inn 9255  df-2 9313  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-grp 13758  df-minusg 13759

This theorem is referenced by:  grplactf1o  13858  eqglact  13978