Theorem gapm 17739

Description: The action of a particular group element is a permutation of the base set. (Contributed by Jeff Hankins, 11-Aug-2009.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)

Hypotheses

Ref	Expression
gapm.1	|- X = ( Base ` G )
gapm.2	|- F = ( x e. Y |-> ( A .(+) x ) )

Assertion

Ref	Expression
gapm	|- ( ( .(+) e. ( G GrpAct Y ) /\ A e. X ) -> F : Y -1-1-onto-> Y )

Distinct variable groups:   x, A   x, G   x, .(+)   x, X   x, Y

Allowed substitution hint:   F( x)

Proof of Theorem gapm

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		gapm.2	. 2 |- F = ( x e. Y |-> ( A .(+) x ) )
2		gapm.1	. . . . 5 |- X = ( Base ` G )
3	2	gaf 17728	. . . 4 |- ( .(+) e. ( G GrpAct Y ) -> .(+) : ( X X. Y ) --> Y )
4	3	ad2antrr 762	. . 3 |- ( ( ( .(+) e. ( G GrpAct Y ) /\ A e. X ) /\ x e. Y ) -> .(+) : ( X X. Y ) --> Y )
5		simplr 792	. . 3 |- ( ( ( .(+) e. ( G GrpAct Y ) /\ A e. X ) /\ x e. Y ) -> A e. X )
6		simpr 477	. . 3 |- ( ( ( .(+) e. ( G GrpAct Y ) /\ A e. X ) /\ x e. Y ) -> x e. Y )
7	4, 5, 6	fovrnd 6806	. 2 |- ( ( ( .(+) e. ( G GrpAct Y ) /\ A e. X ) /\ x e. Y ) -> ( A .(+) x ) e. Y )
8	3	ad2antrr 762	. . 3 |- ( ( ( .(+) e. ( G GrpAct Y ) /\ A e. X ) /\ y e. Y ) -> .(+) : ( X X. Y ) --> Y )
9		gagrp 17725	. . . . 5 |- ( .(+) e. ( G GrpAct Y ) -> G e. Grp )
10	9	ad2antrr 762	. . . 4 |- ( ( ( .(+) e. ( G GrpAct Y ) /\ A e. X ) /\ y e. Y ) -> G e. Grp )
11		simplr 792	. . . 4 |- ( ( ( .(+) e. ( G GrpAct Y ) /\ A e. X ) /\ y e. Y ) -> A e. X )
12		eqid 2622	. . . . 5 |- ( invg ` G ) = ( invg ` G )
13	2, 12	grpinvcl 17467	. . . 4 |- ( ( G e. Grp /\ A e. X ) -> ( ( invg ` G ) ` A ) e. X )
14	10, 11, 13	syl2anc 693	. . 3 |- ( ( ( .(+) e. ( G GrpAct Y ) /\ A e. X ) /\ y e. Y ) -> ( ( invg ` G ) ` A ) e. X )
15		simpr 477	. . 3 |- ( ( ( .(+) e. ( G GrpAct Y ) /\ A e. X ) /\ y e. Y ) -> y e. Y )
16	8, 14, 15	fovrnd 6806	. 2 |- ( ( ( .(+) e. ( G GrpAct Y ) /\ A e. X ) /\ y e. Y ) -> ( ( ( invg ` G ) ` A ) .(+) y ) e. Y )
17		simpll 790	. . . . 5 |- ( ( ( .(+) e. ( G GrpAct Y ) /\ A e. X ) /\ ( x e. Y /\ y e. Y ) ) -> .(+) e. ( G GrpAct Y ) )
18		simplr 792	. . . . 5 |- ( ( ( .(+) e. ( G GrpAct Y ) /\ A e. X ) /\ ( x e. Y /\ y e. Y ) ) -> A e. X )
19		simprl 794	. . . . 5 |- ( ( ( .(+) e. ( G GrpAct Y ) /\ A e. X ) /\ ( x e. Y /\ y e. Y ) ) -> x e. Y )
20		simprr 796	. . . . 5 |- ( ( ( .(+) e. ( G GrpAct Y ) /\ A e. X ) /\ ( x e. Y /\ y e. Y ) ) -> y e. Y )
21	2, 12	gacan 17738	. . . . 5 |- ( ( .(+) e. ( G GrpAct Y ) /\ ( A e. X /\ x e. Y /\ y e. Y ) ) -> ( ( A .(+) x ) = y <-> ( ( ( invg ` G ) ` A ) .(+) y ) = x ) )
22	17, 18, 19, 20, 21	syl13anc 1328	. . . 4 |- ( ( ( .(+) e. ( G GrpAct Y ) /\ A e. X ) /\ ( x e. Y /\ y e. Y ) ) -> ( ( A .(+) x ) = y <-> ( ( ( invg ` G ) ` A ) .(+) y ) = x ) )
23	22	bicomd 213	. . 3 |- ( ( ( .(+) e. ( G GrpAct Y ) /\ A e. X ) /\ ( x e. Y /\ y e. Y ) ) -> ( ( ( ( invg ` G ) ` A ) .(+) y ) = x <-> ( A .(+) x ) = y ) )
24		eqcom 2629	. . 3 |- ( x = ( ( ( invg ` G ) ` A ) .(+) y ) <-> ( ( ( invg ` G ) ` A ) .(+) y ) = x )
25		eqcom 2629	. . 3 |- ( y = ( A .(+) x ) <-> ( A .(+) x ) = y )
26	23, 24, 25	3bitr4g 303	. 2 |- ( ( ( .(+) e. ( G GrpAct Y ) /\ A e. X ) /\ ( x e. Y /\ y e. Y ) ) -> ( x = ( ( ( invg ` G ) ` A ) .(+) y ) <-> y = ( A .(+) x ) ) )
27	1, 7, 16, 26	f1o2d 6887	1 |- ( ( .(+) e. ( G GrpAct Y ) /\ A e. X ) -> F : Y -1-1-onto-> Y )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   |-> cmpt 4729   X. cxp 5112   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Grpcgrp 17422   invgcminusg 17423   GrpAct cga 17722

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  ax-un 6949

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-pw 4160  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-oprab 6654  df-mpt2 6655  df-map 7859  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-ga 17723

This theorem is referenced by:  galactghm  17823