Theorem ga0 17731

Description: The action of a group on the empty set. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)

Assertion

Ref	Expression
ga0	|- ( G e. Grp -> (/) e. ( G GrpAct (/) ) )

Proof of Theorem ga0

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 4790	. . 3 |- (/) e. _V
2	1	jctr 565	. 2 |- ( G e. Grp -> ( G e. Grp /\ (/) e. _V ) )
3		f0 6086	. . . . 5 |- (/) : (/) --> (/)
4		xp0 5552	. . . . . 6 |- ( ( Base ` G ) X. (/) ) = (/)
5	4	feq2i 6037	. . . . 5 |- ( (/) : ( ( Base ` G ) X. (/) ) --> (/) <-> (/) : (/) --> (/) )
6	3, 5	mpbir 221	. . . 4 |- (/) : ( ( Base ` G ) X. (/) ) --> (/)
7		ral0 4076	. . . 4 |- A. x e. (/) ( ( ( 0g ` G ) (/) x ) = x /\ A. y e. ( Base ` G ) A. z e. ( Base ` G ) ( ( y ( +g ` G ) z ) (/) x ) = ( y (/) ( z (/) x ) ) )
8	6, 7	pm3.2i 471	. . 3 |- ( (/) : ( ( Base ` G ) X. (/) ) --> (/) /\ A. x e. (/) ( ( ( 0g ` G ) (/) x ) = x /\ A. y e. ( Base ` G ) A. z e. ( Base ` G ) ( ( y ( +g ` G ) z ) (/) x ) = ( y (/) ( z (/) x ) ) ) )
9	8	a1i 11	. 2 |- ( G e. Grp -> ( (/) : ( ( Base ` G ) X. (/) ) --> (/) /\ A. x e. (/) ( ( ( 0g ` G ) (/) x ) = x /\ A. y e. ( Base ` G ) A. z e. ( Base ` G ) ( ( y ( +g ` G ) z ) (/) x ) = ( y (/) ( z (/) x ) ) ) ) )
10		eqid 2622	. . 3 |- ( Base ` G ) = ( Base ` G )
11		eqid 2622	. . 3 |- ( +g ` G ) = ( +g ` G )
12		eqid 2622	. . 3 |- ( 0g ` G ) = ( 0g ` G )
13	10, 11, 12	isga 17724	. 2 |- ( (/) e. ( G GrpAct (/) ) <-> ( ( G e. Grp /\ (/) e. _V ) /\ ( (/) : ( ( Base ` G ) X. (/) ) --> (/) /\ A. x e. (/) ( ( ( 0g ` G ) (/) x ) = x /\ A. y e. ( Base ` G ) A. z e. ( Base ` G ) ( ( y ( +g ` G ) z ) (/) x ) = ( y (/) ( z (/) x ) ) ) ) ) )
14	2, 9, 13	sylanbrc 698	1 |- ( G e. Grp -> (/) e. ( G GrpAct (/) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   (/)c0 3915   X. cxp 5112   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 16100   Grpcgrp 17422   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-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-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-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-ga 17723

This theorem is referenced by: (None)