Theorem gaid 17732

Description: The trivial action of a group on any set. Each group element corresponds to the identity permutation. (Contributed by Jeff Hankins, 11-Aug-2009.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)

Hypothesis

Ref	Expression
gaid.1	|- X = ( Base ` G )

Assertion

Ref	Expression
gaid	|- ( ( G e. Grp /\ S e. V ) -> ( 2nd |` ( X X. S ) ) e. ( G GrpAct S ) )

Proof of Theorem gaid

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

Step	Hyp	Ref	Expression
1		elex 3212	. . 3 |- ( S e. V -> S e. _V )
2	1	anim2i 593	. 2 |- ( ( G e. Grp /\ S e. V ) -> ( G e. Grp /\ S e. _V ) )
3		gaid.1	. . . . . . . 8 |- X = ( Base ` G )
4		eqid 2622	. . . . . . . 8 |- ( 0g ` G ) = ( 0g ` G )
5	3, 4	grpidcl 17450	. . . . . . 7 |- ( G e. Grp -> ( 0g ` G ) e. X )
6	5	adantr 481	. . . . . 6 |- ( ( G e. Grp /\ S e. V ) -> ( 0g ` G ) e. X )
7		ovres 6800	. . . . . . 7 |- ( ( ( 0g ` G ) e. X /\ x e. S ) -> ( ( 0g ` G ) ( 2nd |` ( X X. S ) ) x ) = ( ( 0g ` G ) 2nd x ) )
8		df-ov 6653	. . . . . . . 8 |- ( ( 0g ` G ) 2nd x ) = ( 2nd ` <. ( 0g ` G ) , x >. )
9		fvex 6201	. . . . . . . . 9 |- ( 0g ` G ) e. _V
10		vex 3203	. . . . . . . . 9 |- x e. _V
11	9, 10	op2nd 7177	. . . . . . . 8 |- ( 2nd ` <. ( 0g ` G ) , x >. ) = x
12	8, 11	eqtri 2644	. . . . . . 7 |- ( ( 0g ` G ) 2nd x ) = x
13	7, 12	syl6eq 2672	. . . . . 6 |- ( ( ( 0g ` G ) e. X /\ x e. S ) -> ( ( 0g ` G ) ( 2nd |` ( X X. S ) ) x ) = x )
14	6, 13	sylan 488	. . . . 5 |- ( ( ( G e. Grp /\ S e. V ) /\ x e. S ) -> ( ( 0g ` G ) ( 2nd |` ( X X. S ) ) x ) = x )
15		simprl 794	. . . . . . . 8 |- ( ( ( ( G e. Grp /\ S e. V ) /\ x e. S ) /\ ( y e. X /\ z e. X ) ) -> y e. X )
16		simplr 792	. . . . . . . 8 |- ( ( ( ( G e. Grp /\ S e. V ) /\ x e. S ) /\ ( y e. X /\ z e. X ) ) -> x e. S )
17		ovres 6800	. . . . . . . . 9 |- ( ( y e. X /\ x e. S ) -> ( y ( 2nd |` ( X X. S ) ) x ) = ( y 2nd x ) )
18		df-ov 6653	. . . . . . . . . 10 |- ( y 2nd x ) = ( 2nd ` <. y , x >. )
19		vex 3203	. . . . . . . . . . 11 |- y e. _V
20	19, 10	op2nd 7177	. . . . . . . . . 10 |- ( 2nd ` <. y , x >. ) = x
21	18, 20	eqtri 2644	. . . . . . . . 9 |- ( y 2nd x ) = x
22	17, 21	syl6eq 2672	. . . . . . . 8 |- ( ( y e. X /\ x e. S ) -> ( y ( 2nd |` ( X X. S ) ) x ) = x )
23	15, 16, 22	syl2anc 693	. . . . . . 7 |- ( ( ( ( G e. Grp /\ S e. V ) /\ x e. S ) /\ ( y e. X /\ z e. X ) ) -> ( y ( 2nd |` ( X X. S ) ) x ) = x )
24		simprr 796	. . . . . . . . 9 |- ( ( ( ( G e. Grp /\ S e. V ) /\ x e. S ) /\ ( y e. X /\ z e. X ) ) -> z e. X )
25		ovres 6800	. . . . . . . . . 10 |- ( ( z e. X /\ x e. S ) -> ( z ( 2nd |` ( X X. S ) ) x ) = ( z 2nd x ) )
26		df-ov 6653	. . . . . . . . . . 11 |- ( z 2nd x ) = ( 2nd ` <. z , x >. )
27		vex 3203	. . . . . . . . . . . 12 |- z e. _V
28	27, 10	op2nd 7177	. . . . . . . . . . 11 |- ( 2nd ` <. z , x >. ) = x
29	26, 28	eqtri 2644	. . . . . . . . . 10 |- ( z 2nd x ) = x
30	25, 29	syl6eq 2672	. . . . . . . . 9 |- ( ( z e. X /\ x e. S ) -> ( z ( 2nd |` ( X X. S ) ) x ) = x )
31	24, 16, 30	syl2anc 693	. . . . . . . 8 |- ( ( ( ( G e. Grp /\ S e. V ) /\ x e. S ) /\ ( y e. X /\ z e. X ) ) -> ( z ( 2nd |` ( X X. S ) ) x ) = x )
32	31	oveq2d 6666	. . . . . . 7 |- ( ( ( ( G e. Grp /\ S e. V ) /\ x e. S ) /\ ( y e. X /\ z e. X ) ) -> ( y ( 2nd |` ( X X. S ) ) ( z ( 2nd |` ( X X. S ) ) x ) ) = ( y ( 2nd |` ( X X. S ) ) x ) )
33		simpll 790	. . . . . . . . 9 |- ( ( ( G e. Grp /\ S e. V ) /\ x e. S ) -> G e. Grp )
34		eqid 2622	. . . . . . . . . . 11 |- ( +g ` G ) = ( +g ` G )
35	3, 34	grpcl 17430	. . . . . . . . . 10 |- ( ( G e. Grp /\ y e. X /\ z e. X ) -> ( y ( +g ` G ) z ) e. X )
36	35	3expb 1266	. . . . . . . . 9 |- ( ( G e. Grp /\ ( y e. X /\ z e. X ) ) -> ( y ( +g ` G ) z ) e. X )
37	33, 36	sylan 488	. . . . . . . 8 |- ( ( ( ( G e. Grp /\ S e. V ) /\ x e. S ) /\ ( y e. X /\ z e. X ) ) -> ( y ( +g ` G ) z ) e. X )
38		ovres 6800	. . . . . . . . 9 |- ( ( ( y ( +g ` G ) z ) e. X /\ x e. S ) -> ( ( y ( +g ` G ) z ) ( 2nd |` ( X X. S ) ) x ) = ( ( y ( +g ` G ) z ) 2nd x ) )
39		df-ov 6653	. . . . . . . . . 10 |- ( ( y ( +g ` G ) z ) 2nd x ) = ( 2nd ` <. ( y ( +g ` G ) z ) , x >. )
40		ovex 6678	. . . . . . . . . . 11 |- ( y ( +g ` G ) z ) e. _V
41	40, 10	op2nd 7177	. . . . . . . . . 10 |- ( 2nd ` <. ( y ( +g ` G ) z ) , x >. ) = x
42	39, 41	eqtri 2644	. . . . . . . . 9 |- ( ( y ( +g ` G ) z ) 2nd x ) = x
43	38, 42	syl6eq 2672	. . . . . . . 8 |- ( ( ( y ( +g ` G ) z ) e. X /\ x e. S ) -> ( ( y ( +g ` G ) z ) ( 2nd |` ( X X. S ) ) x ) = x )
44	37, 16, 43	syl2anc 693	. . . . . . 7 |- ( ( ( ( G e. Grp /\ S e. V ) /\ x e. S ) /\ ( y e. X /\ z e. X ) ) -> ( ( y ( +g ` G ) z ) ( 2nd |` ( X X. S ) ) x ) = x )
45	23, 32, 44	3eqtr4rd 2667	. . . . . 6 |- ( ( ( ( G e. Grp /\ S e. V ) /\ x e. S ) /\ ( y e. X /\ z e. X ) ) -> ( ( y ( +g ` G ) z ) ( 2nd |` ( X X. S ) ) x ) = ( y ( 2nd |` ( X X. S ) ) ( z ( 2nd |` ( X X. S ) ) x ) ) )
46	45	ralrimivva 2971	. . . . 5 |- ( ( ( G e. Grp /\ S e. V ) /\ x e. S ) -> A. y e. X A. z e. X ( ( y ( +g ` G ) z ) ( 2nd |` ( X X. S ) ) x ) = ( y ( 2nd |` ( X X. S ) ) ( z ( 2nd |` ( X X. S ) ) x ) ) )
47	14, 46	jca 554	. . . 4 |- ( ( ( G e. Grp /\ S e. V ) /\ x e. S ) -> ( ( ( 0g ` G ) ( 2nd |` ( X X. S ) ) x ) = x /\ A. y e. X A. z e. X ( ( y ( +g ` G ) z ) ( 2nd |` ( X X. S ) ) x ) = ( y ( 2nd |` ( X X. S ) ) ( z ( 2nd |` ( X X. S ) ) x ) ) ) )
48	47	ralrimiva 2966	. . 3 |- ( ( G e. Grp /\ S e. V ) -> A. x e. S ( ( ( 0g ` G ) ( 2nd |` ( X X. S ) ) x ) = x /\ A. y e. X A. z e. X ( ( y ( +g ` G ) z ) ( 2nd |` ( X X. S ) ) x ) = ( y ( 2nd |` ( X X. S ) ) ( z ( 2nd |` ( X X. S ) ) x ) ) ) )
49		f2ndres 7191	. . 3 |- ( 2nd |` ( X X. S ) ) : ( X X. S ) --> S
50	48, 49	jctil 560	. 2 |- ( ( G e. Grp /\ S e. V ) -> ( ( 2nd |` ( X X. S ) ) : ( X X. S ) --> S /\ A. x e. S ( ( ( 0g ` G ) ( 2nd |` ( X X. S ) ) x ) = x /\ A. y e. X A. z e. X ( ( y ( +g ` G ) z ) ( 2nd |` ( X X. S ) ) x ) = ( y ( 2nd |` ( X X. S ) ) ( z ( 2nd |` ( X X. S ) ) x ) ) ) ) )
51	3, 34, 4	isga 17724	. 2 |- ( ( 2nd |` ( X X. S ) ) e. ( G GrpAct S ) <-> ( ( G e. Grp /\ S e. _V ) /\ ( ( 2nd |` ( X X. S ) ) : ( X X. S ) --> S /\ A. x e. S ( ( ( 0g ` G ) ( 2nd |` ( X X. S ) ) x ) = x /\ A. y e. X A. z e. X ( ( y ( +g ` G ) z ) ( 2nd |` ( X X. S ) ) x ) = ( y ( 2nd |` ( X X. S ) ) ( z ( 2nd |` ( X X. S ) ) x ) ) ) ) ) )
52	2, 50, 51	sylanbrc 698	1 |- ( ( G e. Grp /\ S e. V ) -> ( 2nd |` ( X X. S ) ) e. ( G GrpAct S ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   <.cop 4183   X. cxp 5112   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   2ndc2nd 7167   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-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-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-2nd 7169  df-map 7859  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-ga 17723

This theorem is referenced by: (None)