Theorem eqgen 13091

Description: Each coset is equipotent to the subgroup itself (which is also the coset containing the identity). (Contributed by Mario Carneiro, 20-Sep-2015.)

Hypotheses

Ref	Expression
eqger.x	|- X = ( Base ` G )
eqger.r	|- .~ = ( G ~QG Y )

Assertion

Ref	Expression
eqgen	|- ( ( Y e. (SubGrp ` G ) /\ A e. ( X /. .~ ) ) -> Y ~~ A )

Proof of Theorem eqgen

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

Step	Hyp	Ref	Expression
1		eqid 2177	. 2 |- ( X /. .~ ) = ( X /. .~ )
2		breq2 4009	. 2 |- ( [ x ] .~ = A -> ( Y ~~ [ x ] .~ <-> Y ~~ A ) )
3		simpl 109	. . . 4 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> Y e. (SubGrp ` G ) )
4		subgrcl 13044	. . . . . . 7 |- ( Y e. (SubGrp ` G ) -> G e. Grp )
5		eqger.x	. . . . . . . 8 |- X = ( Base ` G )
6	5	subgss 13039	. . . . . . 7 |- ( Y e. (SubGrp ` G ) -> Y C_ X )
7	4, 6	jca 306	. . . . . 6 |- ( Y e. (SubGrp ` G ) -> ( G e. Grp /\ Y C_ X ) )
8		eqger.r	. . . . . . . 8 |- .~ = ( G ~QG Y )
9		eqid 2177	. . . . . . . 8 |- ( +g ` G ) = ( +g ` G )
10	5, 8, 9	eqglact 13089	. . . . . . 7 |- ( ( G e. Grp /\ Y C_ X /\ x e. X ) -> [ x ] .~ = ( ( z e. X |-> ( x ( +g ` G ) z ) ) " Y ) )
11	10	3expa 1203	. . . . . 6 |- ( ( ( G e. Grp /\ Y C_ X ) /\ x e. X ) -> [ x ] .~ = ( ( z e. X |-> ( x ( +g ` G ) z ) ) " Y ) )
12	7, 11	sylan 283	. . . . 5 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> [ x ] .~ = ( ( z e. X |-> ( x ( +g ` G ) z ) ) " Y ) )
13	5, 8	eqger 13088	. . . . . . . 8 |- ( Y e. (SubGrp ` G ) -> .~ Er X )
14		basfn 12522	. . . . . . . . . 10 |- Base Fn _V
15	4	elexd 2752	. . . . . . . . . 10 |- ( Y e. (SubGrp ` G ) -> G e. _V )
16		funfvex 5534	. . . . . . . . . . 11 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
17	16	funfni 5318	. . . . . . . . . 10 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
18	14, 15, 17	sylancr 414	. . . . . . . . 9 |- ( Y e. (SubGrp ` G ) -> ( Base ` G ) e. _V )
19	5, 18	eqeltrid 2264	. . . . . . . 8 |- ( Y e. (SubGrp ` G ) -> X e. _V )
20		erex 6561	. . . . . . . 8 |- ( .~ Er X -> ( X e. _V -> .~ e. _V ) )
21	13, 19, 20	sylc 62	. . . . . . 7 |- ( Y e. (SubGrp ` G ) -> .~ e. _V )
22		ecexg 6541	. . . . . . 7 |- ( .~ e. _V -> [ x ] .~ e. _V )
23	21, 22	syl 14	. . . . . 6 |- ( Y e. (SubGrp ` G ) -> [ x ] .~ e. _V )
24	23	adantr 276	. . . . 5 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> [ x ] .~ e. _V )
25	12, 24	eqeltrrd 2255	. . . 4 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> ( ( z e. X |-> ( x ( +g ` G ) z ) ) " Y ) e. _V )
26		eqid 2177	. . . . . . . . 9 |- ( y e. X |-> ( z e. X |-> ( y ( +g ` G ) z ) ) ) = ( y e. X |-> ( z e. X |-> ( y ( +g ` G ) z ) ) )
27	26, 5, 9	grplactf1o 12978	. . . . . . . 8 |- ( ( G e. Grp /\ x e. X ) -> ( ( y e. X |-> ( z e. X |-> ( y ( +g ` G ) z ) ) ) ` x ) : X -1-1-onto-> X )
28	26, 5	grplactfval 12976	. . . . . . . . . 10 |- ( x e. X -> ( ( y e. X |-> ( z e. X |-> ( y ( +g ` G ) z ) ) ) ` x ) = ( z e. X |-> ( x ( +g ` G ) z ) ) )
29	28	adantl 277	. . . . . . . . 9 |- ( ( G e. Grp /\ x e. X ) -> ( ( y e. X |-> ( z e. X |-> ( y ( +g ` G ) z ) ) ) ` x ) = ( z e. X |-> ( x ( +g ` G ) z ) ) )
30	29	f1oeq1d 5458	. . . . . . . 8 |- ( ( G e. Grp /\ x e. X ) -> ( ( ( y e. X |-> ( z e. X |-> ( y ( +g ` G ) z ) ) ) ` x ) : X -1-1-onto-> X <-> ( z e. X |-> ( x ( +g ` G ) z ) ) : X -1-1-onto-> X ) )
31	27, 30	mpbid 147	. . . . . . 7 |- ( ( G e. Grp /\ x e. X ) -> ( z e. X |-> ( x ( +g ` G ) z ) ) : X -1-1-onto-> X )
32	4, 31	sylan 283	. . . . . 6 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> ( z e. X |-> ( x ( +g ` G ) z ) ) : X -1-1-onto-> X )
33		f1of1 5462	. . . . . 6 |- ( ( z e. X |-> ( x ( +g ` G ) z ) ) : X -1-1-onto-> X -> ( z e. X |-> ( x ( +g ` G ) z ) ) : X -1-1-> X )
34	32, 33	syl 14	. . . . 5 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> ( z e. X |-> ( x ( +g ` G ) z ) ) : X -1-1-> X )
35	6	adantr 276	. . . . 5 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> Y C_ X )
36		f1ores 5478	. . . . 5 |- ( ( ( z e. X |-> ( x ( +g ` G ) z ) ) : X -1-1-> X /\ Y C_ X ) -> ( ( z e. X |-> ( x ( +g ` G ) z ) ) |` Y ) : Y -1-1-onto-> ( ( z e. X |-> ( x ( +g ` G ) z ) ) " Y ) )
37	34, 35, 36	syl2anc 411	. . . 4 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> ( ( z e. X |-> ( x ( +g ` G ) z ) ) |` Y ) : Y -1-1-onto-> ( ( z e. X |-> ( x ( +g ` G ) z ) ) " Y ) )
38		f1oen2g 6757	. . . 4 |- ( ( Y e. (SubGrp ` G ) /\ ( ( z e. X |-> ( x ( +g ` G ) z ) ) " Y ) e. _V /\ ( ( z e. X |-> ( x ( +g ` G ) z ) ) |` Y ) : Y -1-1-onto-> ( ( z e. X |-> ( x ( +g ` G ) z ) ) " Y ) ) -> Y ~~ ( ( z e. X |-> ( x ( +g ` G ) z ) ) " Y ) )
39	3, 25, 37, 38	syl3anc 1238	. . 3 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> Y ~~ ( ( z e. X |-> ( x ( +g ` G ) z ) ) " Y ) )
40	39, 12	breqtrrd 4033	. 2 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> Y ~~ [ x ] .~ )
41	1, 2, 40	ectocld 6603	1 |- ( ( Y e. (SubGrp ` G ) /\ A e. ( X /. .~ ) ) -> Y ~~ A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   _Vcvv 2739   C_ wss 3131   class class class wbr 4005   |-> cmpt 4066   |` cres 4630   "cima 4631   Fn wfn 5213   -1-1->wf1 5215   -1-1-onto->wf1o 5217   ` cfv 5218  (class class class)co 5877   Er wer 6534   [cec 6535   /.cqs 6536   ~~ cen 6740   Basecbs 12464   +g cplusg 12538   Grpcgrp 12882  SubGrpcsubg 13032   ~_QG cqg 13034

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-in1 614  ax-in2 615  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-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-pre-ltirr 7925  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  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-ne 2348  df-nel 2443  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-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  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-oprab 5881  df-mpo 5882  df-er 6537  df-ec 6539  df-qs 6543  df-en 6743  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-inn 8922  df-2 8980  df-ndx 12467  df-slot 12468  df-base 12470  df-sets 12471  df-iress 12472  df-plusg 12551  df-0g 12712  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-grp 12885  df-minusg 12886  df-subg 13035  df-eqg 13037

This theorem is referenced by: (None)