Theorem eqgen 13877

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 2231	. 2 |- ( X /. .~ ) = ( X /. .~ )
2		breq2 4097	. 2 |- ( [ x ] .~ = A -> ( Y ~~ [ x ] .~ <-> Y ~~ A ) )
3		simpl 109	. . . 4 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> Y e. (SubGrp ` G ) )
4		subgrcl 13829	. . . . . . 7 |- ( Y e. (SubGrp ` G ) -> G e. Grp )
5		eqger.x	. . . . . . . 8 |- X = ( Base ` G )
6	5	subgss 13824	. . . . . . 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 2231	. . . . . . . 8 |- ( +g ` G ) = ( +g ` G )
10	5, 8, 9	eqglact 13875	. . . . . . 7 |- ( ( G e. Grp /\ Y C_ X /\ x e. X ) -> [ x ] .~ = ( ( z e. X |-> ( x ( +g ` G ) z ) ) " Y ) )
11	10	3expa 1230	. . . . . 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 13874	. . . . . . . 8 |- ( Y e. (SubGrp ` G ) -> .~ Er X )
14		basfn 13204	. . . . . . . . . 10 |- Base Fn _V
15	4	elexd 2817	. . . . . . . . . 10 |- ( Y e. (SubGrp ` G ) -> G e. _V )
16		funfvex 5665	. . . . . . . . . . 11 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
17	16	funfni 5439	. . . . . . . . . 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 2318	. . . . . . . 8 |- ( Y e. (SubGrp ` G ) -> X e. _V )
20		erex 6769	. . . . . . . 8 |- ( .~ Er X -> ( X e. _V -> .~ e. _V ) )
21	13, 19, 20	sylc 62	. . . . . . 7 |- ( Y e. (SubGrp ` G ) -> .~ e. _V )
22		ecexg 6749	. . . . . . 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 2309	. . . 4 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> ( ( z e. X |-> ( x ( +g ` G ) z ) ) " Y ) e. _V )
26		eqid 2231	. . . . . . . . 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 13749	. . . . . . . 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 13747	. . . . . . . . . 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 5587	. . . . . . . 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 5591	. . . . . 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 5607	. . . . 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 6971	. . . 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 1274	. . 3 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> Y ~~ ( ( z e. X |-> ( x ( +g ` G ) z ) ) " Y ) )
40	39, 12	breqtrrd 4121	. 2 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> Y ~~ [ x ] .~ )
41	1, 2, 40	ectocld 6813	1 |- ( ( Y e. (SubGrp ` G ) /\ A e. ( X /. .~ ) ) -> Y ~~ A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   _Vcvv 2803   C_ wss 3201   class class class wbr 4093   |-> cmpt 4155   |` cres 4733   "cima 4734   Fn wfn 5328   -1-1->wf1 5330   -1-1-onto->wf1o 5332   ` cfv 5333  (class class class)co 6028   Er wer 6742   [cec 6743   /.cqs 6744   ~~ cen 6950   Basecbs 13145   +g cplusg 13223   Grpcgrp 13646  SubGrpcsubg 13817   ~_QG cqg 13819

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 619  ax-in2 620  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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-pre-ltirr 8187  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-er 6745  df-ec 6747  df-qs 6751  df-en 6953  df-pnf 8258  df-mnf 8259  df-ltxr 8261  df-inn 9186  df-2 9244  df-ndx 13148  df-slot 13149  df-base 13151  df-sets 13152  df-iress 13153  df-plusg 13236  df-0g 13404  df-mgm 13502  df-sgrp 13548  df-mnd 13563  df-grp 13649  df-minusg 13650  df-subg 13820  df-eqg 13822

This theorem is referenced by: (None)