Theorem eqgen 13813

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 4092	. 2 |- ( [ x ] .~ = A -> ( Y ~~ [ x ] .~ <-> Y ~~ A ) )
3		simpl 109	. . . 4 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> Y e. (SubGrp ` G ) )
4		subgrcl 13765	. . . . . . 7 |- ( Y e. (SubGrp ` G ) -> G e. Grp )
5		eqger.x	. . . . . . . 8 |- X = ( Base ` G )
6	5	subgss 13760	. . . . . . 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 13811	. . . . . . 7 |- ( ( G e. Grp /\ Y C_ X /\ x e. X ) -> [ x ] .~ = ( ( z e. X |-> ( x ( +g ` G ) z ) ) " Y ) )
11	10	3expa 1229	. . . . . 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 13810	. . . . . . . 8 |- ( Y e. (SubGrp ` G ) -> .~ Er X )
14		basfn 13140	. . . . . . . . . 10 |- Base Fn _V
15	4	elexd 2816	. . . . . . . . . 10 |- ( Y e. (SubGrp ` G ) -> G e. _V )
16		funfvex 5656	. . . . . . . . . . 11 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
17	16	funfni 5432	. . . . . . . . . 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 6725	. . . . . . . 8 |- ( .~ Er X -> ( X e. _V -> .~ e. _V ) )
21	13, 19, 20	sylc 62	. . . . . . 7 |- ( Y e. (SubGrp ` G ) -> .~ e. _V )
22		ecexg 6705	. . . . . . 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 13685	. . . . . . . 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 13683	. . . . . . . . . 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 5578	. . . . . . . 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 5582	. . . . . 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 5598	. . . . 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 6927	. . . 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 1273	. . 3 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> Y ~~ ( ( z e. X |-> ( x ( +g ` G ) z ) ) " Y ) )
40	39, 12	breqtrrd 4116	. 2 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> Y ~~ [ x ] .~ )
41	1, 2, 40	ectocld 6769	1 |- ( ( Y e. (SubGrp ` G ) /\ A e. ( X /. .~ ) ) -> Y ~~ A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   _Vcvv 2802   C_ wss 3200   class class class wbr 4088   |-> cmpt 4150   |` cres 4727   "cima 4728   Fn wfn 5321   -1-1->wf1 5323   -1-1-onto->wf1o 5325   ` cfv 5326  (class class class)co 6017   Er wer 6698   [cec 6699   /.cqs 6700   ~~ cen 6906   Basecbs 13081   +g cplusg 13159   Grpcgrp 13582  SubGrpcsubg 13753   ~_QG cqg 13755

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-er 6701  df-ec 6703  df-qs 6707  df-en 6909  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-iress 13089  df-plusg 13172  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-minusg 13586  df-subg 13756  df-eqg 13758

This theorem is referenced by: (None)