Theorem eqgen 13563

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 2205	. 2 |- ( X /. .~ ) = ( X /. .~ )
2		breq2 4048	. 2 |- ( [ x ] .~ = A -> ( Y ~~ [ x ] .~ <-> Y ~~ A ) )
3		simpl 109	. . . 4 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> Y e. (SubGrp ` G ) )
4		subgrcl 13515	. . . . . . 7 |- ( Y e. (SubGrp ` G ) -> G e. Grp )
5		eqger.x	. . . . . . . 8 |- X = ( Base ` G )
6	5	subgss 13510	. . . . . . 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 2205	. . . . . . . 8 |- ( +g ` G ) = ( +g ` G )
10	5, 8, 9	eqglact 13561	. . . . . . 7 |- ( ( G e. Grp /\ Y C_ X /\ x e. X ) -> [ x ] .~ = ( ( z e. X |-> ( x ( +g ` G ) z ) ) " Y ) )
11	10	3expa 1206	. . . . . 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 13560	. . . . . . . 8 |- ( Y e. (SubGrp ` G ) -> .~ Er X )
14		basfn 12890	. . . . . . . . . 10 |- Base Fn _V
15	4	elexd 2785	. . . . . . . . . 10 |- ( Y e. (SubGrp ` G ) -> G e. _V )
16		funfvex 5593	. . . . . . . . . . 11 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
17	16	funfni 5376	. . . . . . . . . 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 2292	. . . . . . . 8 |- ( Y e. (SubGrp ` G ) -> X e. _V )
20		erex 6644	. . . . . . . 8 |- ( .~ Er X -> ( X e. _V -> .~ e. _V ) )
21	13, 19, 20	sylc 62	. . . . . . 7 |- ( Y e. (SubGrp ` G ) -> .~ e. _V )
22		ecexg 6624	. . . . . . 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 2283	. . . 4 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> ( ( z e. X |-> ( x ( +g ` G ) z ) ) " Y ) e. _V )
26		eqid 2205	. . . . . . . . 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 13435	. . . . . . . 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 13433	. . . . . . . . . 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 5517	. . . . . . . 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 5521	. . . . . 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 5537	. . . . 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 6846	. . . 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 1250	. . 3 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> Y ~~ ( ( z e. X |-> ( x ( +g ` G ) z ) ) " Y ) )
40	39, 12	breqtrrd 4072	. 2 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> Y ~~ [ x ] .~ )
41	1, 2, 40	ectocld 6688	1 |- ( ( Y e. (SubGrp ` G ) /\ A e. ( X /. .~ ) ) -> Y ~~ A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   _Vcvv 2772   C_ wss 3166   class class class wbr 4044   |-> cmpt 4105   |` cres 4677   "cima 4678   Fn wfn 5266   -1-1->wf1 5268   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5944   Er wer 6617   [cec 6618   /.cqs 6619   ~~ cen 6825   Basecbs 12832   +g cplusg 12909   Grpcgrp 13332  SubGrpcsubg 13503   ~_QG cqg 13505

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-pre-ltirr 8037  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-er 6620  df-ec 6622  df-qs 6626  df-en 6828  df-pnf 8109  df-mnf 8110  df-ltxr 8112  df-inn 9037  df-2 9095  df-ndx 12835  df-slot 12836  df-base 12838  df-sets 12839  df-iress 12840  df-plusg 12922  df-0g 13090  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-grp 13335  df-minusg 13336  df-subg 13506  df-eqg 13508

This theorem is referenced by: (None)