Theorem eqgen 13980

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 2234	. 2 |- ( X /. .~ ) = ( X /. .~ )
2		breq2 4118	. 2 |- ( [ x ] .~ = A -> ( Y ~~ [ x ] .~ <-> Y ~~ A ) )
3		simpl 109	. . . 4 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> Y e. (SubGrp ` G ) )
4		subgrcl 13932	. . . . . . 7 |- ( Y e. (SubGrp ` G ) -> G e. Grp )
5		eqger.x	. . . . . . . 8 |- X = ( Base ` G )
6	5	subgss 13927	. . . . . . 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 2234	. . . . . . . 8 |- ( +g ` G ) = ( +g ` G )
10	5, 8, 9	eqglact 13978	. . . . . . 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 13977	. . . . . . . 8 |- ( Y e. (SubGrp ` G ) -> .~ Er X )
14		basfn 13355	. . . . . . . . . 10 |- Base Fn _V
15	4	elexd 2829	. . . . . . . . . 10 |- ( Y e. (SubGrp ` G ) -> G e. _V )
16		funfvex 5692	. . . . . . . . . . 11 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
17	16	funfni 5463	. . . . . . . . . 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 2321	. . . . . . . 8 |- ( Y e. (SubGrp ` G ) -> X e. _V )
20		erex 6804	. . . . . . . 8 |- ( .~ Er X -> ( X e. _V -> .~ e. _V ) )
21	13, 19, 20	sylc 62	. . . . . . 7 |- ( Y e. (SubGrp ` G ) -> .~ e. _V )
22		ecexg 6784	. . . . . . 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 2312	. . . 4 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> ( ( z e. X |-> ( x ( +g ` G ) z ) ) " Y ) e. _V )
26		eqid 2234	. . . . . . . . 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 13858	. . . . . . . 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 13856	. . . . . . . . . 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 5614	. . . . . . . 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 5618	. . . . . 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 5634	. . . . 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 7007	. . . 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 4142	. 2 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> Y ~~ [ x ] .~ )
41	1, 2, 40	ectocld 6848	1 |- ( ( Y e. (SubGrp ` G ) /\ A e. ( X /. .~ ) ) -> Y ~~ A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   _Vcvv 2815   C_ wss 3214   class class class wbr 4114   |-> cmpt 4176   |` cres 4756   "cima 4757   Fn wfn 5352   -1-1->wf1 5354   -1-1-onto->wf1o 5356   ` cfv 5357  (class class class)co 6058   Er wer 6777   [cec 6778   /.cqs 6779   ~~ cen 6986   Basecbs 13296   +g cplusg 13374   Grpcgrp 13755  SubGrpcsubg 13920   ~_QG cqg 13922

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-er 6780  df-ec 6782  df-qs 6786  df-en 6989  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-iress 13304  df-plusg 13387  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-grp 13758  df-minusg 13759  df-subg 13923  df-eqg 13925

This theorem is referenced by: (None)