Theorem eqgid 13595

Description: The left coset containing the identity is the original subgroup. (Contributed by Mario Carneiro, 20-Sep-2015.)

Hypotheses

Ref	Expression
eqger.x	|- X = ( Base ` G )
eqger.r	|- .~ = ( G ~QG Y )
eqgid.3	|- .0. = ( 0g ` G )

Assertion

Ref	Expression
eqgid	|- ( Y e. (SubGrp ` G ) -> [ .0. ] .~ = Y )

Proof of Theorem eqgid

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		subgrcl 13548	. . . . 5 |- ( Y e. (SubGrp ` G ) -> G e. Grp )
2		eqger.r	. . . . . 6 |- .~ = ( G ~QG Y )
3	2	releqgg 13589	. . . . 5 |- ( ( G e. Grp /\ Y e. (SubGrp ` G ) ) -> Rel .~ )
4	1, 3	mpancom 422	. . . 4 |- ( Y e. (SubGrp ` G ) -> Rel .~ )
5		relelec 6664	. . . 4 |- ( Rel .~ -> ( x e. [ .0. ] .~ <-> .0. .~ x ) )
6	4, 5	syl 14	. . 3 |- ( Y e. (SubGrp ` G ) -> ( x e. [ .0. ] .~ <-> .0. .~ x ) )
7	1	adantr 276	. . . . . . . . 9 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> G e. Grp )
8		eqgid.3	. . . . . . . . . 10 |- .0. = ( 0g ` G )
9		eqid 2205	. . . . . . . . . 10 |- ( invg ` G ) = ( invg ` G )
10	8, 9	grpinvid 13425	. . . . . . . . 9 |- ( G e. Grp -> ( ( invg ` G ) ` .0. ) = .0. )
11	7, 10	syl 14	. . . . . . . 8 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> ( ( invg ` G ) ` .0. ) = .0. )
12	11	oveq1d 5961	. . . . . . 7 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> ( ( ( invg ` G ) ` .0. ) ( +g ` G ) x ) = ( .0. ( +g ` G ) x ) )
13		eqger.x	. . . . . . . . 9 |- X = ( Base ` G )
14		eqid 2205	. . . . . . . . 9 |- ( +g ` G ) = ( +g ` G )
15	13, 14, 8	grplid 13396	. . . . . . . 8 |- ( ( G e. Grp /\ x e. X ) -> ( .0. ( +g ` G ) x ) = x )
16	1, 15	sylan 283	. . . . . . 7 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> ( .0. ( +g ` G ) x ) = x )
17	12, 16	eqtrd 2238	. . . . . 6 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> ( ( ( invg ` G ) ` .0. ) ( +g ` G ) x ) = x )
18	17	eleq1d 2274	. . . . 5 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> ( ( ( ( invg ` G ) ` .0. ) ( +g ` G ) x ) e. Y <-> x e. Y ) )
19	18	pm5.32da 452	. . . 4 |- ( Y e. (SubGrp ` G ) -> ( ( x e. X /\ ( ( ( invg ` G ) ` .0. ) ( +g ` G ) x ) e. Y ) <-> ( x e. X /\ x e. Y ) ) )
20	13	subgss 13543	. . . . 5 |- ( Y e. (SubGrp ` G ) -> Y C_ X )
21	13, 8	grpidcl 13394	. . . . . 6 |- ( G e. Grp -> .0. e. X )
22	1, 21	syl 14	. . . . 5 |- ( Y e. (SubGrp ` G ) -> .0. e. X )
23	13, 9, 14, 2	eqgval 13592	. . . . . . 7 |- ( ( G e. Grp /\ Y C_ X ) -> ( .0. .~ x <-> ( .0. e. X /\ x e. X /\ ( ( ( invg ` G ) ` .0. ) ( +g ` G ) x ) e. Y ) ) )
24		3anass 985	. . . . . . 7 |- ( ( .0. e. X /\ x e. X /\ ( ( ( invg ` G ) ` .0. ) ( +g ` G ) x ) e. Y ) <-> ( .0. e. X /\ ( x e. X /\ ( ( ( invg ` G ) ` .0. ) ( +g ` G ) x ) e. Y ) ) )
25	23, 24	bitrdi 196	. . . . . 6 |- ( ( G e. Grp /\ Y C_ X ) -> ( .0. .~ x <-> ( .0. e. X /\ ( x e. X /\ ( ( ( invg ` G ) ` .0. ) ( +g ` G ) x ) e. Y ) ) ) )
26	25	baibd 925	. . . . 5 |- ( ( ( G e. Grp /\ Y C_ X ) /\ .0. e. X ) -> ( .0. .~ x <-> ( x e. X /\ ( ( ( invg ` G ) ` .0. ) ( +g ` G ) x ) e. Y ) ) )
27	1, 20, 22, 26	syl21anc 1249	. . . 4 |- ( Y e. (SubGrp ` G ) -> ( .0. .~ x <-> ( x e. X /\ ( ( ( invg ` G ) ` .0. ) ( +g ` G ) x ) e. Y ) ) )
28	20	sseld 3192	. . . . 5 |- ( Y e. (SubGrp ` G ) -> ( x e. Y -> x e. X ) )
29	28	pm4.71rd 394	. . . 4 |- ( Y e. (SubGrp ` G ) -> ( x e. Y <-> ( x e. X /\ x e. Y ) ) )
30	19, 27, 29	3bitr4d 220	. . 3 |- ( Y e. (SubGrp ` G ) -> ( .0. .~ x <-> x e. Y ) )
31	6, 30	bitrd 188	. 2 |- ( Y e. (SubGrp ` G ) -> ( x e. [ .0. ] .~ <-> x e. Y ) )
32	31	eqrdv 2203	1 |- ( Y e. (SubGrp ` G ) -> [ .0. ] .~ = Y )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2176   C_ wss 3166   class class class wbr 4045   Rel wrel 4681   ` cfv 5272  (class class class)co 5946   [cec 6620   Basecbs 12865   +g cplusg 12942   0gc0g 13121   Grpcgrp 13365   invgcminusg 13366  SubGrpcsubg 13536   ~_QG cqg 13538

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 4160  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-cnex 8018  ax-resscn 8019  ax-1re 8021  ax-addrcl 8024

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-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-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-ec 6624  df-inn 9039  df-2 9097  df-ndx 12868  df-slot 12869  df-base 12871  df-plusg 12955  df-0g 13123  df-mgm 13221  df-sgrp 13267  df-mnd 13282  df-grp 13368  df-minusg 13369  df-subg 13539  df-eqg 13541

This theorem is referenced by:  eqg0el  13598