Theorem eqgid 13637

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 13590	. . . . 5 |- ( Y e. (SubGrp ` G ) -> G e. Grp )
2		eqger.r	. . . . . 6 |- .~ = ( G ~QG Y )
3	2	releqgg 13631	. . . . 5 |- ( ( G e. Grp /\ Y e. (SubGrp ` G ) ) -> Rel .~ )
4	1, 3	mpancom 422	. . . 4 |- ( Y e. (SubGrp ` G ) -> Rel .~ )
5		relelec 6675	. . . 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 2206	. . . . . . . . . 10 |- ( invg ` G ) = ( invg ` G )
10	8, 9	grpinvid 13467	. . . . . . . . 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 5972	. . . . . . 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 2206	. . . . . . . . 9 |- ( +g ` G ) = ( +g ` G )
15	13, 14, 8	grplid 13438	. . . . . . . 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 2239	. . . . . 6 |- ( ( Y e. (SubGrp ` G ) /\ x e. X ) -> ( ( ( invg ` G ) ` .0. ) ( +g ` G ) x ) = x )
18	17	eleq1d 2275	. . . . 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 13585	. . . . 5 |- ( Y e. (SubGrp ` G ) -> Y C_ X )
21	13, 8	grpidcl 13436	. . . . . 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 13634	. . . . . . 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 3196	. . . . 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 2204	1 |- ( Y e. (SubGrp ` G ) -> [ .0. ] .~ = Y )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2177   C_ wss 3170   class class class wbr 4051   Rel wrel 4688   ` cfv 5280  (class class class)co 5957   [cec 6631   Basecbs 12907   +g cplusg 12984   0gc0g 13163   Grpcgrp 13407   invgcminusg 13408  SubGrpcsubg 13578   ~_QG cqg 13580

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-cnex 8036  ax-resscn 8037  ax-1re 8039  ax-addrcl 8042

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-ec 6635  df-inn 9057  df-2 9115  df-ndx 12910  df-slot 12911  df-base 12913  df-plusg 12997  df-0g 13165  df-mgm 13263  df-sgrp 13309  df-mnd 13324  df-grp 13410  df-minusg 13411  df-subg 13581  df-eqg 13583

This theorem is referenced by:  eqg0el  13640