Theorem eqg0el 13507

Description: Equivalence class of a quotient group for a subgroup. (Contributed by Thierry Arnoux, 15-Jan-2024.)

Hypothesis

Ref	Expression
eqg0el.1	|- .~ = ( G ~QG H )

Assertion

Ref	Expression
eqg0el	|- ( ( G e. Grp /\ H e. (SubGrp ` G ) ) -> ( [ X ] .~ = H <-> X e. H ) )

Proof of Theorem eqg0el

Step	Hyp	Ref	Expression
1		eqid 2204	. . . . . 6 |- ( Base ` G ) = ( Base ` G )
2		eqg0el.1	. . . . . 6 |- .~ = ( G ~QG H )
3	1, 2	eqger 13502	. . . . 5 |- ( H e. (SubGrp ` G ) -> .~ Er ( Base ` G ) )
4	3	adantl 277	. . . 4 |- ( ( G e. Grp /\ H e. (SubGrp ` G ) ) -> .~ Er ( Base ` G ) )
5		eqid 2204	. . . . . 6 |- ( 0g ` G ) = ( 0g ` G )
6	1, 5	grpidcl 13303	. . . . 5 |- ( G e. Grp -> ( 0g ` G ) e. ( Base ` G ) )
7	6	adantr 276	. . . 4 |- ( ( G e. Grp /\ H e. (SubGrp ` G ) ) -> ( 0g ` G ) e. ( Base ` G ) )
8	4, 7	erth 6665	. . 3 |- ( ( G e. Grp /\ H e. (SubGrp ` G ) ) -> ( ( 0g ` G ) .~ X <-> [ ( 0g ` G ) ] .~ = [ X ] .~ ) )
9	1, 2, 5	eqgid 13504	. . . . 5 |- ( H e. (SubGrp ` G ) -> [ ( 0g ` G ) ] .~ = H )
10	9	adantl 277	. . . 4 |- ( ( G e. Grp /\ H e. (SubGrp ` G ) ) -> [ ( 0g ` G ) ] .~ = H )
11	10	eqeq1d 2213	. . 3 |- ( ( G e. Grp /\ H e. (SubGrp ` G ) ) -> ( [ ( 0g ` G ) ] .~ = [ X ] .~ <-> H = [ X ] .~ ) )
12		eqcom 2206	. . . 4 |- ( H = [ X ] .~ <-> [ X ] .~ = H )
13	12	a1i 9	. . 3 |- ( ( G e. Grp /\ H e. (SubGrp ` G ) ) -> ( H = [ X ] .~ <-> [ X ] .~ = H ) )
14	8, 11, 13	3bitrrd 215	. 2 |- ( ( G e. Grp /\ H e. (SubGrp ` G ) ) -> ( [ X ] .~ = H <-> ( 0g ` G ) .~ X ) )
15		errel 6628	. . . 4 |- ( .~ Er ( Base ` G ) -> Rel .~ )
16		relelec 6661	. . . 4 |- ( Rel .~ -> ( X e. [ ( 0g ` G ) ] .~ <-> ( 0g ` G ) .~ X ) )
17	3, 15, 16	3syl 17	. . 3 |- ( H e. (SubGrp ` G ) -> ( X e. [ ( 0g ` G ) ] .~ <-> ( 0g ` G ) .~ X ) )
18	17	adantl 277	. 2 |- ( ( G e. Grp /\ H e. (SubGrp ` G ) ) -> ( X e. [ ( 0g ` G ) ] .~ <-> ( 0g ` G ) .~ X ) )
19	10	eleq2d 2274	. 2 |- ( ( G e. Grp /\ H e. (SubGrp ` G ) ) -> ( X e. [ ( 0g ` G ) ] .~ <-> X e. H ) )
20	14, 18, 19	3bitr2d 216	1 |- ( ( G e. Grp /\ H e. (SubGrp ` G ) ) -> ( [ X ] .~ = H <-> X e. H ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1372   e. wcel 2175   class class class wbr 4043   Rel wrel 4679   ` cfv 5270  (class class class)co 5943   Er wer 6616   [cec 6617   Basecbs 12774   0gc0g 13030   Grpcgrp 13274  SubGrpcsubg 13445   ~_QG cqg 13447

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-addass 8026  ax-i2m1 8029  ax-0lt1 8030  ax-0id 8032  ax-rnegex 8033  ax-pre-ltirr 8036  ax-pre-ltadd 8040

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-er 6619  df-ec 6621  df-pnf 8108  df-mnf 8109  df-ltxr 8111  df-inn 9036  df-2 9094  df-ndx 12777  df-slot 12778  df-base 12780  df-sets 12781  df-iress 12782  df-plusg 12864  df-0g 13032  df-mgm 13130  df-sgrp 13176  df-mnd 13191  df-grp 13277  df-minusg 13278  df-subg 13448  df-eqg 13450

This theorem is referenced by: (None)