Theorem eqg0el 13815

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 2231	. . . . . 6 |- ( Base ` G ) = ( Base ` G )
2		eqg0el.1	. . . . . 6 |- .~ = ( G ~QG H )
3	1, 2	eqger 13810	. . . . 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 2231	. . . . . 6 |- ( 0g ` G ) = ( 0g ` G )
6	1, 5	grpidcl 13611	. . . . 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 6747	. . 3 |- ( ( G e. Grp /\ H e. (SubGrp ` G ) ) -> ( ( 0g ` G ) .~ X <-> [ ( 0g ` G ) ] .~ = [ X ] .~ ) )
9	1, 2, 5	eqgid 13812	. . . . 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 2240	. . 3 |- ( ( G e. Grp /\ H e. (SubGrp ` G ) ) -> ( [ ( 0g ` G ) ] .~ = [ X ] .~ <-> H = [ X ] .~ ) )
12		eqcom 2233	. . . 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 6710	. . . 4 |- ( .~ Er ( Base ` G ) -> Rel .~ )
16		relelec 6743	. . . 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 2301	. 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 1397   e. wcel 2202   class class class wbr 4088   Rel wrel 4730   ` cfv 5326  (class class class)co 6017   Er wer 6698   [cec 6699   Basecbs 13081   0gc0g 13338   Grpcgrp 13582  SubGrpcsubg 13753   ~_QG cqg 13755

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-er 6701  df-ec 6703  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-iress 13089  df-plusg 13172  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-minusg 13586  df-subg 13756  df-eqg 13758

This theorem is referenced by: (None)