Theorem eqg0el 13193

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 2189	. . . . . 6 |- ( Base ` G ) = ( Base ` G )
2		eqg0el.1	. . . . . 6 |- .~ = ( G ~QG H )
3	1, 2	eqger 13188	. . . . 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 2189	. . . . . 6 |- ( 0g ` G ) = ( 0g ` G )
6	1, 5	grpidcl 12996	. . . . 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 6609	. . 3 |- ( ( G e. Grp /\ H e. (SubGrp ` G ) ) -> ( ( 0g ` G ) .~ X <-> [ ( 0g ` G ) ] .~ = [ X ] .~ ) )
9	1, 2, 5	eqgid 13190	. . . . 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 2198	. . 3 |- ( ( G e. Grp /\ H e. (SubGrp ` G ) ) -> ( [ ( 0g ` G ) ] .~ = [ X ] .~ <-> H = [ X ] .~ ) )
12		eqcom 2191	. . . 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 6572	. . . 4 |- ( .~ Er ( Base ` G ) -> Rel .~ )
16		relelec 6605	. . . 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 2259	. 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 1364   e. wcel 2160   class class class wbr 4021   Rel wrel 4652   ` cfv 5238  (class class class)co 5900   Er wer 6560   [cec 6561   Basecbs 12523   0gc0g 12772   Grpcgrp 12968  SubGrpcsubg 13131   ~_QG cqg 13133

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4136  ax-sep 4139  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-cnex 7937  ax-resscn 7938  ax-1cn 7939  ax-1re 7940  ax-icn 7941  ax-addcl 7942  ax-addrcl 7943  ax-mulcl 7944  ax-addcom 7946  ax-addass 7948  ax-i2m1 7951  ax-0lt1 7952  ax-0id 7954  ax-rnegex 7955  ax-pre-ltirr 7958  ax-pre-ltadd 7962

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-iun 3906  df-br 4022  df-opab 4083  df-mpt 4084  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246  df-riota 5855  df-ov 5903  df-oprab 5904  df-mpo 5905  df-er 6563  df-ec 6565  df-pnf 8029  df-mnf 8030  df-ltxr 8032  df-inn 8955  df-2 9013  df-ndx 12526  df-slot 12527  df-base 12529  df-sets 12530  df-iress 12531  df-plusg 12613  df-0g 12774  df-mgm 12843  df-sgrp 12888  df-mnd 12901  df-grp 12971  df-minusg 12972  df-subg 13134  df-eqg 13136

This theorem is referenced by: (None)