Theorem eqg0el 14091

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 2238	. . . . . 6 |- ( Base ` G ) = ( Base ` G )
2		eqg0el.1	. . . . . 6 |- .~ = ( G ~QG H )
3	1, 2	eqger 14086	. . . . 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 2238	. . . . . 6 |- ( 0g ` G ) = ( 0g ` G )
6	1, 5	grpidcl 13893	. . . . 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 6857	. . 3 |- ( ( G e. Grp /\ H e. (SubGrp ` G ) ) -> ( ( 0g ` G ) .~ X <-> [ ( 0g ` G ) ] .~ = [ X ] .~ ) )
9	1, 2, 5	eqgid 14088	. . . . 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 2247	. . 3 |- ( ( G e. Grp /\ H e. (SubGrp ` G ) ) -> ( [ ( 0g ` G ) ] .~ = [ X ] .~ <-> H = [ X ] .~ ) )
12		eqcom 2240	. . . 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 6820	. . . 4 |- ( .~ Er ( Base ` G ) -> Rel .~ )
16		relelec 6853	. . . 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 2308	. 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   class class class wbr 4131   Rel wrel 4781   ` cfv 5380  (class class class)co 6088   Er wer 6808   [cec 6809   Basecbs 13410   0gc0g 13669   Grpcgrp 13864  SubGrpcsubg 14029   ~_QG cqg 14031

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-pre-ltirr 8297  ax-pre-ltadd 8301

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-er 6811  df-ec 6813  df-pnf 8368  df-mnf 8369  df-ltxr 8371  df-inn 9314  df-2 9372  df-ndx 13413  df-slot 13414  df-base 13416  df-sets 13417  df-iress 13418  df-plusg 13503  df-0g 13671  df-mgm 13735  df-sgrp 13776  df-mnd 13789  df-grp 13867  df-minusg 13868  df-subg 14032  df-eqg 14034

This theorem is used by: (None)