Theorem eqg0el 13359

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 2196	. . . . . 6 |- ( Base ` G ) = ( Base ` G )
2		eqg0el.1	. . . . . 6 |- .~ = ( G ~QG H )
3	1, 2	eqger 13354	. . . . 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 2196	. . . . . 6 |- ( 0g ` G ) = ( 0g ` G )
6	1, 5	grpidcl 13161	. . . . 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 6638	. . 3 |- ( ( G e. Grp /\ H e. (SubGrp ` G ) ) -> ( ( 0g ` G ) .~ X <-> [ ( 0g ` G ) ] .~ = [ X ] .~ ) )
9	1, 2, 5	eqgid 13356	. . . . 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 2205	. . 3 |- ( ( G e. Grp /\ H e. (SubGrp ` G ) ) -> ( [ ( 0g ` G ) ] .~ = [ X ] .~ <-> H = [ X ] .~ ) )
12		eqcom 2198	. . . 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 6601	. . . 4 |- ( .~ Er ( Base ` G ) -> Rel .~ )
16		relelec 6634	. . . 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 2266	. 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 2167   class class class wbr 4033   Rel wrel 4668   ` cfv 5258  (class class class)co 5922   Er wer 6589   [cec 6590   Basecbs 12678   0gc0g 12927   Grpcgrp 13132  SubGrpcsubg 13297   ~_QG cqg 13299

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-er 6592  df-ec 6594  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-subg 13300  df-eqg 13302

This theorem is referenced by: (None)