Theorem eqg0el 13809

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 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → ([𝑋] ∼ = 𝐻 ↔ 𝑋 ∈ 𝐻))

Proof of Theorem eqg0el

Step	Hyp	Ref	Expression
1		eqid 2229	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqg0el.1	. . . . . 6 ⊢ ∼ = (𝐺 ~QG 𝐻)
3	1, 2	eqger 13804	. . . . 5 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → ∼ Er (Base‘𝐺))
4	3	adantl 277	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → ∼ Er (Base‘𝐺))
5		eqid 2229	. . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺)
6	1, 5	grpidcl 13605	. . . . 5 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ (Base‘𝐺))
7	6	adantr 276	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → (0g‘𝐺) ∈ (Base‘𝐺))
8	4, 7	erth 6743	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → ((0g‘𝐺) ∼ 𝑋 ↔ [(0g‘𝐺)] ∼ = [𝑋] ∼ ))
9	1, 2, 5	eqgid 13806	. . . . 5 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → [(0g‘𝐺)] ∼ = 𝐻)
10	9	adantl 277	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → [(0g‘𝐺)] ∼ = 𝐻)
11	10	eqeq1d 2238	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → ([(0g‘𝐺)] ∼ = [𝑋] ∼ ↔ 𝐻 = [𝑋] ∼ ))
12		eqcom 2231	. . . 4 ⊢ (𝐻 = [𝑋] ∼ ↔ [𝑋] ∼ = 𝐻)
13	12	a1i 9	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → (𝐻 = [𝑋] ∼ ↔ [𝑋] ∼ = 𝐻))
14	8, 11, 13	3bitrrd 215	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → ([𝑋] ∼ = 𝐻 ↔ (0g‘𝐺) ∼ 𝑋))
15		errel 6706	. . . 4 ⊢ ( ∼ Er (Base‘𝐺) → Rel ∼ )
16		relelec 6739	. . . 4 ⊢ (Rel ∼ → (𝑋 ∈ [(0g‘𝐺)] ∼ ↔ (0g‘𝐺) ∼ 𝑋))
17	3, 15, 16	3syl 17	. . 3 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → (𝑋 ∈ [(0g‘𝐺)] ∼ ↔ (0g‘𝐺) ∼ 𝑋))
18	17	adantl 277	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → (𝑋 ∈ [(0g‘𝐺)] ∼ ↔ (0g‘𝐺) ∼ 𝑋))
19	10	eleq2d 2299	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → (𝑋 ∈ [(0g‘𝐺)] ∼ ↔ 𝑋 ∈ 𝐻))
20	14, 18, 19	3bitr2d 216	1 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → ([𝑋] ∼ = 𝐻 ↔ 𝑋 ∈ 𝐻))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200   class class class wbr 4086  Rel wrel 4728  ‘cfv 5324  (class class class)co 6013   Er wer 6694  [cec 6695  Basecbs 13075  0_gc0g 13332  Grpcgrp 13576  SubGrpcsubg 13747   ~_QG cqg 13749

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-addcom 8125  ax-addass 8127  ax-i2m1 8130  ax-0lt1 8131  ax-0id 8133  ax-rnegex 8134  ax-pre-ltirr 8137  ax-pre-ltadd 8141

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-er 6697  df-ec 6699  df-pnf 8209  df-mnf 8210  df-ltxr 8212  df-inn 9137  df-2 9195  df-ndx 13078  df-slot 13079  df-base 13081  df-sets 13082  df-iress 13083  df-plusg 13166  df-0g 13334  df-mgm 13432  df-sgrp 13478  df-mnd 13493  df-grp 13579  df-minusg 13580  df-subg 13750  df-eqg 13752

This theorem is referenced by: (None)