Theorem eqg0el 13935

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 2232	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqg0el.1	. . . . . 6 ⊢ ∼ = (𝐺 ~QG 𝐻)
3	1, 2	eqger 13930	. . . . 5 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → ∼ Er (Base‘𝐺))
4	3	adantl 277	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → ∼ Er (Base‘𝐺))
5		eqid 2232	. . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺)
6	1, 5	grpidcl 13731	. . . . 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 6812	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → ((0g‘𝐺) ∼ 𝑋 ↔ [(0g‘𝐺)] ∼ = [𝑋] ∼ ))
9	1, 2, 5	eqgid 13932	. . . . 5 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → [(0g‘𝐺)] ∼ = 𝐻)
10	9	adantl 277	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → [(0g‘𝐺)] ∼ = 𝐻)
11	10	eqeq1d 2241	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → ([(0g‘𝐺)] ∼ = [𝑋] ∼ ↔ 𝐻 = [𝑋] ∼ ))
12		eqcom 2234	. . . 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 6775	. . . 4 ⊢ ( ∼ Er (Base‘𝐺) → Rel ∼ )
16		relelec 6808	. . . 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 2302	. 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 1398   ∈ wcel 2203   class class class wbr 4108  Rel wrel 4753  ‘cfv 5351  (class class class)co 6049   Er wer 6763  [cec 6764  Basecbs 13201  0_gc0g 13458  Grpcgrp 13702  SubGrpcsubg 13873   ~_QG cqg 13875

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-addcom 8223  ax-addass 8225  ax-i2m1 8228  ax-0lt1 8229  ax-0id 8231  ax-rnegex 8232  ax-pre-ltirr 8235  ax-pre-ltadd 8239

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-er 6766  df-ec 6768  df-pnf 8306  df-mnf 8307  df-ltxr 8309  df-inn 9234  df-2 9292  df-ndx 13204  df-slot 13205  df-base 13207  df-sets 13208  df-iress 13209  df-plusg 13292  df-0g 13460  df-mgm 13558  df-sgrp 13604  df-mnd 13619  df-grp 13705  df-minusg 13706  df-subg 13876  df-eqg 13878

This theorem is referenced by: (None)