Theorem eqgid 13533

Description: The left coset containing the identity is the original subgroup. (Contributed by Mario Carneiro, 20-Sep-2015.)

Hypotheses

Ref	Expression
eqger.x	⊢ 𝑋 = (Base‘𝐺)
eqger.r	⊢ ∼ = (𝐺 ~QG 𝑌)
eqgid.3	⊢ 0 = (0g‘𝐺)

Assertion

Ref	Expression
eqgid	⊢ (𝑌 ∈ (SubGrp‘𝐺) → [ 0 ] ∼ = 𝑌)

Proof of Theorem eqgid

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		subgrcl 13486	. . . . 5 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
2		eqger.r	. . . . . 6 ⊢ ∼ = (𝐺 ~QG 𝑌)
3	2	releqgg 13527	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ (SubGrp‘𝐺)) → Rel ∼ )
4	1, 3	mpancom 422	. . . 4 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → Rel ∼ )
5		relelec 6661	. . . 4 ⊢ (Rel ∼ → (𝑥 ∈ [ 0 ] ∼ ↔ 0 ∼ 𝑥))
6	4, 5	syl 14	. . 3 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∈ [ 0 ] ∼ ↔ 0 ∼ 𝑥))
7	1	adantr 276	. . . . . . . . 9 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → 𝐺 ∈ Grp)
8		eqgid.3	. . . . . . . . . 10 ⊢ 0 = (0g‘𝐺)
9		eqid 2204	. . . . . . . . . 10 ⊢ (invg‘𝐺) = (invg‘𝐺)
10	8, 9	grpinvid 13363	. . . . . . . . 9 ⊢ (𝐺 ∈ Grp → ((invg‘𝐺)‘ 0 ) = 0 )
11	7, 10	syl 14	. . . . . . . 8 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → ((invg‘𝐺)‘ 0 ) = 0 )
12	11	oveq1d 5958	. . . . . . 7 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → (((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) = ( 0 (+g‘𝐺)𝑥))
13		eqger.x	. . . . . . . . 9 ⊢ 𝑋 = (Base‘𝐺)
14		eqid 2204	. . . . . . . . 9 ⊢ (+g‘𝐺) = (+g‘𝐺)
15	13, 14, 8	grplid 13334	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ( 0 (+g‘𝐺)𝑥) = 𝑥)
16	1, 15	sylan 283	. . . . . . 7 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → ( 0 (+g‘𝐺)𝑥) = 𝑥)
17	12, 16	eqtrd 2237	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → (((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) = 𝑥)
18	17	eleq1d 2273	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → ((((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) ∈ 𝑌 ↔ 𝑥 ∈ 𝑌))
19	18	pm5.32da 452	. . . 4 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → ((𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) ∈ 𝑌) ↔ (𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑌)))
20	13	subgss 13481	. . . . 5 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 ⊆ 𝑋)
21	13, 8	grpidcl 13332	. . . . . 6 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝑋)
22	1, 21	syl 14	. . . . 5 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 0 ∈ 𝑋)
23	13, 9, 14, 2	eqgval 13530	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → ( 0 ∼ 𝑥 ↔ ( 0 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) ∈ 𝑌)))
24		3anass 984	. . . . . . 7 ⊢ (( 0 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) ∈ 𝑌) ↔ ( 0 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) ∈ 𝑌)))
25	23, 24	bitrdi 196	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → ( 0 ∼ 𝑥 ↔ ( 0 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) ∈ 𝑌))))
26	25	baibd 924	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) ∧ 0 ∈ 𝑋) → ( 0 ∼ 𝑥 ↔ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) ∈ 𝑌)))
27	1, 20, 22, 26	syl21anc 1248	. . . 4 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → ( 0 ∼ 𝑥 ↔ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) ∈ 𝑌)))
28	20	sseld 3191	. . . . 5 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∈ 𝑌 → 𝑥 ∈ 𝑋))
29	28	pm4.71rd 394	. . . 4 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∈ 𝑌 ↔ (𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑌)))
30	19, 27, 29	3bitr4d 220	. . 3 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → ( 0 ∼ 𝑥 ↔ 𝑥 ∈ 𝑌))
31	6, 30	bitrd 188	. 2 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∈ [ 0 ] ∼ ↔ 𝑥 ∈ 𝑌))
32	31	eqrdv 2202	1 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → [ 0 ] ∼ = 𝑌)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1372   ∈ wcel 2175   ⊆ wss 3165   class class class wbr 4043  Rel wrel 4679  ‘cfv 5270  (class class class)co 5943  [cec 6617  Basecbs 12803  +_gcplusg 12880  0_gc0g 13059  Grpcgrp 13303  inv_gcminusg 13304  SubGrpcsubg 13474   ~_QG cqg 13476

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1re 8018  ax-addrcl 8021

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-ec 6621  df-inn 9036  df-2 9094  df-ndx 12806  df-slot 12807  df-base 12809  df-plusg 12893  df-0g 13061  df-mgm 13159  df-sgrp 13205  df-mnd 13220  df-grp 13306  df-minusg 13307  df-subg 13477  df-eqg 13479

This theorem is referenced by:  eqg0el  13536