Theorem eqgid 13647

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 13600	. . . . 5 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
2		eqger.r	. . . . . 6 ⊢ ∼ = (𝐺 ~QG 𝑌)
3	2	releqgg 13641	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ (SubGrp‘𝐺)) → Rel ∼ )
4	1, 3	mpancom 422	. . . 4 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → Rel ∼ )
5		relelec 6680	. . . 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 2206	. . . . . . . . . 10 ⊢ (invg‘𝐺) = (invg‘𝐺)
10	8, 9	grpinvid 13477	. . . . . . . . 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 5977	. . . . . . 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 2206	. . . . . . . . 9 ⊢ (+g‘𝐺) = (+g‘𝐺)
15	13, 14, 8	grplid 13448	. . . . . . . 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 2239	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → (((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) = 𝑥)
18	17	eleq1d 2275	. . . . 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 13595	. . . . 5 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 ⊆ 𝑋)
21	13, 8	grpidcl 13446	. . . . . 6 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝑋)
22	1, 21	syl 14	. . . . 5 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 0 ∈ 𝑋)
23	13, 9, 14, 2	eqgval 13644	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → ( 0 ∼ 𝑥 ↔ ( 0 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) ∈ 𝑌)))
24		3anass 985	. . . . . . 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 925	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) ∧ 0 ∈ 𝑋) → ( 0 ∼ 𝑥 ↔ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) ∈ 𝑌)))
27	1, 20, 22, 26	syl21anc 1249	. . . 4 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → ( 0 ∼ 𝑥 ↔ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) ∈ 𝑌)))
28	20	sseld 3196	. . . . 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 2204	1 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → [ 0 ] ∼ = 𝑌)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 981   = wceq 1373   ∈ wcel 2177   ⊆ wss 3170   class class class wbr 4054  Rel wrel 4693  ‘cfv 5285  (class class class)co 5962  [cec 6636  Basecbs 12917  +_gcplusg 12994  0_gc0g 13173  Grpcgrp 13417  inv_gcminusg 13418  SubGrpcsubg 13588   ~_QG cqg 13590

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4170  ax-sep 4173  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-cnex 8046  ax-resscn 8047  ax-1re 8049  ax-addrcl 8052

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-id 4353  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-riota 5917  df-ov 5965  df-oprab 5966  df-mpo 5967  df-ec 6640  df-inn 9067  df-2 9125  df-ndx 12920  df-slot 12921  df-base 12923  df-plusg 13007  df-0g 13175  df-mgm 13273  df-sgrp 13319  df-mnd 13334  df-grp 13420  df-minusg 13421  df-subg 13591  df-eqg 13593

This theorem is referenced by:  eqg0el  13650