Theorem eqgid 14006

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 13959	. . . . 5 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
2		eqger.r	. . . . . 6 ⊢ ∼ = (𝐺 ~QG 𝑌)
3	2	releqgg 14000	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ (SubGrp‘𝐺)) → Rel ∼ )
4	1, 3	mpancom 426	. . . 4 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → Rel ∼ )
5		relelec 6839	. . . 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 2238	. . . . . . . . . 10 ⊢ (invg‘𝐺) = (invg‘𝐺)
10	8, 9	grpinvid 13842	. . . . . . . . 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 6090	. . . . . . 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 2238	. . . . . . . . 9 ⊢ (+g‘𝐺) = (+g‘𝐺)
15	13, 14, 8	grplid 13813	. . . . . . . 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 2271	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → (((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) = 𝑥)
18	17	eleq1d 2307	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → ((((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) ∈ 𝑌 ↔ 𝑥 ∈ 𝑌))
19	18	pm5.32da 456	. . . 4 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → ((𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) ∈ 𝑌) ↔ (𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑌)))
20	13	subgss 13954	. . . . 5 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 ⊆ 𝑋)
21	13, 8	grpidcl 13811	. . . . . 6 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝑋)
22	1, 21	syl 14	. . . . 5 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 0 ∈ 𝑋)
23	13, 9, 14, 2	eqgval 14003	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → ( 0 ∼ 𝑥 ↔ ( 0 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) ∈ 𝑌)))
24		3anass 1013	. . . . . . 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 935	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) ∧ 0 ∈ 𝑋) → ( 0 ∼ 𝑥 ↔ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) ∈ 𝑌)))
27	1, 20, 22, 26	syl21anc 1277	. . . 4 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → ( 0 ∼ 𝑥 ↔ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘ 0 )(+g‘𝐺)𝑥) ∈ 𝑌)))
28	20	sseld 3247	. . . . 5 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∈ 𝑌 → 𝑥 ∈ 𝑋))
29	28	pm4.71rd 398	. . . 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 2236	1 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → [ 0 ] ∼ = 𝑌)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209   ⊆ wss 3220   class class class wbr 4125  Rel wrel 4774  ‘cfv 5372  (class class class)co 6075  [cec 6795  Basecbs 13330  +_gcplusg 13408  0_gc0g 13587  Grpcgrp 13782  inv_gcminusg 13783  SubGrpcsubg 13947   ~_QG cqg 13949

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-ec 6799  df-inn 9284  df-2 9342  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-minusg 13786  df-subg 13950  df-eqg 13952

This theorem is referenced by:  eqg0el  14009