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Theorem eqgid 19169
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 = (0g𝐺)
Assertion
Ref Expression
eqgid (𝑌 ∈ (SubGrp‘𝐺) → [ 0 ] = 𝑌)

Proof of Theorem eqgid
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqger.r . . . . 5 = (𝐺 ~QG 𝑌)
21releqg 19164 . . . 4 Rel
3 relelec 8780 . . . 4 (Rel → (𝑥 ∈ [ 0 ] 0 𝑥))
42, 3ax-mp 5 . . 3 (𝑥 ∈ [ 0 ] 0 𝑥)
5 subgrcl 19120 . . . . . . . . . 10 (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
65adantr 479 . . . . . . . . 9 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → 𝐺 ∈ Grp)
7 eqgid.3 . . . . . . . . . 10 0 = (0g𝐺)
8 eqid 2725 . . . . . . . . . 10 (invg𝐺) = (invg𝐺)
97, 8grpinvid 18989 . . . . . . . . 9 (𝐺 ∈ Grp → ((invg𝐺)‘ 0 ) = 0 )
106, 9syl 17 . . . . . . . 8 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → ((invg𝐺)‘ 0 ) = 0 )
1110oveq1d 7438 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → (((invg𝐺)‘ 0 )(+g𝐺)𝑥) = ( 0 (+g𝐺)𝑥))
12 eqger.x . . . . . . . . 9 𝑋 = (Base‘𝐺)
13 eqid 2725 . . . . . . . . 9 (+g𝐺) = (+g𝐺)
1412, 13, 7grplid 18957 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → ( 0 (+g𝐺)𝑥) = 𝑥)
155, 14sylan 578 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → ( 0 (+g𝐺)𝑥) = 𝑥)
1611, 15eqtrd 2765 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → (((invg𝐺)‘ 0 )(+g𝐺)𝑥) = 𝑥)
1716eleq1d 2810 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → ((((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌𝑥𝑌))
1817pm5.32da 577 . . . 4 (𝑌 ∈ (SubGrp‘𝐺) → ((𝑥𝑋 ∧ (((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌) ↔ (𝑥𝑋𝑥𝑌)))
1912subgss 19116 . . . . 5 (𝑌 ∈ (SubGrp‘𝐺) → 𝑌𝑋)
2012, 7grpidcl 18955 . . . . . 6 (𝐺 ∈ Grp → 0𝑋)
215, 20syl 17 . . . . 5 (𝑌 ∈ (SubGrp‘𝐺) → 0𝑋)
2212, 8, 13, 1eqgval 19166 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → ( 0 𝑥 ↔ ( 0𝑋𝑥𝑋 ∧ (((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌)))
23 3anass 1092 . . . . . . 7 (( 0𝑋𝑥𝑋 ∧ (((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌) ↔ ( 0𝑋 ∧ (𝑥𝑋 ∧ (((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌)))
2422, 23bitrdi 286 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → ( 0 𝑥 ↔ ( 0𝑋 ∧ (𝑥𝑋 ∧ (((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌))))
2524baibd 538 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑌𝑋) ∧ 0𝑋) → ( 0 𝑥 ↔ (𝑥𝑋 ∧ (((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌)))
265, 19, 21, 25syl21anc 836 . . . 4 (𝑌 ∈ (SubGrp‘𝐺) → ( 0 𝑥 ↔ (𝑥𝑋 ∧ (((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌)))
2719sseld 3977 . . . . 5 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥𝑌𝑥𝑋))
2827pm4.71rd 561 . . . 4 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥𝑌 ↔ (𝑥𝑋𝑥𝑌)))
2918, 26, 283bitr4d 310 . . 3 (𝑌 ∈ (SubGrp‘𝐺) → ( 0 𝑥𝑥𝑌))
304, 29bitrid 282 . 2 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∈ [ 0 ] 𝑥𝑌))
3130eqrdv 2723 1 (𝑌 ∈ (SubGrp‘𝐺) → [ 0 ] = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wss 3946   class class class wbr 5152  Rel wrel 5686  cfv 6553  (class class class)co 7423  [cec 8731  Basecbs 17208  +gcplusg 17261  0gc0g 17449  Grpcgrp 18923  invgcminusg 18924  SubGrpcsubg 19109   ~QG cqg 19111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5303  ax-nul 5310  ax-pow 5368  ax-pr 5432  ax-un 7745
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5579  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-riota 7379  df-ov 7426  df-oprab 7427  df-mpo 7428  df-1st 8002  df-2nd 8003  df-ec 8735  df-0g 17451  df-mgm 18628  df-sgrp 18707  df-mnd 18723  df-grp 18926  df-minusg 18927  df-subg 19112  df-eqg 19114
This theorem is referenced by:  eqg0el  19172  ghmqusker  19276  cldsubg  24098  qustgphaus  24110  qsidomlem1  33304  qsidomlem2  33305  qsdrngi  33347
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