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Theorem eqgid 19135
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 19130 . . . 4 Rel
3 relelec 8771 . . . 4 (Rel → (𝑥 ∈ [ 0 ] 0 𝑥))
42, 3ax-mp 5 . . 3 (𝑥 ∈ [ 0 ] 0 𝑥)
5 subgrcl 19086 . . . . . . . . . 10 (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
65adantr 480 . . . . . . . . 9 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → 𝐺 ∈ Grp)
7 eqgid.3 . . . . . . . . . 10 0 = (0g𝐺)
8 eqid 2728 . . . . . . . . . 10 (invg𝐺) = (invg𝐺)
97, 8grpinvid 18956 . . . . . . . . 9 (𝐺 ∈ Grp → ((invg𝐺)‘ 0 ) = 0 )
106, 9syl 17 . . . . . . . 8 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → ((invg𝐺)‘ 0 ) = 0 )
1110oveq1d 7435 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → (((invg𝐺)‘ 0 )(+g𝐺)𝑥) = ( 0 (+g𝐺)𝑥))
12 eqger.x . . . . . . . . 9 𝑋 = (Base‘𝐺)
13 eqid 2728 . . . . . . . . 9 (+g𝐺) = (+g𝐺)
1412, 13, 7grplid 18924 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → ( 0 (+g𝐺)𝑥) = 𝑥)
155, 14sylan 579 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → ( 0 (+g𝐺)𝑥) = 𝑥)
1611, 15eqtrd 2768 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → (((invg𝐺)‘ 0 )(+g𝐺)𝑥) = 𝑥)
1716eleq1d 2814 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → ((((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌𝑥𝑌))
1817pm5.32da 578 . . . 4 (𝑌 ∈ (SubGrp‘𝐺) → ((𝑥𝑋 ∧ (((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌) ↔ (𝑥𝑋𝑥𝑌)))
1912subgss 19082 . . . . 5 (𝑌 ∈ (SubGrp‘𝐺) → 𝑌𝑋)
2012, 7grpidcl 18922 . . . . . 6 (𝐺 ∈ Grp → 0𝑋)
215, 20syl 17 . . . . 5 (𝑌 ∈ (SubGrp‘𝐺) → 0𝑋)
2212, 8, 13, 1eqgval 19132 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → ( 0 𝑥 ↔ ( 0𝑋𝑥𝑋 ∧ (((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌)))
23 3anass 1093 . . . . . . 7 (( 0𝑋𝑥𝑋 ∧ (((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌) ↔ ( 0𝑋 ∧ (𝑥𝑋 ∧ (((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌)))
2422, 23bitrdi 287 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → ( 0 𝑥 ↔ ( 0𝑋 ∧ (𝑥𝑋 ∧ (((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌))))
2524baibd 539 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑌𝑋) ∧ 0𝑋) → ( 0 𝑥 ↔ (𝑥𝑋 ∧ (((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌)))
265, 19, 21, 25syl21anc 837 . . . 4 (𝑌 ∈ (SubGrp‘𝐺) → ( 0 𝑥 ↔ (𝑥𝑋 ∧ (((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌)))
2719sseld 3979 . . . . 5 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥𝑌𝑥𝑋))
2827pm4.71rd 562 . . . 4 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥𝑌 ↔ (𝑥𝑋𝑥𝑌)))
2918, 26, 283bitr4d 311 . . 3 (𝑌 ∈ (SubGrp‘𝐺) → ( 0 𝑥𝑥𝑌))
304, 29bitrid 283 . 2 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∈ [ 0 ] 𝑥𝑌))
3130eqrdv 2726 1 (𝑌 ∈ (SubGrp‘𝐺) → [ 0 ] = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  wss 3947   class class class wbr 5148  Rel wrel 5683  cfv 6548  (class class class)co 7420  [cec 8723  Basecbs 17180  +gcplusg 17233  0gc0g 17421  Grpcgrp 18890  invgcminusg 18891  SubGrpcsubg 19075   ~QG cqg 19077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 7993  df-2nd 7994  df-ec 8727  df-0g 17423  df-mgm 18600  df-sgrp 18679  df-mnd 18695  df-grp 18893  df-minusg 18894  df-subg 19078  df-eqg 19080
This theorem is referenced by:  eqg0el  19138  ghmqusker  19238  cldsubg  24028  qustgphaus  24040  qsidomlem1  33181  qsidomlem2  33182  qsdrngi  33219
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