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Theorem eqgid 18983
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 18978 . . . 4 Rel
3 relelec 8694 . . . 4 (Rel → (𝑥 ∈ [ 0 ] 0 𝑥))
42, 3ax-mp 5 . . 3 (𝑥 ∈ [ 0 ] 0 𝑥)
5 subgrcl 18934 . . . . . . . . . 10 (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
65adantr 482 . . . . . . . . 9 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → 𝐺 ∈ Grp)
7 eqgid.3 . . . . . . . . . 10 0 = (0g𝐺)
8 eqid 2737 . . . . . . . . . 10 (invg𝐺) = (invg𝐺)
97, 8grpinvid 18809 . . . . . . . . 9 (𝐺 ∈ Grp → ((invg𝐺)‘ 0 ) = 0 )
106, 9syl 17 . . . . . . . 8 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → ((invg𝐺)‘ 0 ) = 0 )
1110oveq1d 7373 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → (((invg𝐺)‘ 0 )(+g𝐺)𝑥) = ( 0 (+g𝐺)𝑥))
12 eqger.x . . . . . . . . 9 𝑋 = (Base‘𝐺)
13 eqid 2737 . . . . . . . . 9 (+g𝐺) = (+g𝐺)
1412, 13, 7grplid 18781 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → ( 0 (+g𝐺)𝑥) = 𝑥)
155, 14sylan 581 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → ( 0 (+g𝐺)𝑥) = 𝑥)
1611, 15eqtrd 2777 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → (((invg𝐺)‘ 0 )(+g𝐺)𝑥) = 𝑥)
1716eleq1d 2823 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → ((((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌𝑥𝑌))
1817pm5.32da 580 . . . 4 (𝑌 ∈ (SubGrp‘𝐺) → ((𝑥𝑋 ∧ (((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌) ↔ (𝑥𝑋𝑥𝑌)))
1912subgss 18930 . . . . 5 (𝑌 ∈ (SubGrp‘𝐺) → 𝑌𝑋)
2012, 7grpidcl 18779 . . . . . 6 (𝐺 ∈ Grp → 0𝑋)
215, 20syl 17 . . . . 5 (𝑌 ∈ (SubGrp‘𝐺) → 0𝑋)
2212, 8, 13, 1eqgval 18980 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → ( 0 𝑥 ↔ ( 0𝑋𝑥𝑋 ∧ (((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌)))
23 3anass 1096 . . . . . . 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 541 . . . . 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 3944 . . . . 5 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥𝑌𝑥𝑋))
2827pm4.71rd 564 . . . 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 2735 1 (𝑌 ∈ (SubGrp‘𝐺) → [ 0 ] = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wss 3911   class class class wbr 5106  Rel wrel 5639  cfv 6497  (class class class)co 7358  [cec 8647  Basecbs 17084  +gcplusg 17134  0gc0g 17322  Grpcgrp 18749  invgcminusg 18750  SubGrpcsubg 18923   ~QG cqg 18925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-ec 8651  df-0g 17324  df-mgm 18498  df-sgrp 18547  df-mnd 18558  df-grp 18752  df-minusg 18753  df-subg 18926  df-eqg 18928
This theorem is referenced by:  cldsubg  23465  qustgphaus  23477  eqg0el  32152  ghmqusker  32201  qsidomlem1  32228  qsidomlem2  32229
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