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Theorem eqgid 17847
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 17842 . . . 4 Rel
3 relelec 7937 . . . 4 (Rel → (𝑥 ∈ [ 0 ] 0 𝑥))
42, 3ax-mp 5 . . 3 (𝑥 ∈ [ 0 ] 0 𝑥)
5 subgrcl 17800 . . . . . . . . . 10 (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
65adantr 466 . . . . . . . . 9 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → 𝐺 ∈ Grp)
7 eqgid.3 . . . . . . . . . 10 0 = (0g𝐺)
8 eqid 2771 . . . . . . . . . 10 (invg𝐺) = (invg𝐺)
97, 8grpinvid 17677 . . . . . . . . 9 (𝐺 ∈ Grp → ((invg𝐺)‘ 0 ) = 0 )
106, 9syl 17 . . . . . . . 8 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → ((invg𝐺)‘ 0 ) = 0 )
1110oveq1d 6806 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → (((invg𝐺)‘ 0 )(+g𝐺)𝑥) = ( 0 (+g𝐺)𝑥))
12 eqger.x . . . . . . . . 9 𝑋 = (Base‘𝐺)
13 eqid 2771 . . . . . . . . 9 (+g𝐺) = (+g𝐺)
1412, 13, 7grplid 17653 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → ( 0 (+g𝐺)𝑥) = 𝑥)
155, 14sylan 569 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → ( 0 (+g𝐺)𝑥) = 𝑥)
1611, 15eqtrd 2805 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → (((invg𝐺)‘ 0 )(+g𝐺)𝑥) = 𝑥)
1716eleq1d 2835 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → ((((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌𝑥𝑌))
1817pm5.32da 568 . . . 4 (𝑌 ∈ (SubGrp‘𝐺) → ((𝑥𝑋 ∧ (((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌) ↔ (𝑥𝑋𝑥𝑌)))
1912subgss 17796 . . . . 5 (𝑌 ∈ (SubGrp‘𝐺) → 𝑌𝑋)
2012, 7grpidcl 17651 . . . . . 6 (𝐺 ∈ Grp → 0𝑋)
215, 20syl 17 . . . . 5 (𝑌 ∈ (SubGrp‘𝐺) → 0𝑋)
2212, 8, 13, 1eqgval 17844 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → ( 0 𝑥 ↔ ( 0𝑋𝑥𝑋 ∧ (((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌)))
23 3anass 1080 . . . . . . 7 (( 0𝑋𝑥𝑋 ∧ (((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌) ↔ ( 0𝑋 ∧ (𝑥𝑋 ∧ (((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌)))
2422, 23syl6bb 276 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → ( 0 𝑥 ↔ ( 0𝑋 ∧ (𝑥𝑋 ∧ (((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌))))
2524baibd 529 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑌𝑋) ∧ 0𝑋) → ( 0 𝑥 ↔ (𝑥𝑋 ∧ (((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌)))
265, 19, 21, 25syl21anc 1475 . . . 4 (𝑌 ∈ (SubGrp‘𝐺) → ( 0 𝑥 ↔ (𝑥𝑋 ∧ (((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌)))
2719sseld 3751 . . . . 5 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥𝑌𝑥𝑋))
2827pm4.71rd 552 . . . 4 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥𝑌 ↔ (𝑥𝑋𝑥𝑌)))
2918, 26, 283bitr4d 300 . . 3 (𝑌 ∈ (SubGrp‘𝐺) → ( 0 𝑥𝑥𝑌))
304, 29syl5bb 272 . 2 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∈ [ 0 ] 𝑥𝑌))
3130eqrdv 2769 1 (𝑌 ∈ (SubGrp‘𝐺) → [ 0 ] = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wss 3723   class class class wbr 4786  Rel wrel 5254  cfv 6029  (class class class)co 6791  [cec 7892  Basecbs 16057  +gcplusg 16142  0gc0g 16301  Grpcgrp 17623  invgcminusg 17624  SubGrpcsubg 17789   ~QG cqg 17791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-riota 6752  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-1st 7313  df-2nd 7314  df-ec 7896  df-0g 16303  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-grp 17626  df-minusg 17627  df-subg 17792  df-eqg 17794
This theorem is referenced by:  cldsubg  22127  qustgphaus  22139
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