ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eqgid GIF version

Theorem eqgid 13016
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 subgrcl 12970 . . . . 5 (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
2 eqger.r . . . . . 6 = (𝐺 ~QG 𝑌)
32releqgg 13011 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌 ∈ (SubGrp‘𝐺)) → Rel )
41, 3mpancom 422 . . . 4 (𝑌 ∈ (SubGrp‘𝐺) → Rel )
5 relelec 6572 . . . 4 (Rel → (𝑥 ∈ [ 0 ] 0 𝑥))
64, 5syl 14 . . 3 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∈ [ 0 ] 0 𝑥))
71adantr 276 . . . . . . . . 9 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → 𝐺 ∈ Grp)
8 eqgid.3 . . . . . . . . . 10 0 = (0g𝐺)
9 eqid 2177 . . . . . . . . . 10 (invg𝐺) = (invg𝐺)
108, 9grpinvid 12862 . . . . . . . . 9 (𝐺 ∈ Grp → ((invg𝐺)‘ 0 ) = 0 )
117, 10syl 14 . . . . . . . 8 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → ((invg𝐺)‘ 0 ) = 0 )
1211oveq1d 5887 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → (((invg𝐺)‘ 0 )(+g𝐺)𝑥) = ( 0 (+g𝐺)𝑥))
13 eqger.x . . . . . . . . 9 𝑋 = (Base‘𝐺)
14 eqid 2177 . . . . . . . . 9 (+g𝐺) = (+g𝐺)
1513, 14, 8grplid 12838 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → ( 0 (+g𝐺)𝑥) = 𝑥)
161, 15sylan 283 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → ( 0 (+g𝐺)𝑥) = 𝑥)
1712, 16eqtrd 2210 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → (((invg𝐺)‘ 0 )(+g𝐺)𝑥) = 𝑥)
1817eleq1d 2246 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → ((((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌𝑥𝑌))
1918pm5.32da 452 . . . 4 (𝑌 ∈ (SubGrp‘𝐺) → ((𝑥𝑋 ∧ (((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌) ↔ (𝑥𝑋𝑥𝑌)))
2013subgss 12965 . . . . 5 (𝑌 ∈ (SubGrp‘𝐺) → 𝑌𝑋)
2113, 8grpidcl 12836 . . . . . 6 (𝐺 ∈ Grp → 0𝑋)
221, 21syl 14 . . . . 5 (𝑌 ∈ (SubGrp‘𝐺) → 0𝑋)
2313, 9, 14, 2eqgval 13013 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → ( 0 𝑥 ↔ ( 0𝑋𝑥𝑋 ∧ (((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌)))
24 3anass 982 . . . . . . 7 (( 0𝑋𝑥𝑋 ∧ (((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌) ↔ ( 0𝑋 ∧ (𝑥𝑋 ∧ (((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌)))
2523, 24bitrdi 196 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → ( 0 𝑥 ↔ ( 0𝑋 ∧ (𝑥𝑋 ∧ (((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌))))
2625baibd 923 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑌𝑋) ∧ 0𝑋) → ( 0 𝑥 ↔ (𝑥𝑋 ∧ (((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌)))
271, 20, 22, 26syl21anc 1237 . . . 4 (𝑌 ∈ (SubGrp‘𝐺) → ( 0 𝑥 ↔ (𝑥𝑋 ∧ (((invg𝐺)‘ 0 )(+g𝐺)𝑥) ∈ 𝑌)))
2820sseld 3154 . . . . 5 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥𝑌𝑥𝑋))
2928pm4.71rd 394 . . . 4 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥𝑌 ↔ (𝑥𝑋𝑥𝑌)))
3019, 27, 293bitr4d 220 . . 3 (𝑌 ∈ (SubGrp‘𝐺) → ( 0 𝑥𝑥𝑌))
316, 30bitrd 188 . 2 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 ∈ [ 0 ] 𝑥𝑌))
3231eqrdv 2175 1 (𝑌 ∈ (SubGrp‘𝐺) → [ 0 ] = 𝑌)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 978   = wceq 1353  wcel 2148  wss 3129   class class class wbr 4002  Rel wrel 4630  cfv 5215  (class class class)co 5872  [cec 6530  Basecbs 12454  +gcplusg 12528  0gc0g 12693  Grpcgrp 12809  invgcminusg 12810  SubGrpcsubg 12958   ~QG cqg 12960
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1re 7902  ax-addrcl 7905
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-ec 6534  df-inn 8916  df-2 8974  df-ndx 12457  df-slot 12458  df-base 12460  df-plusg 12541  df-0g 12695  df-mgm 12707  df-sgrp 12740  df-mnd 12750  df-grp 12812  df-minusg 12813  df-subg 12961  df-eqg 12963
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator