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Theorem grpoid 27852
Description: Two ways of saying that an element of a group is the identity element. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpinveu.1 𝑋 = ran 𝐺
grpinveu.2 𝑈 = (GId‘𝐺)
Assertion
Ref Expression
grpoid ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴 = 𝑈 ↔ (𝐴𝐺𝐴) = 𝐴))

Proof of Theorem grpoid
StepHypRef Expression
1 grpinveu.1 . . . . . 6 𝑋 = ran 𝐺
2 grpinveu.2 . . . . . 6 𝑈 = (GId‘𝐺)
31, 2grpoidcl 27846 . . . . 5 (𝐺 ∈ GrpOp → 𝑈𝑋)
41grporcan 27850 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝑈𝑋𝐴𝑋)) → ((𝐴𝐺𝐴) = (𝑈𝐺𝐴) ↔ 𝐴 = 𝑈))
543exp2 1463 . . . . 5 (𝐺 ∈ GrpOp → (𝐴𝑋 → (𝑈𝑋 → (𝐴𝑋 → ((𝐴𝐺𝐴) = (𝑈𝐺𝐴) ↔ 𝐴 = 𝑈)))))
63, 5mpid 44 . . . 4 (𝐺 ∈ GrpOp → (𝐴𝑋 → (𝐴𝑋 → ((𝐴𝐺𝐴) = (𝑈𝐺𝐴) ↔ 𝐴 = 𝑈))))
76pm2.43d 53 . . 3 (𝐺 ∈ GrpOp → (𝐴𝑋 → ((𝐴𝐺𝐴) = (𝑈𝐺𝐴) ↔ 𝐴 = 𝑈)))
87imp 395 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝐴𝐺𝐴) = (𝑈𝐺𝐴) ↔ 𝐴 = 𝑈))
91, 2grpolid 27848 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑈𝐺𝐴) = 𝐴)
109eqeq2d 2775 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝐴𝐺𝐴) = (𝑈𝐺𝐴) ↔ (𝐴𝐺𝐴) = 𝐴))
118, 10bitr3d 272 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴 = 𝑈 ↔ (𝐴𝐺𝐴) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  ran crn 5280  cfv 6070  (class class class)co 6846  GrpOpcgr 27821  GIdcgi 27822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064  ax-un 7151
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-fo 6076  df-fv 6078  df-riota 6807  df-ov 6849  df-grpo 27825  df-gid 27826
This theorem is referenced by:  hhssnv  28598  ghomidOLD  34131
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