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Theorem grpoid 28296
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
grpoinveu.1 𝑋 = ran 𝐺
grpoinveu.2 𝑈 = (GId‘𝐺)
Assertion
Ref Expression
grpoid ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴 = 𝑈 ↔ (𝐴𝐺𝐴) = 𝐴))

Proof of Theorem grpoid
StepHypRef Expression
1 grpoinveu.1 . . . . . 6 𝑋 = ran 𝐺
2 grpoinveu.2 . . . . . 6 𝑈 = (GId‘𝐺)
31, 2grpoidcl 28290 . . . . 5 (𝐺 ∈ GrpOp → 𝑈𝑋)
41grporcan 28294 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝑈𝑋𝐴𝑋)) → ((𝐴𝐺𝐴) = (𝑈𝐺𝐴) ↔ 𝐴 = 𝑈))
543exp2 1350 . . . . 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 409 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝐴𝐺𝐴) = (𝑈𝐺𝐴) ↔ 𝐴 = 𝑈))
91, 2grpolid 28292 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑈𝐺𝐴) = 𝐴)
109eqeq2d 2832 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝐴𝐺𝐴) = (𝑈𝐺𝐴) ↔ (𝐴𝐺𝐴) = 𝐴))
118, 10bitr3d 283 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴 = 𝑈 ↔ (𝐴𝐺𝐴) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1533  wcel 2110  ran crn 5555  cfv 6354  (class class class)co 7155  GrpOpcgr 28265  GIdcgi 28266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fo 6360  df-fv 6362  df-riota 7113  df-ov 7158  df-grpo 28269  df-gid 28270
This theorem is referenced by:  hhssnv  29040  ghomidOLD  35166
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