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Theorem grpoid 27344
 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 27338 . . . . 5 (𝐺 ∈ GrpOp → 𝑈𝑋)
41grporcan 27342 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝑈𝑋𝐴𝑋)) → ((𝐴𝐺𝐴) = (𝑈𝐺𝐴) ↔ 𝐴 = 𝑈))
543exp2 1283 . . . . 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 445 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝐴𝐺𝐴) = (𝑈𝐺𝐴) ↔ 𝐴 = 𝑈))
91, 2grpolid 27340 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑈𝐺𝐴) = 𝐴)
109eqeq2d 2630 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝐴𝐺𝐴) = (𝑈𝐺𝐴) ↔ (𝐴𝐺𝐴) = 𝐴))
118, 10bitr3d 270 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴 = 𝑈 ↔ (𝐴𝐺𝐴) = 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1481   ∈ wcel 1988  ran crn 5105  ‘cfv 5876  (class class class)co 6635  GrpOpcgr 27313  GIdcgi 27314 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fo 5882  df-fv 5884  df-riota 6596  df-ov 6638  df-grpo 27317  df-gid 27318 This theorem is referenced by:  hhssnv  28091  ghomidOLD  33659
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