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Theorem grpoid 29260
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 29254 . . . . 5 (𝐺 ∈ GrpOp → 𝑈𝑋)
41grporcan 29258 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝑈𝑋𝐴𝑋)) → ((𝐴𝐺𝐴) = (𝑈𝐺𝐴) ↔ 𝐴 = 𝑈))
543exp2 1354 . . . . 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 407 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝐴𝐺𝐴) = (𝑈𝐺𝐴) ↔ 𝐴 = 𝑈))
91, 2grpolid 29256 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑈𝐺𝐴) = 𝐴)
109eqeq2d 2748 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝐴𝐺𝐴) = (𝑈𝐺𝐴) ↔ (𝐴𝐺𝐴) = 𝐴))
118, 10bitr3d 280 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴 = 𝑈 ↔ (𝐴𝐺𝐴) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  ran crn 5631  cfv 6491  (class class class)co 7349  GrpOpcgr 29229  GIdcgi 29230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7662
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5528  df-xp 5636  df-rel 5637  df-cnv 5638  df-co 5639  df-dm 5640  df-rn 5641  df-iota 6443  df-fun 6493  df-fn 6494  df-f 6495  df-fo 6497  df-fv 6499  df-riota 7305  df-ov 7352  df-grpo 29233  df-gid 29234
This theorem is referenced by:  hhssnv  30004  ghomidOLD  36243
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