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Theorem grpoid 29248
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 29242 . . . . 5 (𝐺 ∈ GrpOp → 𝑈𝑋)
41grporcan 29246 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝑈𝑋𝐴𝑋)) → ((𝐴𝐺𝐴) = (𝑈𝐺𝐴) ↔ 𝐴 = 𝑈))
543exp2 1355 . . . . 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 408 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝐴𝐺𝐴) = (𝑈𝐺𝐴) ↔ 𝐴 = 𝑈))
91, 2grpolid 29244 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑈𝐺𝐴) = 𝐴)
109eqeq2d 2749 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝐴𝐺𝐴) = (𝑈𝐺𝐴) ↔ (𝐴𝐺𝐴) = 𝐴))
118, 10bitr3d 281 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴 = 𝑈 ↔ (𝐴𝐺𝐴) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  ran crn 5632  cfv 6492  (class class class)co 7350  GrpOpcgr 29217  GIdcgi 29218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383  ax-un 7663
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6444  df-fun 6494  df-fn 6495  df-f 6496  df-fo 6498  df-fv 6500  df-riota 7306  df-ov 7353  df-grpo 29221  df-gid 29222
This theorem is referenced by:  hhssnv  29992  ghomidOLD  36234
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