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Theorem isabloi 29291
Description: Properties that determine an Abelian group operation. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
isabli.1 𝐺 ∈ GrpOp
isabli.2 dom 𝐺 = (𝑋 × 𝑋)
isabli.3 ((𝑥𝑋𝑦𝑋) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))
Assertion
Ref Expression
isabloi 𝐺 ∈ AbelOp
Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝑋,𝑦

Proof of Theorem isabloi
StepHypRef Expression
1 isabli.1 . 2 𝐺 ∈ GrpOp
2 isabli.3 . . 3 ((𝑥𝑋𝑦𝑋) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))
32rgen2 3192 . 2 𝑥𝑋𝑦𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)
4 isabli.2 . . . 4 dom 𝐺 = (𝑋 × 𝑋)
51, 4grporn 29261 . . 3 𝑋 = ran 𝐺
65isablo 29286 . 2 (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
71, 3, 6mpbir2an 709 1 𝐺 ∈ AbelOp
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3062   × cxp 5628  dom cdm 5630  (class class class)co 7349  GrpOpcgr 29229  AbelOpcablo 29284
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-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-ov 7352  df-grpo 29233  df-ablo 29285
This theorem is referenced by:  cnaddabloOLD  29321  hilablo  29900  hhssabloi  30002
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