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Theorem isabloi 27714
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))
32rgen2a 3115 . 2 𝑥𝑋𝑦𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)
4 isabli.2 . . . 4 dom 𝐺 = (𝑋 × 𝑋)
51, 4grporn 27684 . . 3 𝑋 = ran 𝐺
65isablo 27709 . 2 (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
71, 3, 6mpbir2an 993 1 𝐺 ∈ AbelOp
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wral 3050   × cxp 5264  dom cdm 5266  (class class class)co 6813  GrpOpcgr 27652  AbelOpcablo 27707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fo 6055  df-fv 6057  df-ov 6816  df-grpo 27656  df-ablo 27708
This theorem is referenced by:  cnaddabloOLD  27745  hilablo  28326  hhssabloi  28428
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