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Theorem isabloi 29279
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 3193 . 2 𝑥𝑋𝑦𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)
4 isabli.2 . . . 4 dom 𝐺 = (𝑋 × 𝑋)
51, 4grporn 29249 . . 3 𝑋 = ran 𝐺
65isablo 29274 . 2 (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
71, 3, 6mpbir2an 710 1 𝐺 ∈ AbelOp
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3063   × cxp 5629  dom cdm 5631  (class class class)co 7350  GrpOpcgr 29217  AbelOpcablo 29272
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-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-ov 7353  df-grpo 29221  df-ablo 29273
This theorem is referenced by:  cnaddabloOLD  29309  hilablo  29888  hhssabloi  29990
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