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Theorem ablocom 30567
Description: An Abelian group operation is commutative. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
ablcom.1 𝑋 = ran 𝐺
Assertion
Ref Expression
ablocom ((𝐺 ∈ AbelOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))

Proof of Theorem ablocom
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ablcom.1 . . . . 5 𝑋 = ran 𝐺
21isablo 30565 . . . 4 (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
32simprbi 496 . . 3 (𝐺 ∈ AbelOp → ∀𝑥𝑋𝑦𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥))
4 oveq1 7438 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
5 oveq2 7439 . . . . 5 (𝑥 = 𝐴 → (𝑦𝐺𝑥) = (𝑦𝐺𝐴))
64, 5eqeq12d 2753 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐺𝑦) = (𝑦𝐺𝑥) ↔ (𝐴𝐺𝑦) = (𝑦𝐺𝐴)))
7 oveq2 7439 . . . . 5 (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
8 oveq1 7438 . . . . 5 (𝑦 = 𝐵 → (𝑦𝐺𝐴) = (𝐵𝐺𝐴))
97, 8eqeq12d 2753 . . . 4 (𝑦 = 𝐵 → ((𝐴𝐺𝑦) = (𝑦𝐺𝐴) ↔ (𝐴𝐺𝐵) = (𝐵𝐺𝐴)))
106, 9rspc2v 3633 . . 3 ((𝐴𝑋𝐵𝑋) → (∀𝑥𝑋𝑦𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴)))
113, 10syl5com 31 . 2 (𝐺 ∈ AbelOp → ((𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴)))
12113impib 1117 1 ((𝐺 ∈ AbelOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  ran crn 5686  (class class class)co 7431  GrpOpcgr 30508  AbelOpcablo 30563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-cnv 5693  df-dm 5695  df-rn 5696  df-iota 6514  df-fv 6569  df-ov 7434  df-ablo 30564
This theorem is referenced by:  ablo32  30568  ablomuldiv  30571  ablodiv32  30574  nvcom  30640  rngocom  37920  iscringd  38005
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