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Theorem ablocom 27742
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 27740 . . . 4 (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
32simprbi 484 . . 3 (𝐺 ∈ AbelOp → ∀𝑥𝑋𝑦𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥))
4 oveq1 6803 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
5 oveq2 6804 . . . . 5 (𝑥 = 𝐴 → (𝑦𝐺𝑥) = (𝑦𝐺𝐴))
64, 5eqeq12d 2786 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐺𝑦) = (𝑦𝐺𝑥) ↔ (𝐴𝐺𝑦) = (𝑦𝐺𝐴)))
7 oveq2 6804 . . . . 5 (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
8 oveq1 6803 . . . . 5 (𝑦 = 𝐵 → (𝑦𝐺𝐴) = (𝐵𝐺𝐴))
97, 8eqeq12d 2786 . . . 4 (𝑦 = 𝐵 → ((𝐴𝐺𝑦) = (𝑦𝐺𝐴) ↔ (𝐴𝐺𝐵) = (𝐵𝐺𝐴)))
106, 9rspc2v 3472 . . 3 ((𝐴𝑋𝐵𝑋) → (∀𝑥𝑋𝑦𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴)))
113, 10syl5com 31 . 2 (𝐺 ∈ AbelOp → ((𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴)))
12113impib 1108 1 ((𝐺 ∈ AbelOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  ran crn 5251  (class class class)co 6796  GrpOpcgr 27683  AbelOpcablo 27738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-cnv 5258  df-dm 5260  df-rn 5261  df-iota 5993  df-fv 6038  df-ov 6799  df-ablo 27739
This theorem is referenced by:  ablo32  27743  ablomuldiv  27746  ablodiv32  27749  nvcom  27816  rngocom  34044  iscringd  34129
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