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Theorem ablogrpo 26582
Description: An Abelian group operation is a group operation. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)
Assertion
Ref Expression
ablogrpo (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)

Proof of Theorem ablogrpo
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2609 . . 3 ran 𝐺 = ran 𝐺
21isablo 26581 . 2 (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺(𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
32simplbi 474 1 (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  wral 2895  ran crn 5029  (class class class)co 6527  GrpOpcgr 26521  AbelOpcablo 26579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-cnv 5036  df-dm 5038  df-rn 5039  df-iota 5754  df-fv 5798  df-ov 6530  df-ablo 26580
This theorem is referenced by:  ablo32  26584  ablo4  26585  ablomuldiv  26587  ablodivdiv  26588  ablodivdiv4  26589  ablonnncan  26591  ablonncan  26592  ablonnncan1  26593  vcgrp  26607  vcoprnelem  26627  isvcOLD  26630  isvciOLD  26631  cnidOLD  26633  nvgrp  26668  cnnv  26740  cnnvba  26742  cncph  26892  hilid  27236  hhnv  27240  hhba  27242  hhph  27253  hhssabloilem  27336  hhssnv  27339  ablo4pnp  32673  rngogrpo  32703  iscringd  32791
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