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Theorem ablogrpo 27529
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 2651 . . 3 ran 𝐺 = ran 𝐺
21isablo 27528 . 2 (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺(𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
32simplbi 475 1 (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  wral 2941  ran crn 5144  (class class class)co 6690  GrpOpcgr 27471  AbelOpcablo 27526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-cnv 5151  df-dm 5153  df-rn 5154  df-iota 5889  df-fv 5934  df-ov 6693  df-ablo 27527
This theorem is referenced by:  ablo32  27531  ablo4  27532  ablomuldiv  27534  ablodivdiv  27535  ablodivdiv4  27536  ablonnncan  27538  ablonncan  27539  ablonnncan1  27540  vcgrp  27553  isvcOLD  27562  isvciOLD  27563  cnidOLD  27565  nvgrp  27600  cnnv  27660  cnnvba  27662  cncph  27802  hilid  28146  hhnv  28150  hhba  28152  hhph  28163  hhssabloilem  28246  hhssnv  28249  ablo4pnp  33809  rngogrpo  33839  iscringd  33927
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