Theorem ablogrpo 30636

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.

Step	Hyp	Ref	Expression
1		eqid 2737	. . 3 ⊢ ran 𝐺 = ran 𝐺
2	1	isablo 30635	. 2 ⊢ (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
3	2	simplbi 496	1 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ran crn 5626  (class class class)co 7361  GrpOpcgr 30578  AbelOpcablo 30633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-cnv 5633  df-dm 5635  df-rn 5636  df-iota 6449  df-fv 6501  df-ov 7364  df-ablo 30634

This theorem is referenced by:  ablo32  30638  ablo4  30639  ablomuldiv  30641  ablodivdiv  30642  ablodivdiv4  30643  ablonncan  30645  ablonnncan1  30646  vcgrp  30659  isvcOLD  30668  isvciOLD  30669  cnidOLD  30671  nvgrp  30706  cnnv  30766  cnnvba  30768  cncph  30908  hilid  31250  hhnv  31254  hhba  31256  hhph  31267  hhssabloilem  31350  hhssnv  31353  ablo4pnp  38218  rngogrpo  38248  iscringd  38336