Theorem ablogrpo 30534

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 2731	. . 3 ⊢ ran 𝐺 = ran 𝐺
2	1	isablo 30533	. 2 ⊢ (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
3	2	simplbi 497	1 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ran crn 5620  (class class class)co 7352  GrpOpcgr 30476  AbelOpcablo 30531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-cnv 5627  df-dm 5629  df-rn 5630  df-iota 6443  df-fv 6495  df-ov 7355  df-ablo 30532

This theorem is referenced by:  ablo32  30536  ablo4  30537  ablomuldiv  30539  ablodivdiv  30540  ablodivdiv4  30541  ablonncan  30543  ablonnncan1  30544  vcgrp  30557  isvcOLD  30566  isvciOLD  30567  cnidOLD  30569  nvgrp  30604  cnnv  30664  cnnvba  30666  cncph  30806  hilid  31148  hhnv  31152  hhba  31154  hhph  31165  hhssabloilem  31248  hhssnv  31251  ablo4pnp  37926  rngogrpo  37956  iscringd  38044