Theorem ablogrpo 28958

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 2736	. . 3 ⊢ ran 𝐺 = ran 𝐺
2	1	isablo 28957	. 2 ⊢ (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
3	2	simplbi 499	1 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2104  ∀wral 3062  ran crn 5601  (class class class)co 7307  GrpOpcgr 28900  AbelOpcablo 28955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3063  df-rab 3306  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-cnv 5608  df-dm 5610  df-rn 5611  df-iota 6410  df-fv 6466  df-ov 7310  df-ablo 28956

This theorem is referenced by:  ablo32  28960  ablo4  28961  ablomuldiv  28963  ablodivdiv  28964  ablodivdiv4  28965  ablonncan  28967  ablonnncan1  28968  vcgrp  28981  isvcOLD  28990  isvciOLD  28991  cnidOLD  28993  nvgrp  29028  cnnv  29088  cnnvba  29090  cncph  29230  hilid  29572  hhnv  29576  hhba  29578  hhph  29589  hhssabloilem  29672  hhssnv  29675  ablo4pnp  36086  rngogrpo  36116  iscringd  36204