Theorem ablogrpo 31149

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 2761	. . 3 ⊢ ran 𝐺 = ran 𝐺
2	1	isablo 31148	. 2 ⊢ (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
3	2	simplbi 502	1 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ran crn 5652  (class class class)co 7420  GrpOpcgr 31091  AbelOpcablo 31146

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-cnv 5659  df-dm 5661  df-rn 5662  df-iota 6494  df-fv 6546  df-ov 7423  df-ablo 31147

This theorem is used by:  ablo32  31151  ablo4  31152  ablomuldiv  31154  ablodivdiv  31155  ablodivdiv4  31156  ablonncan  31158  ablonnncan1  31159  vcgrp  31172  isvcOLD  31181  isvciOLD  31182  cnidOLD  31184  nvgrp  31219  cnnv  31279  cnnvba  31281  cncph  31421  hilid  31763  hhnv  31767  hhba  31769  hhph  31780  hhssabloilem  31863  hhssnv  31866  ablo4pnp  38814  rngogrpo  38844  iscringd  38932