MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isabld Structured version   Visualization version   GIF version

Theorem isabld 19663
Description: Properties that determine an Abelian group. (Contributed by NM, 6-Aug-2013.)
Hypotheses
Ref Expression
isabld.b (𝜑𝐵 = (Base‘𝐺))
isabld.p (𝜑+ = (+g𝐺))
isabld.g (𝜑𝐺 ∈ Grp)
isabld.c ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
Assertion
Ref Expression
isabld (𝜑𝐺 ∈ Abel)
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐺,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   + (𝑥,𝑦)

Proof of Theorem isabld
StepHypRef Expression
1 isabld.g . 2 (𝜑𝐺 ∈ Grp)
2 isabld.b . . 3 (𝜑𝐵 = (Base‘𝐺))
3 isabld.p . . 3 (𝜑+ = (+g𝐺))
41grpmndd 18832 . . 3 (𝜑𝐺 ∈ Mnd)
5 isabld.c . . 3 ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
62, 3, 4, 5iscmnd 19662 . 2 (𝜑𝐺 ∈ CMnd)
7 isabl 19652 . 2 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
81, 6, 7sylanbrc 584 1 (𝜑𝐺 ∈ Abel)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088   = wceq 1542  wcel 2107  cfv 6544  (class class class)co 7409  Basecbs 17144  +gcplusg 17197  Grpcgrp 18819  CMndccmn 19648  Abelcabl 19649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-grp 18822  df-cmn 19650  df-abl 19651
This theorem is referenced by:  subgabl  19704  gex2abl  19719  cygabl  19759  ringabl  20098  lmodabl  20519  dchrabl  26757  tgrpabl  39622  erngdvlem2N  39860  erngdvlem2-rN  39868
  Copyright terms: Public domain W3C validator