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

Theorem isabl 18129
Description: The predicate "is an Abelian (commutative) group." (Contributed by NM, 17-Oct-2011.)
Assertion
Ref Expression
isabl (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))

Proof of Theorem isabl
StepHypRef Expression
1 df-abl 18128 . 2 Abel = (Grp ∩ CMnd)
21elin2 3784 1 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  wcel 1987  Grpcgrp 17354  CMndccmn 18125  Abelcabl 18126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3191  df-in 3566  df-abl 18128
This theorem is referenced by:  ablgrp  18130  ablcmn  18131  isabl2  18133  ablpropd  18135  isabld  18138  ghmabl  18170  prdsabld  18197  unitabl  18600  tsmsinv  21874  tgptsmscls  21876  tsmsxplem1  21879  tsmsxplem2  21880  abliso  29505  gicabl  37184  2zrngaabl  41258  pgrpgt2nabl  41461
  Copyright terms: Public domain W3C validator