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

Theorem isabli 19002
 Description: Properties that determine an Abelian group. (Contributed by NM, 4-Sep-2011.)
Hypotheses
Ref Expression
isabli.g 𝐺 ∈ Grp
isabli.b 𝐵 = (Base‘𝐺)
isabli.p + = (+g𝐺)
isabli.c ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
Assertion
Ref Expression
isabli 𝐺 ∈ Abel
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐺,𝑦
Allowed substitution hints:   + (𝑥,𝑦)

Proof of Theorem isabli
StepHypRef Expression
1 isabli.g . 2 𝐺 ∈ Grp
2 isabli.c . . 3 ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
32rgen2 3132 . 2 𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)
4 isabli.b . . 3 𝐵 = (Base‘𝐺)
5 isabli.p . . 3 + = (+g𝐺)
64, 5isabl2 18996 . 2 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
71, 3, 6mpbir2an 710 1 𝐺 ∈ Abel
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ‘cfv 6340  (class class class)co 7156  Basecbs 16555  +gcplusg 16637  Grpcgrp 18183  Abelcabl 18988 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-un 3865  df-in 3867  df-ss 3877  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-iota 6299  df-fv 6348  df-ov 7159  df-grp 18186  df-cmn 18989  df-abl 18990 This theorem is referenced by:  cnaddablx  19070  cnaddabl  19071  zaddablx  19074
 Copyright terms: Public domain W3C validator