Theorem isabli 19865

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	⊢ + = (+g‘𝐺)
isabli.c	⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (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

Step	Hyp	Ref	Expression
1		isabli.g	. 2 ⊢ 𝐺 ∈ Grp
2		isabli.c	. . 3 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
3	2	rgen2 3211	. 2 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)
4		isabli.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
5		isabli.p	. . 3 ⊢ + = (+g‘𝐺)
6	4, 5	isabl2 19859	. 2 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
7	1, 3, 6	mpbir2an 723	1 ⊢ 𝐺 ∈ Abel

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ‘cfv 6537  (class class class)co 7411  Basecbs 17268  +_gcplusg 17309  Grpcgrp 18999  Abelcabl 19850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-iota 6493  df-fv 6545  df-ov 7414  df-grp 19002  df-cmn 19851  df-abl 19852

This theorem is referenced by:  cnaddablx  19937  cnaddabl  19938  zaddablx  19941