Theorem isabli 19765

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 3178	. 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 19759	. 2 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
7	1, 3, 6	mpbir2an 712	1 ⊢ 𝐺 ∈ Abel

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ‘cfv 6493  (class class class)co 7361  Basecbs 17173  +_gcplusg 17214  Grpcgrp 18903  Abelcabl 19750

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-iota 6449  df-fv 6501  df-ov 7364  df-grp 18906  df-cmn 19751  df-abl 19752

This theorem is referenced by:  cnaddablx  19837  cnaddabl  19838  zaddablx  19841