Theorem isabli 19829

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 3197	. 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 19823	. 2 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
7	1, 3, 6	mpbir2an 711	1 ⊢ 𝐺 ∈ Abel

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  +_gcplusg 17298  Grpcgrp 18964  Abelcabl 19814

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-grp 18967  df-cmn 19815  df-abl 19816

This theorem is referenced by:  cnaddablx  19901  cnaddabl  19902  zaddablx  19905