Theorem isabli 19625

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 3196	. 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 19619	. 2 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
7	1, 3, 6	mpbir2an 709	1 ⊢ 𝐺 ∈ Abel

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3060  ‘cfv 6529  (class class class)co 7390  Basecbs 17123  +_gcplusg 17176  Grpcgrp 18791  Abelcabl 19610

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3430  df-v 3472  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4520  df-sn 4620  df-pr 4622  df-op 4626  df-uni 4899  df-br 5139  df-iota 6481  df-fv 6537  df-ov 7393  df-grp 18794  df-cmn 19611  df-abl 19612

This theorem is referenced by:  cnaddablx  19693  cnaddabl  19694  zaddablx  19697