Theorem isabl2 19763

Description: The predicate "is an Abelian (commutative) group". (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

Hypotheses

Ref	Expression
iscmn.b	⊢ 𝐵 = (Base‘𝐺)
iscmn.p	⊢ + = (+g‘𝐺)

Assertion

Ref	Expression
isabl2	⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐺,𝑦

Allowed substitution hints:   + (𝑥,𝑦)

Proof of Theorem isabl2

Step	Hyp	Ref	Expression
1		isabl 19757	. 2 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
2		grpmnd 18914	. . . 4 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
3		iscmn.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
4		iscmn.p	. . . . . 6 ⊢ + = (+g‘𝐺)
5	3, 4	iscmn 19762	. . . . 5 ⊢ (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
6	5	baib 540	. . . 4 ⊢ (𝐺 ∈ Mnd → (𝐺 ∈ CMnd ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
7	2, 6	syl 17	. . 3 ⊢ (𝐺 ∈ Grp → (𝐺 ∈ CMnd ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
8	7	pm5.32i 579	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd) ↔ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
9	1, 8	bitri 276	1 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))

Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3054  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  +_gcplusg 17218  Mndcmnd 18700  Grpcgrp 18907  CMndccmn 19753  Abelcabl 19754

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-iota 6448  df-fv 6500  df-ov 7366  df-grp 18910  df-cmn 19755  df-abl 19756

This theorem is referenced by:  isabli  19769  invghm  19806  qusabl  19838  abl1  19839  imasabl  19849  archiabllem1  33281  archiabllem2  33285