Theorem isabld 19707

Description: Properties that determine an Abelian group. (Contributed by NM, 6-Aug-2013.)

Hypotheses

Ref	Expression
isabld.b	⊢ (𝜑 → 𝐵 = (Base‘𝐺))
isabld.p	⊢ (𝜑 → + = (+g‘𝐺))
isabld.g	⊢ (𝜑 → 𝐺 ∈ Grp)
isabld.c	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))

Assertion

Ref	Expression
isabld	⊢ (𝜑 → 𝐺 ∈ Abel)

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

Allowed substitution hints:   + (𝑥,𝑦)

Proof of Theorem isabld

Step	Hyp	Ref	Expression
1		isabld.g	. 2 ⊢ (𝜑 → 𝐺 ∈ Grp)
2		isabld.b	. . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝐺))
3		isabld.p	. . 3 ⊢ (𝜑 → + = (+g‘𝐺))
4	1	grpmndd 18859	. . 3 ⊢ (𝜑 → 𝐺 ∈ Mnd)
5		isabld.c	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
6	2, 3, 4, 5	iscmnd 19706	. 2 ⊢ (𝜑 → 𝐺 ∈ CMnd)
7		isabl 19696	. 2 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
8	1, 6, 7	sylanbrc 583	1 ⊢ (𝜑 → 𝐺 ∈ Abel)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ‘cfv 6481  (class class class)co 7346  Basecbs 17120  +_gcplusg 17161  Grpcgrp 18846  CMndccmn 19692  Abelcabl 19693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489  df-ov 7349  df-grp 18849  df-cmn 19694  df-abl 19695

This theorem is referenced by:  subgabl  19748  gex2abl  19763  cygabl  19803  ringabl  20199  lmodabl  20842  dchrabl  27192  tgrpabl  40860  erngdvlem2N  41098  erngdvlem2-rN  41106