Theorem isabli 12899

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 2561	. 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 12893	. 2 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
7	1, 3, 6	mpbir2an 942	1 ⊢ 𝐺 ∈ Abel

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2146  ∀wral 2453  ‘cfv 5208  (class class class)co 5865  Basecbs 12428  +_gcplusg 12492  Grpcgrp 12738  Abelcabl 12885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-un 3131  df-in 3133  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-iota 5170  df-fv 5216  df-ov 5868  df-grp 12741  df-cmn 12886  df-abl 12887

This theorem is referenced by: (None)