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Theorem isabl2 19809
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 + = (+g𝐺)
Assertion
Ref Expression
isabl2 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐺,𝑦
Allowed substitution hints:   + (𝑥,𝑦)

Proof of Theorem isabl2
StepHypRef Expression
1 isabl 19803 . 2 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
2 grpmnd 18959 . . . 4 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
3 iscmn.b . . . . . 6 𝐵 = (Base‘𝐺)
4 iscmn.p . . . . . 6 + = (+g𝐺)
53, 4iscmn 19808 . . . . 5 (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
65baib 535 . . . 4 (𝐺 ∈ Mnd → (𝐺 ∈ CMnd ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
72, 6syl 17 . . 3 (𝐺 ∈ Grp → (𝐺 ∈ CMnd ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
87pm5.32i 574 . 2 ((𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd) ↔ (𝐺 ∈ Grp ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
91, 8bitri 275 1 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1539  wcel 2107  wral 3060  cfv 6560  (class class class)co 7432  Basecbs 17248  +gcplusg 17298  Mndcmnd 18748  Grpcgrp 18952  CMndccmn 19799  Abelcabl 19800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-ov 7435  df-grp 18955  df-cmn 19801  df-abl 19802
This theorem is referenced by:  isabli  19815  invghm  19852  qusabl  19884  abl1  19885  imasabl  19895  archiabllem1  33201  archiabllem2  33205
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