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Theorem isabl2 18024
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 18020 . 2 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
2 grpmnd 17252 . . . 4 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
3 iscmn.b . . . . . 6 𝐵 = (Base‘𝐺)
4 iscmn.p . . . . . 6 + = (+g𝐺)
53, 4iscmn 18023 . . . . 5 (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
65baib 942 . . . 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 667 . 2 ((𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd) ↔ (𝐺 ∈ Grp ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
91, 8bitri 263 1 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  Mndcmnd 17117  Grpcgrp 17245  CMndccmn 18016  Abelcabl 18017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-grp 17248  df-cmn 18018  df-abl 18019
This theorem is referenced by:  isabli  18030  invghm  18062  qusabl  18091  abl1  18092  archiabllem1  29078  archiabllem2  29082
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