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Theorem isabld 18029
 Description: Properties that determine an Abelian group. (Contributed by NM, 6-Aug-2013.)
Hypotheses
Ref Expression
isabld.b (𝜑𝐵 = (Base‘𝐺))
isabld.p (𝜑+ = (+g𝐺))
isabld.g (𝜑𝐺 ∈ Grp)
isabld.c ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
Assertion
Ref Expression
isabld (𝜑𝐺 ∈ Abel)
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐺,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   + (𝑥,𝑦)

Proof of Theorem isabld
StepHypRef Expression
1 isabld.g . 2 (𝜑𝐺 ∈ Grp)
2 isabld.b . . 3 (𝜑𝐵 = (Base‘𝐺))
3 isabld.p . . 3 (𝜑+ = (+g𝐺))
4 grpmnd 17252 . . . 4 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
51, 4syl 17 . . 3 (𝜑𝐺 ∈ Mnd)
6 isabld.c . . 3 ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
72, 3, 5, 6iscmnd 18028 . 2 (𝜑𝐺 ∈ CMnd)
8 isabl 18020 . 2 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
91, 7, 8sylanbrc 695 1 (𝜑𝐺 ∈ Abel)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ‘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:  subgabl  18064  gex2abl  18077  cygabl  18115  ringabl  18403  lmodabl  18733  dchrabl  24779  tgrpabl  35057  erngdvlem2N  35295  erngdvlem2-rN  35303
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