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Theorem isabld 18413
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 17637 . . . 4 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
51, 4syl 17 . . 3 (𝜑𝐺 ∈ Mnd)
6 isabld.c . . 3 ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
72, 3, 5, 6iscmnd 18412 . 2 (𝜑𝐺 ∈ CMnd)
8 isabl 18404 . 2 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
91, 7, 8sylanbrc 572 1 (𝜑𝐺 ∈ Abel)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1071   = wceq 1631  wcel 2145  cfv 6030  (class class class)co 6796  Basecbs 16064  +gcplusg 16149  Mndcmnd 17502  Grpcgrp 17630  CMndccmn 18400  Abelcabl 18401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-iota 5993  df-fv 6038  df-ov 6799  df-grp 17633  df-cmn 18402  df-abl 18403
This theorem is referenced by:  subgabl  18448  gex2abl  18461  cygabl  18499  ringabl  18788  lmodabl  19120  dchrabl  25200  tgrpabl  36561  erngdvlem2N  36799  erngdvlem2-rN  36807
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