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Theorem isabld 19657
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))
41grpmndd 18828 . . 3 (𝜑𝐺 ∈ Mnd)
5 isabld.c . . 3 ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
62, 3, 4, 5iscmnd 19656 . 2 (𝜑𝐺 ∈ CMnd)
7 isabl 19646 . 2 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
81, 6, 7sylanbrc 583 1 (𝜑𝐺 ∈ Abel)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1541  wcel 2106  cfv 6540  (class class class)co 7405  Basecbs 17140  +gcplusg 17193  Grpcgrp 18815  CMndccmn 19642  Abelcabl 19643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6492  df-fv 6548  df-ov 7408  df-grp 18818  df-cmn 19644  df-abl 19645
This theorem is referenced by:  subgabl  19698  gex2abl  19713  cygabl  19753  ringabl  20091  lmodabl  20511  dchrabl  26746  tgrpabl  39610  erngdvlem2N  39848  erngdvlem2-rN  39856
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