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Theorem isabld 13055
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 12843 . . 3 (𝜑𝐺 ∈ Mnd)
5 isabld.c . . 3 ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
62, 3, 4, 5iscmnd 13054 . 2 (𝜑𝐺 ∈ CMnd)
7 isabl 13045 . 2 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
81, 6, 7sylanbrc 417 1 (𝜑𝐺 ∈ Abel)
Colors of variables: wff set class
Syntax hints:  wi 4  w3a 978   = wceq 1353  wcel 2148  cfv 5216  (class class class)co 5874  Basecbs 12456  +gcplusg 12530  Grpcgrp 12831  CMndccmn 13041  Abelcabl 13042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-un 3133  df-in 3135  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-iota 5178  df-fv 5224  df-ov 5877  df-grp 12834  df-cmn 13043  df-abl 13044
This theorem is referenced by:  ringabl  13168
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