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Theorem ablcom 13111
Description: An Abelian group operation is commutative. (Contributed by NM, 26-Aug-2011.)
Hypotheses
Ref Expression
ablcom.b 𝐵 = (Base‘𝐺)
ablcom.p + = (+g𝐺)
Assertion
Ref Expression
ablcom ((𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Proof of Theorem ablcom
StepHypRef Expression
1 ablcmn 13100 . 2 (𝐺 ∈ Abel → 𝐺 ∈ CMnd)
2 ablcom.b . . 3 𝐵 = (Base‘𝐺)
3 ablcom.p . . 3 + = (+g𝐺)
42, 3cmncom 13110 . 2 ((𝐺 ∈ CMnd ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
51, 4syl3an1 1271 1 ((𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
Colors of variables: wff set class
Syntax hints:  wi 4  w3a 978   = wceq 1353  wcel 2148  cfv 5218  (class class class)co 5877  Basecbs 12464  +gcplusg 12538  CMndccmn 13093  Abelcabl 13094
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 2741  df-un 3135  df-in 3137  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-iota 5180  df-fv 5226  df-ov 5880  df-cmn 13095  df-abl 13096
This theorem is referenced by:  ablinvadd  13118  ablsub2inv  13119  ablsubadd  13120  abladdsub  13123  ablpncan3  13125  ablsub32  13130  ablnnncan  13131  ablsubsub23  13133
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