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Theorem ablcom 18846
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 18835 . 2 (𝐺 ∈ Abel → 𝐺 ∈ CMnd)
2 ablcom.b . . 3 𝐵 = (Base‘𝐺)
3 ablcom.p . . 3 + = (+g𝐺)
42, 3cmncom 18845 . 2 ((𝐺 ∈ CMnd ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
51, 4syl3an1 1157 1 ((𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1081   = wceq 1530  wcel 2106  cfv 6351  (class class class)co 7151  Basecbs 16475  +gcplusg 16557  CMndccmn 18828  Abelcabl 18829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-iota 6311  df-fv 6359  df-ov 7154  df-cmn 18830  df-abl 18831
This theorem is referenced by:  ablinvadd  18852  ablsub2inv  18853  ablsubadd  18854  abladdsub  18857  ablpncan3  18859  ablsub32  18864  ablnnncan  18865  ablsubsub23  18867  eqgabl  18877  subgabl  18878  ablnsg  18889  lsmcomx  18898  qusabl  18907  frgpnabl  18917  ngplcan  23135  clmnegsubdi2  23624  clmvsubval2  23629  ncvspi  23675  r1pid  24668  abliso  30598  lindsunlem  30907  cnaddcom  35976  toycom  35977  lflsub  36071  lfladdcom  36076
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