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Theorem ablcom 19831
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 19819 . 2 (𝐺 ∈ Abel → 𝐺 ∈ CMnd)
2 ablcom.b . . 3 𝐵 = (Base‘𝐺)
3 ablcom.p . . 3 + = (+g𝐺)
42, 3cmncom 19830 . 2 ((𝐺 ∈ CMnd ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
51, 4syl3an1 1162 1 ((𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1536  wcel 2105  cfv 6562  (class class class)co 7430  Basecbs 17244  +gcplusg 17297  CMndccmn 19812  Abelcabl 19813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-12 2174  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-iota 6515  df-fv 6570  df-ov 7433  df-cmn 19814  df-abl 19815
This theorem is referenced by:  ablinvadd  19839  ablsub2inv  19840  ablsubadd  19841  abladdsub  19844  ablsubaddsub  19846  ablpncan3  19848  ablsub32  19853  ablnnncan  19854  ablsubsub23  19856  eqgabl  19866  subgabl  19868  ablnsg  19879  lsmcomx  19888  qusabl  19897  frgpnabl  19907  imasabl  19908  subrngringnsg  20569  ngplcan  24639  clmnegsubdi2  25151  clmvsubval2  25156  ncvspi  25203  r1pid  26214  abliso  33023  r1plmhm  33609  lindsunlem  33651  cnaddcom  38953  toycom  38954  lflsub  39048  lfladdcom  39053
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