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Theorem ablcom 19651
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 19639 . 2 (𝐺 ∈ Abel → 𝐺 ∈ CMnd)
2 ablcom.b . . 3 𝐵 = (Base‘𝐺)
3 ablcom.p . . 3 + = (+g𝐺)
42, 3cmncom 19650 . 2 ((𝐺 ∈ CMnd ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
51, 4syl3an1 1164 1 ((𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088   = wceq 1542  wcel 2107  cfv 6535  (class class class)co 7396  Basecbs 17131  +gcplusg 17184  CMndccmn 19632  Abelcabl 19633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-iota 6487  df-fv 6543  df-ov 7399  df-cmn 19634  df-abl 19635
This theorem is referenced by:  ablinvadd  19658  ablsub2inv  19659  ablsubadd  19660  abladdsub  19663  ablsubaddsub  19665  ablpncan3  19667  ablsub32  19672  ablnnncan  19673  ablsubsub23  19675  eqgabl  19685  subgabl  19687  ablnsg  19698  lsmcomx  19707  qusabl  19716  frgpnabl  19726  imasabl  19727  ngplcan  24089  clmnegsubdi2  24590  clmvsubval2  24595  ncvspi  24642  r1pid  25646  abliso  32168  lindsunlem  32647  cnaddcom  37748  toycom  37749  lflsub  37843  lfladdcom  37848  subrngringnsg  46603
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