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Theorem ablcom 19666
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 19654 . 2 (𝐺 ∈ Abel → 𝐺 ∈ CMnd)
2 ablcom.b . . 3 𝐵 = (Base‘𝐺)
3 ablcom.p . . 3 + = (+g𝐺)
42, 3cmncom 19665 . 2 ((𝐺 ∈ CMnd ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
51, 4syl3an1 1163 1 ((𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1541  wcel 2106  cfv 6543  (class class class)co 7408  Basecbs 17143  +gcplusg 17196  CMndccmn 19647  Abelcabl 19648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7411  df-cmn 19649  df-abl 19650
This theorem is referenced by:  ablinvadd  19674  ablsub2inv  19675  ablsubadd  19676  abladdsub  19679  ablsubaddsub  19681  ablpncan3  19683  ablsub32  19688  ablnnncan  19689  ablsubsub23  19691  eqgabl  19701  subgabl  19703  ablnsg  19714  lsmcomx  19723  qusabl  19732  frgpnabl  19742  imasabl  19743  ngplcan  24119  clmnegsubdi2  24620  clmvsubval2  24625  ncvspi  24672  r1pid  25676  abliso  32192  lindsunlem  32704  cnaddcom  37837  toycom  37838  lflsub  37932  lfladdcom  37937  subrngringnsg  46722
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