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Theorem cmncom 19831
Description: A commutative monoid is commutative. (Contributed by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
ablcom.b 𝐵 = (Base‘𝐺)
ablcom.p + = (+g𝐺)
Assertion
Ref Expression
cmncom ((𝐺 ∈ CMnd ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Proof of Theorem cmncom
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ablcom.b . . . . . 6 𝐵 = (Base‘𝐺)
2 ablcom.p . . . . . 6 + = (+g𝐺)
31, 2iscmn 19822 . . . . 5 (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
43simprbi 496 . . . 4 (𝐺 ∈ CMnd → ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥))
5 rsp2 3275 . . . . 5 (∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) → ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥)))
65imp 406 . . . 4 ((∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
74, 6sylan 580 . . 3 ((𝐺 ∈ CMnd ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
87caovcomg 7628 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
983impb 1114 1 ((𝐺 ∈ CMnd ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  Mndcmnd 18760  CMndccmn 19813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-cmn 19815
This theorem is referenced by:  ablcom  19832  cmn32  19833  cmn4  19834  cmn12  19835  cmnbascntr  19838  rinvmod  19839  mulgnn0di  19858  ghmcmn  19864  subcmn  19870  cntzcmn  19873  prdscmnd  19894  srgcom  20224  srgbinomlem4  20247  csrgbinom  20250  crngcom  20269  ip2di  21677  lply1binom  22330  chfacfscmulgsum  22882  chfacfpmmulgsum  22886  cpmadugsumlemF  22898  cmn246135  33021  cmn145236  33022  gsumwun  33051  omndadd2d  33068  rlocaddval  33255  ofldchr  33324  elrspunsn  33437  evl1deg2  33582  evl1deg3  33583  primrootsunit1  42079  aks6d1c5lem3  42119
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