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Theorem cmncom 19707
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 19698 . . . . 5 (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
43simprbi 495 . . . 4 (𝐺 ∈ CMnd → ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥))
5 rsp2 3272 . . . . 5 (∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) → ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥)))
65imp 405 . . . 4 ((∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
74, 6sylan 578 . . 3 ((𝐺 ∈ CMnd ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
87caovcomg 7604 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
983impb 1113 1 ((𝐺 ∈ CMnd ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1085   = wceq 1539  wcel 2104  wral 3059  cfv 6542  (class class class)co 7411  Basecbs 17148  +gcplusg 17201  Mndcmnd 18659  CMndccmn 19689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-12 2169  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6494  df-fv 6550  df-ov 7414  df-cmn 19691
This theorem is referenced by:  ablcom  19708  cmn32  19709  cmn4  19710  cmn12  19711  cmnbascntr  19714  rinvmod  19715  mulgnn0di  19734  ghmcmn  19740  subcmn  19746  cntzcmn  19749  prdscmnd  19770  srgcom  20100  srgbinomlem4  20123  csrgbinom  20126  crngcom  20145  ip2di  21413  lply1binom  22050  chfacfscmulgsum  22582  chfacfpmmulgsum  22586  cpmadugsumlemF  22598  omndadd2d  32496  ofldchr  32702  elrspunsn  32821
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