MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cmncom Structured version   Visualization version   GIF version

Theorem cmncom 18562
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 18553 . . . . 5 (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
43simprbi 492 . . . 4 (𝐺 ∈ CMnd → ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥))
5 rsp2 3145 . . . . 5 (∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) → ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥)))
65imp 397 . . . 4 ((∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
74, 6sylan 577 . . 3 ((𝐺 ∈ CMnd ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
87caovcomg 7089 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
983impb 1149 1 ((𝐺 ∈ CMnd ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1113   = wceq 1658  wcel 2166  wral 3117  cfv 6123  (class class class)co 6905  Basecbs 16222  +gcplusg 16305  Mndcmnd 17647  CMndccmn 18546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-iota 6086  df-fv 6131  df-ov 6908  df-cmn 18548
This theorem is referenced by:  ablcom  18563  cmn32  18564  cmn4  18565  cmn12  18566  mulgnn0di  18584  ghmcmn  18590  subcmn  18595  cntzcmn  18598  prdscmnd  18617  srgcom  18879  srgbinomlem4  18897  csrgbinom  18900  crngcom  18916  lply1binom  20036  ip2di  20348  chfacfscmulgsum  21035  chfacfpmmulgsum  21039  cpmadugsumlemF  21051  omndadd2d  30253  ofldchr  30359
  Copyright terms: Public domain W3C validator