Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-cmnssmnd Structured version   Visualization version   GIF version

Theorem bj-cmnssmnd 34600
 Description: Commutative monoids are monoids. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-cmnssmnd CMnd ⊆ Mnd

Proof of Theorem bj-cmnssmnd
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cmn 18904 . 2 CMnd = {𝑥 ∈ Mnd ∣ ∀𝑦 ∈ (Base‘𝑥)∀𝑧 ∈ (Base‘𝑥)(𝑦(+g𝑥)𝑧) = (𝑧(+g𝑥)𝑦)}
21ssrab3 4042 1 CMnd ⊆ Mnd
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538  ∀wral 3133   ⊆ wss 3919  ‘cfv 6343  (class class class)co 7145  Basecbs 16479  +gcplusg 16561  Mndcmnd 17907  CMndccmn 18902 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142  df-v 3482  df-in 3926  df-ss 3936  df-cmn 18904 This theorem is referenced by:  bj-cmnssmndel  34601
 Copyright terms: Public domain W3C validator