Theorem bj-cmnssmnd 38194

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.

Step	Hyp	Ref	Expression
1		df-cmn 19996	. 2 ⊢ CMnd = {𝑥 ∈ Mnd ∣ ∀𝑦 ∈ (Base‘𝑥)∀𝑧 ∈ (Base‘𝑥)(𝑦(+g‘𝑥)𝑧) = (𝑧(+g‘𝑥)𝑦)}
2	1	ssrab3 4007	1 ⊢ CMnd ⊆ Mnd

This proof depends on syntax axioms:   = wceq 1569  ∀wral 3076   ⊆ wss 3895  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  +_gcplusg 17428  Mndcmnd 18923  CMndccmn 19994

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-ss 3896  df-cmn 19996

This theorem is used by:  bj-cmnssmndel  38195