Theorem mndsgrp

Description: A monoid is a semigroup. (Contributed by FL, 2-Nov-2009) (Revised by AV, 6-Jan-2020) (Proof shortened by AV, 6-Feb-2020)

	Ref	Expression
Assertion	mndsgrp	⊢ ( 𝐺 ∈ Mnd → 𝐺 ∈ Smgrp )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 )
2		eqid	⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
3	1 2	ismnddef	⊢ ( 𝐺 ∈ Mnd ↔ ( 𝐺 ∈ Smgrp ∧ ∃ 𝑒 ∈ ( Base ‘ 𝐺 ) ∀ 𝑥 ∈ ( Base ‘ 𝐺 ) ( ( 𝑒 ( +g ‘ 𝐺 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ 𝐺 ) 𝑒 ) = 𝑥 ) ) )
4	3	simplbi	⊢ ( 𝐺 ∈ Mnd → 𝐺 ∈ Smgrp )