Theorem mndass

Description: A monoid operation is associative. (Contributed by NM, 14-Aug-2011) (Proof shortened by AV, 8-Feb-2020)

	Ref	Expression
Hypotheses	mndcl.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	mndcl.p	⊢ + = ( +g ‘ 𝐺 )
Assertion	mndass	⊢ ( ( 𝐺 ∈ Mnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) + 𝑍 ) = ( 𝑋 + ( 𝑌 + 𝑍 ) ) )

Proof

Step	Hyp	Ref	Expression
1		mndcl.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		mndcl.p	⊢ + = ( +g ‘ 𝐺 )
3		mndsgrp	⊢ ( 𝐺 ∈ Mnd → 𝐺 ∈ Smgrp )
4	1 2	sgrpass	⊢ ( ( 𝐺 ∈ Smgrp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) + 𝑍 ) = ( 𝑋 + ( 𝑌 + 𝑍 ) ) )
5	3 4	sylan	⊢ ( ( 𝐺 ∈ Mnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) + 𝑍 ) = ( 𝑋 + ( 𝑌 + 𝑍 ) ) )