Theorem slmdass

Description: Semiring left module vector sum is associative. (Contributed by NM, 10-Jan-2014) (Revised by Mario Carneiro, 19-Jun-2014) (Revised by Thierry Arnoux, 1-Apr-2018)

	Ref	Expression
Hypotheses	slmdvacl.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	slmdvacl.a	⊢ + = ( +g ‘ 𝑊 )
Assertion	slmdass	⊢ ( ( 𝑊 ∈ SLMod ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉 ) ) → ( ( 𝑋 + 𝑌 ) + 𝑍 ) = ( 𝑋 + ( 𝑌 + 𝑍 ) ) )

Proof

Step	Hyp	Ref	Expression
1		slmdvacl.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		slmdvacl.a	⊢ + = ( +g ‘ 𝑊 )
3		slmdmnd	⊢ ( 𝑊 ∈ SLMod → 𝑊 ∈ Mnd )
4	1 2	mndass	⊢ ( ( 𝑊 ∈ Mnd ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉 ) ) → ( ( 𝑋 + 𝑌 ) + 𝑍 ) = ( 𝑋 + ( 𝑌 + 𝑍 ) ) )
5	3 4	sylan	⊢ ( ( 𝑊 ∈ SLMod ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉 ) ) → ( ( 𝑋 + 𝑌 ) + 𝑍 ) = ( 𝑋 + ( 𝑌 + 𝑍 ) ) )