Theorem lmodass

Description: Left module vector sum is associative. (Contributed by NM, 10-Jan-2014) (Revised by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	lmodvacl.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	lmodvacl.a	⊢ + = ( +g ‘ 𝑊 )
Assertion	lmodass	⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉 ) ) → ( ( 𝑋 + 𝑌 ) + 𝑍 ) = ( 𝑋 + ( 𝑌 + 𝑍 ) ) )

Proof

Step	Hyp	Ref	Expression
1		lmodvacl.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		lmodvacl.a	⊢ + = ( +g ‘ 𝑊 )
3		lmodgrp	⊢ ( 𝑊 ∈ LMod → 𝑊 ∈ Grp )
4	1 2	grpass	⊢ ( ( 𝑊 ∈ Grp ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉 ) ) → ( ( 𝑋 + 𝑌 ) + 𝑍 ) = ( 𝑋 + ( 𝑌 + 𝑍 ) ) )
5	3 4	sylan	⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉 ) ) → ( ( 𝑋 + 𝑌 ) + 𝑍 ) = ( 𝑋 + ( 𝑌 + 𝑍 ) ) )