Theorem lmod4

Description: Commutative/associative law for left module vector sum. (Contributed by NM, 4-Feb-2014) (Revised by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	lmod4.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	lmod4.p	⊢ + = ( +g ‘ 𝑊 )
Assertion	lmod4	⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ∧ ( 𝑍 ∈ 𝑉 ∧ 𝑈 ∈ 𝑉 ) ) → ( ( 𝑋 + 𝑌 ) + ( 𝑍 + 𝑈 ) ) = ( ( 𝑋 + 𝑍 ) + ( 𝑌 + 𝑈 ) ) )

Proof

Step	Hyp	Ref	Expression
1		lmod4.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		lmod4.p	⊢ + = ( +g ‘ 𝑊 )
3		lmodcmn	⊢ ( 𝑊 ∈ LMod → 𝑊 ∈ CMnd )
4	1 2	cmn4	⊢ ( ( 𝑊 ∈ CMnd ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ∧ ( 𝑍 ∈ 𝑉 ∧ 𝑈 ∈ 𝑉 ) ) → ( ( 𝑋 + 𝑌 ) + ( 𝑍 + 𝑈 ) ) = ( ( 𝑋 + 𝑍 ) + ( 𝑌 + 𝑈 ) ) )
5	3 4	syl3an1	⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ∧ ( 𝑍 ∈ 𝑉 ∧ 𝑈 ∈ 𝑉 ) ) → ( ( 𝑋 + 𝑌 ) + ( 𝑍 + 𝑈 ) ) = ( ( 𝑋 + 𝑍 ) + ( 𝑌 + 𝑈 ) ) )