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Metamath Proof Explorer

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 ) ) )