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Metamath Proof Explorer

Theorem subdii

Description: Distribution of multiplication over subtraction. Theorem I.5 of Apostol p. 18. (Contributed by NM, 26-Nov-1994)

Ref Expression
Hypotheses mulm1.1 ⊢ 𝐴 ∈ ℂ
mulneg.2 ⊢ 𝐵 ∈ ℂ
subdi.3 ⊢ 𝐶 ∈ ℂ
Assertion subdii ( 𝐴 · ( 𝐵 − 𝐶 ) ) = ( ( 𝐴 · 𝐵 ) − ( 𝐴 · 𝐶 ) )

Proof

Step Hyp Ref Expression
1 mulm1.1 ⊢ 𝐴 ∈ ℂ
2 mulneg.2 ⊢ 𝐵 ∈ ℂ
3 subdi.3 ⊢ 𝐶 ∈ ℂ
4 subdi ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 · ( 𝐵 − 𝐶 ) ) = ( ( 𝐴 · 𝐵 ) − ( 𝐴 · 𝐶 ) ) )
5 1 2 3 4 mp3an ⊢ ( 𝐴 · ( 𝐵 − 𝐶 ) ) = ( ( 𝐴 · 𝐵 ) − ( 𝐴 · 𝐶 ) )