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Metamath Proof Explorer

Theorem subdird

Description: Distribution of multiplication over subtraction. Theorem I.5 of Apostol p. 18. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses mulm1d.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
mulnegd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
subdid.3 ⊢ ( 𝜑 → 𝐶 ∈ ℂ )
Assertion subdird ( 𝜑 → ( ( 𝐴 − 𝐵 ) · 𝐶 ) = ( ( 𝐴 · 𝐶 ) − ( 𝐵 · 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 mulm1d.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2 mulnegd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3 subdid.3 ⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4 subdir ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) · 𝐶 ) = ( ( 𝐴 · 𝐶 ) − ( 𝐵 · 𝐶 ) ) )
5 1 2 3 4 syl3anc ⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) · 𝐶 ) = ( ( 𝐴 · 𝐶 ) − ( 𝐵 · 𝐶 ) ) )