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Metamath Proof Explorer

Theorem divdirzi

Description: Distribution of division over addition. (Contributed by NM, 31-Jul-2004)

Ref Expression
Hypotheses divclz.1 ⊢ A ∈ ℂ
divclz.2 ⊢ B ∈ ℂ
divmulz.3 ⊢ C ∈ ℂ
Assertion divdirzi ⊢ C ≠ 0 → A + B C = A C + B C

Proof

Step Hyp Ref Expression
1 divclz.1 ⊢ A ∈ ℂ
2 divclz.2 ⊢ B ∈ ℂ
3 divmulz.3 ⊢ C ∈ ℂ
4 divdir ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ ∧ C ≠ 0 → A + B C = A C + B C
5 1 2 4 mp3an12 ⊢ C ∈ ℂ ∧ C ≠ 0 → A + B C = A C + B C
6 3 5 mpan ⊢ C ≠ 0 → A + B C = A C + B C