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Metamath Proof Explorer

Theorem divdiri

Description: Distribution of division over addition. (Contributed by NM, 16-Feb-1995)

Ref Expression
Hypotheses divclz.1 ⊢ 𝐴 ∈ ℂ
divclz.2 ⊢ 𝐵 ∈ ℂ
divmulz.3 ⊢ 𝐶 ∈ ℂ
divass.4 ⊢ 𝐶 ≠ 0
Assertion divdiri ( ( 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 divass.4 ⊢ 𝐶 ≠ 0
5 1 2 3 divdirzi ⊢ ( 𝐶 ≠ 0 → ( ( 𝐴 + 𝐵 ) / 𝐶 ) = ( ( 𝐴 / 𝐶 ) + ( 𝐵 / 𝐶 ) ) )
6 4 5 ax-mp ⊢ ( ( 𝐴 + 𝐵 ) / 𝐶 ) = ( ( 𝐴 / 𝐶 ) + ( 𝐵 / 𝐶 ) )