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Metamath Proof Explorer

Theorem divsubdird

Description: Distribution of division over subtraction. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses div1d.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
divcld.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
divmuld.3 ⊢ ( 𝜑 → 𝐶 ∈ ℂ )
divassd.4 ⊢ ( 𝜑 → 𝐶 ≠ 0 )
Assertion divsubdird ( 𝜑 → ( ( 𝐴 − 𝐵 ) / 𝐶 ) = ( ( 𝐴 / 𝐶 ) − ( 𝐵 / 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 div1d.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2 divcld.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3 divmuld.3 ⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4 divassd.4 ⊢ ( 𝜑 → 𝐶 ≠ 0 )
5 divsubdir ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐴 − 𝐵 ) / 𝐶 ) = ( ( 𝐴 / 𝐶 ) − ( 𝐵 / 𝐶 ) ) )
6 1 2 3 4 5 syl112anc ⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) / 𝐶 ) = ( ( 𝐴 / 𝐶 ) − ( 𝐵 / 𝐶 ) ) )