Theorem muldivdid

Description: Distribution of division over addition with a multiplication. (Contributed by Thierry Arnoux, 6-Jul-2025)

	Ref	Expression
Hypotheses	muldivdid.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	muldivdid.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	muldivdid.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
	muldivdid.4	⊢ ( 𝜑 → 𝐵 ≠ 0 )
Assertion	muldivdid	⊢ ( 𝜑 → ( ( ( 𝐴 · 𝐵 ) + 𝐶 ) / 𝐵 ) = ( 𝐴 + ( 𝐶 / 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		muldivdid.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		muldivdid.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		muldivdid.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4		muldivdid.4	⊢ ( 𝜑 → 𝐵 ≠ 0 )
5	1 2	mulcomd	⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) )
6	5	oveq1d	⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) + 𝐶 ) = ( ( 𝐵 · 𝐴 ) + 𝐶 ) )
7	6	oveq1d	⊢ ( 𝜑 → ( ( ( 𝐴 · 𝐵 ) + 𝐶 ) / 𝐵 ) = ( ( ( 𝐵 · 𝐴 ) + 𝐶 ) / 𝐵 ) )
8		muldivdir	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( ( 𝐵 · 𝐴 ) + 𝐶 ) / 𝐵 ) = ( 𝐴 + ( 𝐶 / 𝐵 ) ) )
9	1 3 2 4 8	syl112anc	⊢ ( 𝜑 → ( ( ( 𝐵 · 𝐴 ) + 𝐶 ) / 𝐵 ) = ( 𝐴 + ( 𝐶 / 𝐵 ) ) )
10	7 9	eqtrd	⊢ ( 𝜑 → ( ( ( 𝐴 · 𝐵 ) + 𝐶 ) / 𝐵 ) = ( 𝐴 + ( 𝐶 / 𝐵 ) ) )