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Metamath Proof Explorer

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 ) ) )