Theorem muldivdir

Description: Distribution of division over addition with a multiplication. (Contributed by AV, 1-Jul-2021)

	Ref	Expression
Assertion	muldivdir	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( ( 𝐶 · 𝐴 ) + 𝐵 ) / 𝐶 ) = ( 𝐴 + ( 𝐵 / 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		simp3l	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → 𝐶 ∈ ℂ )
2		simp1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → 𝐴 ∈ ℂ )
3	1 2	mulcld	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( 𝐶 · 𝐴 ) ∈ ℂ )
4		divdir	⊢ ( ( ( 𝐶 · 𝐴 ) ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( ( 𝐶 · 𝐴 ) + 𝐵 ) / 𝐶 ) = ( ( ( 𝐶 · 𝐴 ) / 𝐶 ) + ( 𝐵 / 𝐶 ) ) )
5	3 4	syld3an1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( ( 𝐶 · 𝐴 ) + 𝐵 ) / 𝐶 ) = ( ( ( 𝐶 · 𝐴 ) / 𝐶 ) + ( 𝐵 / 𝐶 ) ) )
6		3anass	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ↔ ( 𝐴 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) )
7	6	biimpri	⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) )
8	7	3adant2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) )
9		divcan3	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) → ( ( 𝐶 · 𝐴 ) / 𝐶 ) = 𝐴 )
10	8 9	syl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐶 · 𝐴 ) / 𝐶 ) = 𝐴 )
11	10	oveq1d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( ( 𝐶 · 𝐴 ) / 𝐶 ) + ( 𝐵 / 𝐶 ) ) = ( 𝐴 + ( 𝐵 / 𝐶 ) ) )
12	5 11	eqtrd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( ( 𝐶 · 𝐴 ) + 𝐵 ) / 𝐶 ) = ( 𝐴 + ( 𝐵 / 𝐶 ) ) )