Theorem redivdird

Description: Distribution of division over addition. (Contributed by SN, 9-Apr-2026)

	Ref	Expression
Hypotheses	rediv23d.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	rediv23d.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	rediv23d.c	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
	rediv23d.z	⊢ ( 𝜑 → 𝐶 ≠ 0 )
Assertion	redivdird	⊢ ( 𝜑 → ( ( 𝐴 + 𝐵 ) /ℝ 𝐶 ) = ( ( 𝐴 /ℝ 𝐶 ) + ( 𝐵 /ℝ 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		rediv23d.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		rediv23d.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		rediv23d.c	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
4		rediv23d.z	⊢ ( 𝜑 → 𝐶 ≠ 0 )
5	3	recnd	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
6	1 3 4	sn-redivcld	⊢ ( 𝜑 → ( 𝐴 /ℝ 𝐶 ) ∈ ℝ )
7	6	recnd	⊢ ( 𝜑 → ( 𝐴 /ℝ 𝐶 ) ∈ ℂ )
8	2 3 4	sn-redivcld	⊢ ( 𝜑 → ( 𝐵 /ℝ 𝐶 ) ∈ ℝ )
9	8	recnd	⊢ ( 𝜑 → ( 𝐵 /ℝ 𝐶 ) ∈ ℂ )
10	5 7 9	adddid	⊢ ( 𝜑 → ( 𝐶 · ( ( 𝐴 /ℝ 𝐶 ) + ( 𝐵 /ℝ 𝐶 ) ) ) = ( ( 𝐶 · ( 𝐴 /ℝ 𝐶 ) ) + ( 𝐶 · ( 𝐵 /ℝ 𝐶 ) ) ) )
11	1 3 4	redivcan2d	⊢ ( 𝜑 → ( 𝐶 · ( 𝐴 /ℝ 𝐶 ) ) = 𝐴 )
12	2 3 4	redivcan2d	⊢ ( 𝜑 → ( 𝐶 · ( 𝐵 /ℝ 𝐶 ) ) = 𝐵 )
13	11 12	oveq12d	⊢ ( 𝜑 → ( ( 𝐶 · ( 𝐴 /ℝ 𝐶 ) ) + ( 𝐶 · ( 𝐵 /ℝ 𝐶 ) ) ) = ( 𝐴 + 𝐵 ) )
14	10 13	eqtrd	⊢ ( 𝜑 → ( 𝐶 · ( ( 𝐴 /ℝ 𝐶 ) + ( 𝐵 /ℝ 𝐶 ) ) ) = ( 𝐴 + 𝐵 ) )
15	1 2	readdcld	⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) ∈ ℝ )
16	6 8	readdcld	⊢ ( 𝜑 → ( ( 𝐴 /ℝ 𝐶 ) + ( 𝐵 /ℝ 𝐶 ) ) ∈ ℝ )
17	15 16 3 4	redivmuld	⊢ ( 𝜑 → ( ( ( 𝐴 + 𝐵 ) /ℝ 𝐶 ) = ( ( 𝐴 /ℝ 𝐶 ) + ( 𝐵 /ℝ 𝐶 ) ) ↔ ( 𝐶 · ( ( 𝐴 /ℝ 𝐶 ) + ( 𝐵 /ℝ 𝐶 ) ) ) = ( 𝐴 + 𝐵 ) ) )
18	14 17	mpbird	⊢ ( 𝜑 → ( ( 𝐴 + 𝐵 ) /ℝ 𝐶 ) = ( ( 𝐴 /ℝ 𝐶 ) + ( 𝐵 /ℝ 𝐶 ) ) )