Theorem redivmuld

Description: Relationship between division and multiplication. (Contributed by SN, 25-Nov-2025)

	Ref	Expression
Hypotheses	redivmuld.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	redivmuld.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	redivmuld.c	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
	redivmuld.z	⊢ ( 𝜑 → 𝐶 ≠ 0 )
Assertion	redivmuld	⊢ ( 𝜑 → ( ( 𝐴 /ℝ 𝐶 ) = 𝐵 ↔ ( 𝐶 · 𝐵 ) = 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		redivmuld.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		redivmuld.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		redivmuld.c	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
4		redivmuld.z	⊢ ( 𝜑 → 𝐶 ≠ 0 )
5	1 3 4	redivvald	⊢ ( 𝜑 → ( 𝐴 /ℝ 𝐶 ) = ( ℩ 𝑥 ∈ ℝ ( 𝐶 · 𝑥 ) = 𝐴 ) )
6	5	eqeq1d	⊢ ( 𝜑 → ( ( 𝐴 /ℝ 𝐶 ) = 𝐵 ↔ ( ℩ 𝑥 ∈ ℝ ( 𝐶 · 𝑥 ) = 𝐴 ) = 𝐵 ) )
7	1 3 4	rediveud	⊢ ( 𝜑 → ∃! 𝑥 ∈ ℝ ( 𝐶 · 𝑥 ) = 𝐴 )
8		oveq2	⊢ ( 𝑥 = 𝐵 → ( 𝐶 · 𝑥 ) = ( 𝐶 · 𝐵 ) )
9	8	eqeq1d	⊢ ( 𝑥 = 𝐵 → ( ( 𝐶 · 𝑥 ) = 𝐴 ↔ ( 𝐶 · 𝐵 ) = 𝐴 ) )
10	9	riota2	⊢ ( ( 𝐵 ∈ ℝ ∧ ∃! 𝑥 ∈ ℝ ( 𝐶 · 𝑥 ) = 𝐴 ) → ( ( 𝐶 · 𝐵 ) = 𝐴 ↔ ( ℩ 𝑥 ∈ ℝ ( 𝐶 · 𝑥 ) = 𝐴 ) = 𝐵 ) )
11	2 7 10	syl2anc	⊢ ( 𝜑 → ( ( 𝐶 · 𝐵 ) = 𝐴 ↔ ( ℩ 𝑥 ∈ ℝ ( 𝐶 · 𝑥 ) = 𝐴 ) = 𝐵 ) )
12	6 11	bitr4d	⊢ ( 𝜑 → ( ( 𝐴 /ℝ 𝐶 ) = 𝐵 ↔ ( 𝐶 · 𝐵 ) = 𝐴 ) )