Theorem rdiv

Description: Right-division. (Contributed by BJ, 6-Jun-2019)

	Ref	Expression
Hypotheses	ldiv.a	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	ldiv.b	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	ldiv.c	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
	rdiv.an0	⊢ ( 𝜑 → 𝐴 ≠ 0 )
Assertion	rdiv	⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) = 𝐶 ↔ 𝐵 = ( 𝐶 / 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ldiv.a	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		ldiv.b	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		ldiv.c	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4		rdiv.an0	⊢ ( 𝜑 → 𝐴 ≠ 0 )
5	1 2	mulcomd	⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) )
6	5	eqeq1d	⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) = 𝐶 ↔ ( 𝐵 · 𝐴 ) = 𝐶 ) )
7	2 1 3 4	ldiv	⊢ ( 𝜑 → ( ( 𝐵 · 𝐴 ) = 𝐶 ↔ 𝐵 = ( 𝐶 / 𝐴 ) ) )
8	6 7	bitrd	⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) = 𝐶 ↔ 𝐵 = ( 𝐶 / 𝐴 ) ) )