Theorem mdiv

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

	Ref	Expression
Hypotheses	ldiv.a	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	ldiv.b	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	ldiv.c	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
	mdiv.an0	⊢ ( 𝜑 → 𝐴 ≠ 0 )
	mdiv.bn0	⊢ ( 𝜑 → 𝐵 ≠ 0 )
Assertion	mdiv	⊢ ( 𝜑 → ( 𝐴 = ( 𝐶 / 𝐵 ) ↔ 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		mdiv.an0	⊢ ( 𝜑 → 𝐴 ≠ 0 )
5		mdiv.bn0	⊢ ( 𝜑 → 𝐵 ≠ 0 )
6	1 2 3 5	ldiv	⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) = 𝐶 ↔ 𝐴 = ( 𝐶 / 𝐵 ) ) )
7	1 2 3 4	rdiv	⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) = 𝐶 ↔ 𝐵 = ( 𝐶 / 𝐴 ) ) )
8	6 7	bitr3d	⊢ ( 𝜑 → ( 𝐴 = ( 𝐶 / 𝐵 ) ↔ 𝐵 = ( 𝐶 / 𝐴 ) ) )