Theorem ldiv

Description: Left-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 )
	ldiv.bn0	⊢ ( 𝜑 → 𝐵 ≠ 0 )
Assertion	ldiv	⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) = 𝐶 ↔ 𝐴 = ( 𝐶 / 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ldiv.a	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		ldiv.b	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		ldiv.c	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4		ldiv.bn0	⊢ ( 𝜑 → 𝐵 ≠ 0 )
5		oveq1	⊢ ( ( 𝐴 · 𝐵 ) = 𝐶 → ( ( 𝐴 · 𝐵 ) / 𝐵 ) = ( 𝐶 / 𝐵 ) )
6	1 2 4	divcan4d	⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) / 𝐵 ) = 𝐴 )
7	6	eqeq1d	⊢ ( 𝜑 → ( ( ( 𝐴 · 𝐵 ) / 𝐵 ) = ( 𝐶 / 𝐵 ) ↔ 𝐴 = ( 𝐶 / 𝐵 ) ) )
8	5 7	syl5ib	⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) = 𝐶 → 𝐴 = ( 𝐶 / 𝐵 ) ) )
9		oveq1	⊢ ( 𝐴 = ( 𝐶 / 𝐵 ) → ( 𝐴 · 𝐵 ) = ( ( 𝐶 / 𝐵 ) · 𝐵 ) )
10	3 2 4	divcan1d	⊢ ( 𝜑 → ( ( 𝐶 / 𝐵 ) · 𝐵 ) = 𝐶 )
11	10	eqeq2d	⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) = ( ( 𝐶 / 𝐵 ) · 𝐵 ) ↔ ( 𝐴 · 𝐵 ) = 𝐶 ) )
12	9 11	syl5ib	⊢ ( 𝜑 → ( 𝐴 = ( 𝐶 / 𝐵 ) → ( 𝐴 · 𝐵 ) = 𝐶 ) )
13	8 12	impbid	⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) = 𝐶 ↔ 𝐴 = ( 𝐶 / 𝐵 ) ) )