Theorem diveq1

Description: Equality in terms of unit ratio. (Contributed by Stefan O'Rear, 27-Aug-2015)

	Ref	Expression
Assertion	diveq1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ( 𝐴 / 𝐵 ) = 1 ↔ 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ax-1cn	⊢ 1 ∈ ℂ
2		divmul2	⊢ ( ( 𝐴 ∈ ℂ ∧ 1 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 𝐴 / 𝐵 ) = 1 ↔ 𝐴 = ( 𝐵 · 1 ) ) )
3	1 2	mp3an2	⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 𝐴 / 𝐵 ) = 1 ↔ 𝐴 = ( 𝐵 · 1 ) ) )
4	3	3impb	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ( 𝐴 / 𝐵 ) = 1 ↔ 𝐴 = ( 𝐵 · 1 ) ) )
5		simp2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → 𝐵 ∈ ℂ )
6	5	mulid1d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐵 · 1 ) = 𝐵 )
7	6	eqeq2d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐴 = ( 𝐵 · 1 ) ↔ 𝐴 = 𝐵 ) )
8	4 7	bitrd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ( 𝐴 / 𝐵 ) = 1 ↔ 𝐴 = 𝐵 ) )