Theorem divid

Description: A number divided by itself is one. (Contributed by NM, 1-Aug-2004) (Proof shortened by SN, 9-Jul-2025)

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

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ 𝐴 = 𝐴
2		diveq1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ( 𝐴 / 𝐴 ) = 1 ↔ 𝐴 = 𝐴 ) )
3	1 2	mpbiri	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 𝐴 / 𝐴 ) = 1 )
4	3	3anidm12	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 𝐴 / 𝐴 ) = 1 )