Theorem divid

Description: A number divided by itself is one. (Contributed by NM, 1-Aug-2004)

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

Proof

Step	Hyp	Ref	Expression
1		divrec	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 𝐴 / 𝐴 ) = ( 𝐴 · ( 1 / 𝐴 ) ) )
2	1	3anidm12	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 𝐴 / 𝐴 ) = ( 𝐴 · ( 1 / 𝐴 ) ) )
3		recid	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 𝐴 · ( 1 / 𝐴 ) ) = 1 )
4	2 3	eqtrd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 𝐴 / 𝐴 ) = 1 )