Theorem recidi

Description: Multiplication of a number and its reciprocal. (Contributed by NM, 9-Feb-1995)

	Ref	Expression
Hypotheses	divclz.1	⊢ 𝐴 ∈ ℂ
	reccl.2	⊢ 𝐴 ≠ 0
Assertion	recidi	⊢ ( 𝐴 · ( 1 / 𝐴 ) ) = 1

Proof

Step	Hyp	Ref	Expression
1		divclz.1	⊢ 𝐴 ∈ ℂ
2		reccl.2	⊢ 𝐴 ≠ 0
3	1	recidzi	⊢ ( 𝐴 ≠ 0 → ( 𝐴 · ( 1 / 𝐴 ) ) = 1 )
4	2 3	ax-mp	⊢ ( 𝐴 · ( 1 / 𝐴 ) ) = 1