Theorem recid

Description: Multiplication of a number and its reciprocal. (Contributed by NM, 25-Oct-1999) (Proof shortened by Mario Carneiro, 27-May-2016)

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

Proof

Step	Hyp	Ref	Expression
1		ax-1cn	⊢ 1 ∈ ℂ
2		divcan2	⊢ ( ( 1 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 𝐴 · ( 1 / 𝐴 ) ) = 1 )
3	1 2	mp3an1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 𝐴 · ( 1 / 𝐴 ) ) = 1 )