Theorem divreci

Description: Relationship between division and reciprocal. Theorem I.9 of Apostol p. 18. (Contributed by NM, 9-Feb-1995)

	Ref	Expression
Hypotheses	divclz.1	⊢ 𝐴 ∈ ℂ
	divclz.2	⊢ 𝐵 ∈ ℂ
	divcl.3	⊢ 𝐵 ≠ 0
Assertion	divreci	⊢ ( 𝐴 / 𝐵 ) = ( 𝐴 · ( 1 / 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		divclz.1	⊢ 𝐴 ∈ ℂ
2		divclz.2	⊢ 𝐵 ∈ ℂ
3		divcl.3	⊢ 𝐵 ≠ 0
4	1 2	divreczi	⊢ ( 𝐵 ≠ 0 → ( 𝐴 / 𝐵 ) = ( 𝐴 · ( 1 / 𝐵 ) ) )
5	3 4	ax-mp	⊢ ( 𝐴 / 𝐵 ) = ( 𝐴 · ( 1 / 𝐵 ) )