Theorem divrecd

Description: Relationship between division and reciprocal. Theorem I.9 of Apostol p. 18. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	div1d.1	|- ( ph -> A e. CC )
	divcld.2	|- ( ph -> B e. CC )
	divcld.3	|- ( ph -> B =/= 0 )
Assertion	divrecd	|- ( ph -> ( A / B ) = ( A x. ( 1 / B ) ) )

Proof

Step	Hyp	Ref	Expression
1		div1d.1	|- ( ph -> A e. CC )
2		divcld.2	|- ( ph -> B e. CC )
3		divcld.3	|- ( ph -> B =/= 0 )
4		divrec	|- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( A / B ) = ( A x. ( 1 / B ) ) )
5	1 2 3 4	syl3anc	|- ( ph -> ( A / B ) = ( A x. ( 1 / B ) ) )