Theorem absmuld

Description: Absolute value distributes over multiplication. Proposition 10-3.7(f) of Gleason p. 133. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	abscld.1	|- ( ph -> A e. CC )
	abssubd.2	|- ( ph -> B e. CC )
Assertion	absmuld	|- ( ph -> ( abs ` ( A x. B ) ) = ( ( abs ` A ) x. ( abs ` B ) ) )

Proof

Step	Hyp	Ref	Expression
1		abscld.1	|- ( ph -> A e. CC )
2		abssubd.2	|- ( ph -> B e. CC )
3		absmul	|- ( ( A e. CC /\ B e. CC ) -> ( abs ` ( A x. B ) ) = ( ( abs ` A ) x. ( abs ` B ) ) )
4	1 2 3	syl2anc	|- ( ph -> ( abs ` ( A x. B ) ) = ( ( abs ` A ) x. ( abs ` B ) ) )