Theorem wwlemuld

Description: Natural deduction form of lemul2d . (Contributed by Stanislas Polu, 9-Mar-2020)

	Ref	Expression
Hypotheses	wwlemuld.1	|- ( ph -> A e. RR )
	wwlemuld.2	|- ( ph -> B e. RR )
	wwlemuld.3	|- ( ph -> C e. RR )
	wwlemuld.4	|- ( ph -> ( C x. A ) <_ ( C x. B ) )
	wwlemuld.5	|- ( ph -> 0 < C )
Assertion	wwlemuld	|- ( ph -> A <_ B )

Proof

Step	Hyp	Ref	Expression
1		wwlemuld.1	|- ( ph -> A e. RR )
2		wwlemuld.2	|- ( ph -> B e. RR )
3		wwlemuld.3	|- ( ph -> C e. RR )
4		wwlemuld.4	|- ( ph -> ( C x. A ) <_ ( C x. B ) )
5		wwlemuld.5	|- ( ph -> 0 < C )
6	3 5	elrpd	|- ( ph -> C e. RR+ )
7	1 2 6	lemul2d	|- ( ph -> ( A <_ B <-> ( C x. A ) <_ ( C x. B ) ) )
8	4 7	mpbird	|- ( ph -> A <_ B )