Axiom ax-pre-mulgt0 10013

Description: The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, justified by theorem axpre-mulgt0 9989. Normally new proofs would use axmulgt0 10112. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)

Assertion

Ref	Expression
ax-pre-mulgt0	|- ( ( A e. RR /\ B e. RR ) -> ( ( 0 <RR A /\ 0 <RR B ) -> 0 <RR ( A x. B ) ) )

Detailed syntax breakdown of Axiom ax-pre-mulgt0

Step	Hyp	Ref	Expression
1		cA	. . . 4 class A
2		cr 9935	. . . 4 class RR
3	1, 2	wcel 1990	. . 3 wff A e. RR
4		cB	. . . 4 class B
5	4, 2	wcel 1990	. . 3 wff B e. RR
6	3, 5	wa 384	. 2 wff ( A e. RR /\ B e. RR )
7		cc0 9936	. . . . 5 class 0
8		cltrr 9940	. . . . 5 class <RR
9	7, 1, 8	wbr 4653	. . . 4 wff 0 <RR A
10	7, 4, 8	wbr 4653	. . . 4 wff 0 <RR B
11	9, 10	wa 384	. . 3 wff ( 0 <RR A /\ 0 <RR B )
12		cmul 9941	. . . . 5 class x.
13	1, 4, 12	co 6650	. . . 4 class ( A x. B )
14	7, 13, 8	wbr 4653	. . 3 wff 0 <RR ( A x. B )
15	11, 14	wi 4	. 2 wff ( ( 0 <RR A /\ 0 <RR B ) -> 0 <RR ( A x. B ) )
16	6, 15	wi 4	1 wff ( ( A e. RR /\ B e. RR ) -> ( ( 0 <RR A /\ 0 <RR B ) -> 0 <RR ( A x. B ) ) )

This axiom is referenced by:  axmulgt0  10112