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Axiom ax-pre-mulgt0 6780
 Description: The product of two positive reals is positive. Axiom for real and complex numbers, justified by theorem axpre-mulgt0 6751. (Contributed by NM, 13-Oct-2005.)
Assertion
Ref Expression
ax-pre-mulgt0 ((A B ℝ) → ((0 < A 0 < B) → 0 < (A · B)))

Detailed syntax breakdown of Axiom ax-pre-mulgt0
StepHypRef Expression
1 cA . . . 4 class A
2 cr 6690 . . . 4 class
31, 2wcel 1390 . . 3 wff A
4 cB . . . 4 class B
54, 2wcel 1390 . . 3 wff B
63, 5wa 97 . 2 wff (A B ℝ)
7 cc0 6691 . . . . 5 class 0
8 cltrr 6695 . . . . 5 class <
97, 1, 8wbr 3755 . . . 4 wff 0 < A
107, 4, 8wbr 3755 . . . 4 wff 0 < B
119, 10wa 97 . . 3 wff (0 < A 0 < B)
12 cmul 6696 . . . . 5 class ·
131, 4, 12co 5455 . . . 4 class (A · B)
147, 13, 8wbr 3755 . . 3 wff 0 < (A · B)
1511, 14wi 4 . 2 wff ((0 < A 0 < B) → 0 < (A · B))
166, 15wi 4 1 wff ((A B ℝ) → ((0 < A 0 < B) → 0 < (A · B)))
 Colors of variables: wff set class This axiom is referenced by:  axmulgt0  6868
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