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 ∈ ℝ ∧ B ∈ ℝ) → ((0 <ℝ A ∧ 0 <ℝ B) → 0 <ℝ (A · B)))

Detailed syntax breakdown of Axiom ax-pre-mulgt0

Step	Hyp	Ref	Expression
1		cA	. . . 4 class A
2		cr 6690	. . . 4 class ℝ
3	1, 2	wcel 1390	. . 3 wff A ∈ ℝ
4		cB	. . . 4 class B
5	4, 2	wcel 1390	. . 3 wff B ∈ ℝ
6	3, 5	wa 97	. 2 wff (A ∈ ℝ ∧ B ∈ ℝ)
7		cc0 6691	. . . . 5 class 0
8		cltrr 6695	. . . . 5 class <ℝ
9	7, 1, 8	wbr 3755	. . . 4 wff 0 <ℝ A
10	7, 4, 8	wbr 3755	. . . 4 wff 0 <ℝ B
11	9, 10	wa 97	. . 3 wff (0 <ℝ A ∧ 0 <ℝ B)
12		cmul 6696	. . . . 5 class ·
13	1, 4, 12	co 5455	. . . 4 class (A · B)
14	7, 13, 8	wbr 3755	. . 3 wff 0 <ℝ (A · B)
15	11, 14	wi 4	. 2 wff ((0 <ℝ A ∧ 0 <ℝ B) → 0 <ℝ (A · B))
16	6, 15	wi 4	1 wff ((A ∈ ℝ ∧ B ∈ ℝ) → ((0 <ℝ A ∧ 0 <ℝ B) → 0 <ℝ (A · B)))

This axiom is referenced by:  axmulgt0  6868