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Axiom ax-pre-mulgt0 9869
Description: The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, justified by theorem axpre-mulgt0 9845. Normally new proofs would use axmulgt0 9963. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
Assertion
Ref Expression
ax-pre-mulgt0 ((𝐴 ∈ ℝ ∧ 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 9791 . . . 4 class
31, 2wcel 1976 . . 3 wff 𝐴 ∈ ℝ
4 cB . . . 4 class 𝐵
54, 2wcel 1976 . . 3 wff 𝐵 ∈ ℝ
63, 5wa 382 . 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)
7 cc0 9792 . . . . 5 class 0
8 cltrr 9796 . . . . 5 class <
97, 1, 8wbr 4577 . . . 4 wff 0 < 𝐴
107, 4, 8wbr 4577 . . . 4 wff 0 < 𝐵
119, 10wa 382 . . 3 wff (0 < 𝐴 ∧ 0 < 𝐵)
12 cmul 9797 . . . . 5 class ·
131, 4, 12co 6526 . . . 4 class (𝐴 · 𝐵)
147, 13, 8wbr 4577 . . 3 wff 0 < (𝐴 · 𝐵)
1511, 14wi 4 . 2 wff ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 · 𝐵))
166, 15wi 4 1 wff ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 · 𝐵)))
Colors of variables: wff setvar class
This axiom is referenced by:  axmulgt0  9963
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