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Axiom ax-pre-mulgt0 10051
 Description: The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, justified by theorem axpre-mulgt0 10027. Normally new proofs would use axmulgt0 10150. (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 9973 . . . 4 class
31, 2wcel 2030 . . 3 wff 𝐴 ∈ ℝ
4 cB . . . 4 class 𝐵
54, 2wcel 2030 . . 3 wff 𝐵 ∈ ℝ
63, 5wa 383 . 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)
7 cc0 9974 . . . . 5 class 0
8 cltrr 9978 . . . . 5 class <
97, 1, 8wbr 4685 . . . 4 wff 0 < 𝐴
107, 4, 8wbr 4685 . . . 4 wff 0 < 𝐵
119, 10wa 383 . . 3 wff (0 < 𝐴 ∧ 0 < 𝐵)
12 cmul 9979 . . . . 5 class ·
131, 4, 12co 6690 . . . 4 class (𝐴 · 𝐵)
147, 13, 8wbr 4685 . . 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  10150
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