Axiom ax-pre-mulgt0 11213

Description: The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, justified by Theorem axpre-mulgt0 11189. Normally new proofs would use axmulgt0 11316. (New usage is discouraged.) (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 11135	. . . 4 class ℝ
3	1, 2	wcel 2107	. . 3 wff 𝐴 ∈ ℝ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2107	. . 3 wff 𝐵 ∈ ℝ
6	3, 5	wa 395	. 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)
7		cc0 11136	. . . . 5 class 0
8		cltrr 11140	. . . . 5 class <ℝ
9	7, 1, 8	wbr 5123	. . . 4 wff 0 <ℝ 𝐴
10	7, 4, 8	wbr 5123	. . . 4 wff 0 <ℝ 𝐵
11	9, 10	wa 395	. . 3 wff (0 <ℝ 𝐴 ∧ 0 <ℝ 𝐵)
12		cmul 11141	. . . . 5 class ·
13	1, 4, 12	co 7412	. . . 4 class (𝐴 · 𝐵)
14	7, 13, 8	wbr 5123	. . 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  11316