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Mirrors > Home > ILE Home > Th. List > ax-pre-mulgt0 | GIF version |
Description: The product of two positive reals is positive. Axiom for real and complex numbers, justified by theorem axpre-mulgt0 6771. (Contributed by NM, 13-Oct-2005.) |
Ref | Expression |
---|---|
ax-pre-mulgt0 | ⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → ((0 <ℝ A ∧ 0 <ℝ B) → 0 <ℝ (A · B))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . . 4 class A | |
2 | cr 6710 | . . . 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 6711 | . . . . 5 class 0 | |
8 | cltrr 6715 | . . . . 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 6716 | . . . . 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))) |
Colors of variables: wff set class |
This axiom is referenced by: axmulgt0 6888 |
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