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Mirrors > Home > ILE Home > Th. List > axmulgt0 | GIF version |
Description: The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-mulgt0 7737 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.) |
Ref | Expression |
---|---|
axmulgt0 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 · 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-pre-mulgt0 7737 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 <ℝ 𝐴 ∧ 0 <ℝ 𝐵) → 0 <ℝ (𝐴 · 𝐵))) | |
2 | 0re 7766 | . . . 4 ⊢ 0 ∈ ℝ | |
3 | ltxrlt 7830 | . . . 4 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < 𝐴 ↔ 0 <ℝ 𝐴)) | |
4 | 2, 3 | mpan 420 | . . 3 ⊢ (𝐴 ∈ ℝ → (0 < 𝐴 ↔ 0 <ℝ 𝐴)) |
5 | ltxrlt 7830 | . . . 4 ⊢ ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐵 ↔ 0 <ℝ 𝐵)) | |
6 | 2, 5 | mpan 420 | . . 3 ⊢ (𝐵 ∈ ℝ → (0 < 𝐵 ↔ 0 <ℝ 𝐵)) |
7 | 4, 6 | bi2anan9 595 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < 𝐴 ∧ 0 < 𝐵) ↔ (0 <ℝ 𝐴 ∧ 0 <ℝ 𝐵))) |
8 | remulcl 7748 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ) | |
9 | ltxrlt 7830 | . . 3 ⊢ ((0 ∈ ℝ ∧ (𝐴 · 𝐵) ∈ ℝ) → (0 < (𝐴 · 𝐵) ↔ 0 <ℝ (𝐴 · 𝐵))) | |
10 | 2, 8, 9 | sylancr 410 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < (𝐴 · 𝐵) ↔ 0 <ℝ (𝐴 · 𝐵))) |
11 | 1, 7, 10 | 3imtr4d 202 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 · 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 1480 class class class wbr 3929 (class class class)co 5774 ℝcr 7619 0cc0 7620 <ℝ cltrr 7624 · cmul 7625 < clt 7800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1re 7714 ax-addrcl 7717 ax-mulrcl 7719 ax-rnegex 7729 ax-pre-mulgt0 7737 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-pnf 7802 df-mnf 7803 df-ltxr 7805 |
This theorem is referenced by: mulgt0 7839 mulgt0i 7873 sin02gt0 11470 sinq12gt0 12911 |
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