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Mirrors > Home > MPE Home > Th. List > axmulgt0 | Structured version Visualization version GIF version |
Description: The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-mulgt0 11184 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 11184 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 <ℝ 𝐴 ∧ 0 <ℝ 𝐵) → 0 <ℝ (𝐴 · 𝐵))) | |
2 | 0re 11213 | . . . 4 ⊢ 0 ∈ ℝ | |
3 | ltxrlt 11281 | . . . 4 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < 𝐴 ↔ 0 <ℝ 𝐴)) | |
4 | 2, 3 | mpan 689 | . . 3 ⊢ (𝐴 ∈ ℝ → (0 < 𝐴 ↔ 0 <ℝ 𝐴)) |
5 | ltxrlt 11281 | . . . 4 ⊢ ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐵 ↔ 0 <ℝ 𝐵)) | |
6 | 2, 5 | mpan 689 | . . 3 ⊢ (𝐵 ∈ ℝ → (0 < 𝐵 ↔ 0 <ℝ 𝐵)) |
7 | 4, 6 | bi2anan9 638 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < 𝐴 ∧ 0 < 𝐵) ↔ (0 <ℝ 𝐴 ∧ 0 <ℝ 𝐵))) |
8 | remulcl 11192 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ) | |
9 | ltxrlt 11281 | . . 3 ⊢ ((0 ∈ ℝ ∧ (𝐴 · 𝐵) ∈ ℝ) → (0 < (𝐴 · 𝐵) ↔ 0 <ℝ (𝐴 · 𝐵))) | |
10 | 2, 8, 9 | sylancr 588 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < (𝐴 · 𝐵) ↔ 0 <ℝ (𝐴 · 𝐵))) |
11 | 1, 7, 10 | 3imtr4d 294 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 · 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 class class class wbr 5148 (class class class)co 7406 ℝcr 11106 0cc0 11107 <ℝ cltrr 11111 · cmul 11112 < clt 11245 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-resscn 11164 ax-1cn 11165 ax-addrcl 11168 ax-mulrcl 11170 ax-rnegex 11178 ax-cnre 11180 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11247 df-mnf 11248 df-ltxr 11250 |
This theorem is referenced by: mulgt0 11288 mulgt0i 11343 sin02gt0 16132 sinq12gt0 26009 |
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