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Mirrors > Home > MPE Home > Th. List > mulgt0d | Structured version Visualization version GIF version |
Description: The product of two positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
ltd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
mulgt0d.3 | ⊢ (𝜑 → 0 < 𝐴) |
mulgt0d.4 | ⊢ (𝜑 → 0 < 𝐵) |
Ref | Expression |
---|---|
mulgt0d | ⊢ (𝜑 → 0 < (𝐴 · 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | mulgt0d.3 | . 2 ⊢ (𝜑 → 0 < 𝐴) | |
3 | ltd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
4 | mulgt0d.4 | . 2 ⊢ (𝜑 → 0 < 𝐵) | |
5 | mulgt0 10707 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 < (𝐴 · 𝐵)) | |
6 | 1, 2, 3, 4, 5 | syl22anc 837 | 1 ⊢ (𝜑 → 0 < (𝐴 · 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 class class class wbr 5030 (class class class)co 7135 ℝcr 10525 0cc0 10526 · cmul 10531 < clt 10664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-addrcl 10587 ax-mulrcl 10589 ax-rnegex 10597 ax-cnre 10599 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 |
This theorem is referenced by: recgt0 11475 prodgt0 11476 ltmul1a 11478 prodge0rd 12484 expmulnbnd 13592 itg2monolem3 24356 tangtx 25098 tanregt0 25131 asinsinlem 25477 asinsin 25478 ostth2lem3 26219 xrge0iifhom 31290 unbdqndv2lem2 33962 knoppndvlem14 33977 knoppndvlem18 33981 knoppndvlem19 33982 knoppndvlem21 33984 itg2gt0cn 35112 lcmineqlem15 39331 mulgt0con1d 39583 mulgt0con2d 39584 mulgt0b2d 39585 sn-0lt1 39587 pell14qrmulcl 39804 rmxypos 39888 jm2.27a 39946 stoweidlem1 42643 stoweidlem26 42668 stoweidlem44 42686 stoweidlem49 42691 wallispilem4 42710 stirlinglem6 42721 itscnhlinecirc02plem1 45196 |
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