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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 11323 | . 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 2098 class class class wbr 5149 (class class class)co 7419 ℝcr 11139 0cc0 11140 · cmul 11145 < clt 11280 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-resscn 11197 ax-1cn 11198 ax-addrcl 11201 ax-mulrcl 11203 ax-rnegex 11211 ax-cnre 11213 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-ltxr 11285 |
This theorem is referenced by: recgt0 12093 prodgt0 12094 ltmul1a 12096 prodge0rd 13116 expmulnbnd 14233 itg2monolem3 25726 tangtx 26485 tanregt0 26518 asinsinlem 26868 asinsin 26869 ostth2lem3 27613 xrge0iifhom 33669 unbdqndv2lem2 36116 knoppndvlem14 36131 knoppndvlem18 36135 knoppndvlem19 36136 knoppndvlem21 36138 itg2gt0cn 37279 lcmineqlem15 41646 posbezout 41703 mulgt0con1d 42148 mulgt0con2d 42149 mulgt0b2d 42150 sn-0lt1 42152 pell14qrmulcl 42425 rmxypos 42510 jm2.27a 42568 stoweidlem1 45527 stoweidlem26 45552 stoweidlem44 45570 stoweidlem49 45575 wallispilem4 45594 stirlinglem6 45605 itscnhlinecirc02plem1 48041 |
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