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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 10710 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 < (𝐴 · 𝐵)) | |
6 | 1, 2, 3, 4, 5 | syl22anc 836 | 1 ⊢ (𝜑 → 0 < (𝐴 · 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2108 class class class wbr 5057 (class class class)co 7148 ℝcr 10528 0cc0 10529 · cmul 10534 < clt 10667 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-resscn 10586 ax-1cn 10587 ax-addrcl 10590 ax-mulrcl 10592 ax-rnegex 10600 ax-cnre 10602 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-ltxr 10672 |
This theorem is referenced by: recgt0 11478 prodgt0 11479 ltmul1a 11481 prodge0rd 12488 expmulnbnd 13588 itg2monolem3 24345 tangtx 25083 tanregt0 25115 asinsinlem 25461 asinsin 25462 ostth2lem3 26203 xrge0iifhom 31173 unbdqndv2lem2 33842 knoppndvlem14 33857 knoppndvlem18 33861 knoppndvlem19 33862 knoppndvlem21 33864 itg2gt0cn 34939 sn-0lt1 39237 pell14qrmulcl 39451 rmxypos 39535 jm2.27a 39593 stoweidlem1 42277 stoweidlem26 42302 stoweidlem44 42320 stoweidlem49 42325 wallispilem4 42344 stirlinglem6 42355 itscnhlinecirc02plem1 44760 |
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