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Mirrors > Home > MPE Home > Th. List > divge0d | Structured version Visualization version GIF version |
Description: The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
rpgecld.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
rpgecld.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
divge0d.3 | ⊢ (𝜑 → 0 ≤ 𝐴) |
Ref | Expression |
---|---|
divge0d | ⊢ (𝜑 → 0 ≤ (𝐴 / 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpgecld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | divge0d.3 | . 2 ⊢ (𝜑 → 0 ≤ 𝐴) | |
3 | rpgecld.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
4 | 3 | rpregt0d 12163 | . 2 ⊢ (𝜑 → (𝐵 ∈ ℝ ∧ 0 < 𝐵)) |
5 | divge0 11223 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 ≤ (𝐴 / 𝐵)) | |
6 | 1, 2, 4, 5 | syl21anc 873 | 1 ⊢ (𝜑 → 0 ≤ (𝐴 / 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2166 class class class wbr 4874 (class class class)co 6906 ℝcr 10252 0cc0 10253 < clt 10392 ≤ cle 10393 / cdiv 11010 ℝ+crp 12113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-po 5264 df-so 5265 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-div 11011 df-rp 12114 |
This theorem is referenced by: iseralt 14793 nn0ehalf 15470 nn0oddm1d2 15476 bitsfzo 15531 bitsmod 15532 iserodd 15912 icopnfcnv 23112 logdiflbnd 25135 lgamgulmlem3 25171 chpo1ubb 25584 vmadivsumb 25586 rpvmasumlem 25590 dchrisumlem1 25592 dchrvmasumlem2 25601 rplogsum 25630 dirith2 25631 mulog2sumlem2 25638 vmalogdivsum2 25641 2vmadivsumlem 25643 selbergb 25652 selberg2b 25655 selberg4lem1 25663 pntrlog2bndlem2 25681 pntrlog2bndlem4 25683 pntrlog2bndlem5 25684 pntrlog2bndlem6 25686 pntrlog2bnd 25687 pntibndlem2 25694 ttgcontlem1 26185 sqsscirc1 30500 faclimlem1 32172 knoppndvlem14 33049 itg2addnclem2 34006 geomcau 34098 areaquad 38645 stirlinglem11 41096 stirlinglem12 41097 fourierdlem26 41145 fourierdlem30 41149 fourierdlem47 41165 sge0ad2en 41440 eenglngeehlnmlem2 43290 |
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