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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 12254 | . 2 ⊢ (𝜑 → (𝐵 ∈ ℝ ∧ 0 < 𝐵)) |
5 | divge0 11310 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 ≤ (𝐴 / 𝐵)) | |
6 | 1, 2, 4, 5 | syl21anc 825 | 1 ⊢ (𝜑 → 0 ≤ (𝐴 / 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∈ wcel 2050 class class class wbr 4929 (class class class)co 6976 ℝcr 10334 0cc0 10335 < clt 10474 ≤ cle 10475 / cdiv 11098 ℝ+crp 12204 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-po 5326 df-so 5327 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-rp 12205 |
This theorem is referenced by: iseralt 14902 nn0ehalf 15589 nn0oddm1d2 15596 bitsfzo 15644 bitsmod 15645 iserodd 16028 icopnfcnv 23249 logdiflbnd 25274 lgamgulmlem3 25310 chpo1ubb 25759 vmadivsumb 25761 rpvmasumlem 25765 dchrisumlem1 25767 dchrvmasumlem2 25776 rplogsum 25805 dirith2 25806 mulog2sumlem2 25813 vmalogdivsum2 25816 2vmadivsumlem 25818 selbergb 25827 selberg2b 25830 selberg4lem1 25838 pntrlog2bndlem2 25856 pntrlog2bndlem4 25858 pntrlog2bndlem5 25859 pntrlog2bndlem6 25861 pntrlog2bnd 25862 pntibndlem2 25869 ttgcontlem1 26374 sqsscirc1 30792 faclimlem1 32492 knoppndvlem14 33381 itg2addnclem2 34382 geomcau 34473 areaquad 39216 stirlinglem11 41798 stirlinglem12 41799 fourierdlem26 41847 fourierdlem30 41851 fourierdlem47 41867 sge0ad2en 42142 eenglngeehlnmlem2 44091 |
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