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| Description: The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 27-Sep-1999.) | 
| Ref | Expression | 
|---|---|
| divge0 | ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 ≤ (𝐴 / 𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ge0div 12136 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐵) → (0 ≤ 𝐴 ↔ 0 ≤ (𝐴 / 𝐵))) | |
| 2 | 1 | biimpd 229 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐵) → (0 ≤ 𝐴 → 0 ≤ (𝐴 / 𝐵))) | 
| 3 | 2 | 3exp 1119 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐵 ∈ ℝ → (0 < 𝐵 → (0 ≤ 𝐴 → 0 ≤ (𝐴 / 𝐵))))) | 
| 4 | 3 | com34 91 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐵 ∈ ℝ → (0 ≤ 𝐴 → (0 < 𝐵 → 0 ≤ (𝐴 / 𝐵))))) | 
| 5 | 4 | com23 86 | . 2 ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 → (𝐵 ∈ ℝ → (0 < 𝐵 → 0 ≤ (𝐴 / 𝐵))))) | 
| 6 | 5 | imp43 427 | 1 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 ≤ (𝐴 / 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2107 class class class wbr 5142 (class class class)co 7432 ℝcr 11155 0cc0 11156 < clt 11296 ≤ cle 11297 / cdiv 11921 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 | 
| This theorem is referenced by: mulge0b 12139 ledivp1 12171 divge0i 12178 divge0d 13118 divelunit 13535 adddivflid 13859 fldiv4p1lem1div2 13876 fldiv 13901 modid 13937 modmuladdnn0 13957 expnbnd 14272 sqrtdiv 15305 sqreulem 15399 efcllem 16114 ege2le3 16127 flodddiv4 16453 hashgcdlem 16826 fldivp1 16936 4sqlem14 16997 odmodnn0 19559 prmirredlem 21484 icopnfcnv 24974 lebnumii 24999 nmoleub2lem3 25149 ncvs1 25192 minveclem4 25467 mbfi1fseqlem1 25751 mbfi1fseqlem5 25755 radcnvlem1 26457 cxpaddle 26796 log2tlbnd 26989 birthdaylem3 26997 jensenlem2 27032 amgm 27035 basellem3 27127 ppiub 27249 logfac2 27262 gausslemma2dlem0d 27404 chto1ub 27521 vmadivsum 27527 rpvmasumlem 27532 dchrvmasumlem2 27543 dchrvmasumiflem1 27546 dchrisum0fno1 27556 dchrisum0re 27558 mulog2sumlem2 27580 selberg2lem 27595 pntrmax 27609 pntrsumo1 27610 pntpbnd1 27631 ostth2lem2 27679 axpaschlem 28956 axcontlem2 28981 nv1 30695 siii 30873 minvecolem4 30900 norm1 31269 strlem1 32270 unitdivcld 33901 cvmliftlem2 35292 cvmliftlem10 35300 cvmliftlem13 35302 snmlff 35335 poimirlem29 37657 poimirlem30 37658 poimirlem31 37659 poimirlem32 37660 pellexlem1 42845 pellexlem6 42850 jm2.22 43012 jm2.23 43013 stoweidlem36 46056 stoweidlem38 46058 nn0eo 48454 dignn0flhalf 48544 | 
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