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Mirrors > Home > MPE Home > Th. List > divge0 | Structured version Visualization version GIF version |
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 12085 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐵) → (0 ≤ 𝐴 ↔ 0 ≤ (𝐴 / 𝐵))) | |
2 | 1 | biimpd 228 | . . . . 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 428 | 1 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 ≤ (𝐴 / 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 ∈ wcel 2106 class class class wbr 5148 (class class class)co 7411 ℝcr 11111 0cc0 11112 < clt 11252 ≤ cle 11253 / cdiv 11875 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 |
This theorem is referenced by: mulge0b 12088 ledivp1 12120 divge0i 12127 divge0d 13060 divelunit 13475 adddivflid 13787 fldiv4p1lem1div2 13804 fldiv 13829 modid 13865 modmuladdnn0 13884 expnbnd 14199 sqrtdiv 15216 sqreulem 15310 efcllem 16025 ege2le3 16037 flodddiv4 16360 hashgcdlem 16725 fldivp1 16834 4sqlem14 16895 odmodnn0 19449 prmirredlem 21243 icopnfcnv 24682 lebnumii 24706 nmoleub2lem3 24855 ncvs1 24898 minveclem4 25173 mbfi1fseqlem1 25457 mbfi1fseqlem5 25461 radcnvlem1 26149 cxpaddle 26484 log2tlbnd 26674 birthdaylem3 26682 jensenlem2 26716 amgm 26719 basellem3 26811 ppiub 26931 logfac2 26944 gausslemma2dlem0d 27086 chto1ub 27203 vmadivsum 27209 rpvmasumlem 27214 dchrvmasumlem2 27225 dchrvmasumiflem1 27228 dchrisum0fno1 27238 dchrisum0re 27240 mulog2sumlem2 27262 selberg2lem 27277 pntrmax 27291 pntrsumo1 27292 pntpbnd1 27313 ostth2lem2 27361 axpaschlem 28453 axcontlem2 28478 nv1 30183 siii 30361 minvecolem4 30388 norm1 30757 strlem1 31758 unitdivcld 33167 cvmliftlem2 34563 cvmliftlem10 34571 cvmliftlem13 34573 snmlff 34606 poimirlem29 36820 poimirlem30 36821 poimirlem31 36822 poimirlem32 36823 pellexlem1 41869 pellexlem6 41874 jm2.22 42036 jm2.23 42037 stoweidlem36 45051 stoweidlem38 45053 nn0eo 47302 dignn0flhalf 47392 |
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