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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 11723 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐵) → (0 ≤ 𝐴 ↔ 0 ≤ (𝐴 / 𝐵))) | |
2 | 1 | biimpd 232 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐵) → (0 ≤ 𝐴 → 0 ≤ (𝐴 / 𝐵))) |
3 | 2 | 3exp 1121 | . . . 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 431 | 1 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 ≤ (𝐴 / 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 ∈ wcel 2111 class class class wbr 5067 (class class class)co 7231 ℝcr 10752 0cc0 10753 < clt 10891 ≤ cle 10892 / cdiv 11513 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 ax-un 7541 ax-resscn 10810 ax-1cn 10811 ax-icn 10812 ax-addcl 10813 ax-addrcl 10814 ax-mulcl 10815 ax-mulrcl 10816 ax-mulcom 10817 ax-addass 10818 ax-mulass 10819 ax-distr 10820 ax-i2m1 10821 ax-1ne0 10822 ax-1rid 10823 ax-rnegex 10824 ax-rrecex 10825 ax-cnre 10826 ax-pre-lttri 10827 ax-pre-lttrn 10828 ax-pre-ltadd 10829 ax-pre-mulgt0 10830 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4834 df-br 5068 df-opab 5130 df-mpt 5150 df-id 5469 df-po 5482 df-so 5483 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 df-fv 6405 df-riota 7188 df-ov 7234 df-oprab 7235 df-mpo 7236 df-er 8411 df-en 8647 df-dom 8648 df-sdom 8649 df-pnf 10893 df-mnf 10894 df-xr 10895 df-ltxr 10896 df-le 10897 df-sub 11088 df-neg 11089 df-div 11514 |
This theorem is referenced by: mulge0b 11726 ledivp1 11758 divge0i 11765 divge0d 12692 divelunit 13106 adddivflid 13417 fldiv4p1lem1div2 13434 fldiv 13457 modid 13493 modmuladdnn0 13512 expnbnd 13823 sqrtdiv 14853 sqreulem 14947 efcllem 15663 ege2le3 15675 flodddiv4 15998 hashgcdlem 16365 fldivp1 16474 4sqlem14 16535 odmodnn0 18956 prmirredlem 20483 icopnfcnv 23863 lebnumii 23887 nmoleub2lem3 24036 ncvs1 24078 minveclem4 24353 mbfi1fseqlem1 24637 mbfi1fseqlem5 24641 radcnvlem1 25329 cxpaddle 25662 log2tlbnd 25852 birthdaylem3 25860 jensenlem2 25894 amgm 25897 basellem3 25989 ppiub 26109 logfac2 26122 gausslemma2dlem0d 26264 chto1ub 26381 vmadivsum 26387 rpvmasumlem 26392 dchrvmasumlem2 26403 dchrvmasumiflem1 26406 dchrisum0fno1 26416 dchrisum0re 26418 mulog2sumlem2 26440 selberg2lem 26455 pntrmax 26469 pntrsumo1 26470 pntpbnd1 26491 ostth2lem2 26539 axpaschlem 27055 axcontlem2 27080 nv1 28780 siii 28958 minvecolem4 28985 norm1 29354 strlem1 30355 unitdivcld 31589 cvmliftlem2 32984 cvmliftlem10 32992 cvmliftlem13 32994 snmlff 33027 poimirlem29 35569 poimirlem30 35570 poimirlem31 35571 poimirlem32 35572 pellexlem1 40382 pellexlem6 40387 jm2.22 40548 jm2.23 40549 stoweidlem36 43280 stoweidlem38 43282 nn0eo 45575 dignn0flhalf 45665 |
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