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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 10769 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐵) → (0 ≤ 𝐴 ↔ 0 ≤ (𝐴 / 𝐵))) | |
2 | 1 | biimpd 218 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐵) → (0 ≤ 𝐴 → 0 ≤ (𝐴 / 𝐵))) |
3 | 2 | 3exp 1256 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐵 ∈ ℝ → (0 < 𝐵 → (0 ≤ 𝐴 → 0 ≤ (𝐴 / 𝐵))))) |
4 | 3 | com34 89 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐵 ∈ ℝ → (0 ≤ 𝐴 → (0 < 𝐵 → 0 ≤ (𝐴 / 𝐵))))) |
5 | 4 | com23 84 | . 2 ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 → (𝐵 ∈ ℝ → (0 < 𝐵 → 0 ≤ (𝐴 / 𝐵))))) |
6 | 5 | imp43 619 | 1 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 ≤ (𝐴 / 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 ∈ wcel 1977 class class class wbr 4583 (class class class)co 6549 ℝcr 9814 0cc0 9815 < clt 9953 ≤ cle 9954 / cdiv 10563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 |
This theorem is referenced by: mulge0b 10772 ledivp1 10804 divge0i 10812 divge0d 11788 divelunit 12185 adddivflid 12481 fldiv4p1lem1div2 12498 fldiv 12521 modid 12557 modmuladdnn0 12576 expnbnd 12855 sqrtdiv 13854 sqreulem 13947 iseralt 14263 efcllem 14647 ege2le3 14659 flodddiv4 14975 hashgcdlem 15331 iserodd 15378 fldivp1 15439 4sqlem14 15500 odmodnn0 17782 prmirredlem 19660 icopnfcnv 22549 lebnumii 22573 nmoleub2lem3 22723 ncvs1 22765 minveclem4 23011 mbfi1fseqlem1 23288 mbfi1fseqlem5 23292 radcnvlem1 23971 cxpaddle 24293 leibpilem1 24467 log2tlbnd 24472 birthdaylem3 24480 jensenlem2 24514 amgm 24517 basellem3 24609 ppiub 24729 logfac2 24742 gausslemma2dlem0d 24884 chto1ub 24965 vmadivsum 24971 rpvmasumlem 24976 dchrvmasumlem2 24987 dchrvmasumiflem1 24990 dchrisum0fno1 25000 dchrisum0re 25002 mulog2sumlem2 25024 selberg2lem 25039 pntrmax 25053 pntrsumo1 25054 pntpbnd1 25075 ostth2lem2 25123 axpaschlem 25620 axcontlem2 25645 nv1 26914 siii 27092 minvecolem4 27120 norm1 27490 strlem1 28493 unitdivcld 29275 cvmliftlem2 30522 cvmliftlem10 30530 cvmliftlem13 30532 snmlff 30565 poimirlem29 32608 poimirlem30 32609 poimirlem31 32610 poimirlem32 32611 pellexlem1 36411 pellexlem6 36416 jm2.22 36580 jm2.23 36581 stoweidlem36 38929 stoweidlem38 38931 nn0eo 42116 dignn0flhalf 42210 |
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