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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 12438 | . 2 ⊢ (𝜑 → (𝐵 ∈ ℝ ∧ 0 < 𝐵)) |
5 | divge0 11509 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 ≤ (𝐴 / 𝐵)) | |
6 | 1, 2, 4, 5 | syl21anc 835 | 1 ⊢ (𝜑 → 0 ≤ (𝐴 / 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 class class class wbr 5066 (class class class)co 7156 ℝcr 10536 0cc0 10537 < clt 10675 ≤ cle 10676 / cdiv 11297 ℝ+crp 12390 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-rp 12391 |
This theorem is referenced by: iseralt 15041 nn0ehalf 15729 nn0oddm1d2 15736 bitsfzo 15784 bitsmod 15785 iserodd 16172 icopnfcnv 23546 logdiflbnd 25572 lgamgulmlem3 25608 chpo1ubb 26057 vmadivsumb 26059 rpvmasumlem 26063 dchrisumlem1 26065 dchrvmasumlem2 26074 rplogsum 26103 dirith2 26104 mulog2sumlem2 26111 vmalogdivsum2 26114 2vmadivsumlem 26116 selbergb 26125 selberg2b 26128 selberg4lem1 26136 pntrlog2bndlem2 26154 pntrlog2bndlem4 26156 pntrlog2bndlem5 26157 pntrlog2bndlem6 26159 pntrlog2bnd 26160 pntibndlem2 26167 ttgcontlem1 26671 sqsscirc1 31151 faclimlem1 32975 knoppndvlem14 33864 itg2addnclem2 34959 geomcau 35049 areaquad 39843 stirlinglem11 42389 stirlinglem12 42390 fourierdlem26 42438 fourierdlem30 42442 fourierdlem47 42458 sge0ad2en 42733 eenglngeehlnmlem2 44745 |
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