![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 13081 | . 2 ⊢ (𝜑 → (𝐵 ∈ ℝ ∧ 0 < 𝐵)) |
5 | divge0 12135 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 ≤ (𝐴 / 𝐵)) | |
6 | 1, 2, 4, 5 | syl21anc 838 | 1 ⊢ (𝜑 → 0 ≤ (𝐴 / 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 class class class wbr 5148 (class class class)co 7431 ℝcr 11152 0cc0 11153 < clt 11293 ≤ cle 11294 / cdiv 11918 ℝ+crp 13032 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-rp 13033 |
This theorem is referenced by: iseralt 15718 nn0ehalf 16412 nn0oddm1d2 16419 bitsfzo 16469 bitsmod 16470 iserodd 16869 icopnfcnv 24987 logdiflbnd 27053 lgamgulmlem3 27089 chpo1ubb 27540 vmadivsumb 27542 rpvmasumlem 27546 dchrisumlem1 27548 dchrvmasumlem2 27557 rplogsum 27586 dirith2 27587 mulog2sumlem2 27594 vmalogdivsum2 27597 2vmadivsumlem 27599 selbergb 27608 selberg2b 27611 selberg4lem1 27619 pntrlog2bndlem2 27637 pntrlog2bndlem4 27639 pntrlog2bndlem5 27640 pntrlog2bndlem6 27642 pntrlog2bnd 27643 pntibndlem2 27650 ttgcontlem1 28914 sqsscirc1 33869 faclimlem1 35723 knoppndvlem14 36508 itg2addnclem2 37659 geomcau 37746 3lexlogpow5ineq2 42037 aks4d1p1p7 42056 aks6d1c2lem4 42109 aks6d1c7lem1 42162 areaquad 43205 sqrtcvallem2 43627 sqrtcvallem4 43629 stirlinglem11 46040 stirlinglem12 46041 fourierdlem26 46089 fourierdlem30 46093 fourierdlem47 46109 sge0ad2en 46387 eenglngeehlnmlem2 48588 |
Copyright terms: Public domain | W3C validator |