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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 12429 | . 2 ⊢ (𝜑 → (𝐵 ∈ ℝ ∧ 0 < 𝐵)) |
5 | divge0 11501 | . 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 2108 class class class wbr 5057 (class class class)co 7148 ℝcr 10528 0cc0 10529 < clt 10667 ≤ cle 10668 / cdiv 11289 ℝ+crp 12381 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-rp 12382 |
This theorem is referenced by: iseralt 15033 nn0ehalf 15721 nn0oddm1d2 15728 bitsfzo 15776 bitsmod 15777 iserodd 16164 icopnfcnv 23538 logdiflbnd 25564 lgamgulmlem3 25600 chpo1ubb 26049 vmadivsumb 26051 rpvmasumlem 26055 dchrisumlem1 26057 dchrvmasumlem2 26066 rplogsum 26095 dirith2 26096 mulog2sumlem2 26103 vmalogdivsum2 26106 2vmadivsumlem 26108 selbergb 26117 selberg2b 26120 selberg4lem1 26128 pntrlog2bndlem2 26146 pntrlog2bndlem4 26148 pntrlog2bndlem5 26149 pntrlog2bndlem6 26151 pntrlog2bnd 26152 pntibndlem2 26159 ttgcontlem1 26663 sqsscirc1 31144 faclimlem1 32968 knoppndvlem14 33857 itg2addnclem2 34936 geomcau 35026 areaquad 39814 stirlinglem11 42360 stirlinglem12 42361 fourierdlem26 42409 fourierdlem30 42413 fourierdlem47 42429 sge0ad2en 42704 eenglngeehlnmlem2 44716 |
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