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Mirrors > Home > MPE Home > Th. List > rpge0 | Structured version Visualization version GIF version |
Description: A positive real is greater than or equal to zero. (Contributed by NM, 22-Feb-2008.) |
Ref | Expression |
---|---|
rpge0 | ⊢ (𝐴 ∈ ℝ+ → 0 ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpre 13022 | . 2 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
2 | rpgt0 13026 | . 2 ⊢ (𝐴 ∈ ℝ+ → 0 < 𝐴) | |
3 | 0re 11253 | . . 3 ⊢ 0 ∈ ℝ | |
4 | ltle 11339 | . . 3 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < 𝐴 → 0 ≤ 𝐴)) | |
5 | 3, 4 | mpan 688 | . 2 ⊢ (𝐴 ∈ ℝ → (0 < 𝐴 → 0 ≤ 𝐴)) |
6 | 1, 2, 5 | sylc 65 | 1 ⊢ (𝐴 ∈ ℝ+ → 0 ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 class class class wbr 5149 ℝcr 11144 0cc0 11145 < clt 11285 ≤ cle 11286 ℝ+crp 13014 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-resscn 11202 ax-1cn 11203 ax-addrcl 11206 ax-rnegex 11216 ax-cnre 11218 ax-pre-lttri 11219 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-rp 13015 |
This theorem is referenced by: rprege0 13029 rpge0d 13060 xralrple 13224 xlemul1 13309 infmrp1 13363 01sqrexlem1 15233 rpsqrtcl 15255 divrcnv 15842 ef01bndlem 16172 stdbdmet 24486 reconnlem2 24804 cphsqrtcl3 25176 iscmet3lem3 25279 minveclem3 25418 itg2const2 25732 itg2mulclem 25737 aalioulem2 26330 pige3ALT 26516 argregt0 26606 argrege0 26607 2irrexpq 26727 cxpcn3 26745 cxplim 26969 cxp2lim 26974 divsqrtsumlem 26977 logdiflbnd 26992 basellem4 27081 ppiltx 27174 bposlem8 27289 bposlem9 27290 chebbnd1 27470 mulog2sumlem2 27533 selbergb 27547 selberg2b 27550 nmcexi 31928 nmcopexi 31929 nmcfnexi 31953 sqsscirc1 33660 divsqrtid 34377 logdivsqrle 34433 hgt750lem2 34435 subfacval3 34950 ptrecube 37244 heicant 37279 itg2addnclem 37295 itg2gt0cn 37299 areacirclem1 37332 areacirclem4 37335 areacirc 37337 cntotbnd 37420 xralrple4 44898 xralrple3 44899 fourierdlem103 45740 blenre 47838 itscnhlinecirc02plem3 48048 itscnhlinecirc02p 48049 |
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