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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 12052 | . 2 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
2 | rpgt0 12057 | . 2 ⊢ (𝐴 ∈ ℝ+ → 0 < 𝐴) | |
3 | 0re 10252 | . . 3 ⊢ 0 ∈ ℝ | |
4 | ltle 10338 | . . 3 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < 𝐴 → 0 ≤ 𝐴)) | |
5 | 3, 4 | mpan 708 | . 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 2139 class class class wbr 4804 ℝcr 10147 0cc0 10148 < clt 10286 ≤ cle 10287 ℝ+crp 12045 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-i2m1 10216 ax-1ne0 10217 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6817 df-er 7913 df-en 8124 df-dom 8125 df-sdom 8126 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-rp 12046 |
This theorem is referenced by: rprege0 12060 rpge0d 12089 xralrple 12249 xlemul1 12333 infmrp1 12387 sqrlem1 14202 rpsqrtcl 14224 divrcnv 14803 ef01bndlem 15133 stdbdmet 22542 reconnlem2 22851 cphsqrtcl3 23207 iscmet3lem3 23308 minveclem3 23420 itg2const2 23727 itg2mulclem 23732 aalioulem2 24307 pige3 24489 argregt0 24576 argrege0 24577 cxpcn3 24709 cxplim 24918 cxp2lim 24923 divsqrtsumlem 24926 logdiflbnd 24941 basellem4 25030 ppiltx 25123 bposlem8 25236 bposlem9 25237 chebbnd1 25381 mulog2sumlem2 25444 selbergb 25458 selberg2b 25461 nmcexi 29215 nmcopexi 29216 nmcfnexi 29240 sqsscirc1 30284 divsqrtid 31002 logdivsqrle 31058 hgt750lem2 31060 subfacval3 31499 ptrecube 33740 heicant 33775 itg2addnclem 33792 itg2gt0cn 33796 areacirclem1 33831 areacirclem4 33834 areacirc 33836 cntotbnd 33926 xralrple4 40105 xralrple3 40106 fourierdlem103 40947 blenre 42896 |
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