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| 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 13044 | . 2 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
| 2 | rpgt0 13048 | . 2 ⊢ (𝐴 ∈ ℝ+ → 0 < 𝐴) | |
| 3 | 0re 11264 | . . 3 ⊢ 0 ∈ ℝ | |
| 4 | ltle 11350 | . . 3 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < 𝐴 → 0 ≤ 𝐴)) | |
| 5 | 3, 4 | mpan 690 | . 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 2107 class class class wbr 5142 ℝcr 11155 0cc0 11156 < clt 11296 ≤ cle 11297 ℝ+crp 13035 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-resscn 11213 ax-1cn 11214 ax-addrcl 11217 ax-rnegex 11227 ax-cnre 11229 ax-pre-lttri 11230 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-rp 13036 | 
| This theorem is referenced by: rprege0 13051 rpge0d 13082 xralrple 13248 xlemul1 13333 infmrp1 13387 01sqrexlem1 15282 rpsqrtcl 15304 divrcnv 15889 ef01bndlem 16221 stdbdmet 24530 reconnlem2 24850 cphsqrtcl3 25222 iscmet3lem3 25325 minveclem3 25464 itg2const2 25777 itg2mulclem 25782 aalioulem2 26376 pige3ALT 26563 argregt0 26653 argrege0 26654 2irrexpq 26774 cxpcn3 26792 cxplim 27016 cxp2lim 27021 divsqrtsumlem 27024 logdiflbnd 27039 basellem4 27128 ppiltx 27221 bposlem8 27336 bposlem9 27337 chebbnd1 27517 mulog2sumlem2 27580 selbergb 27594 selberg2b 27597 nmcexi 32046 nmcopexi 32047 nmcfnexi 32071 sqsscirc1 33908 divsqrtid 34610 logdivsqrle 34666 hgt750lem2 34668 subfacval3 35195 ptrecube 37628 heicant 37663 itg2addnclem 37679 itg2gt0cn 37683 areacirclem1 37716 areacirclem4 37719 areacirc 37721 cntotbnd 37804 rpabsid 42361 xralrple4 45389 xralrple3 45390 fourierdlem103 46229 blenre 48500 itscnhlinecirc02plem3 48710 itscnhlinecirc02p 48711 | 
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