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Mirrors > Home > MPE Home > Th. List > rprege0 | Structured version Visualization version GIF version |
Description: A positive real is a nonnegative real number. (Contributed by Mario Carneiro, 31-Jan-2014.) |
Ref | Expression |
---|---|
rprege0 | ⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpre 12385 | . 2 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
2 | rpge0 12390 | . 2 ⊢ (𝐴 ∈ ℝ+ → 0 ≤ 𝐴) | |
3 | 1, 2 | jca 512 | 1 ⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 class class class wbr 5057 ℝcr 10524 0cc0 10525 ≤ cle 10664 ℝ+crp 12377 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-addrcl 10586 ax-rnegex 10596 ax-cnre 10598 ax-pre-lttri 10599 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 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-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-rp 12378 |
This theorem is referenced by: resqrex 14598 sqrtdiv 14613 o1fsum 15156 prmreclem3 16242 aaliou3lem3 24860 pige3ALT 25032 rpcxpcl 25186 cxprec 25196 harmoniclbnd 25513 harmonicbnd4 25515 basellem4 25588 logfaclbnd 25725 logfacrlim 25727 logexprlim 25728 bposlem7 25793 vmadivsum 25985 dchrisum0lem2a 26020 dchrisum0lem2 26021 dchrisum0 26023 mudivsum 26033 mulogsumlem 26034 selberglem2 26049 selberg2lem 26053 pntrsumo1 26068 minvecolem3 28580 ehl2eudis0lt 44641 itsclc0 44686 itsclc0b 44687 |
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