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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 515 | 1 ⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 class class class wbr 5030 ℝcr 10525 0cc0 10526 ≤ cle 10665 ℝ+crp 12377 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-addrcl 10587 ax-rnegex 10597 ax-cnre 10599 ax-pre-lttri 10600 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-rp 12378 |
This theorem is referenced by: resqrex 14602 sqrtdiv 14617 o1fsum 15160 prmreclem3 16244 aaliou3lem3 24940 pige3ALT 25112 rpcxpcl 25267 cxprec 25277 harmoniclbnd 25594 harmonicbnd4 25596 basellem4 25669 logfaclbnd 25806 logfacrlim 25808 logexprlim 25809 bposlem7 25874 vmadivsum 26066 dchrisum0lem2a 26101 dchrisum0lem2 26102 dchrisum0 26104 mudivsum 26114 mulogsumlem 26115 selberglem2 26130 selberg2lem 26134 pntrsumo1 26149 minvecolem3 28659 ehl2eudis0lt 45140 itsclc0 45185 itsclc0b 45186 |
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