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Mirrors > Home > MPE Home > Th. List > rprege0d | Structured version Visualization version GIF version |
Description: A positive real is real and greater or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
rpred.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
Ref | Expression |
---|---|
rprege0d | ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpred.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
2 | 1 | rpred 12065 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
3 | 1 | rpge0d 12069 | . 2 ⊢ (𝜑 → 0 ≤ 𝐴) |
4 | 2, 3 | jca 555 | 1 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 2139 class class class wbr 4804 ℝcr 10127 0cc0 10128 ≤ cle 10267 ℝ+crp 12025 |
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 7114 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-i2m1 10196 ax-1ne0 10197 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 |
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 6816 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-rp 12026 |
This theorem is referenced by: eirrlem 15131 prmreclem3 15824 prmreclem6 15827 cxprec 24631 cxpsqrt 24648 cxpcn3lem 24687 cxplim 24897 cxploglim2 24904 divsqrtsumlem 24905 divsqrtsumo1 24909 fsumharmonic 24937 zetacvg 24940 logfacubnd 25145 logfacbnd3 25147 bposlem1 25208 bposlem4 25211 bposlem7 25214 bposlem9 25216 dchrmusum2 25382 dchrvmasumlem3 25387 dchrisum0flblem2 25397 dchrisum0fno1 25399 dchrisum0lema 25402 dchrisum0lem1b 25403 dchrisum0lem1 25404 dchrisum0lem2a 25405 dchrisum0lem2 25406 dchrisum0lem3 25407 chpdifbndlem2 25442 selberg3lem1 25445 pntrsumo1 25453 pntrlog2bndlem2 25466 pntrlog2bndlem4 25468 pntrlog2bndlem6a 25470 pntpbnd2 25475 pntibndlem2 25479 pntlemb 25485 pntlemg 25486 pntlemh 25487 pntlemn 25488 pntlemr 25490 pntlemj 25491 pntlemf 25493 pntlemk 25494 pntlemo 25495 blocnilem 27968 ubthlem2 28036 minvecolem4 28045 2sqmod 29957 eulerpartlemgc 30733 irrapxlem4 37891 irrapxlem5 37892 stirlinglem3 40796 stirlinglem15 40808 amgmlemALT 43062 |
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