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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 than 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 13077 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
| 3 | 1 | rpge0d 13081 | . 2 ⊢ (𝜑 → 0 ≤ 𝐴) |
| 4 | 2, 3 | jca 511 | 1 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 class class class wbr 5143 ℝcr 11154 0cc0 11155 ≤ cle 11296 ℝ+crp 13034 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-addrcl 11216 ax-rnegex 11226 ax-cnre 11228 ax-pre-lttri 11229 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-rp 13035 |
| This theorem is referenced by: eirrlem 16240 prmreclem3 16956 prmreclem6 16959 cxprec 26728 cxpsqrt 26745 cxpcn3lem 26790 cxplim 27015 cxploglim2 27022 divsqrtsumlem 27023 divsqrtsumo1 27027 fsumharmonic 27055 zetacvg 27058 logfacubnd 27265 logfacbnd3 27267 bposlem1 27328 bposlem4 27331 bposlem7 27334 bposlem9 27336 2sqmod 27480 dchrmusum2 27538 dchrvmasumlem3 27543 dchrisum0flblem2 27553 dchrisum0fno1 27555 dchrisum0lema 27558 dchrisum0lem1b 27559 dchrisum0lem1 27560 dchrisum0lem2a 27561 dchrisum0lem2 27562 dchrisum0lem3 27563 chpdifbndlem2 27598 selberg3lem1 27601 pntrsumo1 27609 pntrlog2bndlem2 27622 pntrlog2bndlem4 27624 pntrlog2bndlem6a 27626 pntpbnd2 27631 pntibndlem2 27635 pntlemb 27641 pntlemg 27642 pntlemh 27643 pntlemn 27644 pntlemr 27646 pntlemj 27647 pntlemf 27649 pntlemk 27650 pntlemo 27651 blocnilem 30823 ubthlem2 30890 minvecolem4 30899 eulerpartlemgc 34364 irrapxlem4 42836 irrapxlem5 42837 stirlinglem3 46091 stirlinglem15 46103 inlinecirc02plem 48707 amgmlemALT 49322 |
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