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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 11748 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
3 | 1 | rpge0d 11752 | . 2 ⊢ (𝜑 → 0 ≤ 𝐴) |
4 | 2, 3 | jca 553 | 1 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 class class class wbr 4583 ℝcr 9814 0cc0 9815 ≤ cle 9954 ℝ+crp 11708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-rp 11709 |
This theorem is referenced by: eirrlem 14771 prmreclem3 15460 prmreclem6 15463 cxprec 24232 cxpsqrt 24249 cxpcn3lem 24288 cxplim 24498 cxploglim2 24505 divsqrtsumlem 24506 divsqrtsumo1 24510 fsumharmonic 24538 zetacvg 24541 logfacubnd 24746 logfacbnd3 24748 bposlem1 24809 bposlem4 24812 bposlem7 24815 bposlem9 24817 dchrmusum2 24983 dchrvmasumlem3 24988 dchrisum0flblem2 24998 dchrisum0fno1 25000 dchrisum0lema 25003 dchrisum0lem1b 25004 dchrisum0lem1 25005 dchrisum0lem2a 25006 dchrisum0lem2 25007 dchrisum0lem3 25008 chpdifbndlem2 25043 selberg3lem1 25046 pntrsumo1 25054 pntrlog2bndlem2 25067 pntrlog2bndlem4 25069 pntrlog2bndlem6a 25071 pntpbnd2 25076 pntibndlem2 25080 pntlemb 25086 pntlemg 25087 pntlemh 25088 pntlemn 25089 pntlemr 25091 pntlemj 25092 pntlemf 25094 pntlemk 25095 pntlemo 25096 blocnilem 27043 ubthlem2 27111 minvecolem4 27120 2sqmod 28979 eulerpartlemgc 29751 irrapxlem4 36407 irrapxlem5 36408 stirlinglem3 38969 stirlinglem15 38981 amgmlemALT 42358 |
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