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| Mirrors > Home > MPE Home > Th. List > elrege0 | Structured version Visualization version GIF version | ||
| Description: The predicate "is a nonnegative real". (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 18-Jun-2014.) |
| Ref | Expression |
|---|---|
| elrege0 | ⊢ (𝐴 ∈ (0[,)+∞) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 0re 11263 | . 2 ⊢ 0 ∈ ℝ | |
| 2 | elicopnf 13485 | . 2 ⊢ (0 ∈ ℝ → (𝐴 ∈ (0[,)+∞) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ∈ (0[,)+∞) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 ℝcr 11154 0cc0 11155 +∞cpnf 11292 ≤ cle 11296 [,)cico 13389 |
| 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-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-addrcl 11216 ax-rnegex 11226 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 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-po 5592 df-so 5593 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-ov 7434 df-oprab 7435 df-mpo 7436 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-ico 13393 |
| This theorem is referenced by: nn0rp0 13495 rge0ssre 13496 0e0icopnf 13498 ge0addcl 13500 ge0mulcl 13501 fsumge0 15831 fprodge0 16029 isabvd 20813 abvge0 20818 nmolb 24738 nmoge0 24742 nmoi 24749 icopnfcnv 24973 cphsqrtcl 25218 tcphcph 25271 cphsscph 25285 ovolfsf 25506 ovolmge0 25512 ovolunlem1a 25531 ovoliunlem1 25537 ovolicc2lem4 25555 ioombl1lem4 25596 uniioombllem2 25618 uniioombllem6 25623 0plef 25707 i1fpos 25741 mbfi1fseqlem1 25750 mbfi1fseqlem3 25752 mbfi1fseqlem4 25753 mbfi1fseqlem5 25754 mbfi1fseqlem6 25755 mbfi1flimlem 25757 itg2const 25775 itg2const2 25776 itg2mulclem 25781 itg2mulc 25782 itg2monolem1 25785 itg2mono 25788 itg2addlem 25793 itg2gt0 25795 itg2cnlem1 25796 itg2cnlem2 25797 itg2cn 25798 iblconst 25853 itgconst 25854 ibladdlem 25855 itgaddlem1 25858 iblabslem 25863 iblabs 25864 iblmulc2 25866 itgmulc2lem1 25867 bddmulibl 25874 bddiblnc 25877 itggt0 25879 itgcn 25880 dvge0 26045 dvle 26046 dvfsumrlim 26072 cxpcn3lem 26790 cxpcn3 26791 resqrtcn 26792 loglesqrt 26804 areaf 27004 areacl 27005 areage0 27006 rlimcnp3 27010 jensenlem2 27031 jensen 27032 amgmlem 27033 amgm 27034 dchrisumlem3 27535 dchrmusumlema 27537 dchrmusum2 27538 dchrvmasumlem2 27542 dchrvmasumiflem1 27545 dchrisum0lema 27558 dchrisum0lem1b 27559 dchrisum0lem1 27560 dchrisum0lem2 27562 axcontlem2 28980 axcontlem7 28985 axcontlem8 28986 axcontlem10 28988 rge0scvg 33948 esumpcvgval 34079 hasheuni 34086 esumcvg 34087 sibfof 34342 mbfposadd 37674 itg2addnclem2 37679 itg2addnclem3 37680 itg2addnc 37681 itg2gt0cn 37682 ibladdnclem 37683 itgaddnclem1 37685 iblabsnclem 37690 iblabsnc 37691 iblmulc2nc 37692 itgmulc2nclem1 37693 itggt0cn 37697 ftc1anclem3 37702 ftc1anclem4 37703 ftc1anclem5 37704 ftc1anclem6 37705 ftc1anclem7 37706 ftc1anclem8 37707 areacirclem2 37716 sge0iunmptlemfi 46428 digvalnn0 48520 nn0digval 48521 dignn0fr 48522 dig2nn1st 48526 digexp 48528 2sphere 48670 itsclc0 48692 itsclc0b 48693 |
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