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| Mirrors > Home > MPE Home > Th. List > rpgt0 | Structured version Visualization version GIF version | ||
| Description: A positive real is greater than zero. (Contributed by FL, 27-Dec-2007.) |
| Ref | Expression |
|---|---|
| rpgt0 | ⊢ (𝐴 ∈ ℝ+ → 0 < 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elrp 13036 | . 2 ⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) | |
| 2 | 1 | simprbi 496 | 1 ⊢ (𝐴 ∈ ℝ+ → 0 < 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 class class class wbr 5143 ℝcr 11154 0cc0 11155 < clt 11295 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-ext 2708 |
| 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-rp 13035 |
| This theorem is referenced by: rpge0 13048 rpgecl 13063 0nrp 13070 rpgt0d 13080 addlelt 13149 0mod 13942 sgnrrp 15130 01sqrexlem2 15282 01sqrexlem4 15284 01sqrexlem6 15286 resqrex 15289 rpsqrtcl 15303 climconst 15579 rlimconst 15580 divrcnv 15888 rprisefaccl 16059 blcntrps 24422 blcntr 24423 stdbdmet 24529 stdbdmopn 24531 prdsxmslem2 24542 metustid 24567 nmoix 24750 metdseq0 24876 lebnumii 24998 itgulm 26451 pilem2 26496 cos02pilt1 26568 tanregt0 26581 logdmnrp 26683 cxple2 26739 asinneg 26929 asin1 26937 reasinsin 26939 atanbndlem 26968 atanbnd 26969 atan1 26971 rlimcnp 27008 chtrpcl 27218 ppiltx 27220 bposlem8 27335 pntlem3 27653 padicabvcxp 27676 0cnop 31998 0cnfn 31999 rpdp2cl 32864 xdivpnfrp 32915 pnfinf 33190 hgt750lem2 34667 taupilem1 37322 itg2gt0cn 37682 areacirclem1 37715 areacirclem4 37718 prdstotbnd 37801 prdsbnd2 37802 aks4d1p1p6 42074 irrapxlem3 42835 neglt 45296 xralrple2 45365 constlimc 45639 0cnv 45757 ioodvbdlimc1lem1 45946 fourierdlem103 46224 fourierdlem104 46225 etransclem18 46267 etransclem46 46295 hoidmvlelem3 46612 |
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