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Mirrors > Home > MPE Home > Th. List > rphalfcld | Structured version Visualization version GIF version |
Description: Closure law for half of a positive real. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
rpred.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
Ref | Expression |
---|---|
rphalfcld | ⊢ (𝜑 → (𝐴 / 2) ∈ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpred.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
2 | rphalfcl 12057 | . 2 ⊢ (𝐴 ∈ ℝ+ → (𝐴 / 2) ∈ ℝ+) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (𝐴 / 2) ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2145 (class class class)co 6792 / cdiv 10886 2c2 11272 ℝ+crp 12031 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-po 5170 df-so 5171 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-2 11281 df-rp 12032 |
This theorem is referenced by: nnesq 13191 rlimuni 14485 climuni 14487 reccn2 14531 iseralt 14619 mertenslem1 14819 mertenslem2 14820 ege2le3 15022 rpcoshcl 15089 sqrt2irrlem 15179 sqrt2irrlemOLD 15180 4sqlem7 15851 ssblex 22449 methaus 22541 met2ndci 22543 metustexhalf 22577 cfilucfil 22580 nlmvscnlem2 22705 nlmvscnlem1 22706 nrginvrcnlem 22711 reperflem 22837 icccmplem2 22842 metdcnlem 22855 metnrmlem2 22879 metnrmlem3 22880 ipcnlem2 23258 ipcnlem1 23259 minveclem3 23415 ovollb2lem 23472 ovolunlem2 23482 uniioombl 23573 itg2cnlem2 23745 itg2cn 23746 lhop1lem 23992 lhop1 23993 aaliou2b 24312 ulmcn 24369 pserdvlem1 24397 pserdv 24399 cxpcn3lem 24705 lgamgulmlem3 24974 lgamucov 24981 ftalem2 25017 bposlem7 25232 bposlem9 25234 lgsquadlem2 25323 chebbnd1lem2 25376 pntibndlem3 25498 pntibnd 25499 pntlemr 25508 lt2addrd 29852 tpr2rico 30294 knoppndvlem17 32852 tan2h 33730 mblfinlem4 33778 sstotbnd2 33901 dstregt0 40008 suplesup 40068 infleinf 40101 lptre2pt 40387 0ellimcdiv 40396 limsupgtlem 40524 ioodvbdlimc1lem2 40662 ioodvbdlimc2lem 40664 stoweidlem62 40793 stirlinglem1 40805 |
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