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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 12757 | . 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 2106 (class class class)co 7275 / cdiv 11632 2c2 12028 ℝ+crp 12730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-2 12036 df-rp 12731 |
This theorem is referenced by: nnesq 13942 rlimuni 15259 climuni 15261 reccn2 15306 iseralt 15396 mertenslem1 15596 mertenslem2 15597 ege2le3 15799 rpcoshcl 15866 sqrt2irrlem 15957 4sqlem7 16645 ssblex 23581 methaus 23676 met2ndci 23678 metustexhalf 23712 cfilucfil 23715 nlmvscnlem2 23849 nlmvscnlem1 23850 nrginvrcnlem 23855 reperflem 23981 icccmplem2 23986 metdcnlem 23999 metnrmlem2 24023 metnrmlem3 24024 ipcnlem2 24408 ipcnlem1 24409 minveclem3 24593 ovollb2lem 24652 ovolunlem2 24662 uniioombl 24753 itg2cnlem2 24927 itg2cn 24928 lhop1lem 25177 lhop1 25178 aaliou2b 25501 ulmcn 25558 pserdvlem1 25586 pserdv 25588 cxpcn3lem 25900 lgamgulmlem3 26180 lgamucov 26187 ftalem2 26223 bposlem7 26438 bposlem9 26440 lgsquadlem2 26529 chebbnd1lem2 26618 pntibndlem3 26740 pntibnd 26741 pntlemr 26750 lt2addrd 31074 tpr2rico 31862 knoppndvlem17 34708 tan2h 35769 mblfinlem4 35817 sstotbnd2 35932 3lexlogpow2ineq2 40067 dstregt0 42820 suplesup 42878 infleinf 42911 lptre2pt 43181 0ellimcdiv 43190 limsupgtlem 43318 ioodvbdlimc1lem2 43473 ioodvbdlimc2lem 43475 stoweidlem62 43603 stirlinglem1 43615 |
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