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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 12255 | . 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 2079 (class class class)co 7007 / cdiv 11134 2c2 11529 ℝ+crp 12228 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-op 4473 df-uni 4740 df-br 4957 df-opab 5019 df-mpt 5036 df-id 5340 df-po 5354 df-so 5355 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-er 8130 df-en 8348 df-dom 8349 df-sdom 8350 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-div 11135 df-2 11537 df-rp 12229 |
This theorem is referenced by: nnesq 13426 rlimuni 14729 climuni 14731 reccn2 14775 iseralt 14863 mertenslem1 15061 mertenslem2 15062 ege2le3 15264 rpcoshcl 15331 sqrt2irrlem 15422 4sqlem7 16097 ssblex 22709 methaus 22801 met2ndci 22803 metustexhalf 22837 cfilucfil 22840 nlmvscnlem2 22965 nlmvscnlem1 22966 nrginvrcnlem 22971 reperflem 23097 icccmplem2 23102 metdcnlem 23115 metnrmlem2 23139 metnrmlem3 23140 ipcnlem2 23518 ipcnlem1 23519 minveclem3 23703 ovollb2lem 23760 ovolunlem2 23770 uniioombl 23861 itg2cnlem2 24034 itg2cn 24035 lhop1lem 24281 lhop1 24282 aaliou2b 24601 ulmcn 24658 pserdvlem1 24686 pserdv 24688 cxpcn3lem 24997 lgamgulmlem3 25278 lgamucov 25285 ftalem2 25321 bposlem7 25536 bposlem9 25538 lgsquadlem2 25627 chebbnd1lem2 25716 pntibndlem3 25838 pntibnd 25839 pntlemr 25848 lt2addrd 30136 tpr2rico 30728 knoppndvlem17 33420 tan2h 34361 mblfinlem4 34409 sstotbnd2 34530 dstregt0 41041 suplesup 41101 infleinf 41134 lptre2pt 41417 0ellimcdiv 41426 limsupgtlem 41554 ioodvbdlimc1lem2 41712 ioodvbdlimc2lem 41714 stoweidlem62 41843 stirlinglem1 41855 |
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