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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 13049 | . 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 2099 (class class class)co 7416 / cdiv 11912 2c2 12313 ℝ+crp 13022 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-po 5586 df-so 5587 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-2 12321 df-rp 13023 |
This theorem is referenced by: nnesq 14239 rlimuni 15547 climuni 15549 reccn2 15594 iseralt 15684 mertenslem1 15883 mertenslem2 15884 ege2le3 16087 rpcoshcl 16154 sqrt2irrlem 16245 4sqlem7 16941 ssblex 24422 methaus 24517 met2ndci 24519 metustexhalf 24553 cfilucfil 24556 nlmvscnlem2 24690 nlmvscnlem1 24691 nrginvrcnlem 24696 reperflem 24822 icccmplem2 24827 metdcnlem 24840 metnrmlem2 24864 metnrmlem3 24865 ipcnlem2 25260 ipcnlem1 25261 minveclem3 25445 ovollb2lem 25505 ovolunlem2 25515 uniioombl 25606 itg2cnlem2 25780 itg2cn 25781 lhop1lem 26034 lhop1 26035 aaliou2b 26366 ulmcn 26425 pserdvlem1 26454 pserdv 26456 cxpcn3lem 26772 lgamgulmlem3 27056 lgamucov 27063 ftalem2 27099 bposlem7 27316 bposlem9 27318 lgsquadlem2 27407 chebbnd1lem2 27496 pntibndlem3 27618 pntibnd 27619 pntlemr 27628 lt2addrd 32658 tpr2rico 33740 knoppndvlem17 36244 tan2h 37326 mblfinlem4 37374 sstotbnd2 37488 3lexlogpow2ineq2 41771 dstregt0 44932 suplesup 44990 infleinf 45023 lptre2pt 45297 0ellimcdiv 45306 limsupgtlem 45434 ioodvbdlimc1lem2 45589 ioodvbdlimc2lem 45591 stoweidlem62 45719 stirlinglem1 45731 |
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