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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 13062 | . 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 2108 (class class class)co 7431 / cdiv 11920 2c2 12321 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-2 12329 df-rp 13035 | 
| This theorem is referenced by: nnesq 14266 rlimuni 15586 climuni 15588 reccn2 15633 iseralt 15721 mertenslem1 15920 mertenslem2 15921 ege2le3 16126 rpcoshcl 16193 sqrt2irrlem 16284 4sqlem7 16982 ssblex 24438 methaus 24533 met2ndci 24535 metustexhalf 24569 cfilucfil 24572 nlmvscnlem2 24706 nlmvscnlem1 24707 nrginvrcnlem 24712 reperflem 24840 icccmplem2 24845 metdcnlem 24858 metnrmlem2 24882 metnrmlem3 24883 ipcnlem2 25278 ipcnlem1 25279 minveclem3 25463 ovollb2lem 25523 ovolunlem2 25533 uniioombl 25624 itg2cnlem2 25797 itg2cn 25798 lhop1lem 26052 lhop1 26053 aaliou2b 26383 ulmcn 26442 pserdvlem1 26471 pserdv 26473 cxpcn3lem 26790 lgamgulmlem3 27074 lgamucov 27081 ftalem2 27117 bposlem7 27334 bposlem9 27336 lgsquadlem2 27425 chebbnd1lem2 27514 pntibndlem3 27636 pntibnd 27637 pntlemr 27646 lt2addrd 32755 tpr2rico 33911 knoppndvlem17 36529 tan2h 37619 mblfinlem4 37667 sstotbnd2 37781 3lexlogpow2ineq2 42060 dstregt0 45293 suplesup 45350 infleinf 45383 lptre2pt 45655 0ellimcdiv 45664 limsupgtlem 45792 ioodvbdlimc1lem2 45947 ioodvbdlimc2lem 45949 stoweidlem62 46077 stirlinglem1 46089 | 
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