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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 12613 | . 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 2110 (class class class)co 7213 / cdiv 11489 2c2 11885 ℝ+crp 12586 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-po 5468 df-so 5469 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-2 11893 df-rp 12587 |
This theorem is referenced by: nnesq 13794 rlimuni 15111 climuni 15113 reccn2 15158 iseralt 15248 mertenslem1 15448 mertenslem2 15449 ege2le3 15651 rpcoshcl 15718 sqrt2irrlem 15809 4sqlem7 16497 ssblex 23326 methaus 23418 met2ndci 23420 metustexhalf 23454 cfilucfil 23457 nlmvscnlem2 23583 nlmvscnlem1 23584 nrginvrcnlem 23589 reperflem 23715 icccmplem2 23720 metdcnlem 23733 metnrmlem2 23757 metnrmlem3 23758 ipcnlem2 24141 ipcnlem1 24142 minveclem3 24326 ovollb2lem 24385 ovolunlem2 24395 uniioombl 24486 itg2cnlem2 24660 itg2cn 24661 lhop1lem 24910 lhop1 24911 aaliou2b 25234 ulmcn 25291 pserdvlem1 25319 pserdv 25321 cxpcn3lem 25633 lgamgulmlem3 25913 lgamucov 25920 ftalem2 25956 bposlem7 26171 bposlem9 26173 lgsquadlem2 26262 chebbnd1lem2 26351 pntibndlem3 26473 pntibnd 26474 pntlemr 26483 lt2addrd 30794 tpr2rico 31576 knoppndvlem17 34445 tan2h 35506 mblfinlem4 35554 sstotbnd2 35669 3lexlogpow2ineq2 39801 dstregt0 42492 suplesup 42551 infleinf 42584 lptre2pt 42856 0ellimcdiv 42865 limsupgtlem 42993 ioodvbdlimc1lem2 43148 ioodvbdlimc2lem 43150 stoweidlem62 43278 stirlinglem1 43290 |
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