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| Mirrors > Home > MPE Home > Th. List > rpreccld | Structured version Visualization version GIF version | ||
| Description: Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.) |
| Ref | Expression |
|---|---|
| rpred.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
| Ref | Expression |
|---|---|
| rpreccld | ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ+) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rpred.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
| 2 | rpreccl 13061 | . 2 ⊢ (𝐴 ∈ ℝ+ → (1 / 𝐴) ∈ ℝ+) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ+) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 (class class class)co 7431 1c1 11156 / cdiv 11920 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-rp 13035 |
| This theorem is referenced by: rprecred 13088 resqrex 15289 rlimno1 15690 supcvg 15892 harmonic 15895 expcnv 15900 eirrlem 16240 prmreclem5 16958 prmreclem6 16959 met1stc 24534 met2ndci 24535 nmoi2 24751 bcthlem5 25362 ovolsca 25550 vitali 25648 ismbf3d 25689 itg2seq 25777 itg2mulclem 25781 itg2mulc 25782 aalioulem3 26376 aaliou3lem8 26387 dvradcnv 26464 tanregt0 26581 divlogrlim 26677 advlogexp 26697 logtayllem 26701 divcxp 26729 cxpcn3lem 26790 loglesqrt 26804 logbrec 26825 ang180lem2 26853 asinlem3 26914 leibpi 26985 rlimcnp 27008 rlimcnp2 27009 efrlim 27012 efrlimOLD 27013 cxplim 27015 cxp2lim 27020 divsqrtsumlem 27023 amgmlem 27033 emcllem2 27040 emcllem4 27042 emcllem5 27043 emcllem6 27044 fsumharmonic 27055 lgamgulmlem5 27076 lgambdd 27080 basellem3 27126 basellem6 27129 logfaclbnd 27266 bclbnd 27324 rplogsumlem2 27529 rpvmasumlem 27531 dchrisum0lem2a 27561 log2sumbnd 27588 logdivbnd 27600 pntlemo 27651 nrt2irr 30492 smcnlem 30716 minvecolem3 30895 minvecolem4 30899 esumdivc 34084 dya2ub 34272 omssubadd 34302 logdivsqrle 34665 iprodgam 35742 faclimlem1 35743 faclimlem3 35745 faclim 35746 iprodfac 35747 poimirlem29 37656 poimirlem30 37657 heiborlem3 37820 heiborlem6 37823 heiborlem8 37825 heibor 37828 irrapxlem4 42836 irrapxlem5 42837 oddfl 45289 xralrple4 45384 xrralrecnnge 45401 ioodvbdlimc1lem2 45947 ioodvbdlimc2lem 45949 stoweid 46078 wallispi 46085 stirlinglem1 46089 stirlinglem6 46094 stirlinglem10 46098 stirlinglem11 46099 dirkertrigeqlem3 46115 dirkercncflem2 46119 iinhoiicc 46689 iunhoiioo 46691 vonioolem2 46696 vonicclem1 46698 eenglngeehlnmlem2 48659 amgmlemALT 49322 young2d 49324 |
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