Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 12405 | . 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 2105 (class class class)co 7145 1c1 10527 / cdiv 11286 ℝ+crp 12379 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-rp 12380 |
This theorem is referenced by: rprecred 12432 resqrex 14600 rlimno1 15000 supcvg 15201 harmonic 15204 expcnv 15209 eirrlem 15547 prmreclem5 16246 prmreclem6 16247 met1stc 23060 met2ndci 23061 nmoi2 23268 bcthlem5 23860 ovolsca 24045 vitali 24143 ismbf3d 24184 itg2seq 24272 itg2mulclem 24276 itg2mulc 24277 aalioulem3 24852 aaliou3lem8 24863 dvradcnv 24938 tanregt0 25050 divlogrlim 25145 advlogexp 25165 logtayllem 25169 divcxp 25197 cxpcn3lem 25255 loglesqrt 25266 logbrec 25287 ang180lem2 25315 asinlem3 25376 leibpi 25448 rlimcnp 25471 rlimcnp2 25472 efrlim 25475 cxplim 25477 cxp2lim 25482 divsqrtsumlem 25485 amgmlem 25495 emcllem2 25502 emcllem4 25504 emcllem5 25505 emcllem6 25506 fsumharmonic 25517 lgamgulmlem5 25538 lgambdd 25542 basellem3 25588 basellem6 25591 logfaclbnd 25726 bclbnd 25784 rplogsumlem2 25989 rpvmasumlem 25991 dchrisum0lem2a 26021 log2sumbnd 26048 logdivbnd 26060 pntlemo 26111 smcnlem 28402 minvecolem3 28581 minvecolem4 28585 esumdivc 31242 dya2ub 31428 omssubadd 31458 logdivsqrle 31821 iprodgam 32872 faclimlem1 32873 faclimlem3 32875 faclim 32876 iprodfac 32877 poimirlem29 34803 poimirlem30 34804 heiborlem3 34974 heiborlem6 34977 heiborlem8 34979 heibor 34982 irrapxlem4 39302 irrapxlem5 39303 oddfl 41423 xralrple4 41521 xrralrecnnge 41542 ioodvbdlimc1lem2 42097 ioodvbdlimc2lem 42099 stoweid 42229 wallispi 42236 stirlinglem1 42240 stirlinglem6 42245 stirlinglem10 42249 stirlinglem11 42250 dirkertrigeqlem3 42266 dirkercncflem2 42270 iinhoiicc 42837 iunhoiioo 42839 vonioolem2 42844 vonicclem1 42846 eenglngeehlnmlem2 44623 amgmlemALT 44802 young2d 44804 |
Copyright terms: Public domain | W3C validator |