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Mirrors > Home > MPE Home > Th. List > nnrecred | Structured version Visualization version GIF version |
Description: The reciprocal of a positive integer is real. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
nnge1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℕ) |
Ref | Expression |
---|---|
nnrecred | ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnge1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℕ) | |
2 | nnrecre 11482 | . 2 ⊢ (𝐴 ∈ ℕ → (1 / 𝐴) ∈ ℝ) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2050 (class class class)co 6976 ℝcr 10334 1c1 10336 / cdiv 11098 ℕcn 11439 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-nn 11440 |
This theorem is referenced by: trireciplem 15077 trirecip 15078 geo2sum 15089 geo2lim 15091 bpolydiflem 15268 ege2le3 15303 eftlub 15322 eirrlem 15417 prmreclem4 16111 prmreclem6 16113 lmnn 23569 bcthlem5 23634 opnmbllem 23905 mbfi1fseqlem4 24022 taylthlem2 24665 logtayl 24944 leibpi 25222 amgmlem 25269 emcllem1 25275 emcllem2 25276 emcllem3 25277 emcllem5 25279 harmoniclbnd 25288 harmonicubnd 25289 harmonicbnd4 25290 fsumharmonic 25291 lgamgulmlem1 25308 lgamgulmlem2 25309 lgamgulmlem3 25310 lgamgulmlem5 25312 lgamucov 25317 ftalem4 25355 ftalem5 25356 basellem6 25365 basellem7 25366 basellem9 25368 chpchtsum 25497 logfaclbnd 25500 rplogsumlem2 25763 rpvmasumlem 25765 dchrmusum2 25772 dchrvmasumlem3 25777 dchrisum0fno1 25789 mulogsumlem 25809 mulogsum 25810 mulog2sumlem1 25812 vmalogdivsum2 25816 logdivbnd 25834 pntrsumo1 25843 pntrlog2bndlem2 25856 pntrlog2bndlem5 25859 pntrlog2bndlem6 25861 pntpbnd2 25865 padicabvf 25909 minvecolem3 28431 minvecolem4 28435 subfacval3 32027 cvmliftlem13 32134 poimirlem29 34368 opnmbllem0 34375 heiborlem7 34543 fltne 38685 irrapxlem4 38824 hashnzfz2 40075 hashnzfzclim 40076 stoweidlem30 41752 stoweidlem38 41760 stoweidlem44 41766 vonioolem1 42399 smflimlem3 42486 amgmlemALT 44277 |
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