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Mirrors > Home > MPE Home > Th. List > nnrecre | Structured version Visualization version GIF version |
Description: The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.) |
Ref | Expression |
---|---|
nnrecre | ⊢ (𝑁 ∈ ℕ → (1 / 𝑁) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 10077 | . 2 ⊢ 1 ∈ ℝ | |
2 | nndivre 11094 | . 2 ⊢ ((1 ∈ ℝ ∧ 𝑁 ∈ ℕ) → (1 / 𝑁) ∈ ℝ) | |
3 | 1, 2 | mpan 706 | 1 ⊢ (𝑁 ∈ ℕ → (1 / 𝑁) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2030 (class class class)co 6690 ℝcr 9973 1c1 9975 / cdiv 10722 ℕcn 11058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 |
This theorem is referenced by: nnrecred 11104 rpnnen1lem5 11856 rpnnen1lem5OLD 11862 fldiv 12699 supcvg 14632 harmonic 14635 rpnnen2lem11 14997 flodddiv4 15184 prmreclem4 15670 prmreclem5 15671 prmreclem6 15672 prmrec 15673 met1stc 22373 pcoass 22870 bcthlem4 23170 vitali 23427 ismbf3d 23466 itg2seq 23554 itg2gt0 23572 plyeq0lem 24011 logtayllem 24450 cxproot 24481 cxpeq 24543 quartlem3 24631 leibpi 24714 emcllem4 24770 emcllem6 24772 basellem6 24857 mulogsumlem 25265 pntpbnd2 25321 ipasslem4 27817 ipasslem5 27818 minvecolem5 27865 subfaclim 31296 faclim 31758 poimirlem29 33568 poimirlem30 33569 xrralrecnnle 39915 xrralrecnnge 39926 iooiinicc 40087 iooiinioc 40101 stirlinglem1 40609 iinhoiicclem 41208 iunhoiioolem 41210 iccvonmbllem 41213 vonioolem1 41215 vonioolem2 41216 vonicclem1 41218 vonicclem2 41219 preimageiingt 41251 preimaleiinlt 41252 salpreimagtge 41255 salpreimaltle 41256 smflimlem6 41305 |
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