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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 11668 | . 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 2105 (class class class)co 7145 ℝcr 10525 1c1 10527 / cdiv 11286 ℕcn 11627 |
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-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 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-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 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-om 7569 df-wrecs 7938 df-recs 7999 df-rdg 8037 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-nn 11628 |
This theorem is referenced by: trireciplem 15207 trirecip 15208 geo2sum 15219 geo2lim 15221 bpolydiflem 15398 ege2le3 15433 eftlub 15452 eirrlem 15547 prmreclem4 16245 prmreclem6 16247 lmnn 23795 bcthlem5 23860 opnmbllem 24131 mbfi1fseqlem4 24248 taylthlem2 24891 logtayl 25170 leibpi 25448 amgmlem 25495 emcllem1 25501 emcllem2 25502 emcllem3 25503 emcllem5 25505 harmoniclbnd 25514 harmonicubnd 25515 harmonicbnd4 25516 fsumharmonic 25517 lgamgulmlem1 25534 lgamgulmlem2 25535 lgamgulmlem3 25536 lgamgulmlem5 25538 lgamucov 25543 ftalem4 25581 ftalem5 25582 basellem6 25591 basellem7 25592 basellem9 25594 chpchtsum 25723 logfaclbnd 25726 rplogsumlem2 25989 rpvmasumlem 25991 dchrmusum2 25998 dchrvmasumlem3 26003 dchrisum0fno1 26015 mulogsumlem 26035 mulogsum 26036 mulog2sumlem1 26038 vmalogdivsum2 26042 logdivbnd 26060 pntrsumo1 26069 pntrlog2bndlem2 26082 pntrlog2bndlem5 26085 pntrlog2bndlem6 26087 pntpbnd2 26091 padicabvf 26135 minvecolem3 28581 minvecolem4 28585 subfacval3 32334 cvmliftlem13 32441 poimirlem29 34803 opnmbllem0 34810 heiborlem7 34978 fltne 39152 irrapxlem4 39302 hashnzfz2 40533 hashnzfzclim 40534 stoweidlem30 42196 stoweidlem38 42204 stoweidlem44 42210 vonioolem1 42843 smflimlem3 42930 amgmlemALT 44802 |
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