Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 10641 | . 2 ⊢ 1 ∈ ℝ | |
2 | nndivre 11679 | . 2 ⊢ ((1 ∈ ℝ ∧ 𝑁 ∈ ℕ) → (1 / 𝑁) ∈ ℝ) | |
3 | 1, 2 | mpan 688 | 1 ⊢ (𝑁 ∈ ℕ → (1 / 𝑁) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 (class class class)co 7156 ℝcr 10536 1c1 10538 / cdiv 11297 ℕcn 11638 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 |
This theorem is referenced by: nnrecred 11689 rpnnen1lem5 12381 fldiv 13229 supcvg 15211 harmonic 15214 rpnnen2lem11 15577 flodddiv4 15764 prmreclem4 16255 prmreclem5 16256 prmreclem6 16257 prmrec 16258 met1stc 23131 pcoass 23628 bcthlem4 23930 vitali 24214 ismbf3d 24255 itg2seq 24343 itg2gt0 24361 plyeq0lem 24800 logtayllem 25242 cxproot 25273 cxpeq 25338 quartlem3 25437 leibpi 25520 emcllem4 25576 emcllem6 25578 basellem6 25663 mulogsumlem 26107 pntpbnd2 26163 ipasslem4 28611 ipasslem5 28612 minvecolem5 28658 subfaclim 32435 faclim 32978 poimirlem29 34936 poimirlem30 34937 xrralrecnnle 41673 xrralrecnnge 41682 iooiinicc 41838 iooiinioc 41852 stirlinglem1 42379 iinhoiicclem 42975 iunhoiioolem 42977 iccvonmbllem 42980 vonioolem1 42982 vonioolem2 42983 vonicclem1 42985 vonicclem2 42986 preimageiingt 43018 preimaleiinlt 43019 salpreimagtge 43022 salpreimaltle 43023 smflimlem6 43072 |
Copyright terms: Public domain | W3C validator |