![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nnrei | Structured version Visualization version GIF version |
Description: A positive integer is a real number. (Contributed by NM, 18-Aug-1999.) |
Ref | Expression |
---|---|
nnre.1 | ⊢ 𝐴 ∈ ℕ |
Ref | Expression |
---|---|
nnrei | ⊢ 𝐴 ∈ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnre.1 | . 2 ⊢ 𝐴 ∈ ℕ | |
2 | nnre 11632 | . 2 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝐴 ∈ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 ℝcr 10525 ℕcn 11625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-i2m1 10594 ax-1ne0 10595 ax-rrecex 10598 ax-cnre 10599 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-nn 11626 |
This theorem is referenced by: nnne0i 11665 numlt 12111 numltc 12112 faclbnd4lem1 13649 ef01bndlem 15529 dvdslelem 15651 divalglem6 15739 pockthi 16233 modsubi 16398 prmlem1 16433 prmlem2 16445 strleun 16583 strle1 16584 oppchomfval 16976 oppcbas 16980 rescco 17094 opprlem 19374 sralem 19942 zlmlem 20210 znbaslem 20230 opsrbaslem 20717 tnglem 23246 log2ublem1 25532 log2ublem2 25533 log2ub 25535 bpos1lem 25866 bposlem8 25875 bposlem9 25876 ttgval 26669 ttglem 26670 cchhllem 26681 slotsbaseefdif 26788 structvtxvallem 26813 lmat22e12 31172 lmat22e21 31173 lmat22e22 31174 ballotlem2 31856 ballotlem5 31867 ballotth 31905 chtvalz 32010 hgt750lem 32032 tgoldbachgt 32044 cnndvlem1 33989 hlhilslem 39234 lcmineqlem 39340 jm2.27dlem2 39951 bgoldbtbndlem1 44323 tgblthelfgott 44333 tgoldbachlt 44334 |
Copyright terms: Public domain | W3C validator |