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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 12201 | . 2 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝐴 ∈ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 ℝcr 11091 ℕcn 12194 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7708 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-i2m1 11160 ax-1ne0 11161 ax-rrecex 11164 ax-cnre 11165 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-ov 7396 df-om 7839 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-nn 12195 |
This theorem is referenced by: nnne0i 12234 numlt 12684 numltc 12685 faclbnd4lem1 14235 ef01bndlem 16109 dvdslelem 16234 divalglem6 16323 pockthi 16822 modsubi 16987 prmlem1 17023 prmlem2 17035 strleun 17072 strle1 17073 basendxnplusgndx 17209 tsetndxnbasendx 17283 plendxnbasendx 17297 dsndxnbasendx 17316 unifndxnbasendx 17326 slotsdifunifndx 17328 slotsdifocndx 17345 oppchomfvalOLD 17641 oppcbasOLD 17646 resccoOLD 17763 opprlemOLD 20108 sralemOLD 20740 zlmlemOLD 21000 znbaslemOLD 21024 opsrbaslemOLD 21533 tnglemOLD 24079 log2ublem1 26378 log2ublem2 26379 log2ub 26381 bpos1lem 26712 bposlem8 26721 bposlem9 26722 slotsinbpsd 27557 slotslnbpsd 27558 lngndxnitvndx 27559 ttgvalOLD 27992 ttglemOLD 27994 cchhllemOLD 28010 basendxnedgfndx 28120 structvtxvallem 28145 lmat22e12 32630 lmat22e21 32631 lmat22e22 32632 ballotlem2 33318 ballotlem5 33329 ballotth 33367 chtvalz 33472 hgt750lem 33494 tgoldbachgt 33506 cnndvlem1 35217 hlhilslemOLD 40615 lcmineqlem 40722 3lexlogpow5ineq1 40724 jm2.27dlem2 41520 bgoldbtbndlem1 46245 tgblthelfgott 46255 tgoldbachlt 46256 |
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