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Mirrors > Home > ILE Home > Th. List > nnrpd | GIF version |
Description: A positive integer is a positive real. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
nnrpd.1 | ⊢ (𝜑 → 𝐴 ∈ ℕ) |
Ref | Expression |
---|---|
nnrpd | ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnrpd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℕ) | |
2 | nnrp 9451 | . 2 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ+) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 ℕcn 8720 ℝ+crp 9441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1re 7714 ax-addrcl 7717 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-xp 4545 df-cnv 4547 df-iota 5088 df-fv 5131 df-ov 5777 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-inn 8721 df-rp 9442 |
This theorem is referenced by: qtri3or 10020 qbtwnrelemcalc 10033 qbtwnre 10034 flqdiv 10094 addmodlteq 10171 nnesq 10411 bcpasc 10512 cvg1nlemcxze 10754 cvg1nlemcau 10756 cvg1nlemres 10757 resqrexlemnmsq 10789 resqrexlemnm 10790 resqrexlemcvg 10791 climrecvg1n 11117 climcvg1nlem 11118 cvgratnnlembern 11292 cvgratnnlemfm 11298 mertenslemi1 11304 mertenslem2 11305 efcllemp 11364 ege2le3 11377 eftlub 11396 effsumlt 11398 efgt1p2 11401 eirraplem 11483 prmind2 11801 sqrt2irrlem 11839 sqrt2irraplemnn 11857 sqrt2irrap 11858 cvgcmp2nlemabs 13227 trilpolemlt1 13234 |
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