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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 9444 | . 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 8713 ℝ+crp 9434 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1re 7707 ax-addrcl 7710 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-ltadd 7729 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-xp 4540 df-cnv 4542 df-iota 5083 df-fv 5126 df-ov 5770 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-inn 8714 df-rp 9435 |
This theorem is referenced by: qtri3or 10013 qbtwnrelemcalc 10026 qbtwnre 10027 flqdiv 10087 addmodlteq 10164 nnesq 10404 bcpasc 10505 cvg1nlemcxze 10747 cvg1nlemcau 10749 cvg1nlemres 10750 resqrexlemnmsq 10782 resqrexlemnm 10783 resqrexlemcvg 10784 climrecvg1n 11110 climcvg1nlem 11111 cvgratnnlembern 11285 cvgratnnlemfm 11291 mertenslemi1 11297 mertenslem2 11298 efcllemp 11353 ege2le3 11366 eftlub 11385 effsumlt 11387 efgt1p2 11390 eirraplem 11472 prmind2 11790 sqrt2irrlem 11828 sqrt2irraplemnn 11846 sqrt2irrap 11847 cvgcmp2nlemabs 13216 trilpolemlt1 13223 |
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