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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 9570 | . 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 2128 ℕcn 8833 ℝ+crp 9560 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 ax-1re 7826 ax-addrcl 7829 ax-0lt1 7838 ax-0id 7840 ax-rnegex 7841 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-ltadd 7848 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-br 3966 df-opab 4026 df-xp 4592 df-cnv 4594 df-iota 5135 df-fv 5178 df-ov 5827 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-inn 8834 df-rp 9561 |
This theorem is referenced by: zgt1rpn0n1 9602 qtri3or 10142 qbtwnrelemcalc 10155 qbtwnre 10156 flqdiv 10220 addmodlteq 10297 nnesq 10537 bcpasc 10640 cvg1nlemcxze 10882 cvg1nlemcau 10884 cvg1nlemres 10885 resqrexlemnmsq 10917 resqrexlemnm 10918 resqrexlemcvg 10919 climrecvg1n 11245 climcvg1nlem 11246 cvgratnnlembern 11420 cvgratnnlemfm 11426 mertenslemi1 11432 mertenslem2 11433 efcllemp 11555 ege2le3 11568 eftlub 11587 effsumlt 11589 efgt1p2 11592 eirraplem 11673 prmind2 11996 sqrt2irrlem 12035 sqrt2irraplemnn 12053 sqrt2irrap 12054 logbrec 13277 logbgcd1irr 13284 logbgcd1irraplemexp 13285 logbgcd1irraplemap 13286 cvgcmp2nlemabs 13603 trilpolemlt1 13612 |
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