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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 9204 | . 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 1439 ℕcn 8483 ℝ+crp 9195 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-cnex 7497 ax-resscn 7498 ax-1re 7500 ax-addrcl 7503 ax-0lt1 7512 ax-0id 7514 ax-rnegex 7515 ax-pre-ltirr 7518 ax-pre-ltwlin 7519 ax-pre-lttrn 7520 ax-pre-ltadd 7522 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-rab 2369 df-v 2622 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-br 3852 df-opab 3906 df-xp 4458 df-cnv 4460 df-iota 4993 df-fv 5036 df-ov 5669 df-pnf 7585 df-mnf 7586 df-xr 7587 df-ltxr 7588 df-le 7589 df-inn 8484 df-rp 9196 |
This theorem is referenced by: qtri3or 9715 qbtwnrelemcalc 9728 qbtwnre 9729 flqdiv 9789 addmodlteq 9866 nnesq 10134 bcpasc 10235 cvg1nlemcxze 10476 cvg1nlemcau 10478 cvg1nlemres 10479 resqrexlemnmsq 10511 resqrexlemnm 10512 resqrexlemcvg 10513 climrecvg1n 10798 climcvg1nlem 10799 cvgratnnlembern 10978 cvgratnnlemfm 10984 mertenslemi1 10990 mertenslem2 10991 efcllemp 11009 ege2le3 11022 eftlub 11041 effsumlt 11043 efgt1p2 11046 eirraplem 11125 prmind2 11441 sqrt2irrlem 11479 sqrt2irraplemnn 11496 sqrt2irrap 11497 |
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