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Mirrors > Home > ILE Home > Th. List > elfznn | GIF version |
Description: A member of a finite set of sequential integers starting at 1 is a positive integer. (Contributed by NM, 24-Aug-2005.) |
Ref | Expression |
---|---|
elfznn | ⊢ (𝐾 ∈ (1...𝑁) → 𝐾 ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzelz 9928 | . 2 ⊢ (𝐾 ∈ (1...𝑁) → 𝐾 ∈ ℤ) | |
2 | elfzle1 9929 | . 2 ⊢ (𝐾 ∈ (1...𝑁) → 1 ≤ 𝐾) | |
3 | elnnz1 9190 | . 2 ⊢ (𝐾 ∈ ℕ ↔ (𝐾 ∈ ℤ ∧ 1 ≤ 𝐾)) | |
4 | 1, 2, 3 | sylanbrc 414 | 1 ⊢ (𝐾 ∈ (1...𝑁) → 𝐾 ∈ ℕ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2128 class class class wbr 3965 (class class class)co 5824 1c1 7733 ≤ cle 7913 ℕcn 8833 ℤcz 9167 ...cfz 9912 |
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-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-addcom 7832 ax-addass 7834 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-0id 7840 ax-rnegex 7841 ax-cnre 7843 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-3or 964 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-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 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-mpt 4027 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-sub 8048 df-neg 8049 df-inn 8834 df-z 9168 df-uz 9440 df-fz 9913 |
This theorem is referenced by: elfz1end 9957 fz1ssnn 9958 fzossnn 10088 bcm1k 10634 bcpasc 10640 seq3coll 10713 summodclem3 11277 summodclem2a 11278 fsum3 11284 isumz 11286 fsumcl2lem 11295 binomlem 11380 arisum2 11396 trireciplem 11397 geo2sum 11411 cvgratnnlemsumlt 11425 prodmodclem3 11472 prodmodclem2a 11473 fprodseq 11480 prod1dc 11483 fzm1ndvds 11748 nninfdcex 11839 phicl 12090 eulerthlemrprm 12104 prmdivdiv 12112 phisum 12115 odzcllem 12117 odzdvds 12120 modprm0 12129 nnmindc 12188 nnminle 12189 cvgcmp2nlemabs 13614 trilpolemlt1 13623 nconstwlpolemgt0 13645 |
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