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Mirrors > Home > MPE Home > Th. List > elfzonn0 | Structured version Visualization version GIF version |
Description: A member of a half-open range of nonnegative integers is a nonnegative integer. (Contributed by Alexander van der Vekens, 21-May-2018.) |
Ref | Expression |
---|---|
elfzonn0 | ⊢ (𝐾 ∈ (0..^𝑁) → 𝐾 ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzouz 12892 | . 2 ⊢ (𝐾 ∈ (0..^𝑁) → 𝐾 ∈ (ℤ≥‘0)) | |
2 | elnn0uz 12132 | . 2 ⊢ (𝐾 ∈ ℕ0 ↔ 𝐾 ∈ (ℤ≥‘0)) | |
3 | 1, 2 | sylibr 235 | 1 ⊢ (𝐾 ∈ (0..^𝑁) → 𝐾 ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2081 ‘cfv 6225 (class class class)co 7016 0cc0 10383 ℕ0cn0 11745 ℤ≥cuz 12093 ..^cfzo 12883 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-nn 11487 df-n0 11746 df-z 11830 df-uz 12094 df-fz 12743 df-fzo 12884 |
This theorem is referenced by: fzo0ssnn0 12968 modsumfzodifsn 13162 resunimafz0 13651 pfxmpt 13876 pfxsuffeqwrdeq 13896 swrdccatin2 13927 repswswrd 13982 pwdif 15056 pwm1geoser 15057 fzo0dvdseq 15506 smueqlem 15672 hashgcdlem 15954 cshwsidrepsw 16256 advlogexp 24919 upgrwlkdvdelem 27204 crctcshwlkn0lem2 27276 crctcshwlkn0lem4 27278 crctcshwlkn0 27286 crctcsh 27289 clwwlkel 27512 clwwlknonex2lem2 27574 eucrctshift 27710 signsplypnf 31437 poimirlem5 34447 poimirlem6 34448 poimirlem7 34449 poimirlem10 34452 poimirlem11 34453 poimirlem12 34454 poimirlem16 34458 poimirlem17 34459 poimirlem19 34461 poimirlem20 34462 poimirlem22 34464 poimirlem23 34465 poimirlem25 34467 poimirlem29 34471 poimirlem30 34472 poimirlem31 34473 frlmvscadiccat 38691 fltnltalem 38790 dvnmul 41789 fourierdlem48 42001 2pwp1prm 43253 nnpw2pb 44148 nn0sumshdiglemA 44180 nn0sumshdiglemB 44181 nn0mullong 44186 |
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