![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fznn0sub | Structured version Visualization version GIF version |
Description: Subtraction closure for a member of a finite set of sequential integers. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
fznn0sub | ⊢ (𝐾 ∈ (𝑀...𝑁) → (𝑁 − 𝐾) ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzuz3 12589 | . 2 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾)) | |
2 | uznn0sub 11959 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (𝑁 − 𝐾) ∈ ℕ0) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐾 ∈ (𝑀...𝑁) → (𝑁 − 𝐾) ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2157 ‘cfv 6099 (class class class)co 6876 − cmin 10554 ℕ0cn0 11576 ℤ≥cuz 11926 ...cfz 12576 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-om 7298 df-1st 7399 df-2nd 7400 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-er 7980 df-en 8194 df-dom 8195 df-sdom 8196 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-nn 11311 df-n0 11577 df-z 11663 df-uz 11927 df-fz 12577 |
This theorem is referenced by: fznn0sub2 12697 bcrpcl 13344 bcm1k 13351 bcp1n 13352 bcval5 13354 bcpasc 13357 permnn 13362 swrdlen 13670 swrdwrdsymb 13697 swrd0swrdOLD 13746 pfxswrd 13747 binomlem 14896 binom1p 14898 mertenslem1 14950 mertens 14952 binomfallfaclem1 15103 binomfallfaclem2 15104 fallfacval4 15107 bcfallfac 15108 bpolycl 15116 bpolysum 15117 bpolydiflem 15118 efaddlem 15156 pcbc 15934 srgbinomlem3 18855 srgbinomlem4 18856 srgbinomlem 18857 coe1mul2 19958 coe1tmmul2 19965 coe1tmmul 19966 cply1mul 19983 lply1binomsc 19996 decpmatmul 20902 pm2mpmhmlem2 20949 chpscmatgsumbin 20974 chpscmatgsummon 20975 coe1mul3 24197 plymullem1 24308 plymullem 24310 coemullem 24344 coemulhi 24348 coemulc 24349 vieta1lem2 24404 aareccl 24419 aalioulem1 24425 dvntaylp 24463 dvntaylp0 24464 birthdaylem2 25028 basellem3 25158 plymulx0 31134 jm2.22 38335 jm2.23 38336 dvnmul 40890 pwdif 42271 ply1mulgsumlem2 42962 ply1mulgsum 42965 |
Copyright terms: Public domain | W3C validator |