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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 12904 | . 2 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾)) | |
2 | uznn0sub 12276 | . 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 2110 ‘cfv 6354 (class class class)co 7155 − cmin 10869 ℕ0cn0 11896 ℤ≥cuz 12242 ...cfz 12891 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 |
This theorem is referenced by: fznn0sub2 13013 bcrpcl 13667 bcm1k 13674 bcp1n 13675 bcval5 13677 bcpasc 13680 permnn 13685 swrdlen 14008 swrdwrdsymb 14023 pfxswrd 14067 binomlem 15183 binom1p 15185 pwdif 15222 mertenslem1 15239 mertens 15241 binomfallfaclem1 15392 binomfallfaclem2 15393 fallfacval4 15396 bcfallfac 15397 bpolycl 15405 bpolysum 15406 bpolydiflem 15407 efaddlem 15445 pcbc 16235 srgbinomlem3 19291 srgbinomlem4 19292 srgbinomlem 19293 coe1mul2 20436 coe1tmmul2 20443 coe1tmmul 20444 cply1mul 20461 lply1binomsc 20474 decpmatmul 21379 pm2mpmhmlem2 21426 chpscmatgsumbin 21451 chpscmatgsummon 21452 coe1mul3 24692 plymullem1 24803 plymullem 24805 coemullem 24839 coemulhi 24843 coemulc 24844 vieta1lem2 24899 aareccl 24914 aalioulem1 24920 dvntaylp 24958 dvntaylp0 24959 birthdaylem2 25529 basellem3 25659 cycpmco2lem5 30772 freshmansdream 30859 plymulx0 31817 jm2.22 39590 jm2.23 39591 dvnmul 42226 ply1mulgsumlem2 44440 ply1mulgsum 44443 |
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