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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 13547 | . 2 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾)) | |
2 | uznn0sub 12908 | . 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 2098 ‘cfv 6553 (class class class)co 7423 − cmin 11490 ℕ0cn0 12519 ℤ≥cuz 12869 ...cfz 13533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7876 df-1st 8002 df-2nd 8003 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-en 8974 df-dom 8975 df-sdom 8976 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-nn 12260 df-n0 12520 df-z 12606 df-uz 12870 df-fz 13534 |
This theorem is referenced by: fznn0sub2 13657 bcrpcl 14320 bcm1k 14327 bcp1n 14328 bcval5 14330 bcpasc 14333 permnn 14338 swrdlen 14650 swrdwrdsymb 14665 pfxswrd 14709 binomlem 15828 binom1p 15830 pwdif 15867 mertenslem1 15883 mertens 15885 binomfallfaclem1 16036 binomfallfaclem2 16037 fallfacval4 16040 bcfallfac 16041 bpolycl 16049 bpolysum 16050 bpolydiflem 16051 efaddlem 16090 pcbc 16897 srgbinomlem3 20206 srgbinomlem4 20207 srgbinomlem 20208 freshmansdream 21564 coe1mul2 22252 coe1tmmul2 22259 coe1tmmul 22260 cply1mul 22279 lply1binomsc 22294 decpmatmul 22757 pm2mpmhmlem2 22804 chpscmatgsumbin 22829 chpscmatgsummon 22830 coe1mul3 26118 plymullem1 26233 plymullem 26235 coemullem 26269 coemulhi 26273 coemulc 26274 vieta1lem2 26331 aareccl 26346 aalioulem1 26352 dvntaylp 26391 dvntaylp0 26392 birthdaylem2 26972 basellem3 27103 cycpmco2lem5 32985 plymulx0 34349 jm2.22 42590 jm2.23 42591 dvnmul 45501 ply1mulgsumlem2 47707 ply1mulgsum 47710 |
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