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Mirrors > Home > MPE Home > Th. List > fznn0sub2 | Structured version Visualization version GIF version |
Description: Subtraction closure for a member of a finite set of sequential nonnegative integers. (Contributed by NM, 26-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
fznn0sub2 | ⊢ (𝐾 ∈ (0...𝑁) → (𝑁 − 𝐾) ∈ (0...𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzle1 12595 | . . 3 ⊢ (𝐾 ∈ (0...𝑁) → 0 ≤ 𝐾) | |
2 | elfzel2 12591 | . . . 4 ⊢ (𝐾 ∈ (0...𝑁) → 𝑁 ∈ ℤ) | |
3 | elfzelz 12593 | . . . 4 ⊢ (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℤ) | |
4 | zre 11667 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
5 | zre 11667 | . . . . 5 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℝ) | |
6 | subge02 10835 | . . . . 5 ⊢ ((𝑁 ∈ ℝ ∧ 𝐾 ∈ ℝ) → (0 ≤ 𝐾 ↔ (𝑁 − 𝐾) ≤ 𝑁)) | |
7 | 4, 5, 6 | syl2an 590 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (0 ≤ 𝐾 ↔ (𝑁 − 𝐾) ≤ 𝑁)) |
8 | 2, 3, 7 | syl2anc 580 | . . 3 ⊢ (𝐾 ∈ (0...𝑁) → (0 ≤ 𝐾 ↔ (𝑁 − 𝐾) ≤ 𝑁)) |
9 | 1, 8 | mpbid 224 | . 2 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁 − 𝐾) ≤ 𝑁) |
10 | fznn0sub 12624 | . . . 4 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁 − 𝐾) ∈ ℕ0) | |
11 | nn0uz 11963 | . . . 4 ⊢ ℕ0 = (ℤ≥‘0) | |
12 | 10, 11 | syl6eleq 2887 | . . 3 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁 − 𝐾) ∈ (ℤ≥‘0)) |
13 | elfz5 12585 | . . 3 ⊢ (((𝑁 − 𝐾) ∈ (ℤ≥‘0) ∧ 𝑁 ∈ ℤ) → ((𝑁 − 𝐾) ∈ (0...𝑁) ↔ (𝑁 − 𝐾) ≤ 𝑁)) | |
14 | 12, 2, 13 | syl2anc 580 | . 2 ⊢ (𝐾 ∈ (0...𝑁) → ((𝑁 − 𝐾) ∈ (0...𝑁) ↔ (𝑁 − 𝐾) ≤ 𝑁)) |
15 | 9, 14 | mpbird 249 | 1 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁 − 𝐾) ∈ (0...𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∈ wcel 2157 class class class wbr 4842 ‘cfv 6100 (class class class)co 6877 ℝcr 10222 0cc0 10223 ≤ cle 10363 − cmin 10555 ℕ0cn0 11577 ℤcz 11663 ℤ≥cuz 11927 ...cfz 12577 |
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 2377 ax-ext 2776 ax-sep 4974 ax-nul 4982 ax-pow 5034 ax-pr 5096 ax-un 7182 ax-cnex 10279 ax-resscn 10280 ax-1cn 10281 ax-icn 10282 ax-addcl 10283 ax-addrcl 10284 ax-mulcl 10285 ax-mulrcl 10286 ax-mulcom 10287 ax-addass 10288 ax-mulass 10289 ax-distr 10290 ax-i2m1 10291 ax-1ne0 10292 ax-1rid 10293 ax-rnegex 10294 ax-rrecex 10295 ax-cnre 10296 ax-pre-lttri 10297 ax-pre-lttrn 10298 ax-pre-ltadd 10299 ax-pre-mulgt0 10300 |
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 2591 df-eu 2609 df-clab 2785 df-cleq 2791 df-clel 2794 df-nfc 2929 df-ne 2971 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3386 df-sbc 3633 df-csb 3728 df-dif 3771 df-un 3773 df-in 3775 df-ss 3782 df-pss 3784 df-nul 4115 df-if 4277 df-pw 4350 df-sn 4368 df-pr 4370 df-tp 4372 df-op 4374 df-uni 4628 df-iun 4711 df-br 4843 df-opab 4905 df-mpt 4922 df-tr 4945 df-id 5219 df-eprel 5224 df-po 5232 df-so 5233 df-fr 5270 df-we 5272 df-xp 5317 df-rel 5318 df-cnv 5319 df-co 5320 df-dm 5321 df-rn 5322 df-res 5323 df-ima 5324 df-pred 5897 df-ord 5943 df-on 5944 df-lim 5945 df-suc 5946 df-iota 6063 df-fun 6102 df-fn 6103 df-f 6104 df-f1 6105 df-fo 6106 df-f1o 6107 df-fv 6108 df-riota 6838 df-ov 6880 df-oprab 6881 df-mpt2 6882 df-om 7299 df-1st 7400 df-2nd 7401 df-wrecs 7644 df-recs 7706 df-rdg 7744 df-er 7981 df-en 8195 df-dom 8196 df-sdom 8197 df-pnf 10364 df-mnf 10365 df-xr 10366 df-ltxr 10367 df-le 10368 df-sub 10557 df-neg 10558 df-nn 11312 df-n0 11578 df-z 11664 df-uz 11928 df-fz 12578 |
This theorem is referenced by: uzsubfz0 12699 bccmpl 13346 pfxlswccat 13760 revcl 13838 revlen 13839 revccat 13843 revrev 13844 2cshwcshw 13907 cshwcshid 13909 revco 13916 fsum0diag2 14850 mertenslem1 14950 cshwshashlem2 16128 taylthlem2 24466 birthdaylem2 25028 basellem3 25158 eleclclwwlknlem2 27372 signstfveq0 31166 signstfveq0OLD 31167 dvnprodlem2 40895 ply1mulgsumlem2 42963 |
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