![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 13536 | . . 3 ⊢ (𝐾 ∈ (0...𝑁) → 0 ≤ 𝐾) | |
2 | elfzel2 13531 | . . . 4 ⊢ (𝐾 ∈ (0...𝑁) → 𝑁 ∈ ℤ) | |
3 | elfzelz 13533 | . . . 4 ⊢ (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℤ) | |
4 | zre 12592 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
5 | zre 12592 | . . . . 5 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℝ) | |
6 | subge02 11760 | . . . . 5 ⊢ ((𝑁 ∈ ℝ ∧ 𝐾 ∈ ℝ) → (0 ≤ 𝐾 ↔ (𝑁 − 𝐾) ≤ 𝑁)) | |
7 | 4, 5, 6 | syl2an 594 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (0 ≤ 𝐾 ↔ (𝑁 − 𝐾) ≤ 𝑁)) |
8 | 2, 3, 7 | syl2anc 582 | . . 3 ⊢ (𝐾 ∈ (0...𝑁) → (0 ≤ 𝐾 ↔ (𝑁 − 𝐾) ≤ 𝑁)) |
9 | 1, 8 | mpbid 231 | . 2 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁 − 𝐾) ≤ 𝑁) |
10 | fznn0sub 13565 | . . . 4 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁 − 𝐾) ∈ ℕ0) | |
11 | nn0uz 12894 | . . . 4 ⊢ ℕ0 = (ℤ≥‘0) | |
12 | 10, 11 | eleqtrdi 2835 | . . 3 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁 − 𝐾) ∈ (ℤ≥‘0)) |
13 | elfz5 13525 | . . 3 ⊢ (((𝑁 − 𝐾) ∈ (ℤ≥‘0) ∧ 𝑁 ∈ ℤ) → ((𝑁 − 𝐾) ∈ (0...𝑁) ↔ (𝑁 − 𝐾) ≤ 𝑁)) | |
14 | 12, 2, 13 | syl2anc 582 | . 2 ⊢ (𝐾 ∈ (0...𝑁) → ((𝑁 − 𝐾) ∈ (0...𝑁) ↔ (𝑁 − 𝐾) ≤ 𝑁)) |
15 | 9, 14 | mpbird 256 | 1 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁 − 𝐾) ∈ (0...𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2098 class class class wbr 5143 ‘cfv 6543 (class class class)co 7416 ℝcr 11137 0cc0 11138 ≤ cle 11279 − cmin 11474 ℕ0cn0 12502 ℤcz 12588 ℤ≥cuz 12852 ...cfz 13516 |
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 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
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 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 |
This theorem is referenced by: uzsubfz0 13641 bccmpl 14300 pfxlswccat 14695 revcl 14743 revlen 14744 revccat 14748 revrev 14749 2cshwcshw 14808 cshwcshid 14810 revco 14817 fsum0diag2 15761 mertenslem1 15862 cshwshashlem2 17065 taylthlem2 26327 taylthlem2OLD 26328 birthdaylem2 26902 basellem3 27033 eleclclwwlknlem2 29915 signstfveq0 34266 revpfxsfxrev 34782 swrdrevpfx 34783 swrdwlk 34793 dvnprodlem2 45398 ply1mulgsumlem2 47567 |
Copyright terms: Public domain | W3C validator |