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Mirrors > Home > MPE Home > Th. List > fznn | Structured version Visualization version GIF version |
Description: Finite set of sequential integers starting at 1. (Contributed by NM, 31-Aug-2011.) (Revised by Mario Carneiro, 18-Jun-2015.) |
Ref | Expression |
---|---|
fznn | ⊢ (𝑁 ∈ ℤ → (𝐾 ∈ (1...𝑁) ↔ (𝐾 ∈ ℕ ∧ 𝐾 ≤ 𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzuzb 12993 | . . 3 ⊢ (𝐾 ∈ (1...𝑁) ↔ (𝐾 ∈ (ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘𝐾))) | |
2 | elnnuz 12365 | . . . 4 ⊢ (𝐾 ∈ ℕ ↔ 𝐾 ∈ (ℤ≥‘1)) | |
3 | 2 | anbi1i 627 | . . 3 ⊢ ((𝐾 ∈ ℕ ∧ 𝑁 ∈ (ℤ≥‘𝐾)) ↔ (𝐾 ∈ (ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘𝐾))) |
4 | 1, 3 | bitr4i 281 | . 2 ⊢ (𝐾 ∈ (1...𝑁) ↔ (𝐾 ∈ ℕ ∧ 𝑁 ∈ (ℤ≥‘𝐾))) |
5 | nnz 12086 | . . . . 5 ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ ℤ) | |
6 | eluz 12339 | . . . . 5 ⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝐾) ↔ 𝐾 ≤ 𝑁)) | |
7 | 5, 6 | sylan 583 | . . . 4 ⊢ ((𝐾 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝐾) ↔ 𝐾 ≤ 𝑁)) |
8 | 7 | ancoms 462 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℕ) → (𝑁 ∈ (ℤ≥‘𝐾) ↔ 𝐾 ≤ 𝑁)) |
9 | 8 | pm5.32da 582 | . 2 ⊢ (𝑁 ∈ ℤ → ((𝐾 ∈ ℕ ∧ 𝑁 ∈ (ℤ≥‘𝐾)) ↔ (𝐾 ∈ ℕ ∧ 𝐾 ≤ 𝑁))) |
10 | 4, 9 | syl5bb 286 | 1 ⊢ (𝑁 ∈ ℤ → (𝐾 ∈ (1...𝑁) ↔ (𝐾 ∈ ℕ ∧ 𝐾 ≤ 𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2113 class class class wbr 5031 ‘cfv 6340 (class class class)co 7171 1c1 10617 ≤ cle 10755 ℕcn 11717 ℤcz 12063 ℤ≥cuz 12325 ...cfz 12982 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7480 ax-cnex 10672 ax-resscn 10673 ax-1cn 10674 ax-icn 10675 ax-addcl 10676 ax-addrcl 10677 ax-mulcl 10678 ax-mulrcl 10679 ax-mulcom 10680 ax-addass 10681 ax-mulass 10682 ax-distr 10683 ax-i2m1 10684 ax-1ne0 10685 ax-1rid 10686 ax-rnegex 10687 ax-rrecex 10688 ax-cnre 10689 ax-pre-lttri 10690 ax-pre-lttrn 10691 ax-pre-ltadd 10692 ax-pre-mulgt0 10693 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3683 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7128 df-ov 7174 df-oprab 7175 df-mpo 7176 df-om 7601 df-1st 7715 df-2nd 7716 df-wrecs 7977 df-recs 8038 df-rdg 8076 df-er 8321 df-en 8557 df-dom 8558 df-sdom 8559 df-pnf 10756 df-mnf 10757 df-xr 10758 df-ltxr 10759 df-le 10760 df-sub 10951 df-neg 10952 df-nn 11718 df-z 12064 df-uz 12326 df-fz 12983 |
This theorem is referenced by: elfz1b 13068 elfz1uz 13069 fznnfl 13322 isercoll 15118 incexc2 15287 dvdsssfz1 15764 prmind2 16127 vdwlem6 16423 prmdvdsprmo 16479 odlem2 18786 gexlem2 18826 gexcl2 18833 efgredlemd 18989 efgredlem 18992 ablfac1eu 19315 ablfaclem3 19329 dvdsflf1o 25924 vmasum 25952 logfac2 25953 lgseisenlem1 26111 lgseisenlem2 26112 lgseisenlem3 26113 lgsquadlem1 26116 lgsquadlem2 26117 2sqlem8 26162 chebbnd1lem1 26205 wwlksnredwwlkn0 27834 clwwlkf 27984 psgnfzto1stlem 30944 smatrcl 31318 reprinfz1 32172 poimirlem13 35410 poimirlem15 35412 lcmineqlem4 39657 |
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