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Mirrors > Home > ILE Home > Th. List > fznn | 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 9793 | . . 3 ⊢ (𝐾 ∈ (1...𝑁) ↔ (𝐾 ∈ (ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘𝐾))) | |
2 | elnnuz 9355 | . . . 4 ⊢ (𝐾 ∈ ℕ ↔ 𝐾 ∈ (ℤ≥‘1)) | |
3 | 2 | anbi1i 453 | . . 3 ⊢ ((𝐾 ∈ ℕ ∧ 𝑁 ∈ (ℤ≥‘𝐾)) ↔ (𝐾 ∈ (ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘𝐾))) |
4 | 1, 3 | bitr4i 186 | . 2 ⊢ (𝐾 ∈ (1...𝑁) ↔ (𝐾 ∈ ℕ ∧ 𝑁 ∈ (ℤ≥‘𝐾))) |
5 | nnz 9066 | . . . . 5 ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ ℤ) | |
6 | eluz 9332 | . . . . 5 ⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝐾) ↔ 𝐾 ≤ 𝑁)) | |
7 | 5, 6 | sylan 281 | . . . 4 ⊢ ((𝐾 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝐾) ↔ 𝐾 ≤ 𝑁)) |
8 | 7 | ancoms 266 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℕ) → (𝑁 ∈ (ℤ≥‘𝐾) ↔ 𝐾 ≤ 𝑁)) |
9 | 8 | pm5.32da 447 | . 2 ⊢ (𝑁 ∈ ℤ → ((𝐾 ∈ ℕ ∧ 𝑁 ∈ (ℤ≥‘𝐾)) ↔ (𝐾 ∈ ℕ ∧ 𝐾 ≤ 𝑁))) |
10 | 4, 9 | syl5bb 191 | 1 ⊢ (𝑁 ∈ ℤ → (𝐾 ∈ (1...𝑁) ↔ (𝐾 ∈ ℕ ∧ 𝐾 ≤ 𝑁))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 1480 class class class wbr 3924 ‘cfv 5118 (class class class)co 5767 1c1 7614 ≤ cle 7794 ℕcn 8713 ℤcz 9047 ℤ≥cuz 9319 ...cfz 9783 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-ltadd 7729 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-inn 8714 df-z 9048 df-uz 9320 df-fz 9784 |
This theorem is referenced by: sumeq2 11121 prodeq2 11319 dvdsssfz1 11539 prmind2 11790 |
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