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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 10085 | . . 3 ⊢ (𝐾 ∈ (1...𝑁) ↔ (𝐾 ∈ (ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘𝐾))) | |
2 | elnnuz 9629 | . . . 4 ⊢ (𝐾 ∈ ℕ ↔ 𝐾 ∈ (ℤ≥‘1)) | |
3 | 2 | anbi1i 458 | . . 3 ⊢ ((𝐾 ∈ ℕ ∧ 𝑁 ∈ (ℤ≥‘𝐾)) ↔ (𝐾 ∈ (ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘𝐾))) |
4 | 1, 3 | bitr4i 187 | . 2 ⊢ (𝐾 ∈ (1...𝑁) ↔ (𝐾 ∈ ℕ ∧ 𝑁 ∈ (ℤ≥‘𝐾))) |
5 | nnz 9336 | . . . . 5 ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ ℤ) | |
6 | eluz 9605 | . . . . 5 ⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝐾) ↔ 𝐾 ≤ 𝑁)) | |
7 | 5, 6 | sylan 283 | . . . 4 ⊢ ((𝐾 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝐾) ↔ 𝐾 ≤ 𝑁)) |
8 | 7 | ancoms 268 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℕ) → (𝑁 ∈ (ℤ≥‘𝐾) ↔ 𝐾 ≤ 𝑁)) |
9 | 8 | pm5.32da 452 | . 2 ⊢ (𝑁 ∈ ℤ → ((𝐾 ∈ ℕ ∧ 𝑁 ∈ (ℤ≥‘𝐾)) ↔ (𝐾 ∈ ℕ ∧ 𝐾 ≤ 𝑁))) |
10 | 4, 9 | bitrid 192 | 1 ⊢ (𝑁 ∈ ℤ → (𝐾 ∈ (1...𝑁) ↔ (𝐾 ∈ ℕ ∧ 𝐾 ≤ 𝑁))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∈ wcel 2164 class class class wbr 4029 ‘cfv 5254 (class class class)co 5918 1c1 7873 ≤ cle 8055 ℕcn 8982 ℤcz 9317 ℤ≥cuz 9592 ...cfz 10074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-addcom 7972 ax-addass 7974 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-0id 7980 ax-rnegex 7981 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-ltadd 7988 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2986 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4324 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-fv 5262 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-inn 8983 df-z 9318 df-uz 9593 df-fz 10075 |
This theorem is referenced by: sumeq2 11502 prodeq2 11700 fprodseq 11726 dvdsssfz1 11994 prmind2 12258 lgseisenlem1 15186 lgseisenlem2 15187 lgseisenlem3 15188 lgsquadlem1 15191 2sqlem8 15210 |
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