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Mathbox for Scott Fenton |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fz0n | Structured version Visualization version GIF version |
Description: The sequence (0...(𝑁 − 1)) is empty iff 𝑁 is zero. (Contributed by Scott Fenton, 16-May-2014.) |
Ref | Expression |
---|---|
fz0n | ⊢ (𝑁 ∈ ℕ0 → ((0...(𝑁 − 1)) = ∅ ↔ 𝑁 = 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0z 12553 | . . 3 ⊢ 0 ∈ ℤ | |
2 | nn0z 12567 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
3 | peano2zm 12589 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑁 − 1) ∈ ℤ) |
5 | fzn 13501 | . . 3 ⊢ ((0 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) → ((𝑁 − 1) < 0 ↔ (0...(𝑁 − 1)) = ∅)) | |
6 | 1, 4, 5 | sylancr 587 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 − 1) < 0 ↔ (0...(𝑁 − 1)) = ∅)) |
7 | elnn0 12458 | . . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
8 | nnge1 12224 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁) | |
9 | nnre 12203 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
10 | 1re 11198 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
11 | subge0 11711 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℝ ∧ 1 ∈ ℝ) → (0 ≤ (𝑁 − 1) ↔ 1 ≤ 𝑁)) | |
12 | 0re 11200 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
13 | resubcl 11508 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℝ ∧ 1 ∈ ℝ) → (𝑁 − 1) ∈ ℝ) | |
14 | lenlt 11276 | . . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (𝑁 − 1) ∈ ℝ) → (0 ≤ (𝑁 − 1) ↔ ¬ (𝑁 − 1) < 0)) | |
15 | 12, 13, 14 | sylancr 587 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℝ ∧ 1 ∈ ℝ) → (0 ≤ (𝑁 − 1) ↔ ¬ (𝑁 − 1) < 0)) |
16 | 11, 15 | bitr3d 280 | . . . . . . 7 ⊢ ((𝑁 ∈ ℝ ∧ 1 ∈ ℝ) → (1 ≤ 𝑁 ↔ ¬ (𝑁 − 1) < 0)) |
17 | 9, 10, 16 | sylancl 586 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (1 ≤ 𝑁 ↔ ¬ (𝑁 − 1) < 0)) |
18 | 8, 17 | mpbid 231 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ¬ (𝑁 − 1) < 0) |
19 | nnne0 12230 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0) | |
20 | 19 | neneqd 2945 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ¬ 𝑁 = 0) |
21 | 18, 20 | 2falsed 376 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) < 0 ↔ 𝑁 = 0)) |
22 | oveq1 7401 | . . . . . . 7 ⊢ (𝑁 = 0 → (𝑁 − 1) = (0 − 1)) | |
23 | df-neg 11431 | . . . . . . 7 ⊢ -1 = (0 − 1) | |
24 | 22, 23 | eqtr4di 2790 | . . . . . 6 ⊢ (𝑁 = 0 → (𝑁 − 1) = -1) |
25 | neg1lt0 12313 | . . . . . 6 ⊢ -1 < 0 | |
26 | 24, 25 | eqbrtrdi 5181 | . . . . 5 ⊢ (𝑁 = 0 → (𝑁 − 1) < 0) |
27 | id 22 | . . . . 5 ⊢ (𝑁 = 0 → 𝑁 = 0) | |
28 | 26, 27 | 2thd 264 | . . . 4 ⊢ (𝑁 = 0 → ((𝑁 − 1) < 0 ↔ 𝑁 = 0)) |
29 | 21, 28 | jaoi 855 | . . 3 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝑁 − 1) < 0 ↔ 𝑁 = 0)) |
30 | 7, 29 | sylbi 216 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 − 1) < 0 ↔ 𝑁 = 0)) |
31 | 6, 30 | bitr3d 280 | 1 ⊢ (𝑁 ∈ ℕ0 → ((0...(𝑁 − 1)) = ∅ ↔ 𝑁 = 0)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ∅c0 4319 class class class wbr 5142 (class class class)co 7394 ℝcr 11093 0cc0 11094 1c1 11095 < clt 11232 ≤ cle 11233 − cmin 11428 -cneg 11429 ℕcn 12196 ℕ0cn0 12456 ℤcz 12542 ...cfz 13468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 ax-cnex 11150 ax-resscn 11151 ax-1cn 11152 ax-icn 11153 ax-addcl 11154 ax-addrcl 11155 ax-mulcl 11156 ax-mulrcl 11157 ax-mulcom 11158 ax-addass 11159 ax-mulass 11160 ax-distr 11161 ax-i2m1 11162 ax-1ne0 11163 ax-1rid 11164 ax-rnegex 11165 ax-rrecex 11166 ax-cnre 11167 ax-pre-lttri 11168 ax-pre-lttrn 11169 ax-pre-ltadd 11170 ax-pre-mulgt0 11171 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7350 df-ov 7397 df-oprab 7398 df-mpo 7399 df-om 7840 df-1st 7959 df-2nd 7960 df-frecs 8250 df-wrecs 8281 df-recs 8355 df-rdg 8394 df-er 8688 df-en 8925 df-dom 8926 df-sdom 8927 df-pnf 11234 df-mnf 11235 df-xr 11236 df-ltxr 11237 df-le 11238 df-sub 11430 df-neg 11431 df-nn 12197 df-n0 12457 df-z 12543 df-uz 12807 df-fz 13469 |
This theorem is referenced by: (None) |
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