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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 12576 | . . 3 ⊢ 0 ∈ ℤ | |
2 | nn0z 12590 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
3 | peano2zm 12612 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑁 − 1) ∈ ℤ) |
5 | fzn 13524 | . . 3 ⊢ ((0 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) → ((𝑁 − 1) < 0 ↔ (0...(𝑁 − 1)) = ∅)) | |
6 | 1, 4, 5 | sylancr 586 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 − 1) < 0 ↔ (0...(𝑁 − 1)) = ∅)) |
7 | elnn0 12481 | . . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
8 | nnge1 12247 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁) | |
9 | nnre 12226 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
10 | 1re 11221 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
11 | subge0 11734 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℝ ∧ 1 ∈ ℝ) → (0 ≤ (𝑁 − 1) ↔ 1 ≤ 𝑁)) | |
12 | 0re 11223 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
13 | resubcl 11531 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℝ ∧ 1 ∈ ℝ) → (𝑁 − 1) ∈ ℝ) | |
14 | lenlt 11299 | . . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (𝑁 − 1) ∈ ℝ) → (0 ≤ (𝑁 − 1) ↔ ¬ (𝑁 − 1) < 0)) | |
15 | 12, 13, 14 | sylancr 586 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℝ ∧ 1 ∈ ℝ) → (0 ≤ (𝑁 − 1) ↔ ¬ (𝑁 − 1) < 0)) |
16 | 11, 15 | bitr3d 281 | . . . . . . 7 ⊢ ((𝑁 ∈ ℝ ∧ 1 ∈ ℝ) → (1 ≤ 𝑁 ↔ ¬ (𝑁 − 1) < 0)) |
17 | 9, 10, 16 | sylancl 585 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (1 ≤ 𝑁 ↔ ¬ (𝑁 − 1) < 0)) |
18 | 8, 17 | mpbid 231 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ¬ (𝑁 − 1) < 0) |
19 | nnne0 12253 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0) | |
20 | 19 | neneqd 2944 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ¬ 𝑁 = 0) |
21 | 18, 20 | 2falsed 376 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) < 0 ↔ 𝑁 = 0)) |
22 | oveq1 7419 | . . . . . . 7 ⊢ (𝑁 = 0 → (𝑁 − 1) = (0 − 1)) | |
23 | df-neg 11454 | . . . . . . 7 ⊢ -1 = (0 − 1) | |
24 | 22, 23 | eqtr4di 2789 | . . . . . 6 ⊢ (𝑁 = 0 → (𝑁 − 1) = -1) |
25 | neg1lt0 12336 | . . . . . 6 ⊢ -1 < 0 | |
26 | 24, 25 | eqbrtrdi 5187 | . . . . 5 ⊢ (𝑁 = 0 → (𝑁 − 1) < 0) |
27 | id 22 | . . . . 5 ⊢ (𝑁 = 0 → 𝑁 = 0) | |
28 | 26, 27 | 2thd 265 | . . . 4 ⊢ (𝑁 = 0 → ((𝑁 − 1) < 0 ↔ 𝑁 = 0)) |
29 | 21, 28 | jaoi 854 | . . 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 281 | 1 ⊢ (𝑁 ∈ ℕ0 → ((0...(𝑁 − 1)) = ∅ ↔ 𝑁 = 0)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 844 = wceq 1540 ∈ wcel 2105 ∅c0 4322 class class class wbr 5148 (class class class)co 7412 ℝcr 11115 0cc0 11116 1c1 11117 < clt 11255 ≤ cle 11256 − cmin 11451 -cneg 11452 ℕcn 12219 ℕ0cn0 12479 ℤcz 12565 ...cfz 13491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 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 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-n0 12480 df-z 12566 df-uz 12830 df-fz 13492 |
This theorem is referenced by: (None) |
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