![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > uzubioo | Structured version Visualization version GIF version |
Description: The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
uzubioo.1 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
uzubioo.2 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
uzubioo.3 | ⊢ (𝜑 → 𝑋 ∈ ℝ) |
Ref | Expression |
---|---|
uzubioo | ⊢ (𝜑 → ∃𝑘 ∈ (𝑋(,)+∞)𝑘 ∈ 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uzubioo.3 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ℝ) | |
2 | 1 | rexrd 10490 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ*) |
3 | pnfxr 10494 | . . . 4 ⊢ +∞ ∈ ℝ* | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → +∞ ∈ ℝ*) |
5 | 1 | ceilcld 41155 | . . . . . 6 ⊢ (𝜑 → (⌈‘𝑋) ∈ ℤ) |
6 | 1zzd 11826 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℤ) | |
7 | 5, 6 | zaddcld 11904 | . . . . 5 ⊢ (𝜑 → ((⌈‘𝑋) + 1) ∈ ℤ) |
8 | 7 | zred 11900 | . . . 4 ⊢ (𝜑 → ((⌈‘𝑋) + 1) ∈ ℝ) |
9 | uzubioo.1 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
10 | 9 | zred 11900 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
11 | 8, 10 | ifcld 4395 | . . 3 ⊢ (𝜑 → if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀) ∈ ℝ) |
12 | 5 | zred 11900 | . . . . 5 ⊢ (𝜑 → (⌈‘𝑋) ∈ ℝ) |
13 | 1 | ceilged 41149 | . . . . 5 ⊢ (𝜑 → 𝑋 ≤ (⌈‘𝑋)) |
14 | 12 | ltp1d 11371 | . . . . 5 ⊢ (𝜑 → (⌈‘𝑋) < ((⌈‘𝑋) + 1)) |
15 | 1, 12, 8, 13, 14 | lelttrd 10598 | . . . 4 ⊢ (𝜑 → 𝑋 < ((⌈‘𝑋) + 1)) |
16 | 10, 8 | max2d 41163 | . . . 4 ⊢ (𝜑 → ((⌈‘𝑋) + 1) ≤ if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀)) |
17 | 1, 8, 11, 15, 16 | ltletrd 10600 | . . 3 ⊢ (𝜑 → 𝑋 < if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀)) |
18 | 11 | ltpnfd 12333 | . . 3 ⊢ (𝜑 → if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀) < +∞) |
19 | 2, 4, 11, 17, 18 | eliood 41202 | . 2 ⊢ (𝜑 → if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀) ∈ (𝑋(,)+∞)) |
20 | uzubioo.2 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
21 | 7, 9 | ifcld 4395 | . . 3 ⊢ (𝜑 → if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀) ∈ ℤ) |
22 | max1 12395 | . . . 4 ⊢ ((𝑀 ∈ ℝ ∧ ((⌈‘𝑋) + 1) ∈ ℝ) → 𝑀 ≤ if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀)) | |
23 | 10, 8, 22 | syl2anc 576 | . . 3 ⊢ (𝜑 → 𝑀 ≤ if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀)) |
24 | 20, 9, 21, 23 | eluzd 41111 | . 2 ⊢ (𝜑 → if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀) ∈ 𝑍) |
25 | eleq1 2854 | . . 3 ⊢ (𝑘 = if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀) → (𝑘 ∈ 𝑍 ↔ if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀) ∈ 𝑍)) | |
26 | 25 | rspcev 3536 | . 2 ⊢ ((if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀) ∈ (𝑋(,)+∞) ∧ if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀) ∈ 𝑍) → ∃𝑘 ∈ (𝑋(,)+∞)𝑘 ∈ 𝑍) |
27 | 19, 24, 26 | syl2anc 576 | 1 ⊢ (𝜑 → ∃𝑘 ∈ (𝑋(,)+∞)𝑘 ∈ 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 ∃wrex 3090 ifcif 4350 class class class wbr 4929 ‘cfv 6188 (class class class)co 6976 ℝcr 10334 1c1 10336 + caddc 10338 +∞cpnf 10471 ℝ*cxr 10473 ≤ cle 10475 ℤcz 11793 ℤ≥cuz 12058 (,)cioo 12554 ⌈cceil 12976 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 ax-pre-sup 10413 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-sup 8701 df-inf 8702 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-n0 11708 df-z 11794 df-uz 12059 df-ioo 12558 df-fl 12977 df-ceil 12978 |
This theorem is referenced by: uzubico 41273 uzubioo2 41274 |
Copyright terms: Public domain | W3C validator |