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Mathbox for Glauco Siliprandi |
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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 11265 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ*) |
3 | pnfxr 11269 | . . . 4 ⊢ +∞ ∈ ℝ* | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → +∞ ∈ ℝ*) |
5 | 1 | ceilcld 13811 | . . . . . 6 ⊢ (𝜑 → (⌈‘𝑋) ∈ ℤ) |
6 | 1zzd 12594 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℤ) | |
7 | 5, 6 | zaddcld 12671 | . . . . 5 ⊢ (𝜑 → ((⌈‘𝑋) + 1) ∈ ℤ) |
8 | 7 | zred 12667 | . . . 4 ⊢ (𝜑 → ((⌈‘𝑋) + 1) ∈ ℝ) |
9 | uzubioo.1 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
10 | 9 | zred 12667 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
11 | 8, 10 | ifcld 4569 | . . 3 ⊢ (𝜑 → if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀) ∈ ℝ) |
12 | 5 | zred 12667 | . . . . 5 ⊢ (𝜑 → (⌈‘𝑋) ∈ ℝ) |
13 | 1 | ceilged 13814 | . . . . 5 ⊢ (𝜑 → 𝑋 ≤ (⌈‘𝑋)) |
14 | 12 | ltp1d 12145 | . . . . 5 ⊢ (𝜑 → (⌈‘𝑋) < ((⌈‘𝑋) + 1)) |
15 | 1, 12, 8, 13, 14 | lelttrd 11373 | . . . 4 ⊢ (𝜑 → 𝑋 < ((⌈‘𝑋) + 1)) |
16 | 10, 8 | max2d 44721 | . . . 4 ⊢ (𝜑 → ((⌈‘𝑋) + 1) ≤ if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀)) |
17 | 1, 8, 11, 15, 16 | ltletrd 11375 | . . 3 ⊢ (𝜑 → 𝑋 < if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀)) |
18 | 11 | ltpnfd 13104 | . . 3 ⊢ (𝜑 → if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀) < +∞) |
19 | 2, 4, 11, 17, 18 | eliood 44764 | . 2 ⊢ (𝜑 → if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀) ∈ (𝑋(,)+∞)) |
20 | uzubioo.2 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
21 | 7, 9 | ifcld 4569 | . . 3 ⊢ (𝜑 → if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀) ∈ ℤ) |
22 | max1 13167 | . . . 4 ⊢ ((𝑀 ∈ ℝ ∧ ((⌈‘𝑋) + 1) ∈ ℝ) → 𝑀 ≤ if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀)) | |
23 | 10, 8, 22 | syl2anc 583 | . . 3 ⊢ (𝜑 → 𝑀 ≤ if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀)) |
24 | 20, 9, 21, 23 | eluzd 44672 | . 2 ⊢ (𝜑 → if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀) ∈ 𝑍) |
25 | eleq1 2815 | . . 3 ⊢ (𝑘 = if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀) → (𝑘 ∈ 𝑍 ↔ if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀) ∈ 𝑍)) | |
26 | 25 | rspcev 3606 | . 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 583 | 1 ⊢ (𝜑 → ∃𝑘 ∈ (𝑋(,)+∞)𝑘 ∈ 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∃wrex 3064 ifcif 4523 class class class wbr 5141 ‘cfv 6536 (class class class)co 7404 ℝcr 11108 1c1 11110 + caddc 11112 +∞cpnf 11246 ℝ*cxr 11248 ≤ cle 11250 ℤcz 12559 ℤ≥cuz 12823 (,)cioo 13327 ⌈cceil 13759 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-inf 9437 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-n0 12474 df-z 12560 df-uz 12824 df-ioo 13331 df-fl 13760 df-ceil 13761 |
This theorem is referenced by: uzubico 44834 uzubioo2 44835 |
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