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Mirrors > Home > MPE Home > Th. List > uzsup | Structured version Visualization version GIF version |
Description: An upper set of integers is unbounded above. (Contributed by Mario Carneiro, 7-May-2016.) |
Ref | Expression |
---|---|
uzsup.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
Ref | Expression |
---|---|
uzsup | ⊢ (𝑀 ∈ ℤ → sup(𝑍, ℝ*, < ) = +∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 485 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ ℝ) → 𝑀 ∈ ℤ) | |
2 | flcl 13166 | . . . . . . . 8 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℤ) | |
3 | 2 | peano2zd 12091 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ → ((⌊‘𝑥) + 1) ∈ ℤ) |
4 | id 22 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℤ) | |
5 | ifcl 4511 | . . . . . . 7 ⊢ ((((⌊‘𝑥) + 1) ∈ ℤ ∧ 𝑀 ∈ ℤ) → if(𝑀 ≤ ((⌊‘𝑥) + 1), ((⌊‘𝑥) + 1), 𝑀) ∈ ℤ) | |
6 | 3, 4, 5 | syl2anr 598 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ ℝ) → if(𝑀 ≤ ((⌊‘𝑥) + 1), ((⌊‘𝑥) + 1), 𝑀) ∈ ℤ) |
7 | zre 11986 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
8 | reflcl 13167 | . . . . . . . 8 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℝ) | |
9 | peano2re 10813 | . . . . . . . 8 ⊢ ((⌊‘𝑥) ∈ ℝ → ((⌊‘𝑥) + 1) ∈ ℝ) | |
10 | 8, 9 | syl 17 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ → ((⌊‘𝑥) + 1) ∈ ℝ) |
11 | max1 12579 | . . . . . . 7 ⊢ ((𝑀 ∈ ℝ ∧ ((⌊‘𝑥) + 1) ∈ ℝ) → 𝑀 ≤ if(𝑀 ≤ ((⌊‘𝑥) + 1), ((⌊‘𝑥) + 1), 𝑀)) | |
12 | 7, 10, 11 | syl2an 597 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ ℝ) → 𝑀 ≤ if(𝑀 ≤ ((⌊‘𝑥) + 1), ((⌊‘𝑥) + 1), 𝑀)) |
13 | eluz2 12250 | . . . . . 6 ⊢ (if(𝑀 ≤ ((⌊‘𝑥) + 1), ((⌊‘𝑥) + 1), 𝑀) ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ if(𝑀 ≤ ((⌊‘𝑥) + 1), ((⌊‘𝑥) + 1), 𝑀) ∈ ℤ ∧ 𝑀 ≤ if(𝑀 ≤ ((⌊‘𝑥) + 1), ((⌊‘𝑥) + 1), 𝑀))) | |
14 | 1, 6, 12, 13 | syl3anbrc 1339 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ ℝ) → if(𝑀 ≤ ((⌊‘𝑥) + 1), ((⌊‘𝑥) + 1), 𝑀) ∈ (ℤ≥‘𝑀)) |
15 | uzsup.1 | . . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
16 | 14, 15 | eleqtrrdi 2924 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ ℝ) → if(𝑀 ≤ ((⌊‘𝑥) + 1), ((⌊‘𝑥) + 1), 𝑀) ∈ 𝑍) |
17 | simpr 487 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ) | |
18 | 10 | adantl 484 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ ℝ) → ((⌊‘𝑥) + 1) ∈ ℝ) |
19 | 6 | zred 12088 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ ℝ) → if(𝑀 ≤ ((⌊‘𝑥) + 1), ((⌊‘𝑥) + 1), 𝑀) ∈ ℝ) |
20 | fllep1 13172 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ≤ ((⌊‘𝑥) + 1)) | |
21 | 20 | adantl 484 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ ℝ) → 𝑥 ≤ ((⌊‘𝑥) + 1)) |
22 | max2 12581 | . . . . . 6 ⊢ ((𝑀 ∈ ℝ ∧ ((⌊‘𝑥) + 1) ∈ ℝ) → ((⌊‘𝑥) + 1) ≤ if(𝑀 ≤ ((⌊‘𝑥) + 1), ((⌊‘𝑥) + 1), 𝑀)) | |
23 | 7, 10, 22 | syl2an 597 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ ℝ) → ((⌊‘𝑥) + 1) ≤ if(𝑀 ≤ ((⌊‘𝑥) + 1), ((⌊‘𝑥) + 1), 𝑀)) |
24 | 17, 18, 19, 21, 23 | letrd 10797 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ ℝ) → 𝑥 ≤ if(𝑀 ≤ ((⌊‘𝑥) + 1), ((⌊‘𝑥) + 1), 𝑀)) |
25 | breq2 5070 | . . . . 5 ⊢ (𝑛 = if(𝑀 ≤ ((⌊‘𝑥) + 1), ((⌊‘𝑥) + 1), 𝑀) → (𝑥 ≤ 𝑛 ↔ 𝑥 ≤ if(𝑀 ≤ ((⌊‘𝑥) + 1), ((⌊‘𝑥) + 1), 𝑀))) | |
26 | 25 | rspcev 3623 | . . . 4 ⊢ ((if(𝑀 ≤ ((⌊‘𝑥) + 1), ((⌊‘𝑥) + 1), 𝑀) ∈ 𝑍 ∧ 𝑥 ≤ if(𝑀 ≤ ((⌊‘𝑥) + 1), ((⌊‘𝑥) + 1), 𝑀)) → ∃𝑛 ∈ 𝑍 𝑥 ≤ 𝑛) |
27 | 16, 24, 26 | syl2anc 586 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ ℝ) → ∃𝑛 ∈ 𝑍 𝑥 ≤ 𝑛) |
28 | 27 | ralrimiva 3182 | . 2 ⊢ (𝑀 ∈ ℤ → ∀𝑥 ∈ ℝ ∃𝑛 ∈ 𝑍 𝑥 ≤ 𝑛) |
29 | uzssz 12265 | . . . . . 6 ⊢ (ℤ≥‘𝑀) ⊆ ℤ | |
30 | 15, 29 | eqsstri 4001 | . . . . 5 ⊢ 𝑍 ⊆ ℤ |
31 | zssre 11989 | . . . . 5 ⊢ ℤ ⊆ ℝ | |
32 | 30, 31 | sstri 3976 | . . . 4 ⊢ 𝑍 ⊆ ℝ |
33 | ressxr 10685 | . . . 4 ⊢ ℝ ⊆ ℝ* | |
34 | 32, 33 | sstri 3976 | . . 3 ⊢ 𝑍 ⊆ ℝ* |
35 | supxrunb1 12713 | . . 3 ⊢ (𝑍 ⊆ ℝ* → (∀𝑥 ∈ ℝ ∃𝑛 ∈ 𝑍 𝑥 ≤ 𝑛 ↔ sup(𝑍, ℝ*, < ) = +∞)) | |
36 | 34, 35 | ax-mp 5 | . 2 ⊢ (∀𝑥 ∈ ℝ ∃𝑛 ∈ 𝑍 𝑥 ≤ 𝑛 ↔ sup(𝑍, ℝ*, < ) = +∞) |
37 | 28, 36 | sylib 220 | 1 ⊢ (𝑀 ∈ ℤ → sup(𝑍, ℝ*, < ) = +∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 ⊆ wss 3936 ifcif 4467 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 supcsup 8904 ℝcr 10536 1c1 10538 + caddc 10540 +∞cpnf 10672 ℝ*cxr 10674 < clt 10675 ≤ cle 10676 ℤcz 11982 ℤ≥cuz 12244 ⌊cfl 13161 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fl 13163 |
This theorem is referenced by: climrecl 14940 climge0 14941 caurcvg 15033 caucvg 15035 mbflimsup 24267 limsupvaluz 42009 ioodvbdlimc1lem2 42237 ioodvbdlimc2lem 42239 |
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