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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > limsup10ex | Structured version Visualization version GIF version |
Description: The superior limit of a function that alternates between two values. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
limsup10ex.1 | β’ πΉ = (π β β β¦ if(2 β₯ π, 0, 1)) |
Ref | Expression |
---|---|
limsup10ex | β’ (lim supβπΉ) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nftru 1806 | . . . 4 β’ β²πβ€ | |
2 | nnex 12214 | . . . . 5 β’ β β V | |
3 | 2 | a1i 11 | . . . 4 β’ (β€ β β β V) |
4 | limsup10ex.1 | . . . . . 6 β’ πΉ = (π β β β¦ if(2 β₯ π, 0, 1)) | |
5 | 0xr 11257 | . . . . . . . 8 β’ 0 β β* | |
6 | 5 | a1i 11 | . . . . . . 7 β’ (π β β β 0 β β*) |
7 | 1xr 11269 | . . . . . . . 8 β’ 1 β β* | |
8 | 7 | a1i 11 | . . . . . . 7 β’ (π β β β 1 β β*) |
9 | 6, 8 | ifcld 4573 | . . . . . 6 β’ (π β β β if(2 β₯ π, 0, 1) β β*) |
10 | 4, 9 | fmpti 7108 | . . . . 5 β’ πΉ:ββΆβ* |
11 | 10 | a1i 11 | . . . 4 β’ (β€ β πΉ:ββΆβ*) |
12 | eqid 2732 | . . . 4 β’ (π β β β¦ sup((πΉ β (π[,)+β)), β*, < )) = (π β β β¦ sup((πΉ β (π[,)+β)), β*, < )) | |
13 | 1, 3, 11, 12 | limsupval3 44394 | . . 3 β’ (β€ β (lim supβπΉ) = inf(ran (π β β β¦ sup((πΉ β (π[,)+β)), β*, < )), β*, < )) |
14 | 13 | mptru 1548 | . 2 β’ (lim supβπΉ) = inf(ran (π β β β¦ sup((πΉ β (π[,)+β)), β*, < )), β*, < ) |
15 | id 22 | . . . . . . . . 9 β’ (π β β β π β β) | |
16 | 4, 15 | limsup10exlem 44474 | . . . . . . . 8 β’ (π β β β (πΉ β (π[,)+β)) = {0, 1}) |
17 | 16 | supeq1d 9437 | . . . . . . 7 β’ (π β β β sup((πΉ β (π[,)+β)), β*, < ) = sup({0, 1}, β*, < )) |
18 | xrltso 13116 | . . . . . . . . 9 β’ < Or β* | |
19 | suppr 9462 | . . . . . . . . 9 β’ (( < Or β* β§ 0 β β* β§ 1 β β*) β sup({0, 1}, β*, < ) = if(1 < 0, 0, 1)) | |
20 | 18, 5, 7, 19 | mp3an 1461 | . . . . . . . 8 β’ sup({0, 1}, β*, < ) = if(1 < 0, 0, 1) |
21 | 0le1 11733 | . . . . . . . . . 10 β’ 0 β€ 1 | |
22 | 0re 11212 | . . . . . . . . . . 11 β’ 0 β β | |
23 | 1re 11210 | . . . . . . . . . . 11 β’ 1 β β | |
24 | 22, 23 | lenlti 11330 | . . . . . . . . . 10 β’ (0 β€ 1 β Β¬ 1 < 0) |
25 | 21, 24 | mpbi 229 | . . . . . . . . 9 β’ Β¬ 1 < 0 |
26 | 25 | iffalsei 4537 | . . . . . . . 8 β’ if(1 < 0, 0, 1) = 1 |
27 | 20, 26 | eqtri 2760 | . . . . . . 7 β’ sup({0, 1}, β*, < ) = 1 |
28 | 17, 27 | eqtrdi 2788 | . . . . . 6 β’ (π β β β sup((πΉ β (π[,)+β)), β*, < ) = 1) |
29 | 28 | mpteq2ia 5250 | . . . . 5 β’ (π β β β¦ sup((πΉ β (π[,)+β)), β*, < )) = (π β β β¦ 1) |
30 | 29 | rneqi 5934 | . . . 4 β’ ran (π β β β¦ sup((πΉ β (π[,)+β)), β*, < )) = ran (π β β β¦ 1) |
31 | eqid 2732 | . . . . . 6 β’ (π β β β¦ 1) = (π β β β¦ 1) | |
32 | ren0 44098 | . . . . . . 7 β’ β β β | |
33 | 32 | a1i 11 | . . . . . 6 β’ (β€ β β β β ) |
34 | 31, 33 | rnmptc 7204 | . . . . 5 β’ (β€ β ran (π β β β¦ 1) = {1}) |
35 | 34 | mptru 1548 | . . . 4 β’ ran (π β β β¦ 1) = {1} |
36 | 30, 35 | eqtri 2760 | . . 3 β’ ran (π β β β¦ sup((πΉ β (π[,)+β)), β*, < )) = {1} |
37 | 36 | infeq1i 9469 | . 2 β’ inf(ran (π β β β¦ sup((πΉ β (π[,)+β)), β*, < )), β*, < ) = inf({1}, β*, < ) |
38 | infsn 9496 | . . 3 β’ (( < Or β* β§ 1 β β*) β inf({1}, β*, < ) = 1) | |
39 | 18, 7, 38 | mp2an 690 | . 2 β’ inf({1}, β*, < ) = 1 |
40 | 14, 37, 39 | 3eqtri 2764 | 1 β’ (lim supβπΉ) = 1 |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 = wceq 1541 β€wtru 1542 β wcel 2106 β wne 2940 Vcvv 3474 β c0 4321 ifcif 4527 {csn 4627 {cpr 4629 class class class wbr 5147 β¦ cmpt 5230 Or wor 5586 ran crn 5676 β cima 5678 βΆwf 6536 βcfv 6540 (class class class)co 7405 supcsup 9431 infcinf 9432 βcr 11105 0cc0 11106 1c1 11107 +βcpnf 11241 β*cxr 11243 < clt 11244 β€ cle 11245 βcn 12208 2c2 12263 [,)cico 13322 lim supclsp 15410 β₯ cdvds 16193 |
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-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
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-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-ico 13326 df-fl 13753 df-ceil 13754 df-limsup 15411 df-dvds 16194 |
This theorem is referenced by: liminfltlimsupex 44483 |
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