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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > liminf10ex | Structured version Visualization version GIF version |
Description: The inferior limit of a function that alternates between two values. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
liminf10ex.1 | β’ πΉ = (π β β β¦ if(2 β₯ π, 0, 1)) |
Ref | Expression |
---|---|
liminf10ex | β’ (lim infβπΉ) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nftru 1798 | . . . 4 β’ β²πβ€ | |
2 | nnex 12243 | . . . . 5 β’ β β V | |
3 | 2 | a1i 11 | . . . 4 β’ (β€ β β β V) |
4 | liminf10ex.1 | . . . . . 6 β’ πΉ = (π β β β¦ if(2 β₯ π, 0, 1)) | |
5 | 0xr 11286 | . . . . . . . 8 β’ 0 β β* | |
6 | 5 | a1i 11 | . . . . . . 7 β’ (π β β β 0 β β*) |
7 | 1xr 11298 | . . . . . . . 8 β’ 1 β β* | |
8 | 7 | a1i 11 | . . . . . . 7 β’ (π β β β 1 β β*) |
9 | 6, 8 | ifcld 4571 | . . . . . 6 β’ (π β β β if(2 β₯ π, 0, 1) β β*) |
10 | 4, 9 | fmpti 7115 | . . . . 5 β’ πΉ:ββΆβ* |
11 | 10 | a1i 11 | . . . 4 β’ (β€ β πΉ:ββΆβ*) |
12 | eqid 2725 | . . . 4 β’ (π β β β¦ inf((πΉ β (π[,)+β)), β*, < )) = (π β β β¦ inf((πΉ β (π[,)+β)), β*, < )) | |
13 | 1, 3, 11, 12 | liminfval5 45212 | . . 3 β’ (β€ β (lim infβπΉ) = sup(ran (π β β β¦ inf((πΉ β (π[,)+β)), β*, < )), β*, < )) |
14 | 13 | mptru 1540 | . 2 β’ (lim infβπΉ) = sup(ran (π β β β¦ inf((πΉ β (π[,)+β)), β*, < )), β*, < ) |
15 | id 22 | . . . . . . . . 9 β’ (π β β β π β β) | |
16 | 4, 15 | limsup10exlem 45219 | . . . . . . . 8 β’ (π β β β (πΉ β (π[,)+β)) = {0, 1}) |
17 | 16 | infeq1d 9495 | . . . . . . 7 β’ (π β β β inf((πΉ β (π[,)+β)), β*, < ) = inf({0, 1}, β*, < )) |
18 | xrltso 13147 | . . . . . . . . 9 β’ < Or β* | |
19 | infpr 9521 | . . . . . . . . 9 β’ (( < Or β* β§ 0 β β* β§ 1 β β*) β inf({0, 1}, β*, < ) = if(0 < 1, 0, 1)) | |
20 | 18, 5, 7, 19 | mp3an 1457 | . . . . . . . 8 β’ inf({0, 1}, β*, < ) = if(0 < 1, 0, 1) |
21 | 0lt1 11761 | . . . . . . . . 9 β’ 0 < 1 | |
22 | 21 | iftruei 4532 | . . . . . . . 8 β’ if(0 < 1, 0, 1) = 0 |
23 | 20, 22 | eqtri 2753 | . . . . . . 7 β’ inf({0, 1}, β*, < ) = 0 |
24 | 17, 23 | eqtrdi 2781 | . . . . . 6 β’ (π β β β inf((πΉ β (π[,)+β)), β*, < ) = 0) |
25 | 24 | mpteq2ia 5247 | . . . . 5 β’ (π β β β¦ inf((πΉ β (π[,)+β)), β*, < )) = (π β β β¦ 0) |
26 | 25 | rneqi 5934 | . . . 4 β’ ran (π β β β¦ inf((πΉ β (π[,)+β)), β*, < )) = ran (π β β β¦ 0) |
27 | eqid 2725 | . . . . . 6 β’ (π β β β¦ 0) = (π β β β¦ 0) | |
28 | ren0 44843 | . . . . . . 7 β’ β β β | |
29 | 28 | a1i 11 | . . . . . 6 β’ (β€ β β β β ) |
30 | 27, 29 | rnmptc 7213 | . . . . 5 β’ (β€ β ran (π β β β¦ 0) = {0}) |
31 | 30 | mptru 1540 | . . . 4 β’ ran (π β β β¦ 0) = {0} |
32 | 26, 31 | eqtri 2753 | . . 3 β’ ran (π β β β¦ inf((πΉ β (π[,)+β)), β*, < )) = {0} |
33 | 32 | supeq1i 9465 | . 2 β’ sup(ran (π β β β¦ inf((πΉ β (π[,)+β)), β*, < )), β*, < ) = sup({0}, β*, < ) |
34 | supsn 9490 | . . 3 β’ (( < Or β* β§ 0 β β*) β sup({0}, β*, < ) = 0) | |
35 | 18, 5, 34 | mp2an 690 | . 2 β’ sup({0}, β*, < ) = 0 |
36 | 14, 33, 35 | 3eqtri 2757 | 1 β’ (lim infβπΉ) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 β€wtru 1534 β wcel 2098 β wne 2930 Vcvv 3463 β c0 4319 ifcif 4525 {csn 4625 {cpr 4627 class class class wbr 5144 β¦ cmpt 5227 Or wor 5584 ran crn 5674 β cima 5676 βΆwf 6539 βcfv 6543 (class class class)co 7413 supcsup 9458 infcinf 9459 βcr 11132 0cc0 11133 1c1 11134 +βcpnf 11270 β*cxr 11272 < clt 11273 βcn 12237 2c2 12292 [,)cico 13353 β₯ cdvds 16225 lim infclsi 45198 |
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 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 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 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-sup 9460 df-inf 9461 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-n0 12498 df-z 12584 df-uz 12848 df-rp 13002 df-ico 13357 df-fl 13784 df-ceil 13785 df-dvds 16226 df-liminf 45199 |
This theorem is referenced by: liminfltlimsupex 45228 |
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