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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > liminfval5 | Structured version Visualization version GIF version |
Description: The inferior limit of an infinite sequence πΉ of extended real numbers. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
limsupval5.1 | β’ β²ππ |
limsupval5.2 | β’ (π β π΄ β π) |
limsupval5.3 | β’ (π β πΉ:π΄βΆβ*) |
limsupval5.4 | β’ πΊ = (π β β β¦ inf((πΉ β (π[,)+β)), β*, < )) |
Ref | Expression |
---|---|
liminfval5 | β’ (π β (lim infβπΉ) = sup(ran πΊ, β*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limsupval5.3 | . . . 4 β’ (π β πΉ:π΄βΆβ*) | |
2 | limsupval5.2 | . . . 4 β’ (π β π΄ β π) | |
3 | 1, 2 | fexd 7224 | . . 3 β’ (π β πΉ β V) |
4 | eqid 2726 | . . . 4 β’ (π β β β¦ inf(((πΉ β (π[,)+β)) β© β*), β*, < )) = (π β β β¦ inf(((πΉ β (π[,)+β)) β© β*), β*, < )) | |
5 | 4 | liminfval 45052 | . . 3 β’ (πΉ β V β (lim infβπΉ) = sup(ran (π β β β¦ inf(((πΉ β (π[,)+β)) β© β*), β*, < )), β*, < )) |
6 | 3, 5 | syl 17 | . 2 β’ (π β (lim infβπΉ) = sup(ran (π β β β¦ inf(((πΉ β (π[,)+β)) β© β*), β*, < )), β*, < )) |
7 | limsupval5.4 | . . . . . 6 β’ πΊ = (π β β β¦ inf((πΉ β (π[,)+β)), β*, < )) | |
8 | 7 | a1i 11 | . . . . 5 β’ (π β πΊ = (π β β β¦ inf((πΉ β (π[,)+β)), β*, < ))) |
9 | limsupval5.1 | . . . . . 6 β’ β²ππ | |
10 | 1 | fimassd 44507 | . . . . . . . . . 10 β’ (π β (πΉ β (π[,)+β)) β β*) |
11 | df-ss 3960 | . . . . . . . . . 10 β’ ((πΉ β (π[,)+β)) β β* β ((πΉ β (π[,)+β)) β© β*) = (πΉ β (π[,)+β))) | |
12 | 10, 11 | sylib 217 | . . . . . . . . 9 β’ (π β ((πΉ β (π[,)+β)) β© β*) = (πΉ β (π[,)+β))) |
13 | 12 | eqcomd 2732 | . . . . . . . 8 β’ (π β (πΉ β (π[,)+β)) = ((πΉ β (π[,)+β)) β© β*)) |
14 | 13 | adantr 480 | . . . . . . 7 β’ ((π β§ π β β) β (πΉ β (π[,)+β)) = ((πΉ β (π[,)+β)) β© β*)) |
15 | 14 | infeq1d 9474 | . . . . . 6 β’ ((π β§ π β β) β inf((πΉ β (π[,)+β)), β*, < ) = inf(((πΉ β (π[,)+β)) β© β*), β*, < )) |
16 | 9, 15 | mpteq2da 5239 | . . . . 5 β’ (π β (π β β β¦ inf((πΉ β (π[,)+β)), β*, < )) = (π β β β¦ inf(((πΉ β (π[,)+β)) β© β*), β*, < ))) |
17 | 8, 16 | eqtr2d 2767 | . . . 4 β’ (π β (π β β β¦ inf(((πΉ β (π[,)+β)) β© β*), β*, < )) = πΊ) |
18 | 17 | rneqd 5931 | . . 3 β’ (π β ran (π β β β¦ inf(((πΉ β (π[,)+β)) β© β*), β*, < )) = ran πΊ) |
19 | 18 | supeq1d 9443 | . 2 β’ (π β sup(ran (π β β β¦ inf(((πΉ β (π[,)+β)) β© β*), β*, < )), β*, < ) = sup(ran πΊ, β*, < )) |
20 | 6, 19 | eqtrd 2766 | 1 β’ (π β (lim infβπΉ) = sup(ran πΊ, β*, < )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β²wnf 1777 β wcel 2098 Vcvv 3468 β© cin 3942 β wss 3943 β¦ cmpt 5224 ran crn 5670 β cima 5672 βΆwf 6533 βcfv 6537 (class class class)co 7405 supcsup 9437 infcinf 9438 βcr 11111 +βcpnf 11249 β*cxr 11251 < clt 11252 [,)cico 13332 lim infclsi 45044 |
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-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-pre-lttri 11186 ax-pre-lttrn 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-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-id 5567 df-po 5581 df-so 5582 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-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-liminf 45045 |
This theorem is referenced by: liminf10ex 45067 |
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