![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
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 7234 | . . 3 β’ (π β πΉ β V) |
4 | eqid 2725 | . . . 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 45209 | . . 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 6738 | . . . . . . . . . 10 β’ (π β (πΉ β (π[,)+β)) β β*) |
11 | dfss2 3958 | . . . . . . . . . 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 2731 | . . . . . . . 8 β’ (π β (πΉ β (π[,)+β)) = ((πΉ β (π[,)+β)) β© β*)) |
14 | 13 | adantr 479 | . . . . . . 7 β’ ((π β§ π β β) β (πΉ β (π[,)+β)) = ((πΉ β (π[,)+β)) β© β*)) |
15 | 14 | infeq1d 9498 | . . . . . 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 5241 | . . . . 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 2766 | . . . 4 β’ (π β (π β β β¦ inf(((πΉ β (π[,)+β)) β© β*), β*, < )) = πΊ) |
18 | 17 | rneqd 5934 | . . 3 β’ (π β ran (π β β β¦ inf(((πΉ β (π[,)+β)) β© β*), β*, < )) = ran πΊ) |
19 | 18 | supeq1d 9467 | . 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 2765 | 1 β’ (π β (lim infβπΉ) = sup(ran πΊ, β*, < )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β²wnf 1777 β wcel 2098 Vcvv 3463 β© cin 3939 β wss 3940 β¦ cmpt 5226 ran crn 5673 β cima 5675 βΆwf 6538 βcfv 6542 (class class class)co 7415 supcsup 9461 infcinf 9462 βcr 11135 +βcpnf 11273 β*cxr 11275 < clt 11276 [,)cico 13356 lim infclsi 45201 |
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 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-pre-lttri 11210 ax-pre-lttrn 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 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-sup 9463 df-inf 9464 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-liminf 45202 |
This theorem is referenced by: liminf10ex 45224 |
Copyright terms: Public domain | W3C validator |