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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > liminfval | Structured version Visualization version GIF version |
Description: The inferior limit of a set πΉ. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
liminfval.1 | β’ πΊ = (π β β β¦ inf(((πΉ β (π[,)+β)) β© β*), β*, < )) |
Ref | Expression |
---|---|
liminfval | β’ (πΉ β π β (lim infβπΉ) = sup(ran πΊ, β*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-liminf 44468 | . 2 β’ lim inf = (π₯ β V β¦ sup(ran (π β β β¦ inf(((π₯ β (π[,)+β)) β© β*), β*, < )), β*, < )) | |
2 | imaeq1 6055 | . . . . . . . 8 β’ (π₯ = πΉ β (π₯ β (π[,)+β)) = (πΉ β (π[,)+β))) | |
3 | 2 | ineq1d 4212 | . . . . . . 7 β’ (π₯ = πΉ β ((π₯ β (π[,)+β)) β© β*) = ((πΉ β (π[,)+β)) β© β*)) |
4 | 3 | infeq1d 9472 | . . . . . 6 β’ (π₯ = πΉ β inf(((π₯ β (π[,)+β)) β© β*), β*, < ) = inf(((πΉ β (π[,)+β)) β© β*), β*, < )) |
5 | 4 | mpteq2dv 5251 | . . . . 5 β’ (π₯ = πΉ β (π β β β¦ inf(((π₯ β (π[,)+β)) β© β*), β*, < )) = (π β β β¦ inf(((πΉ β (π[,)+β)) β© β*), β*, < ))) |
6 | liminfval.1 | . . . . . 6 β’ πΊ = (π β β β¦ inf(((πΉ β (π[,)+β)) β© β*), β*, < )) | |
7 | 6 | a1i 11 | . . . . 5 β’ (π₯ = πΉ β πΊ = (π β β β¦ inf(((πΉ β (π[,)+β)) β© β*), β*, < ))) |
8 | 5, 7 | eqtr4d 2776 | . . . 4 β’ (π₯ = πΉ β (π β β β¦ inf(((π₯ β (π[,)+β)) β© β*), β*, < )) = πΊ) |
9 | 8 | rneqd 5938 | . . 3 β’ (π₯ = πΉ β ran (π β β β¦ inf(((π₯ β (π[,)+β)) β© β*), β*, < )) = ran πΊ) |
10 | 9 | supeq1d 9441 | . 2 β’ (π₯ = πΉ β sup(ran (π β β β¦ inf(((π₯ β (π[,)+β)) β© β*), β*, < )), β*, < ) = sup(ran πΊ, β*, < )) |
11 | elex 3493 | . 2 β’ (πΉ β π β πΉ β V) | |
12 | xrltso 13120 | . . . 4 β’ < Or β* | |
13 | 12 | supex 9458 | . . 3 β’ sup(ran πΊ, β*, < ) β V |
14 | 13 | a1i 11 | . 2 β’ (πΉ β π β sup(ran πΊ, β*, < ) β V) |
15 | 1, 10, 11, 14 | fvmptd3 7022 | 1 β’ (πΉ β π β (lim infβπΉ) = sup(ran πΊ, β*, < )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 Vcvv 3475 β© cin 3948 β¦ cmpt 5232 ran crn 5678 β cima 5680 βcfv 6544 (class class class)co 7409 supcsup 9435 infcinf 9436 βcr 11109 +βcpnf 11245 β*cxr 11247 < clt 11248 [,)cico 13326 lim infclsi 44467 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-pre-lttri 11184 ax-pre-lttrn 11185 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-sup 9437 df-inf 9438 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-liminf 44468 |
This theorem is referenced by: liminfcl 44479 liminfvald 44480 liminfval5 44481 liminfresxr 44483 liminfval2 44484 liminfvalxr 44499 |
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