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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 44458 | . 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 6054 | . . . . . . . 8 β’ (π₯ = πΉ β (π₯ β (π[,)+β)) = (πΉ β (π[,)+β))) | |
3 | 2 | ineq1d 4211 | . . . . . . 7 β’ (π₯ = πΉ β ((π₯ β (π[,)+β)) β© β*) = ((πΉ β (π[,)+β)) β© β*)) |
4 | 3 | infeq1d 9471 | . . . . . 6 β’ (π₯ = πΉ β inf(((π₯ β (π[,)+β)) β© β*), β*, < ) = inf(((πΉ β (π[,)+β)) β© β*), β*, < )) |
5 | 4 | mpteq2dv 5250 | . . . . 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 2775 | . . . 4 β’ (π₯ = πΉ β (π β β β¦ inf(((π₯ β (π[,)+β)) β© β*), β*, < )) = πΊ) |
9 | 8 | rneqd 5937 | . . 3 β’ (π₯ = πΉ β ran (π β β β¦ inf(((π₯ β (π[,)+β)) β© β*), β*, < )) = ran πΊ) |
10 | 9 | supeq1d 9440 | . 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 3492 | . 2 β’ (πΉ β π β πΉ β V) | |
12 | xrltso 13119 | . . . 4 β’ < Or β* | |
13 | 12 | supex 9457 | . . 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 7021 | 1 β’ (πΉ β π β (lim infβπΉ) = sup(ran πΊ, β*, < )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1541 β wcel 2106 Vcvv 3474 β© cin 3947 β¦ cmpt 5231 ran crn 5677 β cima 5679 βcfv 6543 (class class class)co 7408 supcsup 9434 infcinf 9435 βcr 11108 +βcpnf 11244 β*cxr 11246 < clt 11247 [,)cico 13325 lim infclsi 44457 |
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-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-pre-lttri 11183 ax-pre-lttrn 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-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 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-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-inf 9437 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-liminf 44458 |
This theorem is referenced by: liminfcl 44469 liminfvald 44470 liminfval5 44471 liminfresxr 44473 liminfval2 44474 liminfvalxr 44489 |
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