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 43637 | . 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 5994 | . . . . . . . 8 β’ (π₯ = πΉ β (π₯ β (π[,)+β)) = (πΉ β (π[,)+β))) | |
3 | 2 | ineq1d 4158 | . . . . . . 7 β’ (π₯ = πΉ β ((π₯ β (π[,)+β)) β© β*) = ((πΉ β (π[,)+β)) β© β*)) |
4 | 3 | infeq1d 9334 | . . . . . 6 β’ (π₯ = πΉ β inf(((π₯ β (π[,)+β)) β© β*), β*, < ) = inf(((πΉ β (π[,)+β)) β© β*), β*, < )) |
5 | 4 | mpteq2dv 5194 | . . . . 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 2779 | . . . 4 β’ (π₯ = πΉ β (π β β β¦ inf(((π₯ β (π[,)+β)) β© β*), β*, < )) = πΊ) |
9 | 8 | rneqd 5879 | . . 3 β’ (π₯ = πΉ β ran (π β β β¦ inf(((π₯ β (π[,)+β)) β© β*), β*, < )) = ran πΊ) |
10 | 9 | supeq1d 9303 | . 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 3459 | . 2 β’ (πΉ β π β πΉ β V) | |
12 | xrltso 12976 | . . . 4 β’ < Or β* | |
13 | 12 | supex 9320 | . . 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 6954 | 1 β’ (πΉ β π β (lim infβπΉ) = sup(ran πΊ, β*, < )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1540 β wcel 2105 Vcvv 3441 β© cin 3897 β¦ cmpt 5175 ran crn 5621 β cima 5623 βcfv 6479 (class class class)co 7337 supcsup 9297 infcinf 9298 βcr 10971 +βcpnf 11107 β*cxr 11109 < clt 11110 [,)cico 13182 lim infclsi 43636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-pre-lttri 11046 ax-pre-lttrn 11047 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-po 5532 df-so 5533 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-er 8569 df-en 8805 df-dom 8806 df-sdom 8807 df-sup 9299 df-inf 9300 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-liminf 43637 |
This theorem is referenced by: liminfcl 43648 liminfvald 43649 liminfval5 43650 liminfresxr 43652 liminfval2 43653 liminfvalxr 43668 |
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