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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > liminfvalxrmpt | Structured version Visualization version GIF version |
Description: Alternate definition of lim inf when πΉ is an extended real-valued function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
liminfvalxrmpt.1 | β’ β²π₯π |
liminfvalxrmpt.2 | β’ (π β π΄ β π) |
liminfvalxrmpt.3 | β’ ((π β§ π₯ β π΄) β π΅ β β*) |
Ref | Expression |
---|---|
liminfvalxrmpt | β’ (π β (lim infβ(π₯ β π΄ β¦ π΅)) = -π(lim supβ(π₯ β π΄ β¦ -ππ΅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfmpt1 5214 | . . 3 β’ β²π₯(π₯ β π΄ β¦ π΅) | |
2 | liminfvalxrmpt.2 | . . 3 β’ (π β π΄ β π) | |
3 | liminfvalxrmpt.1 | . . . 4 β’ β²π₯π | |
4 | liminfvalxrmpt.3 | . . . 4 β’ ((π β§ π₯ β π΄) β π΅ β β*) | |
5 | 3, 4 | fmptd2f 43547 | . . 3 β’ (π β (π₯ β π΄ β¦ π΅):π΄βΆβ*) |
6 | 1, 2, 5 | liminfvalxr 44110 | . 2 β’ (π β (lim infβ(π₯ β π΄ β¦ π΅)) = -π(lim supβ(π₯ β π΄ β¦ -π((π₯ β π΄ β¦ π΅)βπ₯)))) |
7 | eqidd 2734 | . . . . . . 7 β’ (π β (π₯ β π΄ β¦ π΅) = (π₯ β π΄ β¦ π΅)) | |
8 | 7, 4 | fvmpt2d 6962 | . . . . . 6 β’ ((π β§ π₯ β π΄) β ((π₯ β π΄ β¦ π΅)βπ₯) = π΅) |
9 | 8 | xnegeqd 43758 | . . . . 5 β’ ((π β§ π₯ β π΄) β -π((π₯ β π΄ β¦ π΅)βπ₯) = -ππ΅) |
10 | 3, 9 | mpteq2da 5204 | . . . 4 β’ (π β (π₯ β π΄ β¦ -π((π₯ β π΄ β¦ π΅)βπ₯)) = (π₯ β π΄ β¦ -ππ΅)) |
11 | 10 | fveq2d 6847 | . . 3 β’ (π β (lim supβ(π₯ β π΄ β¦ -π((π₯ β π΄ β¦ π΅)βπ₯))) = (lim supβ(π₯ β π΄ β¦ -ππ΅))) |
12 | 11 | xnegeqd 43758 | . 2 β’ (π β -π(lim supβ(π₯ β π΄ β¦ -π((π₯ β π΄ β¦ π΅)βπ₯))) = -π(lim supβ(π₯ β π΄ β¦ -ππ΅))) |
13 | 6, 12 | eqtrd 2773 | 1 β’ (π β (lim infβ(π₯ β π΄ β¦ π΅)) = -π(lim supβ(π₯ β π΄ β¦ -ππ΅))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β²wnf 1786 β wcel 2107 β¦ cmpt 5189 βcfv 6497 β*cxr 11193 -πcxne 13035 lim supclsp 15358 lim infclsi 44078 |
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-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-sup 9383 df-inf 9384 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-xneg 13038 df-limsup 15359 df-liminf 44079 |
This theorem is referenced by: liminfval4 44116 liminfval3 44117 |
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