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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > limsupval4 | Structured version Visualization version GIF version |
Description: Alternate definition of lim inf when the given a function is eventually extended real-valued. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
limsupval4.x | β’ β²π₯π |
limsupval4.a | β’ (π β π΄ β π) |
limsupval4.m | β’ (π β π β β) |
limsupval4.b | β’ ((π β§ π₯ β (π΄ β© (π[,)+β))) β π΅ β β*) |
Ref | Expression |
---|---|
limsupval4 | β’ (π β (lim supβ(π₯ β π΄ β¦ π΅)) = -π(lim infβ(π₯ β π΄ β¦ -ππ΅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 7438 | . . . . . . . 8 β’ (π[,)+β) β V | |
2 | 1 | inex2 5311 | . . . . . . 7 β’ (π΄ β© (π[,)+β)) β V |
3 | 2 | mptex 7220 | . . . . . 6 β’ (π₯ β (π΄ β© (π[,)+β)) β¦ π΅) β V |
4 | limsupcl 15423 | . . . . . 6 β’ ((π₯ β (π΄ β© (π[,)+β)) β¦ π΅) β V β (lim supβ(π₯ β (π΄ β© (π[,)+β)) β¦ π΅)) β β*) | |
5 | 3, 4 | ax-mp 5 | . . . . 5 β’ (lim supβ(π₯ β (π΄ β© (π[,)+β)) β¦ π΅)) β β* |
6 | 5 | a1i 11 | . . . 4 β’ (π β (lim supβ(π₯ β (π΄ β© (π[,)+β)) β¦ π΅)) β β*) |
7 | 6 | xnegnegd 44729 | . . 3 β’ (π β -π-π(lim supβ(π₯ β (π΄ β© (π[,)+β)) β¦ π΅)) = (lim supβ(π₯ β (π΄ β© (π[,)+β)) β¦ π΅))) |
8 | 7 | eqcomd 2732 | . 2 β’ (π β (lim supβ(π₯ β (π΄ β© (π[,)+β)) β¦ π΅)) = -π-π(lim supβ(π₯ β (π΄ β© (π[,)+β)) β¦ π΅))) |
9 | limsupval4.a | . . 3 β’ (π β π΄ β π) | |
10 | limsupval4.m | . . 3 β’ (π β π β β) | |
11 | eqid 2726 | . . 3 β’ (π[,)+β) = (π[,)+β) | |
12 | 9, 10, 11 | limsupresicompt 45049 | . 2 β’ (π β (lim supβ(π₯ β π΄ β¦ π΅)) = (lim supβ(π₯ β (π΄ β© (π[,)+β)) β¦ π΅))) |
13 | limsupval4.x | . . . . 5 β’ β²π₯π | |
14 | limsupval4.b | . . . . . 6 β’ ((π β§ π₯ β (π΄ β© (π[,)+β))) β π΅ β β*) | |
15 | 14 | xnegcld 13285 | . . . . 5 β’ ((π β§ π₯ β (π΄ β© (π[,)+β))) β -ππ΅ β β*) |
16 | 13, 9, 10, 15 | liminfval3 45083 | . . . 4 β’ (π β (lim infβ(π₯ β π΄ β¦ -ππ΅)) = -π(lim supβ(π₯ β π΄ β¦ -π-ππ΅))) |
17 | 9, 10, 11 | limsupresicompt 45049 | . . . . . 6 β’ (π β (lim supβ(π₯ β π΄ β¦ -π-ππ΅)) = (lim supβ(π₯ β (π΄ β© (π[,)+β)) β¦ -π-ππ΅))) |
18 | 14 | xnegnegd 44729 | . . . . . . . 8 β’ ((π β§ π₯ β (π΄ β© (π[,)+β))) β -π-ππ΅ = π΅) |
19 | 13, 18 | mpteq2da 5239 | . . . . . . 7 β’ (π β (π₯ β (π΄ β© (π[,)+β)) β¦ -π-ππ΅) = (π₯ β (π΄ β© (π[,)+β)) β¦ π΅)) |
20 | 19 | fveq2d 6889 | . . . . . 6 β’ (π β (lim supβ(π₯ β (π΄ β© (π[,)+β)) β¦ -π-ππ΅)) = (lim supβ(π₯ β (π΄ β© (π[,)+β)) β¦ π΅))) |
21 | 17, 20 | eqtrd 2766 | . . . . 5 β’ (π β (lim supβ(π₯ β π΄ β¦ -π-ππ΅)) = (lim supβ(π₯ β (π΄ β© (π[,)+β)) β¦ π΅))) |
22 | 21 | xnegeqd 44724 | . . . 4 β’ (π β -π(lim supβ(π₯ β π΄ β¦ -π-ππ΅)) = -π(lim supβ(π₯ β (π΄ β© (π[,)+β)) β¦ π΅))) |
23 | 16, 22 | eqtrd 2766 | . . 3 β’ (π β (lim infβ(π₯ β π΄ β¦ -ππ΅)) = -π(lim supβ(π₯ β (π΄ β© (π[,)+β)) β¦ π΅))) |
24 | 23 | xnegeqd 44724 | . 2 β’ (π β -π(lim infβ(π₯ β π΄ β¦ -ππ΅)) = -π-π(lim supβ(π₯ β (π΄ β© (π[,)+β)) β¦ π΅))) |
25 | 8, 12, 24 | 3eqtr4d 2776 | 1 β’ (π β (lim supβ(π₯ β π΄ β¦ π΅)) = -π(lim infβ(π₯ β π΄ β¦ -ππ΅))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β²wnf 1777 β wcel 2098 Vcvv 3468 β© cin 3942 β¦ cmpt 5224 βcfv 6537 (class class class)co 7405 βcr 11111 +βcpnf 11249 β*cxr 11251 -πcxne 13095 [,)cico 13332 lim supclsp 15420 lim infclsi 45044 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-q 12937 df-xneg 13098 df-ico 13336 df-limsup 15421 df-liminf 45045 |
This theorem is referenced by: limsupvaluz3 45091 |
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