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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 7391 | . . . . . . . 8 β’ (π[,)+β) β V | |
2 | 1 | inex2 5276 | . . . . . . 7 β’ (π΄ β© (π[,)+β)) β V |
3 | 2 | mptex 7174 | . . . . . 6 β’ (π₯ β (π΄ β© (π[,)+β)) β¦ π΅) β V |
4 | limsupcl 15361 | . . . . . 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 43763 | . . 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 2739 | . 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 2733 | . . 3 β’ (π[,)+β) = (π[,)+β) | |
12 | 9, 10, 11 | limsupresicompt 44083 | . 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 13225 | . . . . 5 β’ ((π β§ π₯ β (π΄ β© (π[,)+β))) β -ππ΅ β β*) |
16 | 13, 9, 10, 15 | liminfval3 44117 | . . . 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 44083 | . . . . . 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 43763 | . . . . . . . 8 β’ ((π β§ π₯ β (π΄ β© (π[,)+β))) β -π-ππ΅ = π΅) |
19 | 13, 18 | mpteq2da 5204 | . . . . . . 7 β’ (π β (π₯ β (π΄ β© (π[,)+β)) β¦ -π-ππ΅) = (π₯ β (π΄ β© (π[,)+β)) β¦ π΅)) |
20 | 19 | fveq2d 6847 | . . . . . 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 2773 | . . . . 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 43758 | . . . 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 2773 | . . 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 43758 | . 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 2783 | 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 397 = wceq 1542 β²wnf 1786 β wcel 2107 Vcvv 3444 β© cin 3910 β¦ cmpt 5189 βcfv 6497 (class class class)co 7358 βcr 11055 +βcpnf 11191 β*cxr 11193 -πcxne 13035 [,)cico 13272 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-pss 3930 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-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 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-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 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-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 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-div 11818 df-nn 12159 df-n0 12419 df-z 12505 df-uz 12769 df-q 12879 df-xneg 13038 df-ico 13276 df-limsup 15359 df-liminf 44079 |
This theorem is referenced by: limsupvaluz3 44125 |
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