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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > liminfgelimsup | Structured version Visualization version GIF version |
Description: The inferior limit is greater than or equal to the superior limit if and only if they are equal. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
liminfgelimsup.1 | β’ (π β πΉ β π) |
liminfgelimsup.2 | β’ (π β βπ β β βπ β (π[,)+β)((πΉ β (π[,)+β)) β© β*) β β ) |
Ref | Expression |
---|---|
liminfgelimsup | β’ (π β ((lim supβπΉ) β€ (lim infβπΉ) β (lim infβπΉ) = (lim supβπΉ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | liminfgelimsup.1 | . . . . 5 β’ (π β πΉ β π) | |
2 | 1 | liminfcld 45081 | . . . 4 β’ (π β (lim infβπΉ) β β*) |
3 | 2 | adantr 480 | . . 3 β’ ((π β§ (lim supβπΉ) β€ (lim infβπΉ)) β (lim infβπΉ) β β*) |
4 | 1 | limsupcld 45001 | . . . 4 β’ (π β (lim supβπΉ) β β*) |
5 | 4 | adantr 480 | . . 3 β’ ((π β§ (lim supβπΉ) β€ (lim infβπΉ)) β (lim supβπΉ) β β*) |
6 | liminfgelimsup.2 | . . . . 5 β’ (π β βπ β β βπ β (π[,)+β)((πΉ β (π[,)+β)) β© β*) β β ) | |
7 | 1, 6 | liminflelimsup 45087 | . . . 4 β’ (π β (lim infβπΉ) β€ (lim supβπΉ)) |
8 | 7 | adantr 480 | . . 3 β’ ((π β§ (lim supβπΉ) β€ (lim infβπΉ)) β (lim infβπΉ) β€ (lim supβπΉ)) |
9 | simpr 484 | . . 3 β’ ((π β§ (lim supβπΉ) β€ (lim infβπΉ)) β (lim supβπΉ) β€ (lim infβπΉ)) | |
10 | 3, 5, 8, 9 | xrletrid 13158 | . 2 β’ ((π β§ (lim supβπΉ) β€ (lim infβπΉ)) β (lim infβπΉ) = (lim supβπΉ)) |
11 | 4 | adantr 480 | . . 3 β’ ((π β§ (lim infβπΉ) = (lim supβπΉ)) β (lim supβπΉ) β β*) |
12 | id 22 | . . . . 5 β’ ((lim infβπΉ) = (lim supβπΉ) β (lim infβπΉ) = (lim supβπΉ)) | |
13 | 12 | eqcomd 2733 | . . . 4 β’ ((lim infβπΉ) = (lim supβπΉ) β (lim supβπΉ) = (lim infβπΉ)) |
14 | 13 | adantl 481 | . . 3 β’ ((π β§ (lim infβπΉ) = (lim supβπΉ)) β (lim supβπΉ) = (lim infβπΉ)) |
15 | 11, 14 | xreqled 44635 | . 2 β’ ((π β§ (lim infβπΉ) = (lim supβπΉ)) β (lim supβπΉ) β€ (lim infβπΉ)) |
16 | 10, 15 | impbida 800 | 1 β’ (π β ((lim supβπΉ) β€ (lim infβπΉ) β (lim infβπΉ) = (lim supβπΉ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1534 β wcel 2099 β wne 2935 βwral 3056 βwrex 3065 β© cin 3943 β c0 4318 class class class wbr 5142 β cima 5675 βcfv 6542 (class class class)co 7414 βcr 11129 +βcpnf 11267 β*cxr 11269 β€ cle 11271 [,)cico 13350 lim supclsp 15438 lim infclsi 45062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-sup 9457 df-inf 9458 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-ico 13354 df-limsup 15439 df-liminf 45063 |
This theorem is referenced by: (None) |
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