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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xlimconst | Structured version Visualization version GIF version |
Description: A constant sequence converges to its value, w.r.t. the standard topology on the extended reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
Ref | Expression |
---|---|
xlimconst.p | β’ β²ππ |
xlimconst.k | β’ β²ππΉ |
xlimconst.m | β’ (π β π β β€) |
xlimconst.z | β’ π = (β€β₯βπ) |
xlimconst.f | β’ (π β πΉ Fn π) |
xlimconst.a | β’ (π β π΄ β β*) |
xlimconst.e | β’ ((π β§ π β π) β (πΉβπ) = π΄) |
Ref | Expression |
---|---|
xlimconst | β’ (π β πΉ~~>*π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xlimconst.p | . . . 4 β’ β²ππ | |
2 | xlimconst.k | . . . 4 β’ β²ππΉ | |
3 | xlimconst.f | . . . 4 β’ (π β πΉ Fn π) | |
4 | xlimconst.a | . . . 4 β’ (π β π΄ β β*) | |
5 | xlimconst.e | . . . 4 β’ ((π β§ π β π) β (πΉβπ) = π΄) | |
6 | 1, 2, 3, 4, 5 | fconst7 44704 | . . 3 β’ (π β πΉ = (π Γ {π΄})) |
7 | letopon 23127 | . . . 4 β’ (ordTopβ β€ ) β (TopOnββ*) | |
8 | xlimconst.m | . . . 4 β’ (π β π β β€) | |
9 | xlimconst.z | . . . . 5 β’ π = (β€β₯βπ) | |
10 | 9 | lmconst 23183 | . . . 4 β’ (((ordTopβ β€ ) β (TopOnββ*) β§ π΄ β β* β§ π β β€) β (π Γ {π΄})(βπ‘β(ordTopβ β€ ))π΄) |
11 | 7, 4, 8, 10 | mp3an2i 1462 | . . 3 β’ (π β (π Γ {π΄})(βπ‘β(ordTopβ β€ ))π΄) |
12 | 6, 11 | eqbrtrd 5165 | . 2 β’ (π β πΉ(βπ‘β(ordTopβ β€ ))π΄) |
13 | df-xlim 45270 | . . 3 β’ ~~>* = (βπ‘β(ordTopβ β€ )) | |
14 | 13 | breqi 5149 | . 2 β’ (πΉ~~>*π΄ β πΉ(βπ‘β(ordTopβ β€ ))π΄) |
15 | 12, 14 | sylibr 233 | 1 β’ (π β πΉ~~>*π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β²wnf 1777 β wcel 2098 β²wnfc 2875 {csn 4624 class class class wbr 5143 Γ cxp 5670 Fn wfn 6538 βcfv 6543 β*cxr 11277 β€ cle 11279 β€cz 12588 β€β₯cuz 12852 ordTopcordt 17480 TopOnctopon 22830 βπ‘clm 23148 ~~>*clsxlim 45269 |
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 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-pre-lttri 11212 ax-pre-lttrn 11213 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 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-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-1o 8485 df-er 8723 df-pm 8846 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fi 9434 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-neg 11477 df-z 12589 df-uz 12853 df-topgen 17424 df-ordt 17482 df-ps 18557 df-tsr 18558 df-top 22814 df-topon 22831 df-bases 22867 df-lm 23151 df-xlim 45270 |
This theorem is referenced by: xlimconst2 45286 |
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