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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 44541 | . . 3 β’ (π β πΉ = (π Γ {π΄})) |
7 | letopon 23064 | . . . 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 23120 | . . . 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 5163 | . 2 β’ (π β πΉ(βπ‘β(ordTopβ β€ ))π΄) |
13 | df-xlim 45107 | . . 3 β’ ~~>* = (βπ‘β(ordTopβ β€ )) | |
14 | 13 | breqi 5147 | . 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 395 = wceq 1533 β²wnf 1777 β wcel 2098 β²wnfc 2877 {csn 4623 class class class wbr 5141 Γ cxp 5667 Fn wfn 6532 βcfv 6537 β*cxr 11251 β€ cle 11253 β€cz 12562 β€β₯cuz 12826 ordTopcordt 17454 TopOnctopon 22767 βπ‘clm 23085 ~~>*clsxlim 45106 |
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-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-pre-lttri 11186 ax-pre-lttrn 11187 |
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-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-int 4944 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-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-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-1o 8467 df-er 8705 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fi 9408 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-neg 11451 df-z 12563 df-uz 12827 df-topgen 17398 df-ordt 17456 df-ps 18531 df-tsr 18532 df-top 22751 df-topon 22768 df-bases 22804 df-lm 23088 df-xlim 45107 |
This theorem is referenced by: xlimconst2 45123 |
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