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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xlimconst2 | Structured version Visualization version GIF version |
Description: A sequence that eventually becomes constant, converges to its constant value (w.r.t. the standard topology on the extended reals). (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
Ref | Expression |
---|---|
xlimconst2.p | β’ β²ππ |
xlimconst2.k | β’ β²ππΉ |
xlimconst2.z | β’ π = (β€β₯βπ) |
xlimconst2.f | β’ (π β πΉ:πβΆβ*) |
xlimconst2.n | β’ (π β π β π) |
xlimconst2.a | β’ (π β π΄ β β*) |
xlimconst2.e | β’ ((π β§ π β (β€β₯βπ)) β (πΉβπ) = π΄) |
Ref | Expression |
---|---|
xlimconst2 | β’ (π β πΉ~~>*π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xlimconst2.p | . . 3 β’ β²ππ | |
2 | xlimconst2.k | . . . 4 β’ β²ππΉ | |
3 | nfcv 2897 | . . . 4 β’ β²π(β€β₯βπ) | |
4 | 2, 3 | nfres 5977 | . . 3 β’ β²π(πΉ βΎ (β€β₯βπ)) |
5 | xlimconst2.z | . . . 4 β’ π = (β€β₯βπ) | |
6 | xlimconst2.n | . . . 4 β’ (π β π β π) | |
7 | 5, 6 | eluzelz2d 44695 | . . 3 β’ (π β π β β€) |
8 | eqid 2726 | . . 3 β’ (β€β₯βπ) = (β€β₯βπ) | |
9 | xlimconst2.f | . . . . 5 β’ (π β πΉ:πβΆβ*) | |
10 | 9 | ffnd 6712 | . . . 4 β’ (π β πΉ Fn π) |
11 | 5, 6 | uzssd2 44699 | . . . 4 β’ (π β (β€β₯βπ) β π) |
12 | 10, 11 | fnssresd 6668 | . . 3 β’ (π β (πΉ βΎ (β€β₯βπ)) Fn (β€β₯βπ)) |
13 | xlimconst2.a | . . 3 β’ (π β π΄ β β*) | |
14 | fvres 6904 | . . . . 5 β’ (π β (β€β₯βπ) β ((πΉ βΎ (β€β₯βπ))βπ) = (πΉβπ)) | |
15 | 14 | adantl 481 | . . . 4 β’ ((π β§ π β (β€β₯βπ)) β ((πΉ βΎ (β€β₯βπ))βπ) = (πΉβπ)) |
16 | xlimconst2.e | . . . 4 β’ ((π β§ π β (β€β₯βπ)) β (πΉβπ) = π΄) | |
17 | 15, 16 | eqtrd 2766 | . . 3 β’ ((π β§ π β (β€β₯βπ)) β ((πΉ βΎ (β€β₯βπ))βπ) = π΄) |
18 | 1, 4, 7, 8, 12, 13, 17 | xlimconst 45113 | . 2 β’ (π β (πΉ βΎ (β€β₯βπ))~~>*π΄) |
19 | 5, 9 | fuzxrpmcn 45116 | . . 3 β’ (π β πΉ β (β* βpm β)) |
20 | 19, 7 | xlimres 45109 | . 2 β’ (π β (πΉ~~>*π΄ β (πΉ βΎ (β€β₯βπ))~~>*π΄)) |
21 | 18, 20 | mpbird 257 | 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 class class class wbr 5141 βΎ cres 5671 βΆwf 6533 βcfv 6537 β*cxr 11251 β€β₯cuz 12826 ~~>*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: climxlim2lem 45133 |
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