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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xlimres | Structured version Visualization version GIF version |
Description: A function converges iff its restriction to an upper integers set converges. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
Ref | Expression |
---|---|
xlimres.1 | β’ (π β πΉ β (β* βpm β)) |
xlimres.2 | β’ (π β π β β€) |
Ref | Expression |
---|---|
xlimres | β’ (π β (πΉ~~>*π΄ β (πΉ βΎ (β€β₯βπ))~~>*π΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | letopon 23060 | . . . 4 β’ (ordTopβ β€ ) β (TopOnββ*) | |
2 | 1 | a1i 11 | . . 3 β’ (π β (ordTopβ β€ ) β (TopOnββ*)) |
3 | xlimres.1 | . . 3 β’ (π β πΉ β (β* βpm β)) | |
4 | xlimres.2 | . . 3 β’ (π β π β β€) | |
5 | 2, 3, 4 | lmres 23155 | . 2 β’ (π β (πΉ(βπ‘β(ordTopβ β€ ))π΄ β (πΉ βΎ (β€β₯βπ))(βπ‘β(ordTopβ β€ ))π΄)) |
6 | df-xlim 45088 | . . 3 β’ ~~>* = (βπ‘β(ordTopβ β€ )) | |
7 | 6 | breqi 5147 | . 2 β’ (πΉ~~>*π΄ β πΉ(βπ‘β(ordTopβ β€ ))π΄) |
8 | 6 | breqi 5147 | . 2 β’ ((πΉ βΎ (β€β₯βπ))~~>*π΄ β (πΉ βΎ (β€β₯βπ))(βπ‘β(ordTopβ β€ ))π΄) |
9 | 5, 7, 8 | 3bitr4g 314 | 1 β’ (π β (πΉ~~>*π΄ β (πΉ βΎ (β€β₯βπ))~~>*π΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β wcel 2098 class class class wbr 5141 βΎ cres 5671 βcfv 6536 (class class class)co 7404 βpm cpm 8820 βcc 11107 β*cxr 11248 β€ cle 11250 β€cz 12559 β€β₯cuz 12823 ordTopcordt 17452 TopOnctopon 22763 βπ‘clm 23081 ~~>*clsxlim 45087 |
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 7721 ax-cnex 11165 ax-resscn 11166 ax-pre-lttri 11183 ax-pre-lttrn 11184 |
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 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-1o 8464 df-er 8702 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fi 9405 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-neg 11448 df-z 12560 df-uz 12824 df-topgen 17396 df-ordt 17454 df-ps 18529 df-tsr 18530 df-top 22747 df-topon 22764 df-bases 22800 df-lm 23084 df-xlim 45088 |
This theorem is referenced by: xlimconst2 45104 xlimclim2lem 45108 climxlim2 45115 xlimresdm 45128 xlimliminflimsup 45131 |
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