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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 23127 | . . . 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 23222 | . 2 β’ (π β (πΉ(βπ‘β(ordTopβ β€ ))π΄ β (πΉ βΎ (β€β₯βπ))(βπ‘β(ordTopβ β€ ))π΄)) |
6 | df-xlim 45209 | . . 3 β’ ~~>* = (βπ‘β(ordTopβ β€ )) | |
7 | 6 | breqi 5156 | . 2 β’ (πΉ~~>*π΄ β πΉ(βπ‘β(ordTopβ β€ ))π΄) |
8 | 6 | breqi 5156 | . 2 β’ ((πΉ βΎ (β€β₯βπ))~~>*π΄ β (πΉ βΎ (β€β₯βπ))(βπ‘β(ordTopβ β€ ))π΄) |
9 | 5, 7, 8 | 3bitr4g 313 | 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 5150 βΎ cres 5682 βcfv 6551 (class class class)co 7424 βpm cpm 8850 βcc 11142 β*cxr 11283 β€ cle 11285 β€cz 12594 β€β₯cuz 12858 ordTopcordt 17486 TopOnctopon 22830 βπ‘clm 23148 ~~>*clsxlim 45208 |
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 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-pre-lttri 11218 ax-pre-lttrn 11219 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-1o 8491 df-er 8729 df-pm 8852 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-fi 9440 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-neg 11483 df-z 12595 df-uz 12859 df-topgen 17430 df-ordt 17488 df-ps 18563 df-tsr 18564 df-top 22814 df-topon 22831 df-bases 22867 df-lm 23151 df-xlim 45209 |
This theorem is referenced by: xlimconst2 45225 xlimclim2lem 45229 climxlim2 45236 xlimresdm 45249 xlimliminflimsup 45252 |
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