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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 22572 | . . . 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 22667 | . 2 β’ (π β (πΉ(βπ‘β(ordTopβ β€ ))π΄ β (πΉ βΎ (β€β₯βπ))(βπ‘β(ordTopβ β€ ))π΄)) |
6 | df-xlim 44146 | . . 3 β’ ~~>* = (βπ‘β(ordTopβ β€ )) | |
7 | 6 | breqi 5112 | . 2 β’ (πΉ~~>*π΄ β πΉ(βπ‘β(ordTopβ β€ ))π΄) |
8 | 6 | breqi 5112 | . 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 2107 class class class wbr 5106 βΎ cres 5636 βcfv 6497 (class class class)co 7358 βpm cpm 8769 βcc 11054 β*cxr 11193 β€ cle 11195 β€cz 12504 β€β₯cuz 12768 ordTopcordt 17386 TopOnctopon 22275 βπ‘clm 22593 ~~>*clsxlim 44145 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-pre-lttri 11130 ax-pre-lttrn 11131 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-1o 8413 df-er 8651 df-pm 8771 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fi 9352 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-neg 11393 df-z 12505 df-uz 12769 df-topgen 17330 df-ordt 17388 df-ps 18460 df-tsr 18461 df-top 22259 df-topon 22276 df-bases 22312 df-lm 22596 df-xlim 44146 |
This theorem is referenced by: xlimconst2 44162 xlimclim2lem 44166 climxlim2 44173 xlimresdm 44186 xlimliminflimsup 44189 |
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