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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > climreeq | Structured version Visualization version GIF version |
Description: If πΉ is a real function, then πΉ converges to π΄ with respect to the standard topology on the reals if and only if it converges to π΄ with respect to the standard topology on complex numbers. In the theorem, π is defined to be convergence w.r.t. the standard topology on the reals and then πΉπ π΄ represents the statement "πΉ converges to π΄, with respect to the standard topology on the reals". Notice that there is no need for the hypothesis that π΄ is a real number. (Contributed by Glauco Siliprandi, 2-Jul-2017.) |
Ref | Expression |
---|---|
climreeq.1 | β’ π = (βπ‘β(topGenβran (,))) |
climreeq.2 | β’ π = (β€β₯βπ) |
climreeq.3 | β’ (π β π β β€) |
climreeq.4 | β’ (π β πΉ:πβΆβ) |
Ref | Expression |
---|---|
climreeq | β’ (π β (πΉπ π΄ β πΉ β π΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climreeq.1 | . . 3 β’ π = (βπ‘β(topGenβran (,))) | |
2 | 1 | breqi 5154 | . 2 β’ (πΉπ π΄ β πΉ(βπ‘β(topGenβran (,)))π΄) |
3 | climreeq.3 | . . . 4 β’ (π β π β β€) | |
4 | climreeq.4 | . . . . 5 β’ (π β πΉ:πβΆβ) | |
5 | ax-resscn 11169 | . . . . . 6 β’ β β β | |
6 | 5 | a1i 11 | . . . . 5 β’ (π β β β β) |
7 | 4, 6 | fssd 6735 | . . . 4 β’ (π β πΉ:πβΆβ) |
8 | eqid 2732 | . . . . 5 β’ (TopOpenββfld) = (TopOpenββfld) | |
9 | climreeq.2 | . . . . 5 β’ π = (β€β₯βπ) | |
10 | 8, 9 | lmclimf 24828 | . . . 4 β’ ((π β β€ β§ πΉ:πβΆβ) β (πΉ(βπ‘β(TopOpenββfld))π΄ β πΉ β π΄)) |
11 | 3, 7, 10 | syl2anc 584 | . . 3 β’ (π β (πΉ(βπ‘β(TopOpenββfld))π΄ β πΉ β π΄)) |
12 | 8 | tgioo2 24326 | . . . . . 6 β’ (topGenβran (,)) = ((TopOpenββfld) βΎt β) |
13 | reex 11203 | . . . . . . 7 β’ β β V | |
14 | 13 | a1i 11 | . . . . . 6 β’ ((π β§ π΄ β β) β β β V) |
15 | 8 | cnfldtop 24307 | . . . . . . 7 β’ (TopOpenββfld) β Top |
16 | 15 | a1i 11 | . . . . . 6 β’ ((π β§ π΄ β β) β (TopOpenββfld) β Top) |
17 | simpr 485 | . . . . . 6 β’ ((π β§ π΄ β β) β π΄ β β) | |
18 | 3 | adantr 481 | . . . . . 6 β’ ((π β§ π΄ β β) β π β β€) |
19 | 4 | adantr 481 | . . . . . 6 β’ ((π β§ π΄ β β) β πΉ:πβΆβ) |
20 | 12, 9, 14, 16, 17, 18, 19 | lmss 22809 | . . . . 5 β’ ((π β§ π΄ β β) β (πΉ(βπ‘β(TopOpenββfld))π΄ β πΉ(βπ‘β(topGenβran (,)))π΄)) |
21 | 20 | pm5.32da 579 | . . . 4 β’ (π β ((π΄ β β β§ πΉ(βπ‘β(TopOpenββfld))π΄) β (π΄ β β β§ πΉ(βπ‘β(topGenβran (,)))π΄))) |
22 | simpr 485 | . . . . 5 β’ ((π΄ β β β§ πΉ(βπ‘β(TopOpenββfld))π΄) β πΉ(βπ‘β(TopOpenββfld))π΄) | |
23 | 3 | adantr 481 | . . . . . . . 8 β’ ((π β§ πΉ(βπ‘β(TopOpenββfld))π΄) β π β β€) |
24 | 11 | biimpa 477 | . . . . . . . 8 β’ ((π β§ πΉ(βπ‘β(TopOpenββfld))π΄) β πΉ β π΄) |
25 | 4 | ffvelcdmda 7086 | . . . . . . . . 9 β’ ((π β§ π β π) β (πΉβπ) β β) |
26 | 25 | adantlr 713 | . . . . . . . 8 β’ (((π β§ πΉ(βπ‘β(TopOpenββfld))π΄) β§ π β π) β (πΉβπ) β β) |
27 | 9, 23, 24, 26 | climrecl 15529 | . . . . . . 7 β’ ((π β§ πΉ(βπ‘β(TopOpenββfld))π΄) β π΄ β β) |
28 | 27 | ex 413 | . . . . . 6 β’ (π β (πΉ(βπ‘β(TopOpenββfld))π΄ β π΄ β β)) |
29 | 28 | ancrd 552 | . . . . 5 β’ (π β (πΉ(βπ‘β(TopOpenββfld))π΄ β (π΄ β β β§ πΉ(βπ‘β(TopOpenββfld))π΄))) |
30 | 22, 29 | impbid2 225 | . . . 4 β’ (π β ((π΄ β β β§ πΉ(βπ‘β(TopOpenββfld))π΄) β πΉ(βπ‘β(TopOpenββfld))π΄)) |
31 | simpr 485 | . . . . 5 β’ ((π΄ β β β§ πΉ(βπ‘β(topGenβran (,)))π΄) β πΉ(βπ‘β(topGenβran (,)))π΄) | |
32 | retopon 24287 | . . . . . . . . 9 β’ (topGenβran (,)) β (TopOnββ) | |
33 | 32 | a1i 11 | . . . . . . . 8 β’ ((π β§ πΉ(βπ‘β(topGenβran (,)))π΄) β (topGenβran (,)) β (TopOnββ)) |
34 | simpr 485 | . . . . . . . 8 β’ ((π β§ πΉ(βπ‘β(topGenβran (,)))π΄) β πΉ(βπ‘β(topGenβran (,)))π΄) | |
35 | lmcl 22808 | . . . . . . . 8 β’ (((topGenβran (,)) β (TopOnββ) β§ πΉ(βπ‘β(topGenβran (,)))π΄) β π΄ β β) | |
36 | 33, 34, 35 | syl2anc 584 | . . . . . . 7 β’ ((π β§ πΉ(βπ‘β(topGenβran (,)))π΄) β π΄ β β) |
37 | 36 | ex 413 | . . . . . 6 β’ (π β (πΉ(βπ‘β(topGenβran (,)))π΄ β π΄ β β)) |
38 | 37 | ancrd 552 | . . . . 5 β’ (π β (πΉ(βπ‘β(topGenβran (,)))π΄ β (π΄ β β β§ πΉ(βπ‘β(topGenβran (,)))π΄))) |
39 | 31, 38 | impbid2 225 | . . . 4 β’ (π β ((π΄ β β β§ πΉ(βπ‘β(topGenβran (,)))π΄) β πΉ(βπ‘β(topGenβran (,)))π΄)) |
40 | 21, 30, 39 | 3bitr3d 308 | . . 3 β’ (π β (πΉ(βπ‘β(TopOpenββfld))π΄ β πΉ(βπ‘β(topGenβran (,)))π΄)) |
41 | 11, 40 | bitr3d 280 | . 2 β’ (π β (πΉ β π΄ β πΉ(βπ‘β(topGenβran (,)))π΄)) |
42 | 2, 41 | bitr4id 289 | 1 β’ (π β (πΉπ π΄ β πΉ β π΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 396 = wceq 1541 β wcel 2106 Vcvv 3474 β wss 3948 class class class wbr 5148 ran crn 5677 βΆwf 6539 βcfv 6543 βcc 11110 βcr 11111 β€cz 12560 β€β₯cuz 12824 (,)cioo 13326 β cli 15430 TopOpenctopn 17369 topGenctg 17385 βfldccnfld 20950 Topctop 22402 TopOnctopon 22419 βπ‘clm 22737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fi 9408 df-sup 9439 df-inf 9440 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-z 12561 df-dec 12680 df-uz 12825 df-q 12935 df-rp 12977 df-xneg 13094 df-xadd 13095 df-xmul 13096 df-ioo 13330 df-fz 13487 df-fl 13759 df-seq 13969 df-exp 14030 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-clim 15434 df-rlim 15435 df-struct 17082 df-slot 17117 df-ndx 17129 df-base 17147 df-plusg 17212 df-mulr 17213 df-starv 17214 df-tset 17218 df-ple 17219 df-ds 17221 df-unif 17222 df-rest 17370 df-topn 17371 df-topgen 17391 df-psmet 20942 df-xmet 20943 df-met 20944 df-bl 20945 df-mopn 20946 df-cnfld 20951 df-top 22403 df-topon 22420 df-topsp 22442 df-bases 22456 df-lm 22740 df-xms 23833 df-ms 23834 |
This theorem is referenced by: xlimclim 44619 stirlingr 44885 |
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