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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 5112 | . 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 11109 | . . . . . 6 β’ β β β | |
6 | 5 | a1i 11 | . . . . 5 β’ (π β β β β) |
7 | 4, 6 | fssd 6687 | . . . 4 β’ (π β πΉ:πβΆβ) |
8 | eqid 2737 | . . . . 5 β’ (TopOpenββfld) = (TopOpenββfld) | |
9 | climreeq.2 | . . . . 5 β’ π = (β€β₯βπ) | |
10 | 8, 9 | lmclimf 24671 | . . . 4 β’ ((π β β€ β§ πΉ:πβΆβ) β (πΉ(βπ‘β(TopOpenββfld))π΄ β πΉ β π΄)) |
11 | 3, 7, 10 | syl2anc 585 | . . 3 β’ (π β (πΉ(βπ‘β(TopOpenββfld))π΄ β πΉ β π΄)) |
12 | 8 | tgioo2 24169 | . . . . . 6 β’ (topGenβran (,)) = ((TopOpenββfld) βΎt β) |
13 | reex 11143 | . . . . . . 7 β’ β β V | |
14 | 13 | a1i 11 | . . . . . 6 β’ ((π β§ π΄ β β) β β β V) |
15 | 8 | cnfldtop 24150 | . . . . . . 7 β’ (TopOpenββfld) β Top |
16 | 15 | a1i 11 | . . . . . 6 β’ ((π β§ π΄ β β) β (TopOpenββfld) β Top) |
17 | simpr 486 | . . . . . 6 β’ ((π β§ π΄ β β) β π΄ β β) | |
18 | 3 | adantr 482 | . . . . . 6 β’ ((π β§ π΄ β β) β π β β€) |
19 | 4 | adantr 482 | . . . . . 6 β’ ((π β§ π΄ β β) β πΉ:πβΆβ) |
20 | 12, 9, 14, 16, 17, 18, 19 | lmss 22652 | . . . . 5 β’ ((π β§ π΄ β β) β (πΉ(βπ‘β(TopOpenββfld))π΄ β πΉ(βπ‘β(topGenβran (,)))π΄)) |
21 | 20 | pm5.32da 580 | . . . 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 486 | . . . . 5 β’ ((π΄ β β β§ πΉ(βπ‘β(TopOpenββfld))π΄) β πΉ(βπ‘β(TopOpenββfld))π΄) | |
23 | 3 | adantr 482 | . . . . . . . 8 β’ ((π β§ πΉ(βπ‘β(TopOpenββfld))π΄) β π β β€) |
24 | 11 | biimpa 478 | . . . . . . . 8 β’ ((π β§ πΉ(βπ‘β(TopOpenββfld))π΄) β πΉ β π΄) |
25 | 4 | ffvelcdmda 7036 | . . . . . . . . 9 β’ ((π β§ π β π) β (πΉβπ) β β) |
26 | 25 | adantlr 714 | . . . . . . . 8 β’ (((π β§ πΉ(βπ‘β(TopOpenββfld))π΄) β§ π β π) β (πΉβπ) β β) |
27 | 9, 23, 24, 26 | climrecl 15466 | . . . . . . 7 β’ ((π β§ πΉ(βπ‘β(TopOpenββfld))π΄) β π΄ β β) |
28 | 27 | ex 414 | . . . . . 6 β’ (π β (πΉ(βπ‘β(TopOpenββfld))π΄ β π΄ β β)) |
29 | 28 | ancrd 553 | . . . . 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 486 | . . . . 5 β’ ((π΄ β β β§ πΉ(βπ‘β(topGenβran (,)))π΄) β πΉ(βπ‘β(topGenβran (,)))π΄) | |
32 | retopon 24130 | . . . . . . . . 9 β’ (topGenβran (,)) β (TopOnββ) | |
33 | 32 | a1i 11 | . . . . . . . 8 β’ ((π β§ πΉ(βπ‘β(topGenβran (,)))π΄) β (topGenβran (,)) β (TopOnββ)) |
34 | simpr 486 | . . . . . . . 8 β’ ((π β§ πΉ(βπ‘β(topGenβran (,)))π΄) β πΉ(βπ‘β(topGenβran (,)))π΄) | |
35 | lmcl 22651 | . . . . . . . 8 β’ (((topGenβran (,)) β (TopOnββ) β§ πΉ(βπ‘β(topGenβran (,)))π΄) β π΄ β β) | |
36 | 33, 34, 35 | syl2anc 585 | . . . . . . 7 β’ ((π β§ πΉ(βπ‘β(topGenβran (,)))π΄) β π΄ β β) |
37 | 36 | ex 414 | . . . . . 6 β’ (π β (πΉ(βπ‘β(topGenβran (,)))π΄ β π΄ β β)) |
38 | 37 | ancrd 553 | . . . . 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 309 | . . 3 β’ (π β (πΉ(βπ‘β(TopOpenββfld))π΄ β πΉ(βπ‘β(topGenβran (,)))π΄)) |
41 | 11, 40 | bitr3d 281 | . 2 β’ (π β (πΉ β π΄ β πΉ(βπ‘β(topGenβran (,)))π΄)) |
42 | 2, 41 | bitr4id 290 | 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 397 = wceq 1542 β wcel 2107 Vcvv 3446 β wss 3911 class class class wbr 5106 ran crn 5635 βΆwf 6493 βcfv 6497 βcc 11050 βcr 11051 β€cz 12500 β€β₯cuz 12764 (,)cioo 13265 β cli 15367 TopOpenctopn 17304 topGenctg 17320 βfldccnfld 20799 Topctop 22245 TopOnctopon 22262 βπ‘clm 22580 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 ax-pre-sup 11130 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 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-tp 4592 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-pred 6254 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-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8649 df-map 8768 df-pm 8769 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fi 9348 df-sup 9379 df-inf 9380 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-div 11814 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-5 12220 df-6 12221 df-7 12222 df-8 12223 df-9 12224 df-n0 12415 df-z 12501 df-dec 12620 df-uz 12765 df-q 12875 df-rp 12917 df-xneg 13034 df-xadd 13035 df-xmul 13036 df-ioo 13269 df-fz 13426 df-fl 13698 df-seq 13908 df-exp 13969 df-cj 14985 df-re 14986 df-im 14987 df-sqrt 15121 df-abs 15122 df-clim 15371 df-rlim 15372 df-struct 17020 df-slot 17055 df-ndx 17067 df-base 17085 df-plusg 17147 df-mulr 17148 df-starv 17149 df-tset 17153 df-ple 17154 df-ds 17156 df-unif 17157 df-rest 17305 df-topn 17306 df-topgen 17326 df-psmet 20791 df-xmet 20792 df-met 20793 df-bl 20794 df-mopn 20795 df-cnfld 20800 df-top 22246 df-topon 22263 df-topsp 22285 df-bases 22299 df-lm 22583 df-xms 23676 df-ms 23677 |
This theorem is referenced by: xlimclim 44072 stirlingr 44338 |
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