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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ellimciota | Structured version Visualization version GIF version |
Description: An explicit value for the limit, when the limit exists at a limit point. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
ellimciota.f | β’ (π β πΉ:π΄βΆβ) |
ellimciota.a | β’ (π β π΄ β β) |
ellimciota.b | β’ (π β π΅ β ((limPtβπΎ)βπ΄)) |
ellimciota.4 | β’ (π β (πΉ limβ π΅) β β ) |
ellimciota.k | β’ πΎ = (TopOpenββfld) |
Ref | Expression |
---|---|
ellimciota | β’ (π β (β©π₯π₯ β (πΉ limβ π΅)) β (πΉ limβ π΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2821 | . . 3 β’ (π₯ = π¦ β (π₯ β (πΉ limβ π΅) β π¦ β (πΉ limβ π΅))) | |
2 | 1 | cbviotavw 6500 | . 2 β’ (β©π₯π₯ β (πΉ limβ π΅)) = (β©π¦π¦ β (πΉ limβ π΅)) |
3 | iotaex 6513 | . . . 4 β’ (β©π¦π¦ β (πΉ limβ π΅)) β V | |
4 | ellimciota.4 | . . . . . 6 β’ (π β (πΉ limβ π΅) β β ) | |
5 | n0 4345 | . . . . . 6 β’ ((πΉ limβ π΅) β β β βπ₯ π₯ β (πΉ limβ π΅)) | |
6 | 4, 5 | sylib 217 | . . . . 5 β’ (π β βπ₯ π₯ β (πΉ limβ π΅)) |
7 | ellimciota.f | . . . . . 6 β’ (π β πΉ:π΄βΆβ) | |
8 | ellimciota.a | . . . . . 6 β’ (π β π΄ β β) | |
9 | ellimciota.b | . . . . . 6 β’ (π β π΅ β ((limPtβπΎ)βπ΄)) | |
10 | ellimciota.k | . . . . . 6 β’ πΎ = (TopOpenββfld) | |
11 | 7, 8, 9, 10 | limcmo 25390 | . . . . 5 β’ (π β β*π₯ π₯ β (πΉ limβ π΅)) |
12 | df-eu 2563 | . . . . 5 β’ (β!π₯ π₯ β (πΉ limβ π΅) β (βπ₯ π₯ β (πΉ limβ π΅) β§ β*π₯ π₯ β (πΉ limβ π΅))) | |
13 | 6, 11, 12 | sylanbrc 583 | . . . 4 β’ (π β β!π₯ π₯ β (πΉ limβ π΅)) |
14 | eleq1 2821 | . . . . 5 β’ (π₯ = (β©π¦π¦ β (πΉ limβ π΅)) β (π₯ β (πΉ limβ π΅) β (β©π¦π¦ β (πΉ limβ π΅)) β (πΉ limβ π΅))) | |
15 | 14 | iota2 6529 | . . . 4 β’ (((β©π¦π¦ β (πΉ limβ π΅)) β V β§ β!π₯ π₯ β (πΉ limβ π΅)) β ((β©π¦π¦ β (πΉ limβ π΅)) β (πΉ limβ π΅) β (β©π₯π₯ β (πΉ limβ π΅)) = (β©π¦π¦ β (πΉ limβ π΅)))) |
16 | 3, 13, 15 | sylancr 587 | . . 3 β’ (π β ((β©π¦π¦ β (πΉ limβ π΅)) β (πΉ limβ π΅) β (β©π₯π₯ β (πΉ limβ π΅)) = (β©π¦π¦ β (πΉ limβ π΅)))) |
17 | 2, 16 | mpbiri 257 | . 2 β’ (π β (β©π¦π¦ β (πΉ limβ π΅)) β (πΉ limβ π΅)) |
18 | 2, 17 | eqeltrid 2837 | 1 β’ (π β (β©π₯π₯ β (πΉ limβ π΅)) β (πΉ limβ π΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1541 βwex 1781 β wcel 2106 β*wmo 2532 β!weu 2562 β wne 2940 Vcvv 3474 β wss 3947 β c0 4321 β©cio 6490 βΆwf 6536 βcfv 6540 (class class class)co 7405 βcc 11104 TopOpenctopn 17363 βfldccnfld 20936 limPtclp 22629 limβ climc 25370 |
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 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
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 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fi 9402 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-icc 13327 df-fz 13481 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-struct 17076 df-slot 17111 df-ndx 17123 df-base 17141 df-plusg 17206 df-mulr 17207 df-starv 17208 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-rest 17364 df-topn 17365 df-topgen 17385 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-cnp 22723 df-haus 22810 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-xms 23817 df-ms 23818 df-limc 25374 |
This theorem is referenced by: fourierdlem94 44902 fourierdlem113 44921 |
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