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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 2826 | . . 3 β’ (π₯ = π¦ β (π₯ β (πΉ limβ π΅) β π¦ β (πΉ limβ π΅))) | |
2 | 1 | cbviotavw 6457 | . 2 β’ (β©π₯π₯ β (πΉ limβ π΅)) = (β©π¦π¦ β (πΉ limβ π΅)) |
3 | iotaex 6470 | . . . 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 4307 | . . . . . 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 25249 | . . . . 5 β’ (π β β*π₯ π₯ β (πΉ limβ π΅)) |
12 | df-eu 2568 | . . . . 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 584 | . . . 4 β’ (π β β!π₯ π₯ β (πΉ limβ π΅)) |
14 | eleq1 2826 | . . . . 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 6486 | . . . 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 588 | . . 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 258 | . 2 β’ (π β (β©π¦π¦ β (πΉ limβ π΅)) β (πΉ limβ π΅)) |
18 | 2, 17 | eqeltrid 2842 | 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 1542 βwex 1782 β wcel 2107 β*wmo 2537 β!weu 2567 β wne 2944 Vcvv 3446 β wss 3911 β c0 4283 β©cio 6447 βΆwf 6493 βcfv 6497 (class class class)co 7358 βcc 11050 TopOpenctopn 17304 βfldccnfld 20799 limPtclp 22488 limβ climc 25229 |
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-iin 4958 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-icc 13272 df-fz 13426 df-seq 13908 df-exp 13969 df-cj 14985 df-re 14986 df-im 14987 df-sqrt 15121 df-abs 15122 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-fbas 20796 df-fg 20797 df-cnfld 20800 df-top 22246 df-topon 22263 df-topsp 22285 df-bases 22299 df-cld 22373 df-ntr 22374 df-cls 22375 df-nei 22452 df-lp 22490 df-cnp 22582 df-haus 22669 df-fil 23200 df-fm 23292 df-flim 23293 df-flf 23294 df-xms 23676 df-ms 23677 df-limc 25233 |
This theorem is referenced by: fourierdlem94 44448 fourierdlem113 44467 |
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