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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > jumpncnp | Structured version Visualization version GIF version |
Description: Jump discontinuity or discontinuity of the first kind: if the left and the right limit don't match, the function is discontinuous at the point. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
jumpncnp.k | β’ πΎ = (TopOpenββfld) |
jumpncnp.a | β’ (π β π΄ β β) |
jumpncnp.3 | β’ π½ = (topGenβran (,)) |
jumpncnp.f | β’ (π β πΉ:π΄βΆβ) |
jumpncnp.b | β’ (π β π΅ β β) |
jumpncnp.lpt1 | β’ (π β π΅ β ((limPtβπ½)β(π΄ β© (-β(,)π΅)))) |
jumpncnp.lpt2 | β’ (π β π΅ β ((limPtβπ½)β(π΄ β© (π΅(,)+β)))) |
jumpncnp.8 | β’ (π β πΏ β ((πΉ βΎ (-β(,)π΅)) limβ π΅)) |
jumpncnp.9 | β’ (π β π β ((πΉ βΎ (π΅(,)+β)) limβ π΅)) |
jumpncnp.lner | β’ (π β πΏ β π ) |
Ref | Expression |
---|---|
jumpncnp | β’ (π β Β¬ πΉ β ((π½ CnP (TopOpenββfld))βπ΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | jumpncnp.k | . . . . 5 β’ πΎ = (TopOpenββfld) | |
2 | jumpncnp.a | . . . . 5 β’ (π β π΄ β β) | |
3 | jumpncnp.3 | . . . . 5 β’ π½ = (topGenβran (,)) | |
4 | jumpncnp.f | . . . . 5 β’ (π β πΉ:π΄βΆβ) | |
5 | jumpncnp.lpt1 | . . . . 5 β’ (π β π΅ β ((limPtβπ½)β(π΄ β© (-β(,)π΅)))) | |
6 | jumpncnp.lpt2 | . . . . 5 β’ (π β π΅ β ((limPtβπ½)β(π΄ β© (π΅(,)+β)))) | |
7 | jumpncnp.8 | . . . . 5 β’ (π β πΏ β ((πΉ βΎ (-β(,)π΅)) limβ π΅)) | |
8 | jumpncnp.9 | . . . . 5 β’ (π β π β ((πΉ βΎ (π΅(,)+β)) limβ π΅)) | |
9 | jumpncnp.lner | . . . . 5 β’ (π β πΏ β π ) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | limclner 45068 | . . . 4 β’ (π β (πΉ limβ π΅) = β ) |
11 | ne0i 4338 | . . . . 5 β’ ((πΉβπ΅) β (πΉ limβ π΅) β (πΉ limβ π΅) β β ) | |
12 | 11 | necon2bi 2968 | . . . 4 β’ ((πΉ limβ π΅) = β β Β¬ (πΉβπ΅) β (πΉ limβ π΅)) |
13 | 10, 12 | syl 17 | . . 3 β’ (π β Β¬ (πΉβπ΅) β (πΉ limβ π΅)) |
14 | 13 | intnand 487 | . 2 β’ (π β Β¬ (πΉ:ββΆβ β§ (πΉβπ΅) β (πΉ limβ π΅))) |
15 | ax-resscn 11203 | . . 3 β’ β β β | |
16 | jumpncnp.b | . . 3 β’ (π β π΅ β β) | |
17 | eqid 2728 | . . . 4 β’ (TopOpenββfld) = (TopOpenββfld) | |
18 | 17 | tgioo2 24739 | . . . . 5 β’ (topGenβran (,)) = ((TopOpenββfld) βΎt β) |
19 | 3, 18 | eqtri 2756 | . . . 4 β’ π½ = ((TopOpenββfld) βΎt β) |
20 | 17, 19 | cnplimc 25836 | . . 3 β’ ((β β β β§ π΅ β β) β (πΉ β ((π½ CnP (TopOpenββfld))βπ΅) β (πΉ:ββΆβ β§ (πΉβπ΅) β (πΉ limβ π΅)))) |
21 | 15, 16, 20 | sylancr 585 | . 2 β’ (π β (πΉ β ((π½ CnP (TopOpenββfld))βπ΅) β (πΉ:ββΆβ β§ (πΉβπ΅) β (πΉ limβ π΅)))) |
22 | 14, 21 | mtbird 324 | 1 β’ (π β Β¬ πΉ β ((π½ CnP (TopOpenββfld))βπ΅)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 β wne 2937 β© cin 3948 β wss 3949 β c0 4326 ran crn 5683 βΎ cres 5684 βΆwf 6549 βcfv 6553 (class class class)co 7426 βcc 11144 βcr 11145 +βcpnf 11283 -βcmnf 11284 (,)cioo 13364 βΎt crest 17409 TopOpenctopn 17410 topGenctg 17426 βfldccnfld 21286 limPtclp 23058 CnP ccnp 23149 limβ climc 25811 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-pm 8854 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fi 9442 df-sup 9473 df-inf 9474 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13368 df-fz 13525 df-seq 14007 df-exp 14067 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-struct 17123 df-slot 17158 df-ndx 17170 df-base 17188 df-plusg 17253 df-mulr 17254 df-starv 17255 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-rest 17411 df-topn 17412 df-topgen 17432 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-cnfld 21287 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-cld 22943 df-ntr 22944 df-cls 22945 df-nei 23022 df-lp 23060 df-cnp 23152 df-xms 24246 df-ms 24247 df-limc 25815 |
This theorem is referenced by: fouriersw 45648 |
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