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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 44921 | . . . 4 β’ (π β (πΉ limβ π΅) = β ) |
11 | ne0i 4329 | . . . . 5 β’ ((πΉβπ΅) β (πΉ limβ π΅) β (πΉ limβ π΅) β β ) | |
12 | 11 | necon2bi 2965 | . . . 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 488 | . 2 β’ (π β Β¬ (πΉ:ββΆβ β§ (πΉβπ΅) β (πΉ limβ π΅))) |
15 | ax-resscn 11166 | . . 3 β’ β β β | |
16 | jumpncnp.b | . . 3 β’ (π β π΅ β β) | |
17 | eqid 2726 | . . . 4 β’ (TopOpenββfld) = (TopOpenββfld) | |
18 | 17 | tgioo2 24669 | . . . . 5 β’ (topGenβran (,)) = ((TopOpenββfld) βΎt β) |
19 | 3, 18 | eqtri 2754 | . . . 4 β’ π½ = ((TopOpenββfld) βΎt β) |
20 | 17, 19 | cnplimc 25766 | . . 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 586 | . 2 β’ (π β (πΉ β ((π½ CnP (TopOpenββfld))βπ΅) β (πΉ:ββΆβ β§ (πΉβπ΅) β (πΉ limβ π΅)))) |
22 | 14, 21 | mtbird 325 | 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 395 = wceq 1533 β wcel 2098 β wne 2934 β© cin 3942 β wss 3943 β c0 4317 ran crn 5670 βΎ cres 5671 βΆwf 6532 βcfv 6536 (class class class)co 7404 βcc 11107 βcr 11108 +βcpnf 11246 -βcmnf 11247 (,)cioo 13327 βΎt crest 17372 TopOpenctopn 17373 topGenctg 17389 βfldccnfld 21235 limPtclp 22988 CnP ccnp 23079 limβ climc 25741 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fi 9405 df-sup 9436 df-inf 9437 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-ioo 13331 df-fz 13488 df-seq 13970 df-exp 14030 df-cj 15049 df-re 15050 df-im 15051 df-sqrt 15185 df-abs 15186 df-struct 17086 df-slot 17121 df-ndx 17133 df-base 17151 df-plusg 17216 df-mulr 17217 df-starv 17218 df-tset 17222 df-ple 17223 df-ds 17225 df-unif 17226 df-rest 17374 df-topn 17375 df-topgen 17395 df-psmet 21227 df-xmet 21228 df-met 21229 df-bl 21230 df-mopn 21231 df-cnfld 21236 df-top 22746 df-topon 22763 df-topsp 22785 df-bases 22799 df-cld 22873 df-ntr 22874 df-cls 22875 df-nei 22952 df-lp 22990 df-cnp 23082 df-xms 24176 df-ms 24177 df-limc 25745 |
This theorem is referenced by: fouriersw 45501 |
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