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 | ⊢ 𝐾 = (TopOpen‘ℂfld) |
jumpncnp.a | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
jumpncnp.3 | ⊢ 𝐽 = (topGen‘ran (,)) |
jumpncnp.f | ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
jumpncnp.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
jumpncnp.lpt1 | ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (-∞(,)𝐵)))) |
jumpncnp.lpt2 | ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (𝐵(,)+∞)))) |
jumpncnp.8 | ⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝐵)) limℂ 𝐵)) |
jumpncnp.9 | ⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝐵(,)+∞)) limℂ 𝐵)) |
jumpncnp.lner | ⊢ (𝜑 → 𝐿 ≠ 𝑅) |
Ref | Expression |
---|---|
jumpncnp | ⊢ (𝜑 → ¬ 𝐹 ∈ ((𝐽 CnP (TopOpen‘ℂfld))‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | jumpncnp.k | . . . . 5 ⊢ 𝐾 = (TopOpen‘ℂfld) | |
2 | jumpncnp.a | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
3 | jumpncnp.3 | . . . . 5 ⊢ 𝐽 = (topGen‘ran (,)) | |
4 | jumpncnp.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) | |
5 | jumpncnp.lpt1 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (-∞(,)𝐵)))) | |
6 | jumpncnp.lpt2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (𝐵(,)+∞)))) | |
7 | jumpncnp.8 | . . . . 5 ⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝐵)) limℂ 𝐵)) | |
8 | jumpncnp.9 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝐵(,)+∞)) limℂ 𝐵)) | |
9 | jumpncnp.lner | . . . . 5 ⊢ (𝜑 → 𝐿 ≠ 𝑅) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | limclner 41808 | . . . 4 ⊢ (𝜑 → (𝐹 limℂ 𝐵) = ∅) |
11 | ne0i 4297 | . . . . 5 ⊢ ((𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵) → (𝐹 limℂ 𝐵) ≠ ∅) | |
12 | 11 | necon2bi 3043 | . . . 4 ⊢ ((𝐹 limℂ 𝐵) = ∅ → ¬ (𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵)) |
13 | 10, 12 | syl 17 | . . 3 ⊢ (𝜑 → ¬ (𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵)) |
14 | 13 | intnand 489 | . 2 ⊢ (𝜑 → ¬ (𝐹:ℝ⟶ℂ ∧ (𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵))) |
15 | ax-resscn 10582 | . . 3 ⊢ ℝ ⊆ ℂ | |
16 | jumpncnp.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
17 | eqid 2818 | . . . 4 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
18 | 17 | tgioo2 23338 | . . . . 5 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
19 | 3, 18 | eqtri 2841 | . . . 4 ⊢ 𝐽 = ((TopOpen‘ℂfld) ↾t ℝ) |
20 | 17, 19 | cnplimc 24412 | . . 3 ⊢ ((ℝ ⊆ ℂ ∧ 𝐵 ∈ ℝ) → (𝐹 ∈ ((𝐽 CnP (TopOpen‘ℂfld))‘𝐵) ↔ (𝐹:ℝ⟶ℂ ∧ (𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵)))) |
21 | 15, 16, 20 | sylancr 587 | . 2 ⊢ (𝜑 → (𝐹 ∈ ((𝐽 CnP (TopOpen‘ℂfld))‘𝐵) ↔ (𝐹:ℝ⟶ℂ ∧ (𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵)))) |
22 | 14, 21 | mtbird 326 | 1 ⊢ (𝜑 → ¬ 𝐹 ∈ ((𝐽 CnP (TopOpen‘ℂfld))‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∩ cin 3932 ⊆ wss 3933 ∅c0 4288 ran crn 5549 ↾ cres 5550 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 ℝcr 10524 +∞cpnf 10660 -∞cmnf 10661 (,)cioo 12726 ↾t crest 16682 TopOpenctopn 16683 topGenctg 16699 ℂfldccnfld 20473 limPtclp 21670 CnP ccnp 21761 limℂ climc 24387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fi 8863 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-fz 12881 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-plusg 16566 df-mulr 16567 df-starv 16568 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-rest 16684 df-topn 16685 df-topgen 16705 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-cnfld 20474 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cld 21555 df-ntr 21556 df-cls 21557 df-nei 21634 df-lp 21672 df-cnp 21764 df-xms 22857 df-ms 22858 df-limc 24391 |
This theorem is referenced by: fouriersw 42393 |
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