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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 | ⊢ 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 41361 | . . . 4 ⊢ (𝜑 → (𝐹 limℂ 𝐵) = ∅) |
11 | ne0i 4187 | . . . . 5 ⊢ ((𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵) → (𝐹 limℂ 𝐵) ≠ ∅) | |
12 | 11 | necon2bi 2998 | . . . 4 ⊢ ((𝐹 limℂ 𝐵) = ∅ → ¬ (𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵)) |
13 | 10, 12 | syl 17 | . . 3 ⊢ (𝜑 → ¬ (𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵)) |
14 | 13 | intnand 481 | . 2 ⊢ (𝜑 → ¬ (𝐹:ℝ⟶ℂ ∧ (𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵))) |
15 | ax-resscn 10392 | . . 3 ⊢ ℝ ⊆ ℂ | |
16 | jumpncnp.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
17 | eqid 2779 | . . . 4 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
18 | 17 | tgioo2 23114 | . . . . 5 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
19 | 3, 18 | eqtri 2803 | . . . 4 ⊢ 𝐽 = ((TopOpen‘ℂfld) ↾t ℝ) |
20 | 17, 19 | cnplimc 24188 | . . 3 ⊢ ((ℝ ⊆ ℂ ∧ 𝐵 ∈ ℝ) → (𝐹 ∈ ((𝐽 CnP (TopOpen‘ℂfld))‘𝐵) ↔ (𝐹:ℝ⟶ℂ ∧ (𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵)))) |
21 | 15, 16, 20 | sylancr 578 | . 2 ⊢ (𝜑 → (𝐹 ∈ ((𝐽 CnP (TopOpen‘ℂfld))‘𝐵) ↔ (𝐹:ℝ⟶ℂ ∧ (𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵)))) |
22 | 14, 21 | mtbird 317 | 1 ⊢ (𝜑 → ¬ 𝐹 ∈ ((𝐽 CnP (TopOpen‘ℂfld))‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ≠ wne 2968 ∩ cin 3829 ⊆ wss 3830 ∅c0 4179 ran crn 5408 ↾ cres 5409 ⟶wf 6184 ‘cfv 6188 (class class class)co 6976 ℂcc 10333 ℝcr 10334 +∞cpnf 10471 -∞cmnf 10472 (,)cioo 12554 ↾t crest 16550 TopOpenctopn 16551 topGenctg 16567 ℂfldccnfld 20247 limPtclp 21446 CnP ccnp 21537 limℂ climc 24163 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 ax-pre-sup 10413 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-iin 4795 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-oadd 7909 df-er 8089 df-map 8208 df-pm 8209 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-fi 8670 df-sup 8701 df-inf 8702 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-nn 11440 df-2 11503 df-3 11504 df-4 11505 df-5 11506 df-6 11507 df-7 11508 df-8 11509 df-9 11510 df-n0 11708 df-z 11794 df-dec 11912 df-uz 12059 df-q 12163 df-rp 12205 df-xneg 12324 df-xadd 12325 df-xmul 12326 df-ioo 12558 df-fz 12709 df-seq 13185 df-exp 13245 df-cj 14319 df-re 14320 df-im 14321 df-sqrt 14455 df-abs 14456 df-struct 16341 df-ndx 16342 df-slot 16343 df-base 16345 df-plusg 16434 df-mulr 16435 df-starv 16436 df-tset 16440 df-ple 16441 df-ds 16443 df-unif 16444 df-rest 16552 df-topn 16553 df-topgen 16573 df-psmet 20239 df-xmet 20240 df-met 20241 df-bl 20242 df-mopn 20243 df-cnfld 20248 df-top 21206 df-topon 21223 df-topsp 21245 df-bases 21258 df-cld 21331 df-ntr 21332 df-cls 21333 df-nei 21410 df-lp 21448 df-cnp 21540 df-xms 22633 df-ms 22634 df-limc 24167 |
This theorem is referenced by: fouriersw 41945 |
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