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 41952 | . . . 4 ⊢ (𝜑 → (𝐹 limℂ 𝐵) = ∅) |
11 | ne0i 4300 | . . . . 5 ⊢ ((𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵) → (𝐹 limℂ 𝐵) ≠ ∅) | |
12 | 11 | necon2bi 3046 | . . . 4 ⊢ ((𝐹 limℂ 𝐵) = ∅ → ¬ (𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵)) |
13 | 10, 12 | syl 17 | . . 3 ⊢ (𝜑 → ¬ (𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵)) |
14 | 13 | intnand 491 | . 2 ⊢ (𝜑 → ¬ (𝐹:ℝ⟶ℂ ∧ (𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵))) |
15 | ax-resscn 10594 | . . 3 ⊢ ℝ ⊆ ℂ | |
16 | jumpncnp.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
17 | eqid 2821 | . . . 4 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
18 | 17 | tgioo2 23411 | . . . . 5 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
19 | 3, 18 | eqtri 2844 | . . . 4 ⊢ 𝐽 = ((TopOpen‘ℂfld) ↾t ℝ) |
20 | 17, 19 | cnplimc 24485 | . . 3 ⊢ ((ℝ ⊆ ℂ ∧ 𝐵 ∈ ℝ) → (𝐹 ∈ ((𝐽 CnP (TopOpen‘ℂfld))‘𝐵) ↔ (𝐹:ℝ⟶ℂ ∧ (𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵)))) |
21 | 15, 16, 20 | sylancr 589 | . 2 ⊢ (𝜑 → (𝐹 ∈ ((𝐽 CnP (TopOpen‘ℂfld))‘𝐵) ↔ (𝐹:ℝ⟶ℂ ∧ (𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵)))) |
22 | 14, 21 | mtbird 327 | 1 ⊢ (𝜑 → ¬ 𝐹 ∈ ((𝐽 CnP (TopOpen‘ℂfld))‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 ran crn 5556 ↾ cres 5557 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 +∞cpnf 10672 -∞cmnf 10673 (,)cioo 12739 ↾t crest 16694 TopOpenctopn 16695 topGenctg 16711 ℂfldccnfld 20545 limPtclp 21742 CnP ccnp 21833 limℂ climc 24460 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fi 8875 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-fz 12894 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-plusg 16578 df-mulr 16579 df-starv 16580 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-rest 16696 df-topn 16697 df-topgen 16717 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-cnp 21836 df-xms 22930 df-ms 22931 df-limc 24464 |
This theorem is referenced by: fouriersw 42536 |
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