Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > limclr | Structured version Visualization version GIF version |
Description: For a limit point, both from the left and from the right, of the domain, the limit of the function exits only if the left and the right limits are equal. In this case, the three limits coincide. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
limclr.k | ⊢ 𝐾 = (TopOpen‘ℂfld) |
limclr.a | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
limclr.j | ⊢ 𝐽 = (topGen‘ran (,)) |
limclr.f | ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
limclr.lp1 | ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (-∞(,)𝐵)))) |
limclr.lp2 | ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (𝐵(,)+∞)))) |
limclr.l | ⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝐵)) limℂ 𝐵)) |
limclr.r | ⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝐵(,)+∞)) limℂ 𝐵)) |
Ref | Expression |
---|---|
limclr | ⊢ (𝜑 → (((𝐹 limℂ 𝐵) ≠ ∅ ↔ 𝐿 = 𝑅) ∧ (𝐿 = 𝑅 → 𝐿 ∈ (𝐹 limℂ 𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neqne 3021 | . . . . . 6 ⊢ (¬ 𝐿 = 𝑅 → 𝐿 ≠ 𝑅) | |
2 | limclr.k | . . . . . . . 8 ⊢ 𝐾 = (TopOpen‘ℂfld) | |
3 | limclr.a | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
4 | 3 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐿 ≠ 𝑅) → 𝐴 ⊆ ℝ) |
5 | limclr.j | . . . . . . . 8 ⊢ 𝐽 = (topGen‘ran (,)) | |
6 | limclr.f | . . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) | |
7 | 6 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐿 ≠ 𝑅) → 𝐹:𝐴⟶ℂ) |
8 | limclr.lp1 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (-∞(,)𝐵)))) | |
9 | 8 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐿 ≠ 𝑅) → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (-∞(,)𝐵)))) |
10 | limclr.lp2 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (𝐵(,)+∞)))) | |
11 | 10 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐿 ≠ 𝑅) → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (𝐵(,)+∞)))) |
12 | limclr.l | . . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝐵)) limℂ 𝐵)) | |
13 | 12 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐿 ≠ 𝑅) → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝐵)) limℂ 𝐵)) |
14 | limclr.r | . . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝐵(,)+∞)) limℂ 𝐵)) | |
15 | 14 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐿 ≠ 𝑅) → 𝑅 ∈ ((𝐹 ↾ (𝐵(,)+∞)) limℂ 𝐵)) |
16 | simpr 485 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐿 ≠ 𝑅) → 𝐿 ≠ 𝑅) | |
17 | 2, 4, 5, 7, 9, 11, 13, 15, 16 | limclner 41808 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐿 ≠ 𝑅) → (𝐹 limℂ 𝐵) = ∅) |
18 | nne 3017 | . . . . . . 7 ⊢ (¬ (𝐹 limℂ 𝐵) ≠ ∅ ↔ (𝐹 limℂ 𝐵) = ∅) | |
19 | 17, 18 | sylibr 235 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐿 ≠ 𝑅) → ¬ (𝐹 limℂ 𝐵) ≠ ∅) |
20 | 1, 19 | sylan2 592 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐿 = 𝑅) → ¬ (𝐹 limℂ 𝐵) ≠ ∅) |
21 | 20 | ex 413 | . . . 4 ⊢ (𝜑 → (¬ 𝐿 = 𝑅 → ¬ (𝐹 limℂ 𝐵) ≠ ∅)) |
22 | 21 | con4d 115 | . . 3 ⊢ (𝜑 → ((𝐹 limℂ 𝐵) ≠ ∅ → 𝐿 = 𝑅)) |
23 | 3 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐿 = 𝑅) → 𝐴 ⊆ ℝ) |
24 | 6 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐿 = 𝑅) → 𝐹:𝐴⟶ℂ) |
25 | retop 23297 | . . . . . . . . . 10 ⊢ (topGen‘ran (,)) ∈ Top | |
26 | 5, 25 | eqeltri 2906 | . . . . . . . . 9 ⊢ 𝐽 ∈ Top |
27 | inss2 4203 | . . . . . . . . . 10 ⊢ (𝐴 ∩ (-∞(,)𝐵)) ⊆ (-∞(,)𝐵) | |
28 | ioossre 12786 | . . . . . . . . . 10 ⊢ (-∞(,)𝐵) ⊆ ℝ | |
29 | 27, 28 | sstri 3973 | . . . . . . . . 9 ⊢ (𝐴 ∩ (-∞(,)𝐵)) ⊆ ℝ |
30 | uniretop 23298 | . . . . . . . . . . 11 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
31 | 5 | eqcomi 2827 | . . . . . . . . . . . 12 ⊢ (topGen‘ran (,)) = 𝐽 |
32 | 31 | unieqi 4839 | . . . . . . . . . . 11 ⊢ ∪ (topGen‘ran (,)) = ∪ 𝐽 |
33 | 30, 32 | eqtri 2841 | . . . . . . . . . 10 ⊢ ℝ = ∪ 𝐽 |
34 | 33 | lpss 21678 | . . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ (𝐴 ∩ (-∞(,)𝐵)) ⊆ ℝ) → ((limPt‘𝐽)‘(𝐴 ∩ (-∞(,)𝐵))) ⊆ ℝ) |
35 | 26, 29, 34 | mp2an 688 | . . . . . . . 8 ⊢ ((limPt‘𝐽)‘(𝐴 ∩ (-∞(,)𝐵))) ⊆ ℝ |
36 | 35, 8 | sseldi 3962 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
37 | 36 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐿 = 𝑅) → 𝐵 ∈ ℝ) |
38 | 12 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐿 = 𝑅) → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝐵)) limℂ 𝐵)) |
39 | 14 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐿 = 𝑅) → 𝑅 ∈ ((𝐹 ↾ (𝐵(,)+∞)) limℂ 𝐵)) |
40 | simpr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐿 = 𝑅) → 𝐿 = 𝑅) | |
41 | 2, 23, 5, 24, 37, 38, 39, 40 | limcleqr 41801 | . . . . 5 ⊢ ((𝜑 ∧ 𝐿 = 𝑅) → 𝐿 ∈ (𝐹 limℂ 𝐵)) |
42 | 41 | ne0d 4298 | . . . 4 ⊢ ((𝜑 ∧ 𝐿 = 𝑅) → (𝐹 limℂ 𝐵) ≠ ∅) |
43 | 42 | ex 413 | . . 3 ⊢ (𝜑 → (𝐿 = 𝑅 → (𝐹 limℂ 𝐵) ≠ ∅)) |
44 | 22, 43 | impbid 213 | . 2 ⊢ (𝜑 → ((𝐹 limℂ 𝐵) ≠ ∅ ↔ 𝐿 = 𝑅)) |
45 | 41 | ex 413 | . 2 ⊢ (𝜑 → (𝐿 = 𝑅 → 𝐿 ∈ (𝐹 limℂ 𝐵))) |
46 | 44, 45 | jca 512 | 1 ⊢ (𝜑 → (((𝐹 limℂ 𝐵) ≠ ∅ ↔ 𝐿 = 𝑅) ∧ (𝐿 = 𝑅 → 𝐿 ∈ (𝐹 limℂ 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 ∪ cuni 4830 ran crn 5549 ↾ cres 5550 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 ℝcr 10524 +∞cpnf 10660 -∞cmnf 10661 (,)cioo 12726 TopOpenctopn 16683 topGenctg 16699 ℂfldccnfld 20473 Topctop 21429 limPtclp 21670 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: (None) |
Copyright terms: Public domain | W3C validator |