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Mathbox for Glauco Siliprandi |
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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 2945 | . . . . . 6 ⊢ (¬ 𝐿 = 𝑅 → 𝐿 ≠ 𝑅) | |
2 | limclr.k | . . . . . . . 8 ⊢ 𝐾 = (TopOpen‘ℂfld) | |
3 | limclr.a | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
4 | 3 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐿 ≠ 𝑅) → 𝐴 ⊆ ℝ) |
5 | limclr.j | . . . . . . . 8 ⊢ 𝐽 = (topGen‘ran (,)) | |
6 | limclr.f | . . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) | |
7 | 6 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐿 ≠ 𝑅) → 𝐹:𝐴⟶ℂ) |
8 | limclr.lp1 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (-∞(,)𝐵)))) | |
9 | 8 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐿 ≠ 𝑅) → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (-∞(,)𝐵)))) |
10 | limclr.lp2 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (𝐵(,)+∞)))) | |
11 | 10 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐿 ≠ 𝑅) → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (𝐵(,)+∞)))) |
12 | limclr.l | . . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝐵)) limℂ 𝐵)) | |
13 | 12 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐿 ≠ 𝑅) → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝐵)) limℂ 𝐵)) |
14 | limclr.r | . . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝐵(,)+∞)) limℂ 𝐵)) | |
15 | 14 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐿 ≠ 𝑅) → 𝑅 ∈ ((𝐹 ↾ (𝐵(,)+∞)) limℂ 𝐵)) |
16 | simpr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐿 ≠ 𝑅) → 𝐿 ≠ 𝑅) | |
17 | 2, 4, 5, 7, 9, 11, 13, 15, 16 | limclner 45606 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐿 ≠ 𝑅) → (𝐹 limℂ 𝐵) = ∅) |
18 | nne 2941 | . . . . . . 7 ⊢ (¬ (𝐹 limℂ 𝐵) ≠ ∅ ↔ (𝐹 limℂ 𝐵) = ∅) | |
19 | 17, 18 | sylibr 234 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐿 ≠ 𝑅) → ¬ (𝐹 limℂ 𝐵) ≠ ∅) |
20 | 1, 19 | sylan2 593 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐿 = 𝑅) → ¬ (𝐹 limℂ 𝐵) ≠ ∅) |
21 | 20 | ex 412 | . . . 4 ⊢ (𝜑 → (¬ 𝐿 = 𝑅 → ¬ (𝐹 limℂ 𝐵) ≠ ∅)) |
22 | 21 | con4d 115 | . . 3 ⊢ (𝜑 → ((𝐹 limℂ 𝐵) ≠ ∅ → 𝐿 = 𝑅)) |
23 | 3 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐿 = 𝑅) → 𝐴 ⊆ ℝ) |
24 | 6 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐿 = 𝑅) → 𝐹:𝐴⟶ℂ) |
25 | retop 24797 | . . . . . . . . . 10 ⊢ (topGen‘ran (,)) ∈ Top | |
26 | 5, 25 | eqeltri 2834 | . . . . . . . . 9 ⊢ 𝐽 ∈ Top |
27 | inss2 4245 | . . . . . . . . . 10 ⊢ (𝐴 ∩ (-∞(,)𝐵)) ⊆ (-∞(,)𝐵) | |
28 | ioossre 13444 | . . . . . . . . . 10 ⊢ (-∞(,)𝐵) ⊆ ℝ | |
29 | 27, 28 | sstri 4004 | . . . . . . . . 9 ⊢ (𝐴 ∩ (-∞(,)𝐵)) ⊆ ℝ |
30 | uniretop 24798 | . . . . . . . . . . 11 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
31 | 5 | eqcomi 2743 | . . . . . . . . . . . 12 ⊢ (topGen‘ran (,)) = 𝐽 |
32 | 31 | unieqi 4923 | . . . . . . . . . . 11 ⊢ ∪ (topGen‘ran (,)) = ∪ 𝐽 |
33 | 30, 32 | eqtri 2762 | . . . . . . . . . 10 ⊢ ℝ = ∪ 𝐽 |
34 | 33 | lpss 23165 | . . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ (𝐴 ∩ (-∞(,)𝐵)) ⊆ ℝ) → ((limPt‘𝐽)‘(𝐴 ∩ (-∞(,)𝐵))) ⊆ ℝ) |
35 | 26, 29, 34 | mp2an 692 | . . . . . . . 8 ⊢ ((limPt‘𝐽)‘(𝐴 ∩ (-∞(,)𝐵))) ⊆ ℝ |
36 | 35, 8 | sselid 3992 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
37 | 36 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐿 = 𝑅) → 𝐵 ∈ ℝ) |
38 | 12 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐿 = 𝑅) → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝐵)) limℂ 𝐵)) |
39 | 14 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐿 = 𝑅) → 𝑅 ∈ ((𝐹 ↾ (𝐵(,)+∞)) limℂ 𝐵)) |
40 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐿 = 𝑅) → 𝐿 = 𝑅) | |
41 | 2, 23, 5, 24, 37, 38, 39, 40 | limcleqr 45599 | . . . . 5 ⊢ ((𝜑 ∧ 𝐿 = 𝑅) → 𝐿 ∈ (𝐹 limℂ 𝐵)) |
42 | 41 | ne0d 4347 | . . . 4 ⊢ ((𝜑 ∧ 𝐿 = 𝑅) → (𝐹 limℂ 𝐵) ≠ ∅) |
43 | 42 | ex 412 | . . 3 ⊢ (𝜑 → (𝐿 = 𝑅 → (𝐹 limℂ 𝐵) ≠ ∅)) |
44 | 22, 43 | impbid 212 | . 2 ⊢ (𝜑 → ((𝐹 limℂ 𝐵) ≠ ∅ ↔ 𝐿 = 𝑅)) |
45 | 41 | ex 412 | . 2 ⊢ (𝜑 → (𝐿 = 𝑅 → 𝐿 ∈ (𝐹 limℂ 𝐵))) |
46 | 44, 45 | jca 511 | 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 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∩ cin 3961 ⊆ wss 3962 ∅c0 4338 ∪ cuni 4911 ran crn 5689 ↾ cres 5690 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 ℝcr 11151 +∞cpnf 11289 -∞cmnf 11290 (,)cioo 13383 TopOpenctopn 17467 topGenctg 17483 ℂfldccnfld 21381 Topctop 22914 limPtclp 23157 limℂ climc 25911 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-map 8866 df-pm 8867 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fi 9448 df-sup 9479 df-inf 9480 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ioo 13387 df-fz 13544 df-seq 14039 df-exp 14099 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-struct 17180 df-slot 17215 df-ndx 17227 df-base 17245 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-rest 17468 df-topn 17469 df-topgen 17489 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cld 23042 df-ntr 23043 df-cls 23044 df-nei 23121 df-lp 23159 df-cnp 23251 df-xms 24345 df-ms 24346 df-limc 25915 |
This theorem is referenced by: (None) |
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