| 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 2948 | . . . . . 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 45666 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐿 ≠ 𝑅) → (𝐹 limℂ 𝐵) = ∅) |
| 18 | nne 2944 | . . . . . . 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 24782 | . . . . . . . . . 10 ⊢ (topGen‘ran (,)) ∈ Top | |
| 26 | 5, 25 | eqeltri 2837 | . . . . . . . . 9 ⊢ 𝐽 ∈ Top |
| 27 | inss2 4238 | . . . . . . . . . 10 ⊢ (𝐴 ∩ (-∞(,)𝐵)) ⊆ (-∞(,)𝐵) | |
| 28 | ioossre 13448 | . . . . . . . . . 10 ⊢ (-∞(,)𝐵) ⊆ ℝ | |
| 29 | 27, 28 | sstri 3993 | . . . . . . . . 9 ⊢ (𝐴 ∩ (-∞(,)𝐵)) ⊆ ℝ |
| 30 | uniretop 24783 | . . . . . . . . . . 11 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
| 31 | 5 | eqcomi 2746 | . . . . . . . . . . . 12 ⊢ (topGen‘ran (,)) = 𝐽 |
| 32 | 31 | unieqi 4919 | . . . . . . . . . . 11 ⊢ ∪ (topGen‘ran (,)) = ∪ 𝐽 |
| 33 | 30, 32 | eqtri 2765 | . . . . . . . . . 10 ⊢ ℝ = ∪ 𝐽 |
| 34 | 33 | lpss 23150 | . . . . . . . . 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 3981 | . . . . . . 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 45659 | . . . . 5 ⊢ ((𝜑 ∧ 𝐿 = 𝑅) → 𝐿 ∈ (𝐹 limℂ 𝐵)) |
| 42 | 41 | ne0d 4342 | . . . 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 1540 ∈ wcel 2108 ≠ wne 2940 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 ∪ cuni 4907 ran crn 5686 ↾ cres 5687 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 +∞cpnf 11292 -∞cmnf 11293 (,)cioo 13387 TopOpenctopn 17466 topGenctg 17482 ℂfldccnfld 21364 Topctop 22899 limPtclp 23142 limℂ climc 25897 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fi 9451 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-fz 13548 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-rest 17467 df-topn 17468 df-topgen 17488 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-cnp 23236 df-xms 24330 df-ms 24331 df-limc 25901 |
| This theorem is referenced by: (None) |
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