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Mirrors > Home > MPE Home > Th. List > limcmo | Structured version Visualization version GIF version |
Description: If 𝐵 is a limit point of the domain of the function 𝐹, then there is at most one limit value of 𝐹 at 𝐵. (Contributed by Mario Carneiro, 25-Dec-2016.) |
Ref | Expression |
---|---|
limcflf.f | ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
limcflf.a | ⊢ (𝜑 → 𝐴 ⊆ ℂ) |
limcflf.b | ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐾)‘𝐴)) |
limcflf.k | ⊢ 𝐾 = (TopOpen‘ℂfld) |
Ref | Expression |
---|---|
limcmo | ⊢ (𝜑 → ∃*𝑥 𝑥 ∈ (𝐹 limℂ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limcflf.k | . . . 4 ⊢ 𝐾 = (TopOpen‘ℂfld) | |
2 | 1 | cnfldhaus 24745 | . . 3 ⊢ 𝐾 ∈ Haus |
3 | limcflf.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) | |
4 | limcflf.a | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℂ) | |
5 | limcflf.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐾)‘𝐴)) | |
6 | eqid 2725 | . . . 4 ⊢ (𝐴 ∖ {𝐵}) = (𝐴 ∖ {𝐵}) | |
7 | eqid 2725 | . . . 4 ⊢ (((nei‘𝐾)‘{𝐵}) ↾t (𝐴 ∖ {𝐵})) = (((nei‘𝐾)‘{𝐵}) ↾t (𝐴 ∖ {𝐵})) | |
8 | 3, 4, 5, 1, 6, 7 | limcflflem 25853 | . . 3 ⊢ (𝜑 → (((nei‘𝐾)‘{𝐵}) ↾t (𝐴 ∖ {𝐵})) ∈ (Fil‘(𝐴 ∖ {𝐵}))) |
9 | difss 4128 | . . . 4 ⊢ (𝐴 ∖ {𝐵}) ⊆ 𝐴 | |
10 | fssres 6763 | . . . 4 ⊢ ((𝐹:𝐴⟶ℂ ∧ (𝐴 ∖ {𝐵}) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ∖ {𝐵})):(𝐴 ∖ {𝐵})⟶ℂ) | |
11 | 3, 9, 10 | sylancl 584 | . . 3 ⊢ (𝜑 → (𝐹 ↾ (𝐴 ∖ {𝐵})):(𝐴 ∖ {𝐵})⟶ℂ) |
12 | 1 | cnfldtopon 24743 | . . . . 5 ⊢ 𝐾 ∈ (TopOn‘ℂ) |
13 | 12 | toponunii 22862 | . . . 4 ⊢ ℂ = ∪ 𝐾 |
14 | 13 | hausflf 23945 | . . 3 ⊢ ((𝐾 ∈ Haus ∧ (((nei‘𝐾)‘{𝐵}) ↾t (𝐴 ∖ {𝐵})) ∈ (Fil‘(𝐴 ∖ {𝐵})) ∧ (𝐹 ↾ (𝐴 ∖ {𝐵})):(𝐴 ∖ {𝐵})⟶ℂ) → ∃*𝑥 𝑥 ∈ ((𝐾 fLimf (((nei‘𝐾)‘{𝐵}) ↾t (𝐴 ∖ {𝐵})))‘(𝐹 ↾ (𝐴 ∖ {𝐵})))) |
15 | 2, 8, 11, 14 | mp3an2i 1462 | . 2 ⊢ (𝜑 → ∃*𝑥 𝑥 ∈ ((𝐾 fLimf (((nei‘𝐾)‘{𝐵}) ↾t (𝐴 ∖ {𝐵})))‘(𝐹 ↾ (𝐴 ∖ {𝐵})))) |
16 | 3, 4, 5, 1, 6, 7 | limcflf 25854 | . . . 4 ⊢ (𝜑 → (𝐹 limℂ 𝐵) = ((𝐾 fLimf (((nei‘𝐾)‘{𝐵}) ↾t (𝐴 ∖ {𝐵})))‘(𝐹 ↾ (𝐴 ∖ {𝐵})))) |
17 | 16 | eleq2d 2811 | . . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐹 limℂ 𝐵) ↔ 𝑥 ∈ ((𝐾 fLimf (((nei‘𝐾)‘{𝐵}) ↾t (𝐴 ∖ {𝐵})))‘(𝐹 ↾ (𝐴 ∖ {𝐵}))))) |
18 | 17 | mobidv 2537 | . 2 ⊢ (𝜑 → (∃*𝑥 𝑥 ∈ (𝐹 limℂ 𝐵) ↔ ∃*𝑥 𝑥 ∈ ((𝐾 fLimf (((nei‘𝐾)‘{𝐵}) ↾t (𝐴 ∖ {𝐵})))‘(𝐹 ↾ (𝐴 ∖ {𝐵}))))) |
19 | 15, 18 | mpbird 256 | 1 ⊢ (𝜑 → ∃*𝑥 𝑥 ∈ (𝐹 limℂ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∃*wmo 2526 ∖ cdif 3941 ⊆ wss 3944 {csn 4630 ↾ cres 5680 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 ℂcc 11138 ↾t crest 17405 TopOpenctopn 17406 ℂfldccnfld 21296 neicnei 23045 limPtclp 23082 Hauscha 23256 Filcfil 23793 fLimf cflf 23883 limℂ climc 25835 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fi 9436 df-sup 9467 df-inf 9468 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-q 12966 df-rp 13010 df-xneg 13127 df-xadd 13128 df-xmul 13129 df-icc 13366 df-fz 13520 df-seq 14003 df-exp 14063 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-struct 17119 df-slot 17154 df-ndx 17166 df-base 17184 df-plusg 17249 df-mulr 17250 df-starv 17251 df-tset 17255 df-ple 17256 df-ds 17258 df-unif 17259 df-rest 17407 df-topn 17408 df-topgen 17428 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-fbas 21293 df-fg 21294 df-cnfld 21297 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22893 df-cld 22967 df-ntr 22968 df-cls 22969 df-nei 23046 df-lp 23084 df-cnp 23176 df-haus 23263 df-fil 23794 df-fm 23886 df-flim 23887 df-flf 23888 df-xms 24270 df-ms 24271 df-limc 25839 |
This theorem is referenced by: perfdvf 25876 ellimciota 45140 |
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