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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 | . . . . 5 ⊢ 𝐾 = (TopOpen‘ℂfld) | |
2 | 1 | cnfldhaus 22996 | . . . 4 ⊢ 𝐾 ∈ Haus |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐾 ∈ Haus) |
4 | limcflf.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) | |
5 | limcflf.a | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℂ) | |
6 | limcflf.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐾)‘𝐴)) | |
7 | eqid 2778 | . . . 4 ⊢ (𝐴 ∖ {𝐵}) = (𝐴 ∖ {𝐵}) | |
8 | eqid 2778 | . . . 4 ⊢ (((nei‘𝐾)‘{𝐵}) ↾t (𝐴 ∖ {𝐵})) = (((nei‘𝐾)‘{𝐵}) ↾t (𝐴 ∖ {𝐵})) | |
9 | 4, 5, 6, 1, 7, 8 | limcflflem 24081 | . . 3 ⊢ (𝜑 → (((nei‘𝐾)‘{𝐵}) ↾t (𝐴 ∖ {𝐵})) ∈ (Fil‘(𝐴 ∖ {𝐵}))) |
10 | difss 3960 | . . . 4 ⊢ (𝐴 ∖ {𝐵}) ⊆ 𝐴 | |
11 | fssres 6320 | . . . 4 ⊢ ((𝐹:𝐴⟶ℂ ∧ (𝐴 ∖ {𝐵}) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ∖ {𝐵})):(𝐴 ∖ {𝐵})⟶ℂ) | |
12 | 4, 10, 11 | sylancl 580 | . . 3 ⊢ (𝜑 → (𝐹 ↾ (𝐴 ∖ {𝐵})):(𝐴 ∖ {𝐵})⟶ℂ) |
13 | 1 | cnfldtopon 22994 | . . . . 5 ⊢ 𝐾 ∈ (TopOn‘ℂ) |
14 | 13 | toponunii 21128 | . . . 4 ⊢ ℂ = ∪ 𝐾 |
15 | 14 | hausflf 22209 | . . 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})))) |
16 | 3, 9, 12, 15 | syl3anc 1439 | . 2 ⊢ (𝜑 → ∃*𝑥 𝑥 ∈ ((𝐾 fLimf (((nei‘𝐾)‘{𝐵}) ↾t (𝐴 ∖ {𝐵})))‘(𝐹 ↾ (𝐴 ∖ {𝐵})))) |
17 | 4, 5, 6, 1, 7, 8 | limcflf 24082 | . . . 4 ⊢ (𝜑 → (𝐹 limℂ 𝐵) = ((𝐾 fLimf (((nei‘𝐾)‘{𝐵}) ↾t (𝐴 ∖ {𝐵})))‘(𝐹 ↾ (𝐴 ∖ {𝐵})))) |
18 | 17 | eleq2d 2845 | . . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐹 limℂ 𝐵) ↔ 𝑥 ∈ ((𝐾 fLimf (((nei‘𝐾)‘{𝐵}) ↾t (𝐴 ∖ {𝐵})))‘(𝐹 ↾ (𝐴 ∖ {𝐵}))))) |
19 | 18 | mobidv 2564 | . 2 ⊢ (𝜑 → (∃*𝑥 𝑥 ∈ (𝐹 limℂ 𝐵) ↔ ∃*𝑥 𝑥 ∈ ((𝐾 fLimf (((nei‘𝐾)‘{𝐵}) ↾t (𝐴 ∖ {𝐵})))‘(𝐹 ↾ (𝐴 ∖ {𝐵}))))) |
20 | 16, 19 | mpbird 249 | 1 ⊢ (𝜑 → ∃*𝑥 𝑥 ∈ (𝐹 limℂ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ∃*wmo 2549 ∖ cdif 3789 ⊆ wss 3792 {csn 4398 ↾ cres 5357 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 ℂcc 10270 ↾t crest 16467 TopOpenctopn 16468 ℂfldccnfld 20142 neicnei 21309 limPtclp 21346 Hauscha 21520 Filcfil 22057 fLimf cflf 22147 limℂ climc 24063 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fi 8605 df-sup 8636 df-inf 8637 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-icc 12494 df-fz 12644 df-seq 13120 df-exp 13179 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-plusg 16351 df-mulr 16352 df-starv 16353 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-rest 16469 df-topn 16470 df-topgen 16490 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-fbas 20139 df-fg 20140 df-cnfld 20143 df-top 21106 df-topon 21123 df-topsp 21145 df-bases 21158 df-cld 21231 df-ntr 21232 df-cls 21233 df-nei 21310 df-lp 21348 df-cnp 21440 df-haus 21527 df-fil 22058 df-fm 22150 df-flim 22151 df-flf 22152 df-xms 22533 df-ms 22534 df-limc 24067 |
This theorem is referenced by: perfdvf 24104 ellimciota 40758 |
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