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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 23944 | . . 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 2740 | . . . 4 ⊢ (𝐴 ∖ {𝐵}) = (𝐴 ∖ {𝐵}) | |
7 | eqid 2740 | . . . 4 ⊢ (((nei‘𝐾)‘{𝐵}) ↾t (𝐴 ∖ {𝐵})) = (((nei‘𝐾)‘{𝐵}) ↾t (𝐴 ∖ {𝐵})) | |
8 | 3, 4, 5, 1, 6, 7 | limcflflem 25040 | . . 3 ⊢ (𝜑 → (((nei‘𝐾)‘{𝐵}) ↾t (𝐴 ∖ {𝐵})) ∈ (Fil‘(𝐴 ∖ {𝐵}))) |
9 | difss 4071 | . . . 4 ⊢ (𝐴 ∖ {𝐵}) ⊆ 𝐴 | |
10 | fssres 6637 | . . . 4 ⊢ ((𝐹:𝐴⟶ℂ ∧ (𝐴 ∖ {𝐵}) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ∖ {𝐵})):(𝐴 ∖ {𝐵})⟶ℂ) | |
11 | 3, 9, 10 | sylancl 586 | . . 3 ⊢ (𝜑 → (𝐹 ↾ (𝐴 ∖ {𝐵})):(𝐴 ∖ {𝐵})⟶ℂ) |
12 | 1 | cnfldtopon 23942 | . . . . 5 ⊢ 𝐾 ∈ (TopOn‘ℂ) |
13 | 12 | toponunii 22061 | . . . 4 ⊢ ℂ = ∪ 𝐾 |
14 | 13 | hausflf 23144 | . . 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 1465 | . 2 ⊢ (𝜑 → ∃*𝑥 𝑥 ∈ ((𝐾 fLimf (((nei‘𝐾)‘{𝐵}) ↾t (𝐴 ∖ {𝐵})))‘(𝐹 ↾ (𝐴 ∖ {𝐵})))) |
16 | 3, 4, 5, 1, 6, 7 | limcflf 25041 | . . . 4 ⊢ (𝜑 → (𝐹 limℂ 𝐵) = ((𝐾 fLimf (((nei‘𝐾)‘{𝐵}) ↾t (𝐴 ∖ {𝐵})))‘(𝐹 ↾ (𝐴 ∖ {𝐵})))) |
17 | 16 | eleq2d 2826 | . . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐹 limℂ 𝐵) ↔ 𝑥 ∈ ((𝐾 fLimf (((nei‘𝐾)‘{𝐵}) ↾t (𝐴 ∖ {𝐵})))‘(𝐹 ↾ (𝐴 ∖ {𝐵}))))) |
18 | 17 | mobidv 2551 | . 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 1542 ∈ wcel 2110 ∃*wmo 2540 ∖ cdif 3889 ⊆ wss 3892 {csn 4567 ↾ cres 5591 ⟶wf 6427 ‘cfv 6431 (class class class)co 7269 ℂcc 10868 ↾t crest 17127 TopOpenctopn 17128 ℂfldccnfld 20593 neicnei 22244 limPtclp 22281 Hauscha 22455 Filcfil 22992 fLimf cflf 23082 limℂ climc 25022 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 ax-pre-sup 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-1st 7822 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-er 8479 df-map 8598 df-pm 8599 df-en 8715 df-dom 8716 df-sdom 8717 df-fin 8718 df-fi 9146 df-sup 9177 df-inf 9178 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-div 11631 df-nn 11972 df-2 12034 df-3 12035 df-4 12036 df-5 12037 df-6 12038 df-7 12039 df-8 12040 df-9 12041 df-n0 12232 df-z 12318 df-dec 12435 df-uz 12580 df-q 12686 df-rp 12728 df-xneg 12845 df-xadd 12846 df-xmul 12847 df-icc 13083 df-fz 13237 df-seq 13718 df-exp 13779 df-cj 14806 df-re 14807 df-im 14808 df-sqrt 14942 df-abs 14943 df-struct 16844 df-slot 16879 df-ndx 16891 df-base 16909 df-plusg 16971 df-mulr 16972 df-starv 16973 df-tset 16977 df-ple 16978 df-ds 16980 df-unif 16981 df-rest 17129 df-topn 17130 df-topgen 17150 df-psmet 20585 df-xmet 20586 df-met 20587 df-bl 20588 df-mopn 20589 df-fbas 20590 df-fg 20591 df-cnfld 20594 df-top 22039 df-topon 22056 df-topsp 22078 df-bases 22092 df-cld 22166 df-ntr 22167 df-cls 22168 df-nei 22245 df-lp 22283 df-cnp 22375 df-haus 22462 df-fil 22993 df-fm 23085 df-flim 23086 df-flf 23087 df-xms 23469 df-ms 23470 df-limc 25026 |
This theorem is referenced by: perfdvf 25063 ellimciota 43124 |
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