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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 23395 | . . 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 2823 | . . . 4 ⊢ (𝐴 ∖ {𝐵}) = (𝐴 ∖ {𝐵}) | |
7 | eqid 2823 | . . . 4 ⊢ (((nei‘𝐾)‘{𝐵}) ↾t (𝐴 ∖ {𝐵})) = (((nei‘𝐾)‘{𝐵}) ↾t (𝐴 ∖ {𝐵})) | |
8 | 3, 4, 5, 1, 6, 7 | limcflflem 24480 | . . 3 ⊢ (𝜑 → (((nei‘𝐾)‘{𝐵}) ↾t (𝐴 ∖ {𝐵})) ∈ (Fil‘(𝐴 ∖ {𝐵}))) |
9 | difss 4110 | . . . 4 ⊢ (𝐴 ∖ {𝐵}) ⊆ 𝐴 | |
10 | fssres 6546 | . . . 4 ⊢ ((𝐹:𝐴⟶ℂ ∧ (𝐴 ∖ {𝐵}) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ∖ {𝐵})):(𝐴 ∖ {𝐵})⟶ℂ) | |
11 | 3, 9, 10 | sylancl 588 | . . 3 ⊢ (𝜑 → (𝐹 ↾ (𝐴 ∖ {𝐵})):(𝐴 ∖ {𝐵})⟶ℂ) |
12 | 1 | cnfldtopon 23393 | . . . . 5 ⊢ 𝐾 ∈ (TopOn‘ℂ) |
13 | 12 | toponunii 21526 | . . . 4 ⊢ ℂ = ∪ 𝐾 |
14 | 13 | hausflf 22607 | . . 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 24481 | . . . 4 ⊢ (𝜑 → (𝐹 limℂ 𝐵) = ((𝐾 fLimf (((nei‘𝐾)‘{𝐵}) ↾t (𝐴 ∖ {𝐵})))‘(𝐹 ↾ (𝐴 ∖ {𝐵})))) |
17 | 16 | eleq2d 2900 | . . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐹 limℂ 𝐵) ↔ 𝑥 ∈ ((𝐾 fLimf (((nei‘𝐾)‘{𝐵}) ↾t (𝐴 ∖ {𝐵})))‘(𝐹 ↾ (𝐴 ∖ {𝐵}))))) |
18 | 17 | mobidv 2633 | . 2 ⊢ (𝜑 → (∃*𝑥 𝑥 ∈ (𝐹 limℂ 𝐵) ↔ ∃*𝑥 𝑥 ∈ ((𝐾 fLimf (((nei‘𝐾)‘{𝐵}) ↾t (𝐴 ∖ {𝐵})))‘(𝐹 ↾ (𝐴 ∖ {𝐵}))))) |
19 | 15, 18 | mpbird 259 | 1 ⊢ (𝜑 → ∃*𝑥 𝑥 ∈ (𝐹 limℂ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∃*wmo 2620 ∖ cdif 3935 ⊆ wss 3938 {csn 4569 ↾ cres 5559 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 ↾t crest 16696 TopOpenctopn 16697 ℂfldccnfld 20547 neicnei 21707 limPtclp 21744 Hauscha 21918 Filcfil 22455 fLimf cflf 22545 limℂ climc 24462 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fi 8877 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-icc 12748 df-fz 12896 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-plusg 16580 df-mulr 16581 df-starv 16582 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-rest 16698 df-topn 16699 df-topgen 16719 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-lp 21746 df-cnp 21838 df-haus 21925 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-xms 22932 df-ms 22933 df-limc 24466 |
This theorem is referenced by: perfdvf 24503 ellimciota 41902 |
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