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Mirrors > Home > MPE Home > Th. List > limcflflem | Structured version Visualization version GIF version |
Description: Lemma for limcflf 24484. (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) |
limcflf.c | ⊢ 𝐶 = (𝐴 ∖ {𝐵}) |
limcflf.l | ⊢ 𝐿 = (((nei‘𝐾)‘{𝐵}) ↾t 𝐶) |
Ref | Expression |
---|---|
limcflflem | ⊢ (𝜑 → 𝐿 ∈ (Fil‘𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limcflf.l | . 2 ⊢ 𝐿 = (((nei‘𝐾)‘{𝐵}) ↾t 𝐶) | |
2 | limcflf.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐾)‘𝐴)) | |
3 | limcflf.k | . . . . . . 7 ⊢ 𝐾 = (TopOpen‘ℂfld) | |
4 | 3 | cnfldtop 23389 | . . . . . 6 ⊢ 𝐾 ∈ Top |
5 | limcflf.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℂ) | |
6 | 3 | cnfldtopon 23388 | . . . . . . . 8 ⊢ 𝐾 ∈ (TopOn‘ℂ) |
7 | 6 | toponunii 21521 | . . . . . . 7 ⊢ ℂ = ∪ 𝐾 |
8 | 7 | islp 21745 | . . . . . 6 ⊢ ((𝐾 ∈ Top ∧ 𝐴 ⊆ ℂ) → (𝐵 ∈ ((limPt‘𝐾)‘𝐴) ↔ 𝐵 ∈ ((cls‘𝐾)‘(𝐴 ∖ {𝐵})))) |
9 | 4, 5, 8 | sylancr 590 | . . . . 5 ⊢ (𝜑 → (𝐵 ∈ ((limPt‘𝐾)‘𝐴) ↔ 𝐵 ∈ ((cls‘𝐾)‘(𝐴 ∖ {𝐵})))) |
10 | 2, 9 | mpbid 235 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ((cls‘𝐾)‘(𝐴 ∖ {𝐵}))) |
11 | limcflf.c | . . . . 5 ⊢ 𝐶 = (𝐴 ∖ {𝐵}) | |
12 | 11 | fveq2i 6648 | . . . 4 ⊢ ((cls‘𝐾)‘𝐶) = ((cls‘𝐾)‘(𝐴 ∖ {𝐵})) |
13 | 10, 12 | eleqtrrdi 2901 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ((cls‘𝐾)‘𝐶)) |
14 | difss 4059 | . . . . . 6 ⊢ (𝐴 ∖ {𝐵}) ⊆ 𝐴 | |
15 | 11, 14 | eqsstri 3949 | . . . . 5 ⊢ 𝐶 ⊆ 𝐴 |
16 | 15, 5 | sstrid 3926 | . . . 4 ⊢ (𝜑 → 𝐶 ⊆ ℂ) |
17 | 7 | lpss 21747 | . . . . . 6 ⊢ ((𝐾 ∈ Top ∧ 𝐴 ⊆ ℂ) → ((limPt‘𝐾)‘𝐴) ⊆ ℂ) |
18 | 4, 5, 17 | sylancr 590 | . . . . 5 ⊢ (𝜑 → ((limPt‘𝐾)‘𝐴) ⊆ ℂ) |
19 | 18, 2 | sseldd 3916 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
20 | trnei 22497 | . . . 4 ⊢ ((𝐾 ∈ (TopOn‘ℂ) ∧ 𝐶 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → (𝐵 ∈ ((cls‘𝐾)‘𝐶) ↔ (((nei‘𝐾)‘{𝐵}) ↾t 𝐶) ∈ (Fil‘𝐶))) | |
21 | 6, 16, 19, 20 | mp3an2i 1463 | . . 3 ⊢ (𝜑 → (𝐵 ∈ ((cls‘𝐾)‘𝐶) ↔ (((nei‘𝐾)‘{𝐵}) ↾t 𝐶) ∈ (Fil‘𝐶))) |
22 | 13, 21 | mpbid 235 | . 2 ⊢ (𝜑 → (((nei‘𝐾)‘{𝐵}) ↾t 𝐶) ∈ (Fil‘𝐶)) |
23 | 1, 22 | eqeltrid 2894 | 1 ⊢ (𝜑 → 𝐿 ∈ (Fil‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 ∖ cdif 3878 ⊆ wss 3881 {csn 4525 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ↾t crest 16686 TopOpenctopn 16687 ℂfldccnfld 20091 Topctop 21498 TopOnctopon 21515 clsccl 21623 neicnei 21702 limPtclp 21739 Filcfil 22450 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-fz 12886 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-plusg 16570 df-mulr 16571 df-starv 16572 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-rest 16688 df-topn 16689 df-topgen 16709 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-fil 22451 df-xms 22927 df-ms 22928 |
This theorem is referenced by: limcflf 24484 limcmo 24485 |
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