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Mirrors > Home > MPE Home > Th. List > limcflflem | Structured version Visualization version GIF version |
Description: Lemma for limcflf 25643. (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 24533 | . . . . . 6 ⊢ 𝐾 ∈ Top |
5 | limcflf.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℂ) | |
6 | 3 | cnfldtopon 24532 | . . . . . . . 8 ⊢ 𝐾 ∈ (TopOn‘ℂ) |
7 | 6 | toponunii 22651 | . . . . . . 7 ⊢ ℂ = ∪ 𝐾 |
8 | 7 | islp 22877 | . . . . . 6 ⊢ ((𝐾 ∈ Top ∧ 𝐴 ⊆ ℂ) → (𝐵 ∈ ((limPt‘𝐾)‘𝐴) ↔ 𝐵 ∈ ((cls‘𝐾)‘(𝐴 ∖ {𝐵})))) |
9 | 4, 5, 8 | sylancr 586 | . . . . 5 ⊢ (𝜑 → (𝐵 ∈ ((limPt‘𝐾)‘𝐴) ↔ 𝐵 ∈ ((cls‘𝐾)‘(𝐴 ∖ {𝐵})))) |
10 | 2, 9 | mpbid 231 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ((cls‘𝐾)‘(𝐴 ∖ {𝐵}))) |
11 | limcflf.c | . . . . 5 ⊢ 𝐶 = (𝐴 ∖ {𝐵}) | |
12 | 11 | fveq2i 6894 | . . . 4 ⊢ ((cls‘𝐾)‘𝐶) = ((cls‘𝐾)‘(𝐴 ∖ {𝐵})) |
13 | 10, 12 | eleqtrrdi 2843 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ((cls‘𝐾)‘𝐶)) |
14 | difss 4131 | . . . . . 6 ⊢ (𝐴 ∖ {𝐵}) ⊆ 𝐴 | |
15 | 11, 14 | eqsstri 4016 | . . . . 5 ⊢ 𝐶 ⊆ 𝐴 |
16 | 15, 5 | sstrid 3993 | . . . 4 ⊢ (𝜑 → 𝐶 ⊆ ℂ) |
17 | 7 | lpss 22879 | . . . . . 6 ⊢ ((𝐾 ∈ Top ∧ 𝐴 ⊆ ℂ) → ((limPt‘𝐾)‘𝐴) ⊆ ℂ) |
18 | 4, 5, 17 | sylancr 586 | . . . . 5 ⊢ (𝜑 → ((limPt‘𝐾)‘𝐴) ⊆ ℂ) |
19 | 18, 2 | sseldd 3983 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
20 | trnei 23629 | . . . 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 1465 | . . 3 ⊢ (𝜑 → (𝐵 ∈ ((cls‘𝐾)‘𝐶) ↔ (((nei‘𝐾)‘{𝐵}) ↾t 𝐶) ∈ (Fil‘𝐶))) |
22 | 13, 21 | mpbid 231 | . 2 ⊢ (𝜑 → (((nei‘𝐾)‘{𝐵}) ↾t 𝐶) ∈ (Fil‘𝐶)) |
23 | 1, 22 | eqeltrid 2836 | 1 ⊢ (𝜑 → 𝐿 ∈ (Fil‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1540 ∈ wcel 2105 ∖ cdif 3945 ⊆ wss 3948 {csn 4628 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 ℂcc 11114 ↾t crest 17373 TopOpenctopn 17374 ℂfldccnfld 21148 Topctop 22628 TopOnctopon 22645 clsccl 22755 neicnei 22834 limPtclp 22871 Filcfil 23582 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9443 df-inf 9444 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-q 12940 df-rp 12982 df-xneg 13099 df-xadd 13100 df-xmul 13101 df-fz 13492 df-seq 13974 df-exp 14035 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-struct 17087 df-slot 17122 df-ndx 17134 df-base 17152 df-plusg 17217 df-mulr 17218 df-starv 17219 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-rest 17375 df-topn 17376 df-topgen 17396 df-psmet 21140 df-xmet 21141 df-met 21142 df-bl 21143 df-mopn 21144 df-fbas 21145 df-cnfld 21149 df-top 22629 df-topon 22646 df-topsp 22668 df-bases 22682 df-cld 22756 df-ntr 22757 df-cls 22758 df-nei 22835 df-lp 22873 df-fil 23583 df-xms 24059 df-ms 24060 |
This theorem is referenced by: limcflf 25643 limcmo 25644 |
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