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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnllysconn | Structured version Visualization version GIF version |
Description: The topology of the complex numbers is locally simply connected. (Contributed by Mario Carneiro, 2-Mar-2015.) |
Ref | Expression |
---|---|
cnllysconn.j | ⊢ 𝐽 = (TopOpen‘ℂfld) |
Ref | Expression |
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cnllysconn | ⊢ 𝐽 ∈ Locally SConn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnllysconn.j | . . 3 ⊢ 𝐽 = (TopOpen‘ℂfld) | |
2 | 1 | cnfldtop 22587 | . 2 ⊢ 𝐽 ∈ Top |
3 | cnxmet 22576 | . . . . 5 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ) | |
4 | 1 | cnfldtopn 22585 | . . . . . 6 ⊢ 𝐽 = (MetOpen‘(abs ∘ − )) |
5 | 4 | mopni2 22298 | . . . . 5 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) → ∃𝑟 ∈ ℝ+ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥) |
6 | 3, 5 | mp3an1 1411 | . . . 4 ⊢ ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) → ∃𝑟 ∈ ℝ+ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥) |
7 | 3 | a1i 11 | . . . . . . 7 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (abs ∘ − ) ∈ (∞Met‘ℂ)) |
8 | 1 | cnfldtopon 22586 | . . . . . . . . 9 ⊢ 𝐽 ∈ (TopOn‘ℂ) |
9 | simpll 790 | . . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑥 ∈ 𝐽) | |
10 | toponss 20731 | . . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ ℂ) | |
11 | 8, 9, 10 | sylancr 695 | . . . . . . . 8 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑥 ⊆ ℂ) |
12 | simplr 792 | . . . . . . . 8 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑦 ∈ 𝑥) | |
13 | 11, 12 | sseldd 3604 | . . . . . . 7 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑦 ∈ ℂ) |
14 | rpxr 11840 | . . . . . . . 8 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ*) | |
15 | 14 | ad2antrl 764 | . . . . . . 7 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑟 ∈ ℝ*) |
16 | 4 | blopn 22305 | . . . . . . 7 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑦 ∈ ℂ ∧ 𝑟 ∈ ℝ*) → (𝑦(ball‘(abs ∘ − ))𝑟) ∈ 𝐽) |
17 | 7, 13, 15, 16 | syl3anc 1326 | . . . . . 6 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑦(ball‘(abs ∘ − ))𝑟) ∈ 𝐽) |
18 | simprr 796 | . . . . . . 7 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥) | |
19 | vex 3203 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
20 | 19 | elpw2 4828 | . . . . . . 7 ⊢ ((𝑦(ball‘(abs ∘ − ))𝑟) ∈ 𝒫 𝑥 ↔ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥) |
21 | 18, 20 | sylibr 224 | . . . . . 6 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑦(ball‘(abs ∘ − ))𝑟) ∈ 𝒫 𝑥) |
22 | 17, 21 | elind 3798 | . . . . 5 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑦(ball‘(abs ∘ − ))𝑟) ∈ (𝐽 ∩ 𝒫 𝑥)) |
23 | simprl 794 | . . . . . 6 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑟 ∈ ℝ+) | |
24 | blcntr 22218 | . . . . . 6 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑦 ∈ ℂ ∧ 𝑟 ∈ ℝ+) → 𝑦 ∈ (𝑦(ball‘(abs ∘ − ))𝑟)) | |
25 | 7, 13, 23, 24 | syl3anc 1326 | . . . . 5 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑦 ∈ (𝑦(ball‘(abs ∘ − ))𝑟)) |
26 | eqid 2622 | . . . . . . 7 ⊢ (𝑦(ball‘(abs ∘ − ))𝑟) = (𝑦(ball‘(abs ∘ − ))𝑟) | |
27 | eqid 2622 | . . . . . . 7 ⊢ (𝐽 ↾t (𝑦(ball‘(abs ∘ − ))𝑟)) = (𝐽 ↾t (𝑦(ball‘(abs ∘ − ))𝑟)) | |
28 | 1, 26, 27 | blsconn 31226 | . . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑟 ∈ ℝ*) → (𝐽 ↾t (𝑦(ball‘(abs ∘ − ))𝑟)) ∈ SConn) |
29 | 13, 15, 28 | syl2anc 693 | . . . . 5 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝐽 ↾t (𝑦(ball‘(abs ∘ − ))𝑟)) ∈ SConn) |
30 | eleq2 2690 | . . . . . . 7 ⊢ (𝑢 = (𝑦(ball‘(abs ∘ − ))𝑟) → (𝑦 ∈ 𝑢 ↔ 𝑦 ∈ (𝑦(ball‘(abs ∘ − ))𝑟))) | |
31 | oveq2 6658 | . . . . . . . 8 ⊢ (𝑢 = (𝑦(ball‘(abs ∘ − ))𝑟) → (𝐽 ↾t 𝑢) = (𝐽 ↾t (𝑦(ball‘(abs ∘ − ))𝑟))) | |
32 | 31 | eleq1d 2686 | . . . . . . 7 ⊢ (𝑢 = (𝑦(ball‘(abs ∘ − ))𝑟) → ((𝐽 ↾t 𝑢) ∈ SConn ↔ (𝐽 ↾t (𝑦(ball‘(abs ∘ − ))𝑟)) ∈ SConn)) |
33 | 30, 32 | anbi12d 747 | . . . . . 6 ⊢ (𝑢 = (𝑦(ball‘(abs ∘ − ))𝑟) → ((𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ SConn) ↔ (𝑦 ∈ (𝑦(ball‘(abs ∘ − ))𝑟) ∧ (𝐽 ↾t (𝑦(ball‘(abs ∘ − ))𝑟)) ∈ SConn))) |
34 | 33 | rspcev 3309 | . . . . 5 ⊢ (((𝑦(ball‘(abs ∘ − ))𝑟) ∈ (𝐽 ∩ 𝒫 𝑥) ∧ (𝑦 ∈ (𝑦(ball‘(abs ∘ − ))𝑟) ∧ (𝐽 ↾t (𝑦(ball‘(abs ∘ − ))𝑟)) ∈ SConn)) → ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ SConn)) |
35 | 22, 25, 29, 34 | syl12anc 1324 | . . . 4 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ SConn)) |
36 | 6, 35 | rexlimddv 3035 | . . 3 ⊢ ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) → ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ SConn)) |
37 | 36 | rgen2 2975 | . 2 ⊢ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ SConn) |
38 | islly 21271 | . 2 ⊢ (𝐽 ∈ Locally SConn ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ SConn))) | |
39 | 2, 37, 38 | mpbir2an 955 | 1 ⊢ 𝐽 ∈ Locally SConn |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 ∩ cin 3573 ⊆ wss 3574 𝒫 cpw 4158 ∘ ccom 5118 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝ*cxr 10073 − cmin 10266 ℝ+crp 11832 abscabs 13974 ↾t crest 16081 TopOpenctopn 16082 ∞Metcxmt 19731 ballcbl 19733 ℂfldccnfld 19746 Topctop 20698 TopOnctopon 20715 Locally clly 21267 SConncsconn 31202 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cn 21031 df-cnp 21032 df-lly 21269 df-tx 21365 df-hmeo 21558 df-xms 22125 df-ms 22126 df-tms 22127 df-ii 22680 df-htpy 22769 df-phtpy 22770 df-phtpc 22791 df-pconn 31203 df-sconn 31204 |
This theorem is referenced by: (None) |
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