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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnllyscon | 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 |
---|---|
cnllyscon.j | ⊢ 𝐽 = (TopOpen‘ℂfld) |
Ref | Expression |
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cnllyscon | ⊢ 𝐽 ∈ Locally SCon |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnllyscon.j | . . 3 ⊢ 𝐽 = (TopOpen‘ℂfld) | |
2 | 1 | cnfldtop 22397 | . 2 ⊢ 𝐽 ∈ Top |
3 | cnxmet 22386 | . . . . 5 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ) | |
4 | 1 | cnfldtopn 22395 | . . . . . 6 ⊢ 𝐽 = (MetOpen‘(abs ∘ − )) |
5 | 4 | mopni2 22108 | . . . . 5 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) → ∃𝑟 ∈ ℝ+ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥) |
6 | 3, 5 | mp3an1 1403 | . . . 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 22396 | . . . . . . . . 9 ⊢ 𝐽 ∈ (TopOn‘ℂ) |
9 | simpll 786 | . . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑥 ∈ 𝐽) | |
10 | toponss 20544 | . . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ ℂ) | |
11 | 8, 9, 10 | sylancr 694 | . . . . . . . 8 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑥 ⊆ ℂ) |
12 | simplr 788 | . . . . . . . 8 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑦 ∈ 𝑥) | |
13 | 11, 12 | sseldd 3569 | . . . . . . 7 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑦 ∈ ℂ) |
14 | rpxr 11716 | . . . . . . . 8 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ*) | |
15 | 14 | ad2antrl 760 | . . . . . . 7 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑟 ∈ ℝ*) |
16 | 4 | blopn 22115 | . . . . . . 7 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑦 ∈ ℂ ∧ 𝑟 ∈ ℝ*) → (𝑦(ball‘(abs ∘ − ))𝑟) ∈ 𝐽) |
17 | 7, 13, 15, 16 | syl3anc 1318 | . . . . . 6 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑦(ball‘(abs ∘ − ))𝑟) ∈ 𝐽) |
18 | simprr 792 | . . . . . . 7 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥) | |
19 | vex 3176 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
20 | 19 | elpw2 4755 | . . . . . . 7 ⊢ ((𝑦(ball‘(abs ∘ − ))𝑟) ∈ 𝒫 𝑥 ↔ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥) |
21 | 18, 20 | sylibr 223 | . . . . . 6 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑦(ball‘(abs ∘ − ))𝑟) ∈ 𝒫 𝑥) |
22 | 17, 21 | elind 3760 | . . . . 5 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑦(ball‘(abs ∘ − ))𝑟) ∈ (𝐽 ∩ 𝒫 𝑥)) |
23 | simprl 790 | . . . . . 6 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑟 ∈ ℝ+) | |
24 | blcntr 22028 | . . . . . 6 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑦 ∈ ℂ ∧ 𝑟 ∈ ℝ+) → 𝑦 ∈ (𝑦(ball‘(abs ∘ − ))𝑟)) | |
25 | 7, 13, 23, 24 | syl3anc 1318 | . . . . 5 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑦 ∈ (𝑦(ball‘(abs ∘ − ))𝑟)) |
26 | eqid 2610 | . . . . . . 7 ⊢ (𝑦(ball‘(abs ∘ − ))𝑟) = (𝑦(ball‘(abs ∘ − ))𝑟) | |
27 | eqid 2610 | . . . . . . 7 ⊢ (𝐽 ↾t (𝑦(ball‘(abs ∘ − ))𝑟)) = (𝐽 ↾t (𝑦(ball‘(abs ∘ − ))𝑟)) | |
28 | 1, 26, 27 | blscon 30480 | . . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑟 ∈ ℝ*) → (𝐽 ↾t (𝑦(ball‘(abs ∘ − ))𝑟)) ∈ SCon) |
29 | 13, 15, 28 | syl2anc 691 | . . . . 5 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝐽 ↾t (𝑦(ball‘(abs ∘ − ))𝑟)) ∈ SCon) |
30 | eleq2 2677 | . . . . . . 7 ⊢ (𝑢 = (𝑦(ball‘(abs ∘ − ))𝑟) → (𝑦 ∈ 𝑢 ↔ 𝑦 ∈ (𝑦(ball‘(abs ∘ − ))𝑟))) | |
31 | oveq2 6557 | . . . . . . . 8 ⊢ (𝑢 = (𝑦(ball‘(abs ∘ − ))𝑟) → (𝐽 ↾t 𝑢) = (𝐽 ↾t (𝑦(ball‘(abs ∘ − ))𝑟))) | |
32 | 31 | eleq1d 2672 | . . . . . . 7 ⊢ (𝑢 = (𝑦(ball‘(abs ∘ − ))𝑟) → ((𝐽 ↾t 𝑢) ∈ SCon ↔ (𝐽 ↾t (𝑦(ball‘(abs ∘ − ))𝑟)) ∈ SCon)) |
33 | 30, 32 | anbi12d 743 | . . . . . 6 ⊢ (𝑢 = (𝑦(ball‘(abs ∘ − ))𝑟) → ((𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ SCon) ↔ (𝑦 ∈ (𝑦(ball‘(abs ∘ − ))𝑟) ∧ (𝐽 ↾t (𝑦(ball‘(abs ∘ − ))𝑟)) ∈ SCon))) |
34 | 33 | rspcev 3282 | . . . . 5 ⊢ (((𝑦(ball‘(abs ∘ − ))𝑟) ∈ (𝐽 ∩ 𝒫 𝑥) ∧ (𝑦 ∈ (𝑦(ball‘(abs ∘ − ))𝑟) ∧ (𝐽 ↾t (𝑦(ball‘(abs ∘ − ))𝑟)) ∈ SCon)) → ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ SCon)) |
35 | 22, 25, 29, 34 | syl12anc 1316 | . . . 4 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ SCon)) |
36 | 6, 35 | rexlimddv 3017 | . . 3 ⊢ ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) → ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ SCon)) |
37 | 36 | rgen2 2958 | . 2 ⊢ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ SCon) |
38 | islly 21081 | . 2 ⊢ (𝐽 ∈ Locally SCon ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ SCon))) | |
39 | 2, 37, 38 | mpbir2an 957 | 1 ⊢ 𝐽 ∈ Locally SCon |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 ∩ cin 3539 ⊆ wss 3540 𝒫 cpw 4108 ∘ ccom 5042 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝ*cxr 9952 − cmin 10145 ℝ+crp 11708 abscabs 13822 ↾t crest 15904 TopOpenctopn 15905 ∞Metcxmt 19552 ballcbl 19554 ℂfldccnfld 19567 Topctop 20517 TopOnctopon 20518 Locally clly 21077 SConcscon 30456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-icc 12053 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cn 20841 df-cnp 20842 df-lly 21079 df-tx 21175 df-hmeo 21368 df-xms 21935 df-ms 21936 df-tms 21937 df-ii 22488 df-htpy 22577 df-phtpy 22578 df-phtpc 22599 df-pcon 30457 df-scon 30458 |
This theorem is referenced by: (None) |
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