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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 23392 | . 2 ⊢ 𝐽 ∈ Top |
3 | cnxmet 23381 | . . . . 5 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ) | |
4 | 1 | cnfldtopn 23390 | . . . . . 6 ⊢ 𝐽 = (MetOpen‘(abs ∘ − )) |
5 | 4 | mopni2 23103 | . . . . 5 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) → ∃𝑟 ∈ ℝ+ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥) |
6 | 3, 5 | mp3an1 1444 | . . . 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 23391 | . . . . . . . . 9 ⊢ 𝐽 ∈ (TopOn‘ℂ) |
9 | simpll 765 | . . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑥 ∈ 𝐽) | |
10 | toponss 21535 | . . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ ℂ) | |
11 | 8, 9, 10 | sylancr 589 | . . . . . . . 8 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑥 ⊆ ℂ) |
12 | simplr 767 | . . . . . . . 8 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑦 ∈ 𝑥) | |
13 | 11, 12 | sseldd 3968 | . . . . . . 7 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑦 ∈ ℂ) |
14 | rpxr 12399 | . . . . . . . 8 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ*) | |
15 | 14 | ad2antrl 726 | . . . . . . 7 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑟 ∈ ℝ*) |
16 | 4 | blopn 23110 | . . . . . . 7 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑦 ∈ ℂ ∧ 𝑟 ∈ ℝ*) → (𝑦(ball‘(abs ∘ − ))𝑟) ∈ 𝐽) |
17 | 7, 13, 15, 16 | syl3anc 1367 | . . . . . 6 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑦(ball‘(abs ∘ − ))𝑟) ∈ 𝐽) |
18 | simprr 771 | . . . . . . 7 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥) | |
19 | vex 3497 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
20 | 19 | elpw2 5248 | . . . . . . 7 ⊢ ((𝑦(ball‘(abs ∘ − ))𝑟) ∈ 𝒫 𝑥 ↔ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥) |
21 | 18, 20 | sylibr 236 | . . . . . 6 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑦(ball‘(abs ∘ − ))𝑟) ∈ 𝒫 𝑥) |
22 | 17, 21 | elind 4171 | . . . . 5 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝑦(ball‘(abs ∘ − ))𝑟) ∈ (𝐽 ∩ 𝒫 𝑥)) |
23 | simprl 769 | . . . . . 6 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑟 ∈ ℝ+) | |
24 | blcntr 23023 | . . . . . 6 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑦 ∈ ℂ ∧ 𝑟 ∈ ℝ+) → 𝑦 ∈ (𝑦(ball‘(abs ∘ − ))𝑟)) | |
25 | 7, 13, 23, 24 | syl3anc 1367 | . . . . 5 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → 𝑦 ∈ (𝑦(ball‘(abs ∘ − ))𝑟)) |
26 | eqid 2821 | . . . . . . 7 ⊢ (𝑦(ball‘(abs ∘ − ))𝑟) = (𝑦(ball‘(abs ∘ − ))𝑟) | |
27 | eqid 2821 | . . . . . . 7 ⊢ (𝐽 ↾t (𝑦(ball‘(abs ∘ − ))𝑟)) = (𝐽 ↾t (𝑦(ball‘(abs ∘ − ))𝑟)) | |
28 | 1, 26, 27 | blsconn 32491 | . . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑟 ∈ ℝ*) → (𝐽 ↾t (𝑦(ball‘(abs ∘ − ))𝑟)) ∈ SConn) |
29 | 13, 15, 28 | syl2anc 586 | . . . . 5 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑟 ∈ ℝ+ ∧ (𝑦(ball‘(abs ∘ − ))𝑟) ⊆ 𝑥)) → (𝐽 ↾t (𝑦(ball‘(abs ∘ − ))𝑟)) ∈ SConn) |
30 | eleq2 2901 | . . . . . . 7 ⊢ (𝑢 = (𝑦(ball‘(abs ∘ − ))𝑟) → (𝑦 ∈ 𝑢 ↔ 𝑦 ∈ (𝑦(ball‘(abs ∘ − ))𝑟))) | |
31 | oveq2 7164 | . . . . . . . 8 ⊢ (𝑢 = (𝑦(ball‘(abs ∘ − ))𝑟) → (𝐽 ↾t 𝑢) = (𝐽 ↾t (𝑦(ball‘(abs ∘ − ))𝑟))) | |
32 | 31 | eleq1d 2897 | . . . . . . 7 ⊢ (𝑢 = (𝑦(ball‘(abs ∘ − ))𝑟) → ((𝐽 ↾t 𝑢) ∈ SConn ↔ (𝐽 ↾t (𝑦(ball‘(abs ∘ − ))𝑟)) ∈ SConn)) |
33 | 30, 32 | anbi12d 632 | . . . . . 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 3623 | . . . . 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 834 | . . . 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 3291 | . . 3 ⊢ ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) → ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ SConn)) |
37 | 36 | rgen2 3203 | . 2 ⊢ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ SConn) |
38 | islly 22076 | . 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 709 | 1 ⊢ 𝐽 ∈ Locally SConn |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 ∩ cin 3935 ⊆ wss 3936 𝒫 cpw 4539 ∘ ccom 5559 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝ*cxr 10674 − cmin 10870 ℝ+crp 12390 abscabs 14593 ↾t crest 16694 TopOpenctopn 16695 ∞Metcxmet 20530 ballcbl 20532 ℂfldccnfld 20545 Topctop 21501 TopOnctopon 21518 Locally clly 22072 SConncsconn 32467 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-icc 12746 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cn 21835 df-cnp 21836 df-lly 22074 df-tx 22170 df-hmeo 22363 df-xms 22930 df-ms 22931 df-tms 22932 df-ii 23485 df-htpy 23574 df-phtpy 23575 df-phtpc 23596 df-pconn 32468 df-sconn 32469 |
This theorem is referenced by: (None) |
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