Mathbox for Mario Carneiro |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > blsconn | Structured version Visualization version GIF version |
Description: An open ball in the complex numbers is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.) |
Ref | Expression |
---|---|
blsconn.j | ⊢ 𝐽 = (TopOpen‘ℂfld) |
blsconn.s | ⊢ 𝑆 = (𝑃(ball‘(abs ∘ − ))𝑅) |
blsconn.k | ⊢ 𝐾 = (𝐽 ↾t 𝑆) |
Ref | Expression |
---|---|
blsconn | ⊢ ((𝑃 ∈ ℂ ∧ 𝑅 ∈ ℝ*) → 𝐾 ∈ SConn) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | blsconn.s | . . 3 ⊢ 𝑆 = (𝑃(ball‘(abs ∘ − ))𝑅) | |
2 | cnxmet 23375 | . . . 4 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ) | |
3 | blssm 23022 | . . . 4 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑃 ∈ ℂ ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘(abs ∘ − ))𝑅) ⊆ ℂ) | |
4 | 2, 3 | mp3an1 1444 | . . 3 ⊢ ((𝑃 ∈ ℂ ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘(abs ∘ − ))𝑅) ⊆ ℂ) |
5 | 1, 4 | eqsstrid 4014 | . 2 ⊢ ((𝑃 ∈ ℂ ∧ 𝑅 ∈ ℝ*) → 𝑆 ⊆ ℂ) |
6 | 1 | blcvx 23400 | . 2 ⊢ (((𝑃 ∈ ℂ ∧ 𝑅 ∈ ℝ*) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑡 ∈ (0[,]1))) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆) |
7 | blsconn.j | . 2 ⊢ 𝐽 = (TopOpen‘ℂfld) | |
8 | blsconn.k | . 2 ⊢ 𝐾 = (𝐽 ↾t 𝑆) | |
9 | 5, 6, 7, 8 | cvxsconn 32485 | 1 ⊢ ((𝑃 ∈ ℂ ∧ 𝑅 ∈ ℝ*) → 𝐾 ∈ SConn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ∘ ccom 5553 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 ℝ*cxr 10668 − cmin 10864 abscabs 14587 ↾t crest 16688 TopOpenctopn 16689 ∞Metcxmet 20524 ballcbl 20526 ℂfldccnfld 20539 SConncsconn 32462 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-icc 12739 df-fz 12887 df-fzo 13028 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-rest 16690 df-topn 16691 df-0g 16709 df-gsum 16710 df-topgen 16711 df-pt 16712 df-prds 16715 df-xrs 16769 df-qtop 16774 df-imas 16775 df-xps 16777 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-mulg 18219 df-cntz 18441 df-cmn 18902 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-cnfld 20540 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-cn 21829 df-cnp 21830 df-tx 22164 df-hmeo 22357 df-xms 22924 df-ms 22925 df-tms 22926 df-ii 23479 df-htpy 23568 df-phtpy 23569 df-phtpc 23590 df-pconn 32463 df-sconn 32464 |
This theorem is referenced by: cnllysconn 32487 |
Copyright terms: Public domain | W3C validator |