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Mirrors > Home > MPE Home > Th. List > Mathboxes > rellysconn | Structured version Visualization version GIF version |
Description: The real numbers are locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.) |
Ref | Expression |
---|---|
rellysconn | ⊢ (topGen‘ran (,)) ∈ Locally SConn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | retop 24109 | . 2 ⊢ (topGen‘ran (,)) ∈ Top | |
2 | tg2 22299 | . . . 4 ⊢ ((𝑥 ∈ (topGen‘ran (,)) ∧ 𝑦 ∈ 𝑥) → ∃𝑧 ∈ ran (,)(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥)) | |
3 | retopbas 24108 | . . . . . . . . . 10 ⊢ ran (,) ∈ TopBases | |
4 | bastg 22300 | . . . . . . . . . 10 ⊢ (ran (,) ∈ TopBases → ran (,) ⊆ (topGen‘ran (,))) | |
5 | 3, 4 | ax-mp 5 | . . . . . . . . 9 ⊢ ran (,) ⊆ (topGen‘ran (,)) |
6 | simprl 769 | . . . . . . . . 9 ⊢ (((𝑥 ∈ (topGen‘ran (,)) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ran (,) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))) → 𝑧 ∈ ran (,)) | |
7 | 5, 6 | sselid 3940 | . . . . . . . 8 ⊢ (((𝑥 ∈ (topGen‘ran (,)) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ran (,) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))) → 𝑧 ∈ (topGen‘ran (,))) |
8 | simprrr 780 | . . . . . . . . 9 ⊢ (((𝑥 ∈ (topGen‘ran (,)) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ran (,) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))) → 𝑧 ⊆ 𝑥) | |
9 | velpw 4563 | . . . . . . . . 9 ⊢ (𝑧 ∈ 𝒫 𝑥 ↔ 𝑧 ⊆ 𝑥) | |
10 | 8, 9 | sylibr 233 | . . . . . . . 8 ⊢ (((𝑥 ∈ (topGen‘ran (,)) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ran (,) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))) → 𝑧 ∈ 𝒫 𝑥) |
11 | 7, 10 | elind 4152 | . . . . . . 7 ⊢ (((𝑥 ∈ (topGen‘ran (,)) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ran (,) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))) → 𝑧 ∈ ((topGen‘ran (,)) ∩ 𝒫 𝑥)) |
12 | simprrl 779 | . . . . . . 7 ⊢ (((𝑥 ∈ (topGen‘ran (,)) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ran (,) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))) → 𝑦 ∈ 𝑧) | |
13 | ioof 13356 | . . . . . . . . . 10 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
14 | ffn 6665 | . . . . . . . . . 10 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*)) | |
15 | ovelrn 7526 | . . . . . . . . . 10 ⊢ ((,) Fn (ℝ* × ℝ*) → (𝑧 ∈ ran (,) ↔ ∃𝑎 ∈ ℝ* ∃𝑏 ∈ ℝ* 𝑧 = (𝑎(,)𝑏))) | |
16 | 13, 14, 15 | mp2b 10 | . . . . . . . . 9 ⊢ (𝑧 ∈ ran (,) ↔ ∃𝑎 ∈ ℝ* ∃𝑏 ∈ ℝ* 𝑧 = (𝑎(,)𝑏)) |
17 | oveq2 7361 | . . . . . . . . . . . 12 ⊢ (𝑧 = (𝑎(,)𝑏) → ((topGen‘ran (,)) ↾t 𝑧) = ((topGen‘ran (,)) ↾t (𝑎(,)𝑏))) | |
18 | ioosconn 33710 | . . . . . . . . . . . 12 ⊢ ((topGen‘ran (,)) ↾t (𝑎(,)𝑏)) ∈ SConn | |
19 | 17, 18 | eqeltrdi 2846 | . . . . . . . . . . 11 ⊢ (𝑧 = (𝑎(,)𝑏) → ((topGen‘ran (,)) ↾t 𝑧) ∈ SConn) |
20 | 19 | rexlimivw 3146 | . . . . . . . . . 10 ⊢ (∃𝑏 ∈ ℝ* 𝑧 = (𝑎(,)𝑏) → ((topGen‘ran (,)) ↾t 𝑧) ∈ SConn) |
21 | 20 | rexlimivw 3146 | . . . . . . . . 9 ⊢ (∃𝑎 ∈ ℝ* ∃𝑏 ∈ ℝ* 𝑧 = (𝑎(,)𝑏) → ((topGen‘ran (,)) ↾t 𝑧) ∈ SConn) |
22 | 16, 21 | sylbi 216 | . . . . . . . 8 ⊢ (𝑧 ∈ ran (,) → ((topGen‘ran (,)) ↾t 𝑧) ∈ SConn) |
23 | 22 | ad2antrl 726 | . . . . . . 7 ⊢ (((𝑥 ∈ (topGen‘ran (,)) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ran (,) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))) → ((topGen‘ran (,)) ↾t 𝑧) ∈ SConn) |
24 | 11, 12, 23 | jca32 516 | . . . . . 6 ⊢ (((𝑥 ∈ (topGen‘ran (,)) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ran (,) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))) → (𝑧 ∈ ((topGen‘ran (,)) ∩ 𝒫 𝑥) ∧ (𝑦 ∈ 𝑧 ∧ ((topGen‘ran (,)) ↾t 𝑧) ∈ SConn))) |
25 | 24 | ex 413 | . . . . 5 ⊢ ((𝑥 ∈ (topGen‘ran (,)) ∧ 𝑦 ∈ 𝑥) → ((𝑧 ∈ ran (,) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥)) → (𝑧 ∈ ((topGen‘ran (,)) ∩ 𝒫 𝑥) ∧ (𝑦 ∈ 𝑧 ∧ ((topGen‘ran (,)) ↾t 𝑧) ∈ SConn)))) |
26 | 25 | reximdv2 3159 | . . . 4 ⊢ ((𝑥 ∈ (topGen‘ran (,)) ∧ 𝑦 ∈ 𝑥) → (∃𝑧 ∈ ran (,)(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥) → ∃𝑧 ∈ ((topGen‘ran (,)) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑧 ∧ ((topGen‘ran (,)) ↾t 𝑧) ∈ SConn))) |
27 | 2, 26 | mpd 15 | . . 3 ⊢ ((𝑥 ∈ (topGen‘ran (,)) ∧ 𝑦 ∈ 𝑥) → ∃𝑧 ∈ ((topGen‘ran (,)) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑧 ∧ ((topGen‘ran (,)) ↾t 𝑧) ∈ SConn)) |
28 | 27 | rgen2 3192 | . 2 ⊢ ∀𝑥 ∈ (topGen‘ran (,))∀𝑦 ∈ 𝑥 ∃𝑧 ∈ ((topGen‘ran (,)) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑧 ∧ ((topGen‘ran (,)) ↾t 𝑧) ∈ SConn) |
29 | islly 22803 | . 2 ⊢ ((topGen‘ran (,)) ∈ Locally SConn ↔ ((topGen‘ran (,)) ∈ Top ∧ ∀𝑥 ∈ (topGen‘ran (,))∀𝑦 ∈ 𝑥 ∃𝑧 ∈ ((topGen‘ran (,)) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑧 ∧ ((topGen‘ran (,)) ↾t 𝑧) ∈ SConn))) | |
30 | 1, 28, 29 | mpbir2an 709 | 1 ⊢ (topGen‘ran (,)) ∈ Locally SConn |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3062 ∃wrex 3071 ∩ cin 3907 ⊆ wss 3908 𝒫 cpw 4558 × cxp 5629 ran crn 5632 Fn wfn 6488 ⟶wf 6489 ‘cfv 6493 (class class class)co 7353 ℝcr 11046 ℝ*cxr 11184 (,)cioo 13256 ↾t crest 17294 topGenctg 17311 Topctop 22226 TopBasesctb 22279 Locally clly 22799 SConncsconn 33683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 ax-pre-sup 11125 ax-addf 11126 ax-mulf 11127 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7613 df-om 7799 df-1st 7917 df-2nd 7918 df-supp 8089 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-1o 8408 df-2o 8409 df-er 8644 df-map 8763 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9302 df-fi 9343 df-sup 9374 df-inf 9375 df-oi 9442 df-card 9871 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-div 11809 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12410 df-z 12496 df-dec 12615 df-uz 12760 df-q 12866 df-rp 12908 df-xneg 13025 df-xadd 13026 df-xmul 13027 df-ioo 13260 df-ico 13262 df-icc 13263 df-fz 13417 df-fzo 13560 df-seq 13899 df-exp 13960 df-hash 14223 df-cj 14976 df-re 14977 df-im 14978 df-sqrt 15112 df-abs 15113 df-struct 17011 df-sets 17028 df-slot 17046 df-ndx 17058 df-base 17076 df-ress 17105 df-plusg 17138 df-mulr 17139 df-starv 17140 df-sca 17141 df-vsca 17142 df-ip 17143 df-tset 17144 df-ple 17145 df-ds 17147 df-unif 17148 df-hom 17149 df-cco 17150 df-rest 17296 df-topn 17297 df-0g 17315 df-gsum 17316 df-topgen 17317 df-pt 17318 df-prds 17321 df-xrs 17376 df-qtop 17381 df-imas 17382 df-xps 17384 df-mre 17458 df-mrc 17459 df-acs 17461 df-mgm 18489 df-sgrp 18538 df-mnd 18549 df-submnd 18594 df-mulg 18864 df-cntz 19088 df-cmn 19555 df-psmet 20773 df-xmet 20774 df-met 20775 df-bl 20776 df-mopn 20777 df-cnfld 20782 df-top 22227 df-topon 22244 df-topsp 22266 df-bases 22280 df-cld 22354 df-cn 22562 df-cnp 22563 df-conn 22747 df-lly 22801 df-tx 22897 df-hmeo 23090 df-xms 23657 df-ms 23658 df-tms 23659 df-ii 24224 df-htpy 24317 df-phtpy 24318 df-phtpc 24339 df-pconn 33684 df-sconn 33685 |
This theorem is referenced by: (None) |
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