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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 23297 | . 2 ⊢ (topGen‘ran (,)) ∈ Top | |
2 | tg2 21501 | . . . 4 ⊢ ((𝑥 ∈ (topGen‘ran (,)) ∧ 𝑦 ∈ 𝑥) → ∃𝑧 ∈ ran (,)(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥)) | |
3 | retopbas 23296 | . . . . . . . . . 10 ⊢ ran (,) ∈ TopBases | |
4 | bastg 21502 | . . . . . . . . . 10 ⊢ (ran (,) ∈ TopBases → ran (,) ⊆ (topGen‘ran (,))) | |
5 | 3, 4 | ax-mp 5 | . . . . . . . . 9 ⊢ ran (,) ⊆ (topGen‘ran (,)) |
6 | simprl 767 | . . . . . . . . 9 ⊢ (((𝑥 ∈ (topGen‘ran (,)) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ran (,) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))) → 𝑧 ∈ ran (,)) | |
7 | 5, 6 | sseldi 3962 | . . . . . . . 8 ⊢ (((𝑥 ∈ (topGen‘ran (,)) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ran (,) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))) → 𝑧 ∈ (topGen‘ran (,))) |
8 | simprrr 778 | . . . . . . . . 9 ⊢ (((𝑥 ∈ (topGen‘ran (,)) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ran (,) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))) → 𝑧 ⊆ 𝑥) | |
9 | velpw 4543 | . . . . . . . . 9 ⊢ (𝑧 ∈ 𝒫 𝑥 ↔ 𝑧 ⊆ 𝑥) | |
10 | 8, 9 | sylibr 235 | . . . . . . . 8 ⊢ (((𝑥 ∈ (topGen‘ran (,)) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ran (,) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))) → 𝑧 ∈ 𝒫 𝑥) |
11 | 7, 10 | elind 4168 | . . . . . . 7 ⊢ (((𝑥 ∈ (topGen‘ran (,)) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ran (,) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))) → 𝑧 ∈ ((topGen‘ran (,)) ∩ 𝒫 𝑥)) |
12 | simprrl 777 | . . . . . . 7 ⊢ (((𝑥 ∈ (topGen‘ran (,)) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ran (,) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))) → 𝑦 ∈ 𝑧) | |
13 | ioof 12823 | . . . . . . . . . 10 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
14 | ffn 6507 | . . . . . . . . . 10 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*)) | |
15 | ovelrn 7313 | . . . . . . . . . 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 7153 | . . . . . . . . . . . 12 ⊢ (𝑧 = (𝑎(,)𝑏) → ((topGen‘ran (,)) ↾t 𝑧) = ((topGen‘ran (,)) ↾t (𝑎(,)𝑏))) | |
18 | ioosconn 32391 | . . . . . . . . . . . 12 ⊢ ((topGen‘ran (,)) ↾t (𝑎(,)𝑏)) ∈ SConn | |
19 | 17, 18 | syl6eqel 2918 | . . . . . . . . . . 11 ⊢ (𝑧 = (𝑎(,)𝑏) → ((topGen‘ran (,)) ↾t 𝑧) ∈ SConn) |
20 | 19 | rexlimivw 3279 | . . . . . . . . . 10 ⊢ (∃𝑏 ∈ ℝ* 𝑧 = (𝑎(,)𝑏) → ((topGen‘ran (,)) ↾t 𝑧) ∈ SConn) |
21 | 20 | rexlimivw 3279 | . . . . . . . . 9 ⊢ (∃𝑎 ∈ ℝ* ∃𝑏 ∈ ℝ* 𝑧 = (𝑎(,)𝑏) → ((topGen‘ran (,)) ↾t 𝑧) ∈ SConn) |
22 | 16, 21 | sylbi 218 | . . . . . . . 8 ⊢ (𝑧 ∈ ran (,) → ((topGen‘ran (,)) ↾t 𝑧) ∈ SConn) |
23 | 22 | ad2antrl 724 | . . . . . . 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 3268 | . . . 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 3200 | . 2 ⊢ ∀𝑥 ∈ (topGen‘ran (,))∀𝑦 ∈ 𝑥 ∃𝑧 ∈ ((topGen‘ran (,)) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑧 ∧ ((topGen‘ran (,)) ↾t 𝑧) ∈ SConn) |
29 | islly 22004 | . 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 707 | 1 ⊢ (topGen‘ran (,)) ∈ Locally SConn |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ∃wrex 3136 ∩ cin 3932 ⊆ wss 3933 𝒫 cpw 4535 × cxp 5546 ran crn 5549 Fn wfn 6343 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ℝcr 10524 ℝ*cxr 10662 (,)cioo 12726 ↾t crest 16682 topGenctg 16699 Topctop 21429 TopBasesctb 21481 Locally clly 22000 SConncsconn 32364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12881 df-fzo 13022 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-rest 16684 df-topn 16685 df-0g 16703 df-gsum 16704 df-topgen 16705 df-pt 16706 df-prds 16709 df-xrs 16763 df-qtop 16768 df-imas 16769 df-xps 16771 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-mulg 18163 df-cntz 18385 df-cmn 18837 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-cnfld 20474 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cld 21555 df-cn 21763 df-cnp 21764 df-conn 21948 df-lly 22002 df-tx 22098 df-hmeo 22291 df-xms 22857 df-ms 22858 df-tms 22859 df-ii 23412 df-htpy 23501 df-phtpy 23502 df-phtpc 23523 df-pconn 32365 df-sconn 32366 |
This theorem is referenced by: (None) |
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