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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 23631 | . 2 ⊢ (topGen‘ran (,)) ∈ Top | |
2 | tg2 21834 | . . . 4 ⊢ ((𝑥 ∈ (topGen‘ran (,)) ∧ 𝑦 ∈ 𝑥) → ∃𝑧 ∈ ran (,)(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥)) | |
3 | retopbas 23630 | . . . . . . . . . 10 ⊢ ran (,) ∈ TopBases | |
4 | bastg 21835 | . . . . . . . . . 10 ⊢ (ran (,) ∈ TopBases → ran (,) ⊆ (topGen‘ran (,))) | |
5 | 3, 4 | ax-mp 5 | . . . . . . . . 9 ⊢ ran (,) ⊆ (topGen‘ran (,)) |
6 | simprl 771 | . . . . . . . . 9 ⊢ (((𝑥 ∈ (topGen‘ran (,)) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ran (,) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))) → 𝑧 ∈ ran (,)) | |
7 | 5, 6 | sseldi 3889 | . . . . . . . 8 ⊢ (((𝑥 ∈ (topGen‘ran (,)) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ran (,) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))) → 𝑧 ∈ (topGen‘ran (,))) |
8 | simprrr 782 | . . . . . . . . 9 ⊢ (((𝑥 ∈ (topGen‘ran (,)) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ran (,) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))) → 𝑧 ⊆ 𝑥) | |
9 | velpw 4508 | . . . . . . . . 9 ⊢ (𝑧 ∈ 𝒫 𝑥 ↔ 𝑧 ⊆ 𝑥) | |
10 | 8, 9 | sylibr 237 | . . . . . . . 8 ⊢ (((𝑥 ∈ (topGen‘ran (,)) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ran (,) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))) → 𝑧 ∈ 𝒫 𝑥) |
11 | 7, 10 | elind 4098 | . . . . . . 7 ⊢ (((𝑥 ∈ (topGen‘ran (,)) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ran (,) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))) → 𝑧 ∈ ((topGen‘ran (,)) ∩ 𝒫 𝑥)) |
12 | simprrl 781 | . . . . . . 7 ⊢ (((𝑥 ∈ (topGen‘ran (,)) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ran (,) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))) → 𝑦 ∈ 𝑧) | |
13 | ioof 13018 | . . . . . . . . . 10 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
14 | ffn 6534 | . . . . . . . . . 10 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*)) | |
15 | ovelrn 7373 | . . . . . . . . . 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 7210 | . . . . . . . . . . . 12 ⊢ (𝑧 = (𝑎(,)𝑏) → ((topGen‘ran (,)) ↾t 𝑧) = ((topGen‘ran (,)) ↾t (𝑎(,)𝑏))) | |
18 | ioosconn 32894 | . . . . . . . . . . . 12 ⊢ ((topGen‘ran (,)) ↾t (𝑎(,)𝑏)) ∈ SConn | |
19 | 17, 18 | eqeltrdi 2842 | . . . . . . . . . . 11 ⊢ (𝑧 = (𝑎(,)𝑏) → ((topGen‘ran (,)) ↾t 𝑧) ∈ SConn) |
20 | 19 | rexlimivw 3194 | . . . . . . . . . 10 ⊢ (∃𝑏 ∈ ℝ* 𝑧 = (𝑎(,)𝑏) → ((topGen‘ran (,)) ↾t 𝑧) ∈ SConn) |
21 | 20 | rexlimivw 3194 | . . . . . . . . 9 ⊢ (∃𝑎 ∈ ℝ* ∃𝑏 ∈ ℝ* 𝑧 = (𝑎(,)𝑏) → ((topGen‘ran (,)) ↾t 𝑧) ∈ SConn) |
22 | 16, 21 | sylbi 220 | . . . . . . . 8 ⊢ (𝑧 ∈ ran (,) → ((topGen‘ran (,)) ↾t 𝑧) ∈ SConn) |
23 | 22 | ad2antrl 728 | . . . . . . 7 ⊢ (((𝑥 ∈ (topGen‘ran (,)) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ran (,) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))) → ((topGen‘ran (,)) ↾t 𝑧) ∈ SConn) |
24 | 11, 12, 23 | jca32 519 | . . . . . 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 416 | . . . . 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 3183 | . . . 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 3117 | . 2 ⊢ ∀𝑥 ∈ (topGen‘ran (,))∀𝑦 ∈ 𝑥 ∃𝑧 ∈ ((topGen‘ran (,)) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑧 ∧ ((topGen‘ran (,)) ↾t 𝑧) ∈ SConn) |
29 | islly 22337 | . 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 711 | 1 ⊢ (topGen‘ran (,)) ∈ Locally SConn |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3054 ∃wrex 3055 ∩ cin 3856 ⊆ wss 3857 𝒫 cpw 4503 × cxp 5538 ran crn 5541 Fn wfn 6364 ⟶wf 6365 ‘cfv 6369 (class class class)co 7202 ℝcr 10711 ℝ*cxr 10849 (,)cioo 12918 ↾t crest 16897 topGenctg 16914 Topctop 21762 TopBasesctb 21814 Locally clly 22333 SConncsconn 32867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 ax-addf 10791 ax-mulf 10792 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-iin 4897 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-of 7458 df-om 7634 df-1st 7750 df-2nd 7751 df-supp 7893 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-2o 8192 df-er 8380 df-map 8499 df-ixp 8568 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-fsupp 8975 df-fi 9016 df-sup 9047 df-inf 9048 df-oi 9115 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-dec 12277 df-uz 12422 df-q 12528 df-rp 12570 df-xneg 12687 df-xadd 12688 df-xmul 12689 df-ioo 12922 df-ico 12924 df-icc 12925 df-fz 13079 df-fzo 13222 df-seq 13558 df-exp 13619 df-hash 13880 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-starv 16782 df-sca 16783 df-vsca 16784 df-ip 16785 df-tset 16786 df-ple 16787 df-ds 16789 df-unif 16790 df-hom 16791 df-cco 16792 df-rest 16899 df-topn 16900 df-0g 16918 df-gsum 16919 df-topgen 16920 df-pt 16921 df-prds 16924 df-xrs 16979 df-qtop 16984 df-imas 16985 df-xps 16987 df-mre 17061 df-mrc 17062 df-acs 17064 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-submnd 18191 df-mulg 18461 df-cntz 18683 df-cmn 19144 df-psmet 20327 df-xmet 20328 df-met 20329 df-bl 20330 df-mopn 20331 df-cnfld 20336 df-top 21763 df-topon 21780 df-topsp 21802 df-bases 21815 df-cld 21888 df-cn 22096 df-cnp 22097 df-conn 22281 df-lly 22335 df-tx 22431 df-hmeo 22624 df-xms 23190 df-ms 23191 df-tms 23192 df-ii 23746 df-htpy 23839 df-phtpy 23840 df-phtpc 23861 df-pconn 32868 df-sconn 32869 |
This theorem is referenced by: (None) |
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