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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 23372 | . 2 ⊢ (topGen‘ran (,)) ∈ Top | |
2 | tg2 21575 | . . . 4 ⊢ ((𝑥 ∈ (topGen‘ran (,)) ∧ 𝑦 ∈ 𝑥) → ∃𝑧 ∈ ran (,)(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥)) | |
3 | retopbas 23371 | . . . . . . . . . 10 ⊢ ran (,) ∈ TopBases | |
4 | bastg 21576 | . . . . . . . . . 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 | sseldi 3967 | . . . . . . . 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 4546 | . . . . . . . . 9 ⊢ (𝑧 ∈ 𝒫 𝑥 ↔ 𝑧 ⊆ 𝑥) | |
10 | 8, 9 | sylibr 236 | . . . . . . . 8 ⊢ (((𝑥 ∈ (topGen‘ran (,)) ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ ran (,) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))) → 𝑧 ∈ 𝒫 𝑥) |
11 | 7, 10 | elind 4173 | . . . . . . 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 12838 | . . . . . . . . . 10 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
14 | ffn 6516 | . . . . . . . . . 10 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*)) | |
15 | ovelrn 7326 | . . . . . . . . . 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 7166 | . . . . . . . . . . . 12 ⊢ (𝑧 = (𝑎(,)𝑏) → ((topGen‘ran (,)) ↾t 𝑧) = ((topGen‘ran (,)) ↾t (𝑎(,)𝑏))) | |
18 | ioosconn 32496 | . . . . . . . . . . . 12 ⊢ ((topGen‘ran (,)) ↾t (𝑎(,)𝑏)) ∈ SConn | |
19 | 17, 18 | eqeltrdi 2923 | . . . . . . . . . . 11 ⊢ (𝑧 = (𝑎(,)𝑏) → ((topGen‘ran (,)) ↾t 𝑧) ∈ SConn) |
20 | 19 | rexlimivw 3284 | . . . . . . . . . 10 ⊢ (∃𝑏 ∈ ℝ* 𝑧 = (𝑎(,)𝑏) → ((topGen‘ran (,)) ↾t 𝑧) ∈ SConn) |
21 | 20 | rexlimivw 3284 | . . . . . . . . 9 ⊢ (∃𝑎 ∈ ℝ* ∃𝑏 ∈ ℝ* 𝑧 = (𝑎(,)𝑏) → ((topGen‘ran (,)) ↾t 𝑧) ∈ SConn) |
22 | 16, 21 | sylbi 219 | . . . . . . . 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 518 | . . . . . 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 415 | . . . . 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 3273 | . . . 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 3205 | . 2 ⊢ ∀𝑥 ∈ (topGen‘ran (,))∀𝑦 ∈ 𝑥 ∃𝑧 ∈ ((topGen‘ran (,)) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑧 ∧ ((topGen‘ran (,)) ↾t 𝑧) ∈ SConn) |
29 | islly 22078 | . 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 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 ∩ cin 3937 ⊆ wss 3938 𝒫 cpw 4541 × cxp 5555 ran crn 5558 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 ℝ*cxr 10676 (,)cioo 12741 ↾t crest 16696 topGenctg 16713 Topctop 21503 TopBasesctb 21555 Locally clly 22074 SConncsconn 32469 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-xrs 16777 df-qtop 16782 df-imas 16783 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-mulg 18227 df-cntz 18449 df-cmn 18910 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-cn 21837 df-cnp 21838 df-conn 22022 df-lly 22076 df-tx 22172 df-hmeo 22365 df-xms 22932 df-ms 22933 df-tms 22934 df-ii 23487 df-htpy 23576 df-phtpy 23577 df-phtpc 23598 df-pconn 32470 df-sconn 32471 |
This theorem is referenced by: (None) |
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