Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dya2iocnei | Structured version Visualization version GIF version |
Description: For any point of an open set of the usual topology on (ℝ × ℝ) there is a closed-below open-above dyadic rational square which contains that point and is entirely in the open set. (Contributed by Thierry Arnoux, 21-Sep-2017.) |
Ref | Expression |
---|---|
sxbrsiga.0 | ⊢ 𝐽 = (topGen‘ran (,)) |
dya2ioc.1 | ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) |
dya2ioc.2 | ⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣)) |
Ref | Expression |
---|---|
dya2iocnei | ⊢ ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elunii 4842 | . . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐴 ∈ (𝐽 ×t 𝐽)) → 𝑋 ∈ ∪ (𝐽 ×t 𝐽)) | |
2 | 1 | ancoms 461 | . . 3 ⊢ ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ ∪ (𝐽 ×t 𝐽)) |
3 | sxbrsiga.0 | . . . 4 ⊢ 𝐽 = (topGen‘ran (,)) | |
4 | 3 | tpr2uni 31148 | . . 3 ⊢ ∪ (𝐽 ×t 𝐽) = (ℝ × ℝ) |
5 | 2, 4 | eleqtrdi 2923 | . 2 ⊢ ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ (ℝ × ℝ)) |
6 | eqid 2821 | . . 3 ⊢ (𝑢 ∈ ℝ, 𝑣 ∈ ℝ ↦ (𝑢 + (i · 𝑣))) = (𝑢 ∈ ℝ, 𝑣 ∈ ℝ ↦ (𝑢 + (i · 𝑣))) | |
7 | eqid 2821 | . . 3 ⊢ ran (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓)) = ran (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓)) | |
8 | 3, 6, 7 | tpr2rico 31155 | . 2 ⊢ ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∃𝑟 ∈ ran (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓))(𝑋 ∈ 𝑟 ∧ 𝑟 ⊆ 𝐴)) |
9 | anass 471 | . . . . 5 ⊢ (((𝑟 ∈ ran (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓)) ∧ 𝑋 ∈ 𝑟) ∧ 𝑟 ⊆ 𝐴) ↔ (𝑟 ∈ ran (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓)) ∧ (𝑋 ∈ 𝑟 ∧ 𝑟 ⊆ 𝐴))) | |
10 | dya2ioc.1 | . . . . . . . . 9 ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) | |
11 | dya2ioc.2 | . . . . . . . . 9 ⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣)) | |
12 | 3, 10, 11, 7 | dya2iocnrect 31539 | . . . . . . . 8 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝑟 ∈ ran (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓)) ∧ 𝑋 ∈ 𝑟) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑟)) |
13 | 12 | 3expb 1116 | . . . . . . 7 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ (𝑟 ∈ ran (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓)) ∧ 𝑋 ∈ 𝑟)) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑟)) |
14 | 13 | anim1i 616 | . . . . . 6 ⊢ (((𝑋 ∈ (ℝ × ℝ) ∧ (𝑟 ∈ ran (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓)) ∧ 𝑋 ∈ 𝑟)) ∧ 𝑟 ⊆ 𝐴) → (∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑟) ∧ 𝑟 ⊆ 𝐴)) |
15 | 14 | anasss 469 | . . . . 5 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ ((𝑟 ∈ ran (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓)) ∧ 𝑋 ∈ 𝑟) ∧ 𝑟 ⊆ 𝐴)) → (∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑟) ∧ 𝑟 ⊆ 𝐴)) |
16 | 9, 15 | sylan2br 596 | . . . 4 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ (𝑟 ∈ ran (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓)) ∧ (𝑋 ∈ 𝑟 ∧ 𝑟 ⊆ 𝐴))) → (∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑟) ∧ 𝑟 ⊆ 𝐴)) |
17 | r19.41v 3347 | . . . . 5 ⊢ (∃𝑏 ∈ ran 𝑅((𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑟) ∧ 𝑟 ⊆ 𝐴) ↔ (∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑟) ∧ 𝑟 ⊆ 𝐴)) | |
18 | simpll 765 | . . . . . . 7 ⊢ (((𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑟) ∧ 𝑟 ⊆ 𝐴) → 𝑋 ∈ 𝑏) | |
19 | simplr 767 | . . . . . . . 8 ⊢ (((𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑟) ∧ 𝑟 ⊆ 𝐴) → 𝑏 ⊆ 𝑟) | |
20 | simpr 487 | . . . . . . . 8 ⊢ (((𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑟) ∧ 𝑟 ⊆ 𝐴) → 𝑟 ⊆ 𝐴) | |
21 | 19, 20 | sstrd 3976 | . . . . . . 7 ⊢ (((𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑟) ∧ 𝑟 ⊆ 𝐴) → 𝑏 ⊆ 𝐴) |
22 | 18, 21 | jca 514 | . . . . . 6 ⊢ (((𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑟) ∧ 𝑟 ⊆ 𝐴) → (𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴)) |
23 | 22 | reximi 3243 | . . . . 5 ⊢ (∃𝑏 ∈ ran 𝑅((𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑟) ∧ 𝑟 ⊆ 𝐴) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴)) |
24 | 17, 23 | sylbir 237 | . . . 4 ⊢ ((∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑟) ∧ 𝑟 ⊆ 𝐴) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴)) |
25 | 16, 24 | syl 17 | . . 3 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ (𝑟 ∈ ran (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓)) ∧ (𝑋 ∈ 𝑟 ∧ 𝑟 ⊆ 𝐴))) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴)) |
26 | 25 | rexlimdvaa 3285 | . 2 ⊢ (𝑋 ∈ (ℝ × ℝ) → (∃𝑟 ∈ ran (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓))(𝑋 ∈ 𝑟 ∧ 𝑟 ⊆ 𝐴) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴))) |
27 | 5, 8, 26 | sylc 65 | 1 ⊢ ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 ⊆ wss 3935 ∪ cuni 4837 × cxp 5552 ran crn 5555 ‘cfv 6354 (class class class)co 7155 ∈ cmpo 7157 ℝcr 10535 1c1 10537 ici 10538 + caddc 10539 · cmul 10541 / cdiv 11296 2c2 11691 ℤcz 11980 (,)cioo 12737 [,)cico 12739 ↑cexp 13428 topGenctg 16710 ×t ctx 22167 |
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 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 ax-addf 10615 ax-mulf 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 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 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-pm 8408 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-fi 8874 df-sup 8905 df-inf 8906 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-q 12348 df-rp 12389 df-xneg 12506 df-xadd 12507 df-xmul 12508 df-ioo 12741 df-ioc 12742 df-ico 12743 df-icc 12744 df-fz 12892 df-fzo 13033 df-fl 13161 df-mod 13237 df-seq 13369 df-exp 13429 df-fac 13633 df-bc 13662 df-hash 13690 df-shft 14425 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-limsup 14827 df-clim 14844 df-rlim 14845 df-sum 15042 df-ef 15420 df-sin 15422 df-cos 15423 df-pi 15425 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-starv 16579 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-unif 16587 df-hom 16588 df-cco 16589 df-rest 16695 df-topn 16696 df-0g 16714 df-gsum 16715 df-topgen 16716 df-pt 16717 df-prds 16720 df-xrs 16774 df-qtop 16779 df-imas 16780 df-xps 16782 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-submnd 17956 df-mulg 18224 df-cntz 18446 df-cmn 18907 df-psmet 20536 df-xmet 20537 df-met 20538 df-bl 20539 df-mopn 20540 df-fbas 20541 df-fg 20542 df-cnfld 20545 df-refld 20748 df-top 21501 df-topon 21518 df-topsp 21540 df-bases 21553 df-cld 21626 df-ntr 21627 df-cls 21628 df-nei 21705 df-lp 21743 df-perf 21744 df-cn 21834 df-cnp 21835 df-haus 21922 df-cmp 21994 df-tx 22169 df-hmeo 22362 df-fil 22453 df-fm 22545 df-flim 22546 df-flf 22547 df-fcls 22548 df-xms 22929 df-ms 22930 df-tms 22931 df-cncf 23485 df-cfil 23857 df-cmet 23859 df-cms 23937 df-limc 24463 df-dv 24464 df-log 25139 df-cxp 25140 df-logb 25342 |
This theorem is referenced by: dya2iocuni 31541 |
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