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Mirrors > Home > MPE Home > Th. List > Mathboxes > qndenserrnopnlem | Structured version Visualization version GIF version |
Description: n-dimensional rational numbers are dense in the space of n-dimensional real numbers, with respect to the n-dimensional standard topology. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Ref | Expression |
---|---|
qndenserrnopnlem.i | ⊢ (𝜑 → 𝐼 ∈ Fin) |
qndenserrnopnlem.j | ⊢ 𝐽 = (TopOpen‘(ℝ^‘𝐼)) |
qndenserrnopnlem.v | ⊢ (𝜑 → 𝑉 ∈ 𝐽) |
qndenserrnopnlem.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
qndenserrnopnlem.d | ⊢ 𝐷 = (dist‘(ℝ^‘𝐼)) |
Ref | Expression |
---|---|
qndenserrnopnlem | ⊢ (𝜑 → ∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qndenserrnopnlem.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ Fin) | |
2 | qndenserrnopnlem.d | . . . . . 6 ⊢ 𝐷 = (dist‘(ℝ^‘𝐼)) | |
3 | 2 | rrxmetfi 23942 | . . . . 5 ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘(ℝ ↑m 𝐼))) |
4 | 1, 3 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ (Met‘(ℝ ↑m 𝐼))) |
5 | metxmet 22871 | . . . 4 ⊢ (𝐷 ∈ (Met‘(ℝ ↑m 𝐼)) → 𝐷 ∈ (∞Met‘(ℝ ↑m 𝐼))) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘(ℝ ↑m 𝐼))) |
7 | qndenserrnopnlem.v | . . . . 5 ⊢ (𝜑 → 𝑉 ∈ 𝐽) | |
8 | qndenserrnopnlem.j | . . . . 5 ⊢ 𝐽 = (TopOpen‘(ℝ^‘𝐼)) | |
9 | 7, 8 | eleqtrdi 2920 | . . . 4 ⊢ (𝜑 → 𝑉 ∈ (TopOpen‘(ℝ^‘𝐼))) |
10 | 1 | rrxtopnfi 42449 | . . . . 5 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝐼)) = (MetOpen‘(𝑓 ∈ (ℝ ↑m 𝐼), 𝑔 ∈ (ℝ ↑m 𝐼) ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))))) |
11 | 2 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 𝐷 = (dist‘(ℝ^‘𝐼))) |
12 | eqid 2818 | . . . . . . . . 9 ⊢ (ℝ^‘𝐼) = (ℝ^‘𝐼) | |
13 | eqid 2818 | . . . . . . . . 9 ⊢ (ℝ ↑m 𝐼) = (ℝ ↑m 𝐼) | |
14 | 12, 13 | rrxdsfi 23941 | . . . . . . . 8 ⊢ (𝐼 ∈ Fin → (dist‘(ℝ^‘𝐼)) = (𝑓 ∈ (ℝ ↑m 𝐼), 𝑔 ∈ (ℝ ↑m 𝐼) ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) |
15 | 1, 14 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (dist‘(ℝ^‘𝐼)) = (𝑓 ∈ (ℝ ↑m 𝐼), 𝑔 ∈ (ℝ ↑m 𝐼) ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) |
16 | 11, 15 | eqtr2d 2854 | . . . . . 6 ⊢ (𝜑 → (𝑓 ∈ (ℝ ↑m 𝐼), 𝑔 ∈ (ℝ ↑m 𝐼) ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))) = 𝐷) |
17 | 16 | fveq2d 6667 | . . . . 5 ⊢ (𝜑 → (MetOpen‘(𝑓 ∈ (ℝ ↑m 𝐼), 𝑔 ∈ (ℝ ↑m 𝐼) ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) = (MetOpen‘𝐷)) |
18 | 10, 17 | eqtrd 2853 | . . . 4 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝐼)) = (MetOpen‘𝐷)) |
19 | 9, 18 | eleqtrd 2912 | . . 3 ⊢ (𝜑 → 𝑉 ∈ (MetOpen‘𝐷)) |
20 | qndenserrnopnlem.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
21 | eqid 2818 | . . . 4 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷) | |
22 | 21 | mopni2 23030 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘(ℝ ↑m 𝐼)) ∧ 𝑉 ∈ (MetOpen‘𝐷) ∧ 𝑋 ∈ 𝑉) → ∃𝑒 ∈ ℝ+ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) |
23 | 6, 19, 20, 22 | syl3anc 1363 | . 2 ⊢ (𝜑 → ∃𝑒 ∈ ℝ+ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) |
24 | 1 | 3ad2ant1 1125 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) → 𝐼 ∈ Fin) |
25 | rrxtps 42448 | . . . . . . . . . . . 12 ⊢ (𝐼 ∈ Fin → (ℝ^‘𝐼) ∈ TopSp) | |
26 | 1, 25 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → (ℝ^‘𝐼) ∈ TopSp) |
27 | eqid 2818 | . . . . . . . . . . . 12 ⊢ (Base‘(ℝ^‘𝐼)) = (Base‘(ℝ^‘𝐼)) | |
28 | 27, 8 | istps 21470 | . . . . . . . . . . 11 ⊢ ((ℝ^‘𝐼) ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘(ℝ^‘𝐼)))) |
29 | 26, 28 | sylib 219 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(Base‘(ℝ^‘𝐼)))) |
30 | 1, 12, 27 | rrxbasefi 23940 | . . . . . . . . . . 11 ⊢ (𝜑 → (Base‘(ℝ^‘𝐼)) = (ℝ ↑m 𝐼)) |
31 | 30 | fveq2d 6667 | . . . . . . . . . 10 ⊢ (𝜑 → (TopOn‘(Base‘(ℝ^‘𝐼))) = (TopOn‘(ℝ ↑m 𝐼))) |
32 | 29, 31 | eleqtrd 2912 | . . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(ℝ ↑m 𝐼))) |
33 | toponss 21463 | . . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘(ℝ ↑m 𝐼)) ∧ 𝑉 ∈ 𝐽) → 𝑉 ⊆ (ℝ ↑m 𝐼)) | |
34 | 32, 7, 33 | syl2anc 584 | . . . . . . . 8 ⊢ (𝜑 → 𝑉 ⊆ (ℝ ↑m 𝐼)) |
35 | 34, 20 | sseldd 3965 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (ℝ ↑m 𝐼)) |
36 | 35 | 3ad2ant1 1125 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) → 𝑋 ∈ (ℝ ↑m 𝐼)) |
37 | simp2 1129 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) → 𝑒 ∈ ℝ+) | |
38 | 24, 36, 2, 37 | qndenserrnbl 42457 | . . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) → ∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝑒)) |
39 | ssel 3958 | . . . . . . . 8 ⊢ ((𝑋(ball‘𝐷)𝑒) ⊆ 𝑉 → (𝑦 ∈ (𝑋(ball‘𝐷)𝑒) → 𝑦 ∈ 𝑉)) | |
40 | 39 | adantr 481 | . . . . . . 7 ⊢ (((𝑋(ball‘𝐷)𝑒) ⊆ 𝑉 ∧ 𝑦 ∈ (ℚ ↑m 𝐼)) → (𝑦 ∈ (𝑋(ball‘𝐷)𝑒) → 𝑦 ∈ 𝑉)) |
41 | 40 | 3ad2antl3 1179 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ+ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) ∧ 𝑦 ∈ (ℚ ↑m 𝐼)) → (𝑦 ∈ (𝑋(ball‘𝐷)𝑒) → 𝑦 ∈ 𝑉)) |
42 | 41 | reximdva 3271 | . . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) → (∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝑒) → ∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ 𝑉)) |
43 | 38, 42 | mpd 15 | . . . 4 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) → ∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ 𝑉) |
44 | 43 | 3exp 1111 | . . 3 ⊢ (𝜑 → (𝑒 ∈ ℝ+ → ((𝑋(ball‘𝐷)𝑒) ⊆ 𝑉 → ∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ 𝑉))) |
45 | 44 | rexlimdv 3280 | . 2 ⊢ (𝜑 → (∃𝑒 ∈ ℝ+ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉 → ∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ 𝑉)) |
46 | 23, 45 | mpd 15 | 1 ⊢ (𝜑 → ∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∃wrex 3136 ⊆ wss 3933 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 ↑m cmap 8395 Fincfn 8497 ℝcr 10524 − cmin 10858 2c2 11680 ℚcq 12336 ℝ+crp 12377 ↑cexp 13417 √csqrt 14580 Σcsu 15030 Basecbs 16471 distcds 16562 TopOpenctopn 16683 ∞Metcxmet 20458 Metcmet 20459 ballcbl 20460 MetOpencmopn 20463 TopOnctopon 21446 TopSpctps 21468 ℝ^crrx 23913 |
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-inf2 9092 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-fal 1541 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-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-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 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-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-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-clim 14833 df-sum 15031 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-prds 16709 df-pws 16711 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-subg 18214 df-ghm 18294 df-cntz 18385 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-rnghom 19396 df-drng 19433 df-field 19434 df-subrg 19462 df-abv 19517 df-staf 19545 df-srng 19546 df-lmod 19565 df-lss 19633 df-lmhm 19723 df-lvec 19804 df-sra 19873 df-rgmod 19874 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-cnfld 20474 df-refld 20677 df-phl 20698 df-dsmm 20804 df-frlm 20819 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-xms 22857 df-ms 22858 df-nm 23119 df-ngp 23120 df-tng 23121 df-nrg 23122 df-nlm 23123 df-clm 23594 df-cph 23699 df-tcph 23700 df-rrx 23915 |
This theorem is referenced by: qndenserrnopn 42460 |
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