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Mathbox for Glauco Siliprandi |
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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 23618 | . . . . 5 ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘(ℝ ↑𝑚 𝐼))) |
4 | 1, 3 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ (Met‘(ℝ ↑𝑚 𝐼))) |
5 | metxmet 22547 | . . . 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 | syl6eleq 2869 | . . . 4 ⊢ (𝜑 → 𝑉 ∈ (TopOpen‘(ℝ^‘𝐼))) |
10 | 1 | rrxtopnfi 41435 | . . . . 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 2778 | . . . . . . . . 9 ⊢ (ℝ^‘𝐼) = (ℝ^‘𝐼) | |
13 | eqid 2778 | . . . . . . . . 9 ⊢ (ℝ ↑𝑚 𝐼) = (ℝ ↑𝑚 𝐼) | |
14 | 12, 13 | rrxdsfi 23617 | . . . . . . . 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 2815 | . . . . . 6 ⊢ (𝜑 → (𝑓 ∈ (ℝ ↑𝑚 𝐼), 𝑔 ∈ (ℝ ↑𝑚 𝐼) ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))) = 𝐷) |
17 | 16 | fveq2d 6450 | . . . . 5 ⊢ (𝜑 → (MetOpen‘(𝑓 ∈ (ℝ ↑𝑚 𝐼), 𝑔 ∈ (ℝ ↑𝑚 𝐼) ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) = (MetOpen‘𝐷)) |
18 | 10, 17 | eqtrd 2814 | . . . 4 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝐼)) = (MetOpen‘𝐷)) |
19 | 9, 18 | eleqtrd 2861 | . . 3 ⊢ (𝜑 → 𝑉 ∈ (MetOpen‘𝐷)) |
20 | qndenserrnopnlem.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
21 | eqid 2778 | . . . 4 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷) | |
22 | 21 | mopni2 22706 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘(ℝ ↑𝑚 𝐼)) ∧ 𝑉 ∈ (MetOpen‘𝐷) ∧ 𝑋 ∈ 𝑉) → ∃𝑒 ∈ ℝ+ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) |
23 | 6, 19, 20, 22 | syl3anc 1439 | . 2 ⊢ (𝜑 → ∃𝑒 ∈ ℝ+ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) |
24 | 1 | 3ad2ant1 1124 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) → 𝐼 ∈ Fin) |
25 | rrxtps 41434 | . . . . . . . . . . . 12 ⊢ (𝐼 ∈ Fin → (ℝ^‘𝐼) ∈ TopSp) | |
26 | 1, 25 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → (ℝ^‘𝐼) ∈ TopSp) |
27 | eqid 2778 | . . . . . . . . . . . 12 ⊢ (Base‘(ℝ^‘𝐼)) = (Base‘(ℝ^‘𝐼)) | |
28 | 27, 8 | istps 21146 | . . . . . . . . . . 11 ⊢ ((ℝ^‘𝐼) ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘(ℝ^‘𝐼)))) |
29 | 26, 28 | sylib 210 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(Base‘(ℝ^‘𝐼)))) |
30 | 1, 12, 27 | rrxbasefi 23616 | . . . . . . . . . . 11 ⊢ (𝜑 → (Base‘(ℝ^‘𝐼)) = (ℝ ↑𝑚 𝐼)) |
31 | 30 | fveq2d 6450 | . . . . . . . . . 10 ⊢ (𝜑 → (TopOn‘(Base‘(ℝ^‘𝐼))) = (TopOn‘(ℝ ↑𝑚 𝐼))) |
32 | 29, 31 | eleqtrd 2861 | . . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(ℝ ↑𝑚 𝐼))) |
33 | toponss 21139 | . . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘(ℝ ↑𝑚 𝐼)) ∧ 𝑉 ∈ 𝐽) → 𝑉 ⊆ (ℝ ↑𝑚 𝐼)) | |
34 | 32, 7, 33 | syl2anc 579 | . . . . . . . 8 ⊢ (𝜑 → 𝑉 ⊆ (ℝ ↑𝑚 𝐼)) |
35 | 34, 20 | sseldd 3822 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (ℝ ↑𝑚 𝐼)) |
36 | 35 | 3ad2ant1 1124 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) → 𝑋 ∈ (ℝ ↑𝑚 𝐼)) |
37 | simp2 1128 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) → 𝑒 ∈ ℝ+) | |
38 | 24, 36, 2, 37 | qndenserrnbl 41443 | . . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) → ∃𝑦 ∈ (ℚ ↑𝑚 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝑒)) |
39 | ssel 3815 | . . . . . . . 8 ⊢ ((𝑋(ball‘𝐷)𝑒) ⊆ 𝑉 → (𝑦 ∈ (𝑋(ball‘𝐷)𝑒) → 𝑦 ∈ 𝑉)) | |
40 | 39 | adantr 474 | . . . . . . 7 ⊢ (((𝑋(ball‘𝐷)𝑒) ⊆ 𝑉 ∧ 𝑦 ∈ (ℚ ↑𝑚 𝐼)) → (𝑦 ∈ (𝑋(ball‘𝐷)𝑒) → 𝑦 ∈ 𝑉)) |
41 | 40 | 3ad2antl3 1195 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ+ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) ∧ 𝑦 ∈ (ℚ ↑𝑚 𝐼)) → (𝑦 ∈ (𝑋(ball‘𝐷)𝑒) → 𝑦 ∈ 𝑉)) |
42 | 41 | reximdva 3198 | . . . . 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 1109 | . . 3 ⊢ (𝜑 → (𝑒 ∈ ℝ+ → ((𝑋(ball‘𝐷)𝑒) ⊆ 𝑉 → ∃𝑦 ∈ (ℚ ↑𝑚 𝐼)𝑦 ∈ 𝑉))) |
45 | 44 | rexlimdv 3212 | . 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 1071 = wceq 1601 ∈ wcel 2107 ∃wrex 3091 ⊆ wss 3792 ‘cfv 6135 (class class class)co 6922 ↦ cmpt2 6924 ↑𝑚 cmap 8140 Fincfn 8241 ℝcr 10271 − cmin 10606 2c2 11430 ℚcq 12095 ℝ+crp 12137 ↑cexp 13178 √csqrt 14380 Σcsu 14824 Basecbs 16255 distcds 16347 TopOpenctopn 16468 ∞Metcxmet 20127 Metcmet 20128 ballcbl 20129 MetOpencmopn 20132 TopOnctopon 21122 TopSpctps 21144 ℝ^crrx 23589 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 ax-mulf 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-tpos 7634 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-ioo 12491 df-ico 12493 df-fz 12644 df-fzo 12785 df-seq 13120 df-exp 13179 df-hash 13436 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-clim 14627 df-sum 14825 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-hom 16362 df-cco 16363 df-rest 16469 df-topn 16470 df-0g 16488 df-gsum 16489 df-topgen 16490 df-prds 16494 df-pws 16496 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-mhm 17721 df-submnd 17722 df-grp 17812 df-minusg 17813 df-sbg 17814 df-subg 17975 df-ghm 18042 df-cntz 18133 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-cring 18937 df-oppr 19010 df-dvdsr 19028 df-unit 19029 df-invr 19059 df-dvr 19070 df-rnghom 19104 df-drng 19141 df-field 19142 df-subrg 19170 df-abv 19209 df-staf 19237 df-srng 19238 df-lmod 19257 df-lss 19325 df-lmhm 19417 df-lvec 19498 df-sra 19569 df-rgmod 19570 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-cnfld 20143 df-refld 20348 df-phl 20369 df-dsmm 20475 df-frlm 20490 df-top 21106 df-topon 21123 df-topsp 21145 df-bases 21158 df-xms 22533 df-ms 22534 df-nm 22795 df-ngp 22796 df-tng 22797 df-nrg 22798 df-nlm 22799 df-clm 23270 df-cph 23375 df-tcph 23376 df-rrx 23591 |
This theorem is referenced by: qndenserrnopn 41446 |
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