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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 39183 | . . . . 5 ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘(ℝ ↑𝑚 𝐼))) |
4 | 1, 3 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ (Met‘(ℝ ↑𝑚 𝐼))) |
5 | metxmet 21949 | . . . 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 2698 | . . . 4 ⊢ (𝜑 → 𝑉 ∈ (TopOpen‘(ℝ^‘𝐼))) |
10 | 1 | rrxtopnfi 39182 | . . . . 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 2610 | . . . . . . . . 9 ⊢ (ℝ^‘𝐼) = (ℝ^‘𝐼) | |
13 | eqid 2610 | . . . . . . . . 9 ⊢ (ℝ ↑𝑚 𝐼) = (ℝ ↑𝑚 𝐼) | |
14 | 12, 13 | rrxdsfi 39181 | . . . . . . . 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 2645 | . . . . . 6 ⊢ (𝜑 → (𝑓 ∈ (ℝ ↑𝑚 𝐼), 𝑔 ∈ (ℝ ↑𝑚 𝐼) ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))) = 𝐷) |
17 | 16 | fveq2d 6107 | . . . . 5 ⊢ (𝜑 → (MetOpen‘(𝑓 ∈ (ℝ ↑𝑚 𝐼), 𝑔 ∈ (ℝ ↑𝑚 𝐼) ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) = (MetOpen‘𝐷)) |
18 | 10, 17 | eqtrd 2644 | . . . 4 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝐼)) = (MetOpen‘𝐷)) |
19 | 9, 18 | eleqtrd 2690 | . . 3 ⊢ (𝜑 → 𝑉 ∈ (MetOpen‘𝐷)) |
20 | qndenserrnopnlem.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
21 | eqid 2610 | . . . 4 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷) | |
22 | 21 | mopni2 22108 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘(ℝ ↑𝑚 𝐼)) ∧ 𝑉 ∈ (MetOpen‘𝐷) ∧ 𝑋 ∈ 𝑉) → ∃𝑒 ∈ ℝ+ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) |
23 | 6, 19, 20, 22 | syl3anc 1318 | . 2 ⊢ (𝜑 → ∃𝑒 ∈ ℝ+ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) |
24 | 1 | 3ad2ant1 1075 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) → 𝐼 ∈ Fin) |
25 | rrxtps 39180 | . . . . . . . . . . . 12 ⊢ (𝐼 ∈ Fin → (ℝ^‘𝐼) ∈ TopSp) | |
26 | 1, 25 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → (ℝ^‘𝐼) ∈ TopSp) |
27 | eqid 2610 | . . . . . . . . . . . 12 ⊢ (Base‘(ℝ^‘𝐼)) = (Base‘(ℝ^‘𝐼)) | |
28 | 27, 8 | istps 20551 | . . . . . . . . . . 11 ⊢ ((ℝ^‘𝐼) ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘(ℝ^‘𝐼)))) |
29 | 26, 28 | sylib 207 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(Base‘(ℝ^‘𝐼)))) |
30 | 1, 12, 27 | rrxbasefi 39179 | . . . . . . . . . . 11 ⊢ (𝜑 → (Base‘(ℝ^‘𝐼)) = (ℝ ↑𝑚 𝐼)) |
31 | 30 | fveq2d 6107 | . . . . . . . . . 10 ⊢ (𝜑 → (TopOn‘(Base‘(ℝ^‘𝐼))) = (TopOn‘(ℝ ↑𝑚 𝐼))) |
32 | 29, 31 | eleqtrd 2690 | . . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(ℝ ↑𝑚 𝐼))) |
33 | toponss 20544 | . . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘(ℝ ↑𝑚 𝐼)) ∧ 𝑉 ∈ 𝐽) → 𝑉 ⊆ (ℝ ↑𝑚 𝐼)) | |
34 | 32, 7, 33 | syl2anc 691 | . . . . . . . 8 ⊢ (𝜑 → 𝑉 ⊆ (ℝ ↑𝑚 𝐼)) |
35 | 34, 20 | sseldd 3569 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (ℝ ↑𝑚 𝐼)) |
36 | 35 | 3ad2ant1 1075 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) → 𝑋 ∈ (ℝ ↑𝑚 𝐼)) |
37 | simp2 1055 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) → 𝑒 ∈ ℝ+) | |
38 | 24, 36, 2, 37 | qndenserrnbl 39191 | . . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) → ∃𝑦 ∈ (ℚ ↑𝑚 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝑒)) |
39 | ssel 3562 | . . . . . . . 8 ⊢ ((𝑋(ball‘𝐷)𝑒) ⊆ 𝑉 → (𝑦 ∈ (𝑋(ball‘𝐷)𝑒) → 𝑦 ∈ 𝑉)) | |
40 | 39 | adantr 480 | . . . . . . 7 ⊢ (((𝑋(ball‘𝐷)𝑒) ⊆ 𝑉 ∧ 𝑦 ∈ (ℚ ↑𝑚 𝐼)) → (𝑦 ∈ (𝑋(ball‘𝐷)𝑒) → 𝑦 ∈ 𝑉)) |
41 | 40 | 3ad2antl3 1218 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ+ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) ∧ 𝑦 ∈ (ℚ ↑𝑚 𝐼)) → (𝑦 ∈ (𝑋(ball‘𝐷)𝑒) → 𝑦 ∈ 𝑉)) |
42 | 41 | reximdva 3000 | . . . . 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 1256 | . . 3 ⊢ (𝜑 → (𝑒 ∈ ℝ+ → ((𝑋(ball‘𝐷)𝑒) ⊆ 𝑉 → ∃𝑦 ∈ (ℚ ↑𝑚 𝐼)𝑦 ∈ 𝑉))) |
45 | 44 | rexlimdv 3012 | . 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 1031 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ⊆ wss 3540 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ↑𝑚 cmap 7744 Fincfn 7841 ℝcr 9814 − cmin 10145 2c2 10947 ℚcq 11664 ℝ+crp 11708 ↑cexp 12722 √csqrt 13821 Σcsu 14264 Basecbs 15695 distcds 15777 TopOpenctopn 15905 ∞Metcxmt 19552 Metcme 19553 ballcbl 19554 MetOpencmopn 19557 TopOnctopon 20518 TopSpctps 20519 ℝ^crrx 22979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ico 12052 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-prds 15931 df-pws 15933 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-subg 17414 df-ghm 17481 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-rnghom 18538 df-drng 18572 df-field 18573 df-subrg 18601 df-abv 18640 df-staf 18668 df-srng 18669 df-lmod 18688 df-lss 18754 df-lmhm 18843 df-lvec 18924 df-sra 18993 df-rgmod 18994 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-cnfld 19568 df-refld 19770 df-phl 19790 df-dsmm 19895 df-frlm 19910 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-xms 21935 df-ms 21936 df-nm 22197 df-ngp 22198 df-tng 22199 df-nrg 22200 df-nlm 22201 df-clm 22671 df-cph 22776 df-tch 22777 df-rrx 22981 |
This theorem is referenced by: qndenserrnopn 39194 |
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