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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 25460 | . . . . 5 ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘(ℝ ↑m 𝐼))) |
4 | 1, 3 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ (Met‘(ℝ ↑m 𝐼))) |
5 | metxmet 24360 | . . . 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 2849 | . . . 4 ⊢ (𝜑 → 𝑉 ∈ (TopOpen‘(ℝ^‘𝐼))) |
10 | 1 | rrxtopnfi 46243 | . . . . 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 2735 | . . . . . . . . 9 ⊢ (ℝ^‘𝐼) = (ℝ^‘𝐼) | |
13 | eqid 2735 | . . . . . . . . 9 ⊢ (ℝ ↑m 𝐼) = (ℝ ↑m 𝐼) | |
14 | 12, 13 | rrxdsfi 25459 | . . . . . . . 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 2776 | . . . . . 6 ⊢ (𝜑 → (𝑓 ∈ (ℝ ↑m 𝐼), 𝑔 ∈ (ℝ ↑m 𝐼) ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))) = 𝐷) |
17 | 16 | fveq2d 6911 | . . . . 5 ⊢ (𝜑 → (MetOpen‘(𝑓 ∈ (ℝ ↑m 𝐼), 𝑔 ∈ (ℝ ↑m 𝐼) ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) = (MetOpen‘𝐷)) |
18 | 10, 17 | eqtrd 2775 | . . . 4 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝐼)) = (MetOpen‘𝐷)) |
19 | 9, 18 | eleqtrd 2841 | . . 3 ⊢ (𝜑 → 𝑉 ∈ (MetOpen‘𝐷)) |
20 | qndenserrnopnlem.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
21 | eqid 2735 | . . . 4 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷) | |
22 | 21 | mopni2 24522 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘(ℝ ↑m 𝐼)) ∧ 𝑉 ∈ (MetOpen‘𝐷) ∧ 𝑋 ∈ 𝑉) → ∃𝑒 ∈ ℝ+ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) |
23 | 6, 19, 20, 22 | syl3anc 1370 | . 2 ⊢ (𝜑 → ∃𝑒 ∈ ℝ+ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) |
24 | 1 | 3ad2ant1 1132 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) → 𝐼 ∈ Fin) |
25 | rrxtps 46242 | . . . . . . . . . . . 12 ⊢ (𝐼 ∈ Fin → (ℝ^‘𝐼) ∈ TopSp) | |
26 | 1, 25 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → (ℝ^‘𝐼) ∈ TopSp) |
27 | eqid 2735 | . . . . . . . . . . . 12 ⊢ (Base‘(ℝ^‘𝐼)) = (Base‘(ℝ^‘𝐼)) | |
28 | 27, 8 | istps 22956 | . . . . . . . . . . 11 ⊢ ((ℝ^‘𝐼) ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘(ℝ^‘𝐼)))) |
29 | 26, 28 | sylib 218 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(Base‘(ℝ^‘𝐼)))) |
30 | 1, 12, 27 | rrxbasefi 25458 | . . . . . . . . . . 11 ⊢ (𝜑 → (Base‘(ℝ^‘𝐼)) = (ℝ ↑m 𝐼)) |
31 | 30 | fveq2d 6911 | . . . . . . . . . 10 ⊢ (𝜑 → (TopOn‘(Base‘(ℝ^‘𝐼))) = (TopOn‘(ℝ ↑m 𝐼))) |
32 | 29, 31 | eleqtrd 2841 | . . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(ℝ ↑m 𝐼))) |
33 | toponss 22949 | . . . . . . . . 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 3996 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (ℝ ↑m 𝐼)) |
36 | 35 | 3ad2ant1 1132 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) → 𝑋 ∈ (ℝ ↑m 𝐼)) |
37 | simp2 1136 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) → 𝑒 ∈ ℝ+) | |
38 | 24, 36, 2, 37 | qndenserrnbl 46251 | . . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) → ∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝑒)) |
39 | ssel 3989 | . . . . . . . 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 1186 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ+ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) ∧ 𝑦 ∈ (ℚ ↑m 𝐼)) → (𝑦 ∈ (𝑋(ball‘𝐷)𝑒) → 𝑦 ∈ 𝑉)) |
42 | 41 | reximdva 3166 | . . . . 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 1118 | . . 3 ⊢ (𝜑 → (𝑒 ∈ ℝ+ → ((𝑋(ball‘𝐷)𝑒) ⊆ 𝑉 → ∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ 𝑉))) |
45 | 44 | rexlimdv 3151 | . 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 1086 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 ⊆ wss 3963 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 ↑m cmap 8865 Fincfn 8984 ℝcr 11152 − cmin 11490 2c2 12319 ℚcq 12988 ℝ+crp 13032 ↑cexp 14099 √csqrt 15269 Σcsu 15719 Basecbs 17245 distcds 17307 TopOpenctopn 17468 ∞Metcxmet 21367 Metcmet 21368 ballcbl 21369 MetOpencmopn 21372 TopOnctopon 22932 TopSpctps 22954 ℝ^crrx 25431 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ico 13390 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-prds 17494 df-pws 17496 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-ghm 19244 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-rhm 20489 df-subrng 20563 df-subrg 20587 df-drng 20748 df-field 20749 df-abv 20827 df-staf 20857 df-srng 20858 df-lmod 20877 df-lss 20948 df-lmhm 21039 df-lvec 21120 df-sra 21190 df-rgmod 21191 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-cnfld 21383 df-refld 21641 df-phl 21662 df-dsmm 21770 df-frlm 21785 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-xms 24346 df-ms 24347 df-nm 24611 df-ngp 24612 df-tng 24613 df-nrg 24614 df-nlm 24615 df-clm 25110 df-cph 25216 df-tcph 25217 df-rrx 25433 |
This theorem is referenced by: qndenserrnopn 46254 |
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