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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 25260 | . . . . 5 ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘(ℝ ↑m 𝐼))) |
4 | 1, 3 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ (Met‘(ℝ ↑m 𝐼))) |
5 | metxmet 24160 | . . . 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 2842 | . . . 4 ⊢ (𝜑 → 𝑉 ∈ (TopOpen‘(ℝ^‘𝐼))) |
10 | 1 | rrxtopnfi 45462 | . . . . 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 2731 | . . . . . . . . 9 ⊢ (ℝ^‘𝐼) = (ℝ^‘𝐼) | |
13 | eqid 2731 | . . . . . . . . 9 ⊢ (ℝ ↑m 𝐼) = (ℝ ↑m 𝐼) | |
14 | 12, 13 | rrxdsfi 25259 | . . . . . . . 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 2772 | . . . . . 6 ⊢ (𝜑 → (𝑓 ∈ (ℝ ↑m 𝐼), 𝑔 ∈ (ℝ ↑m 𝐼) ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))) = 𝐷) |
17 | 16 | fveq2d 6895 | . . . . 5 ⊢ (𝜑 → (MetOpen‘(𝑓 ∈ (ℝ ↑m 𝐼), 𝑔 ∈ (ℝ ↑m 𝐼) ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) = (MetOpen‘𝐷)) |
18 | 10, 17 | eqtrd 2771 | . . . 4 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝐼)) = (MetOpen‘𝐷)) |
19 | 9, 18 | eleqtrd 2834 | . . 3 ⊢ (𝜑 → 𝑉 ∈ (MetOpen‘𝐷)) |
20 | qndenserrnopnlem.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
21 | eqid 2731 | . . . 4 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷) | |
22 | 21 | mopni2 24322 | . . 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 45461 | . . . . . . . . . . . 12 ⊢ (𝐼 ∈ Fin → (ℝ^‘𝐼) ∈ TopSp) | |
26 | 1, 25 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → (ℝ^‘𝐼) ∈ TopSp) |
27 | eqid 2731 | . . . . . . . . . . . 12 ⊢ (Base‘(ℝ^‘𝐼)) = (Base‘(ℝ^‘𝐼)) | |
28 | 27, 8 | istps 22756 | . . . . . . . . . . 11 ⊢ ((ℝ^‘𝐼) ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘(ℝ^‘𝐼)))) |
29 | 26, 28 | sylib 217 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(Base‘(ℝ^‘𝐼)))) |
30 | 1, 12, 27 | rrxbasefi 25258 | . . . . . . . . . . 11 ⊢ (𝜑 → (Base‘(ℝ^‘𝐼)) = (ℝ ↑m 𝐼)) |
31 | 30 | fveq2d 6895 | . . . . . . . . . 10 ⊢ (𝜑 → (TopOn‘(Base‘(ℝ^‘𝐼))) = (TopOn‘(ℝ ↑m 𝐼))) |
32 | 29, 31 | eleqtrd 2834 | . . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(ℝ ↑m 𝐼))) |
33 | toponss 22749 | . . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘(ℝ ↑m 𝐼)) ∧ 𝑉 ∈ 𝐽) → 𝑉 ⊆ (ℝ ↑m 𝐼)) | |
34 | 32, 7, 33 | syl2anc 583 | . . . . . . . 8 ⊢ (𝜑 → 𝑉 ⊆ (ℝ ↑m 𝐼)) |
35 | 34, 20 | sseldd 3983 | . . . . . . 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 45470 | . . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) → ∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝑒)) |
39 | ssel 3975 | . . . . . . . 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 3167 | . . . . 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 3152 | . 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 1540 ∈ wcel 2105 ∃wrex 3069 ⊆ wss 3948 ‘cfv 6543 (class class class)co 7412 ∈ cmpo 7414 ↑m cmap 8826 Fincfn 8945 ℝcr 11115 − cmin 11451 2c2 12274 ℚcq 12939 ℝ+crp 12981 ↑cexp 14034 √csqrt 15187 Σcsu 15639 Basecbs 17151 distcds 17213 TopOpenctopn 17374 ∞Metcxmet 21218 Metcmet 21219 ballcbl 21220 MetOpencmopn 21223 TopOnctopon 22732 TopSpctps 22754 ℝ^crrx 25231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 ax-addf 11195 ax-mulf 11196 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-tpos 8217 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-sup 9443 df-inf 9444 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-q 12940 df-rp 12982 df-xneg 13099 df-xadd 13100 df-xmul 13101 df-ioo 13335 df-ico 13337 df-fz 13492 df-fzo 13635 df-seq 13974 df-exp 14035 df-hash 14298 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-clim 15439 df-sum 15640 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-rest 17375 df-topn 17376 df-0g 17394 df-gsum 17395 df-topgen 17396 df-prds 17400 df-pws 17402 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-mhm 18711 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19046 df-ghm 19135 df-cntz 19229 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-cring 20137 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-dvr 20299 df-rhm 20370 df-subrng 20442 df-subrg 20467 df-drng 20585 df-field 20586 df-abv 20656 df-staf 20684 df-srng 20685 df-lmod 20704 df-lss 20775 df-lmhm 20866 df-lvec 20947 df-sra 21019 df-rgmod 21020 df-psmet 21225 df-xmet 21226 df-met 21227 df-bl 21228 df-mopn 21229 df-cnfld 21234 df-refld 21468 df-phl 21489 df-dsmm 21597 df-frlm 21612 df-top 22716 df-topon 22733 df-topsp 22755 df-bases 22769 df-xms 24146 df-ms 24147 df-nm 24411 df-ngp 24412 df-tng 24413 df-nrg 24414 df-nlm 24415 df-clm 24910 df-cph 25016 df-tcph 25017 df-rrx 25233 |
This theorem is referenced by: qndenserrnopn 45473 |
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