Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > qndenserrnbl | 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 |
---|---|
qndenserrnbl.i | ⊢ (𝜑 → 𝐼 ∈ Fin) |
qndenserrnbl.x | ⊢ (𝜑 → 𝑋 ∈ (ℝ ↑m 𝐼)) |
qndenserrnbl.d | ⊢ 𝐷 = (dist‘(ℝ^‘𝐼)) |
qndenserrnbl.e | ⊢ (𝜑 → 𝐸 ∈ ℝ+) |
Ref | Expression |
---|---|
qndenserrnbl | ⊢ (𝜑 → ∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5244 | . . . . . 6 ⊢ ∅ ∈ V | |
2 | 1 | snid 4605 | . . . . 5 ⊢ ∅ ∈ {∅} |
3 | 2 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ ∈ {∅}) |
4 | oveq2 7321 | . . . . . 6 ⊢ (𝐼 = ∅ → (ℚ ↑m 𝐼) = (ℚ ↑m ∅)) | |
5 | qex 12771 | . . . . . . . 8 ⊢ ℚ ∈ V | |
6 | mapdm0 8676 | . . . . . . . 8 ⊢ (ℚ ∈ V → (ℚ ↑m ∅) = {∅}) | |
7 | 5, 6 | ax-mp 5 | . . . . . . 7 ⊢ (ℚ ↑m ∅) = {∅} |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (𝐼 = ∅ → (ℚ ↑m ∅) = {∅}) |
9 | 4, 8 | eqtr2d 2778 | . . . . 5 ⊢ (𝐼 = ∅ → {∅} = (ℚ ↑m 𝐼)) |
10 | 9 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝐼 = ∅) → {∅} = (ℚ ↑m 𝐼)) |
11 | 3, 10 | eleqtrd 2840 | . . 3 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ ∈ (ℚ ↑m 𝐼)) |
12 | qndenserrnbl.i | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ Fin) | |
13 | qndenserrnbl.d | . . . . . . . . 9 ⊢ 𝐷 = (dist‘(ℝ^‘𝐼)) | |
14 | 13 | rrxmetfi 24647 | . . . . . . . 8 ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘(ℝ ↑m 𝐼))) |
15 | 12, 14 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (Met‘(ℝ ↑m 𝐼))) |
16 | metxmet 23558 | . . . . . . 7 ⊢ (𝐷 ∈ (Met‘(ℝ ↑m 𝐼)) → 𝐷 ∈ (∞Met‘(ℝ ↑m 𝐼))) | |
17 | 15, 16 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘(ℝ ↑m 𝐼))) |
18 | 17 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐼 = ∅) → 𝐷 ∈ (∞Met‘(ℝ ↑m 𝐼))) |
19 | qndenserrnbl.x | . . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (ℝ ↑m 𝐼)) | |
20 | 19 | adantr 481 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐼 = ∅) → 𝑋 ∈ (ℝ ↑m 𝐼)) |
21 | oveq2 7321 | . . . . . . . . . . 11 ⊢ (𝐼 = ∅ → (ℝ ↑m 𝐼) = (ℝ ↑m ∅)) | |
22 | reex 11032 | . . . . . . . . . . . . 13 ⊢ ℝ ∈ V | |
23 | mapdm0 8676 | . . . . . . . . . . . . 13 ⊢ (ℝ ∈ V → (ℝ ↑m ∅) = {∅}) | |
24 | 22, 23 | ax-mp 5 | . . . . . . . . . . . 12 ⊢ (ℝ ↑m ∅) = {∅} |
25 | 24 | a1i 11 | . . . . . . . . . . 11 ⊢ (𝐼 = ∅ → (ℝ ↑m ∅) = {∅}) |
26 | 21, 25 | eqtrd 2777 | . . . . . . . . . 10 ⊢ (𝐼 = ∅ → (ℝ ↑m 𝐼) = {∅}) |
27 | 26 | adantl 482 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐼 = ∅) → (ℝ ↑m 𝐼) = {∅}) |
28 | 20, 27 | eleqtrd 2840 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐼 = ∅) → 𝑋 ∈ {∅}) |
29 | elsng 4583 | . . . . . . . . . 10 ⊢ (𝑋 ∈ (ℝ ↑m 𝐼) → (𝑋 ∈ {∅} ↔ 𝑋 = ∅)) | |
30 | 19, 29 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → (𝑋 ∈ {∅} ↔ 𝑋 = ∅)) |
31 | 30 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐼 = ∅) → (𝑋 ∈ {∅} ↔ 𝑋 = ∅)) |
32 | 28, 31 | mpbid 231 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐼 = ∅) → 𝑋 = ∅) |
33 | 32 | eqcomd 2743 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ = 𝑋) |
34 | 33, 20 | eqeltrd 2838 | . . . . 5 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ ∈ (ℝ ↑m 𝐼)) |
35 | qndenserrnbl.e | . . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ℝ+) | |
36 | 35 | rpxrd 12843 | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ*) |
37 | 35 | rpgt0d 12845 | . . . . . . 7 ⊢ (𝜑 → 0 < 𝐸) |
38 | 36, 37 | jca 512 | . . . . . 6 ⊢ (𝜑 → (𝐸 ∈ ℝ* ∧ 0 < 𝐸)) |
39 | 38 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐼 = ∅) → (𝐸 ∈ ℝ* ∧ 0 < 𝐸)) |
40 | xblcntr 23635 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘(ℝ ↑m 𝐼)) ∧ ∅ ∈ (ℝ ↑m 𝐼) ∧ (𝐸 ∈ ℝ* ∧ 0 < 𝐸)) → ∅ ∈ (∅(ball‘𝐷)𝐸)) | |
41 | 18, 34, 39, 40 | syl3anc 1370 | . . . 4 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ ∈ (∅(ball‘𝐷)𝐸)) |
42 | 33 | oveq1d 7328 | . . . 4 ⊢ ((𝜑 ∧ 𝐼 = ∅) → (∅(ball‘𝐷)𝐸) = (𝑋(ball‘𝐷)𝐸)) |
43 | 41, 42 | eleqtrd 2840 | . . 3 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ ∈ (𝑋(ball‘𝐷)𝐸)) |
44 | eleq1 2825 | . . . 4 ⊢ (𝑦 = ∅ → (𝑦 ∈ (𝑋(ball‘𝐷)𝐸) ↔ ∅ ∈ (𝑋(ball‘𝐷)𝐸))) | |
45 | 44 | rspcev 3570 | . . 3 ⊢ ((∅ ∈ (ℚ ↑m 𝐼) ∧ ∅ ∈ (𝑋(ball‘𝐷)𝐸)) → ∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸)) |
46 | 11, 43, 45 | syl2anc 584 | . 2 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸)) |
47 | 12 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐼 = ∅) → 𝐼 ∈ Fin) |
48 | neqne 2949 | . . . 4 ⊢ (¬ 𝐼 = ∅ → 𝐼 ≠ ∅) | |
49 | 48 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐼 = ∅) → 𝐼 ≠ ∅) |
50 | 19 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐼 = ∅) → 𝑋 ∈ (ℝ ↑m 𝐼)) |
51 | 35 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐼 = ∅) → 𝐸 ∈ ℝ+) |
52 | 47, 49, 50, 13, 51 | qndenserrnbllem 44079 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐼 = ∅) → ∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸)) |
53 | 46, 52 | pm2.61dan 810 | 1 ⊢ (𝜑 → ∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ≠ wne 2941 ∃wrex 3071 Vcvv 3441 ∅c0 4266 {csn 4569 class class class wbr 5085 ‘cfv 6463 (class class class)co 7313 ↑m cmap 8661 Fincfn 8779 ℝcr 10940 0cc0 10941 ℝ*cxr 11078 < clt 11079 ℚcq 12758 ℝ+crp 12800 distcds 17038 ∞Metcxmet 20653 Metcmet 20654 ballcbl 20655 ℝ^crrx 24618 |
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 2708 ax-rep 5222 ax-sep 5236 ax-nul 5243 ax-pow 5301 ax-pr 5365 ax-un 7626 ax-inf2 9467 ax-cnex 10997 ax-resscn 10998 ax-1cn 10999 ax-icn 11000 ax-addcl 11001 ax-addrcl 11002 ax-mulcl 11003 ax-mulrcl 11004 ax-mulcom 11005 ax-addass 11006 ax-mulass 11007 ax-distr 11008 ax-i2m1 11009 ax-1ne0 11010 ax-1rid 11011 ax-rnegex 11012 ax-rrecex 11013 ax-cnre 11014 ax-pre-lttri 11015 ax-pre-lttrn 11016 ax-pre-ltadd 11017 ax-pre-mulgt0 11018 ax-pre-sup 11019 ax-addf 11020 ax-mulf 11021 |
This theorem depends on definitions: df-bi 206 df-an 397 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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3350 df-reu 3351 df-rab 3405 df-v 3443 df-sbc 3726 df-csb 3842 df-dif 3899 df-un 3901 df-in 3903 df-ss 3913 df-pss 3915 df-nul 4267 df-if 4470 df-pw 4545 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4849 df-int 4891 df-iun 4937 df-br 5086 df-opab 5148 df-mpt 5169 df-tr 5203 df-id 5505 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5560 df-se 5561 df-we 5562 df-xp 5611 df-rel 5612 df-cnv 5613 df-co 5614 df-dm 5615 df-rn 5616 df-res 5617 df-ima 5618 df-pred 6222 df-ord 6289 df-on 6290 df-lim 6291 df-suc 6292 df-iota 6415 df-fun 6465 df-fn 6466 df-f 6467 df-f1 6468 df-fo 6469 df-f1o 6470 df-fv 6471 df-isom 6472 df-riota 7270 df-ov 7316 df-oprab 7317 df-mpo 7318 df-of 7571 df-om 7756 df-1st 7874 df-2nd 7875 df-supp 8023 df-tpos 8087 df-frecs 8142 df-wrecs 8173 df-recs 8247 df-rdg 8286 df-1o 8342 df-er 8544 df-map 8663 df-ixp 8732 df-en 8780 df-dom 8781 df-sdom 8782 df-fin 8783 df-fsupp 9197 df-sup 9269 df-inf 9270 df-oi 9337 df-card 9765 df-pnf 11081 df-mnf 11082 df-xr 11083 df-ltxr 11084 df-le 11085 df-sub 11277 df-neg 11278 df-div 11703 df-nn 12044 df-2 12106 df-3 12107 df-4 12108 df-5 12109 df-6 12110 df-7 12111 df-8 12112 df-9 12113 df-n0 12304 df-z 12390 df-dec 12508 df-uz 12653 df-q 12759 df-rp 12801 df-xadd 12919 df-ioo 13153 df-ico 13155 df-fz 13310 df-fzo 13453 df-seq 13792 df-exp 13853 df-hash 14115 df-cj 14879 df-re 14880 df-im 14881 df-sqrt 15015 df-abs 15016 df-clim 15266 df-sum 15467 df-struct 16915 df-sets 16932 df-slot 16950 df-ndx 16962 df-base 16980 df-ress 17009 df-plusg 17042 df-mulr 17043 df-starv 17044 df-sca 17045 df-vsca 17046 df-ip 17047 df-tset 17048 df-ple 17049 df-ds 17051 df-unif 17052 df-hom 17053 df-cco 17054 df-0g 17219 df-gsum 17220 df-prds 17225 df-pws 17227 df-mgm 18393 df-sgrp 18442 df-mnd 18453 df-mhm 18497 df-grp 18647 df-minusg 18648 df-sbg 18649 df-subg 18819 df-ghm 18899 df-cntz 18990 df-cmn 19455 df-abl 19456 df-mgp 19788 df-ur 19805 df-ring 19852 df-cring 19853 df-oppr 19929 df-dvdsr 19950 df-unit 19951 df-invr 19981 df-dvr 19992 df-rnghom 20026 df-drng 20064 df-field 20065 df-subrg 20093 df-staf 20176 df-srng 20177 df-lmod 20196 df-lss 20265 df-sra 20505 df-rgmod 20506 df-psmet 20660 df-xmet 20661 df-met 20662 df-bl 20663 df-cnfld 20669 df-refld 20881 df-dsmm 21010 df-frlm 21025 df-nm 23809 df-tng 23811 df-tcph 24404 df-rrx 24620 |
This theorem is referenced by: qndenserrnopnlem 44082 |
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