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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 5231 | . . . . . 6 ⊢ ∅ ∈ V | |
2 | 1 | snid 4597 | . . . . 5 ⊢ ∅ ∈ {∅} |
3 | 2 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ ∈ {∅}) |
4 | oveq2 7283 | . . . . . 6 ⊢ (𝐼 = ∅ → (ℚ ↑m 𝐼) = (ℚ ↑m ∅)) | |
5 | qex 12701 | . . . . . . . 8 ⊢ ℚ ∈ V | |
6 | mapdm0 8630 | . . . . . . . 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 2779 | . . . . 5 ⊢ (𝐼 = ∅ → {∅} = (ℚ ↑m 𝐼)) |
10 | 9 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝐼 = ∅) → {∅} = (ℚ ↑m 𝐼)) |
11 | 3, 10 | eleqtrd 2841 | . . 3 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ ∈ (ℚ ↑m 𝐼)) |
12 | qndenserrnbl.i | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ Fin) | |
13 | qndenserrnbl.d | . . . . . . . . 9 ⊢ 𝐷 = (dist‘(ℝ^‘𝐼)) | |
14 | 13 | rrxmetfi 24576 | . . . . . . . 8 ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘(ℝ ↑m 𝐼))) |
15 | 12, 14 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (Met‘(ℝ ↑m 𝐼))) |
16 | metxmet 23487 | . . . . . . 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 7283 | . . . . . . . . . . 11 ⊢ (𝐼 = ∅ → (ℝ ↑m 𝐼) = (ℝ ↑m ∅)) | |
22 | reex 10962 | . . . . . . . . . . . . 13 ⊢ ℝ ∈ V | |
23 | mapdm0 8630 | . . . . . . . . . . . . 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 2778 | . . . . . . . . . 10 ⊢ (𝐼 = ∅ → (ℝ ↑m 𝐼) = {∅}) |
27 | 26 | adantl 482 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐼 = ∅) → (ℝ ↑m 𝐼) = {∅}) |
28 | 20, 27 | eleqtrd 2841 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐼 = ∅) → 𝑋 ∈ {∅}) |
29 | elsng 4575 | . . . . . . . . . 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 2744 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ = 𝑋) |
34 | 33, 20 | eqeltrd 2839 | . . . . 5 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ ∈ (ℝ ↑m 𝐼)) |
35 | qndenserrnbl.e | . . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ℝ+) | |
36 | 35 | rpxrd 12773 | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ*) |
37 | 35 | rpgt0d 12775 | . . . . . . 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 23564 | . . . . 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 7290 | . . . 4 ⊢ ((𝜑 ∧ 𝐼 = ∅) → (∅(ball‘𝐷)𝐸) = (𝑋(ball‘𝐷)𝐸)) |
43 | 41, 42 | eleqtrd 2841 | . . 3 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ ∈ (𝑋(ball‘𝐷)𝐸)) |
44 | eleq1 2826 | . . . 4 ⊢ (𝑦 = ∅ → (𝑦 ∈ (𝑋(ball‘𝐷)𝐸) ↔ ∅ ∈ (𝑋(ball‘𝐷)𝐸))) | |
45 | 44 | rspcev 3561 | . . 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 2951 | . . . 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 43835 | . 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 1539 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3065 Vcvv 3432 ∅c0 4256 {csn 4561 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 ↑m cmap 8615 Fincfn 8733 ℝcr 10870 0cc0 10871 ℝ*cxr 11008 < clt 11009 ℚcq 12688 ℝ+crp 12730 distcds 16971 ∞Metcxmet 20582 Metcmet 20583 ballcbl 20584 ℝ^crrx 24547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-tpos 8042 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xadd 12849 df-ioo 13083 df-ico 13085 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-sum 15398 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-0g 17152 df-gsum 17153 df-prds 17158 df-pws 17160 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-grp 18580 df-minusg 18581 df-sbg 18582 df-subg 18752 df-ghm 18832 df-cntz 18923 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-cring 19786 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-dvr 19925 df-rnghom 19959 df-drng 19993 df-field 19994 df-subrg 20022 df-staf 20105 df-srng 20106 df-lmod 20125 df-lss 20194 df-sra 20434 df-rgmod 20435 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-cnfld 20598 df-refld 20810 df-dsmm 20939 df-frlm 20954 df-nm 23738 df-tng 23740 df-tcph 24333 df-rrx 24549 |
This theorem is referenced by: qndenserrnopnlem 43838 |
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