Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > qndenserrn | 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 |
---|---|
qndenserrn.i | ⊢ (𝜑 → 𝐼 ∈ Fin) |
qndenserrn.j | ⊢ 𝐽 = (TopOpen‘(ℝ^‘𝐼)) |
Ref | Expression |
---|---|
qndenserrn | ⊢ (𝜑 → ((cls‘𝐽)‘(ℚ ↑m 𝐼)) = (ℝ ↑m 𝐼)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qndenserrn.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ Fin) | |
2 | qndenserrn.j | . . . . . 6 ⊢ 𝐽 = (TopOpen‘(ℝ^‘𝐼)) | |
3 | 2 | rrxtop 43459 | . . . . 5 ⊢ (𝐼 ∈ Fin → 𝐽 ∈ Top) |
4 | 1, 3 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐽 ∈ Top) |
5 | reex 10803 | . . . . . . 7 ⊢ ℝ ∈ V | |
6 | qssre 12538 | . . . . . . 7 ⊢ ℚ ⊆ ℝ | |
7 | mapss 8559 | . . . . . . 7 ⊢ ((ℝ ∈ V ∧ ℚ ⊆ ℝ) → (ℚ ↑m 𝐼) ⊆ (ℝ ↑m 𝐼)) | |
8 | 5, 6, 7 | mp2an 692 | . . . . . 6 ⊢ (ℚ ↑m 𝐼) ⊆ (ℝ ↑m 𝐼) |
9 | 8 | a1i 11 | . . . . 5 ⊢ (𝜑 → (ℚ ↑m 𝐼) ⊆ (ℝ ↑m 𝐼)) |
10 | eqid 2734 | . . . . . . . 8 ⊢ (ℝ^‘𝐼) = (ℝ^‘𝐼) | |
11 | eqid 2734 | . . . . . . . 8 ⊢ (Base‘(ℝ^‘𝐼)) = (Base‘(ℝ^‘𝐼)) | |
12 | 1, 10, 11 | rrxbasefi 24279 | . . . . . . 7 ⊢ (𝜑 → (Base‘(ℝ^‘𝐼)) = (ℝ ↑m 𝐼)) |
13 | 12 | eqcomd 2740 | . . . . . 6 ⊢ (𝜑 → (ℝ ↑m 𝐼) = (Base‘(ℝ^‘𝐼))) |
14 | rrxtps 43456 | . . . . . . 7 ⊢ (𝐼 ∈ Fin → (ℝ^‘𝐼) ∈ TopSp) | |
15 | eqid 2734 | . . . . . . . 8 ⊢ (TopOpen‘(ℝ^‘𝐼)) = (TopOpen‘(ℝ^‘𝐼)) | |
16 | 11, 15 | tpsuni 21805 | . . . . . . 7 ⊢ ((ℝ^‘𝐼) ∈ TopSp → (Base‘(ℝ^‘𝐼)) = ∪ (TopOpen‘(ℝ^‘𝐼))) |
17 | 1, 14, 16 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → (Base‘(ℝ^‘𝐼)) = ∪ (TopOpen‘(ℝ^‘𝐼))) |
18 | 2 | unieqi 4822 | . . . . . . . 8 ⊢ ∪ 𝐽 = ∪ (TopOpen‘(ℝ^‘𝐼)) |
19 | 18 | eqcomi 2743 | . . . . . . 7 ⊢ ∪ (TopOpen‘(ℝ^‘𝐼)) = ∪ 𝐽 |
20 | 19 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ∪ (TopOpen‘(ℝ^‘𝐼)) = ∪ 𝐽) |
21 | 13, 17, 20 | 3eqtrd 2778 | . . . . 5 ⊢ (𝜑 → (ℝ ↑m 𝐼) = ∪ 𝐽) |
22 | 9, 21 | sseqtrd 3931 | . . . 4 ⊢ (𝜑 → (ℚ ↑m 𝐼) ⊆ ∪ 𝐽) |
23 | eqid 2734 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
24 | 23 | clsss3 21928 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ (ℚ ↑m 𝐼) ⊆ ∪ 𝐽) → ((cls‘𝐽)‘(ℚ ↑m 𝐼)) ⊆ ∪ 𝐽) |
25 | 4, 22, 24 | syl2anc 587 | . . 3 ⊢ (𝜑 → ((cls‘𝐽)‘(ℚ ↑m 𝐼)) ⊆ ∪ 𝐽) |
26 | 21 | eqcomd 2740 | . . 3 ⊢ (𝜑 → ∪ 𝐽 = (ℝ ↑m 𝐼)) |
27 | 25, 26 | sseqtrd 3931 | . 2 ⊢ (𝜑 → ((cls‘𝐽)‘(ℚ ↑m 𝐼)) ⊆ (ℝ ↑m 𝐼)) |
28 | 1 | ad2antrr 726 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → 𝐼 ∈ Fin) |
29 | id 22 | . . . . . . . . . . . 12 ⊢ (𝑣 ∈ 𝐽 → 𝑣 ∈ 𝐽) | |
30 | 29, 2 | eleqtrdi 2844 | . . . . . . . . . . 11 ⊢ (𝑣 ∈ 𝐽 → 𝑣 ∈ (TopOpen‘(ℝ^‘𝐼))) |
31 | 30 | ad2antlr 727 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → 𝑣 ∈ (TopOpen‘(ℝ^‘𝐼))) |
32 | ne0i 4239 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑣 → 𝑣 ≠ ∅) | |
33 | 32 | adantl 485 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → 𝑣 ≠ ∅) |
34 | 28, 15, 31, 33 | qndenserrnopn 43468 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → ∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ 𝑣) |
35 | df-rex 3060 | . . . . . . . . 9 ⊢ (∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ 𝑣 ↔ ∃𝑦(𝑦 ∈ (ℚ ↑m 𝐼) ∧ 𝑦 ∈ 𝑣)) | |
36 | 34, 35 | sylib 221 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → ∃𝑦(𝑦 ∈ (ℚ ↑m 𝐼) ∧ 𝑦 ∈ 𝑣)) |
37 | simpr 488 | . . . . . . . . . . 11 ⊢ ((𝑦 ∈ (ℚ ↑m 𝐼) ∧ 𝑦 ∈ 𝑣) → 𝑦 ∈ 𝑣) | |
38 | simpl 486 | . . . . . . . . . . 11 ⊢ ((𝑦 ∈ (ℚ ↑m 𝐼) ∧ 𝑦 ∈ 𝑣) → 𝑦 ∈ (ℚ ↑m 𝐼)) | |
39 | 37, 38 | elind 4098 | . . . . . . . . . 10 ⊢ ((𝑦 ∈ (ℚ ↑m 𝐼) ∧ 𝑦 ∈ 𝑣) → 𝑦 ∈ (𝑣 ∩ (ℚ ↑m 𝐼))) |
40 | 39 | a1i 11 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → ((𝑦 ∈ (ℚ ↑m 𝐼) ∧ 𝑦 ∈ 𝑣) → 𝑦 ∈ (𝑣 ∩ (ℚ ↑m 𝐼)))) |
41 | 40 | eximdv 1925 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → (∃𝑦(𝑦 ∈ (ℚ ↑m 𝐼) ∧ 𝑦 ∈ 𝑣) → ∃𝑦 𝑦 ∈ (𝑣 ∩ (ℚ ↑m 𝐼)))) |
42 | 36, 41 | mpd 15 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → ∃𝑦 𝑦 ∈ (𝑣 ∩ (ℚ ↑m 𝐼))) |
43 | n0 4251 | . . . . . . 7 ⊢ ((𝑣 ∩ (ℚ ↑m 𝐼)) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝑣 ∩ (ℚ ↑m 𝐼))) | |
44 | 42, 43 | sylibr 237 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → (𝑣 ∩ (ℚ ↑m 𝐼)) ≠ ∅) |
45 | 44 | ex 416 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐽) → (𝑥 ∈ 𝑣 → (𝑣 ∩ (ℚ ↑m 𝐼)) ≠ ∅)) |
46 | 45 | adantlr 715 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℝ ↑m 𝐼)) ∧ 𝑣 ∈ 𝐽) → (𝑥 ∈ 𝑣 → (𝑣 ∩ (ℚ ↑m 𝐼)) ≠ ∅)) |
47 | 46 | ralrimiva 3098 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑m 𝐼)) → ∀𝑣 ∈ 𝐽 (𝑥 ∈ 𝑣 → (𝑣 ∩ (ℚ ↑m 𝐼)) ≠ ∅)) |
48 | 4 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑m 𝐼)) → 𝐽 ∈ Top) |
49 | 22 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑m 𝐼)) → (ℚ ↑m 𝐼) ⊆ ∪ 𝐽) |
50 | simpr 488 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑m 𝐼)) → 𝑥 ∈ (ℝ ↑m 𝐼)) | |
51 | 21 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑m 𝐼)) → (ℝ ↑m 𝐼) = ∪ 𝐽) |
52 | 50, 51 | eleqtrd 2836 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑m 𝐼)) → 𝑥 ∈ ∪ 𝐽) |
53 | 23 | elcls 21942 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ (ℚ ↑m 𝐼) ⊆ ∪ 𝐽 ∧ 𝑥 ∈ ∪ 𝐽) → (𝑥 ∈ ((cls‘𝐽)‘(ℚ ↑m 𝐼)) ↔ ∀𝑣 ∈ 𝐽 (𝑥 ∈ 𝑣 → (𝑣 ∩ (ℚ ↑m 𝐼)) ≠ ∅))) |
54 | 48, 49, 52, 53 | syl3anc 1373 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑m 𝐼)) → (𝑥 ∈ ((cls‘𝐽)‘(ℚ ↑m 𝐼)) ↔ ∀𝑣 ∈ 𝐽 (𝑥 ∈ 𝑣 → (𝑣 ∩ (ℚ ↑m 𝐼)) ≠ ∅))) |
55 | 47, 54 | mpbird 260 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑m 𝐼)) → 𝑥 ∈ ((cls‘𝐽)‘(ℚ ↑m 𝐼))) |
56 | 27, 55 | eqelssd 3912 | 1 ⊢ (𝜑 → ((cls‘𝐽)‘(ℚ ↑m 𝐼)) = (ℝ ↑m 𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∃wex 1787 ∈ wcel 2110 ≠ wne 2935 ∀wral 3054 ∃wrex 3055 Vcvv 3401 ∩ cin 3856 ⊆ wss 3857 ∅c0 4227 ∪ cuni 4809 ‘cfv 6369 (class class class)co 7202 ↑m cmap 8497 Fincfn 8615 ℝcr 10711 ℚcq 12527 Basecbs 16684 TopOpenctopn 16898 Topctop 21762 TopSpctps 21801 clsccl 21887 ℝ^crrx 24252 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-inf2 9245 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 ax-addf 10791 ax-mulf 10792 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-iin 4897 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-of 7458 df-om 7634 df-1st 7750 df-2nd 7751 df-supp 7893 df-tpos 7957 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-map 8499 df-ixp 8568 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-fsupp 8975 df-sup 9047 df-inf 9048 df-oi 9115 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-dec 12277 df-uz 12422 df-q 12528 df-rp 12570 df-xneg 12687 df-xadd 12688 df-xmul 12689 df-ioo 12922 df-ico 12924 df-fz 13079 df-fzo 13222 df-seq 13558 df-exp 13619 df-hash 13880 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-clim 15032 df-sum 15233 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-starv 16782 df-sca 16783 df-vsca 16784 df-ip 16785 df-tset 16786 df-ple 16787 df-ds 16789 df-unif 16790 df-hom 16791 df-cco 16792 df-rest 16899 df-topn 16900 df-0g 16918 df-gsum 16919 df-topgen 16920 df-prds 16924 df-pws 16926 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-mhm 18190 df-submnd 18191 df-grp 18340 df-minusg 18341 df-sbg 18342 df-subg 18512 df-ghm 18592 df-cntz 18683 df-cmn 19144 df-abl 19145 df-mgp 19477 df-ur 19489 df-ring 19536 df-cring 19537 df-oppr 19613 df-dvdsr 19631 df-unit 19632 df-invr 19662 df-dvr 19673 df-rnghom 19707 df-drng 19741 df-field 19742 df-subrg 19770 df-abv 19825 df-staf 19853 df-srng 19854 df-lmod 19873 df-lss 19941 df-lmhm 20031 df-lvec 20112 df-sra 20181 df-rgmod 20182 df-psmet 20327 df-xmet 20328 df-met 20329 df-bl 20330 df-mopn 20331 df-cnfld 20336 df-refld 20539 df-phl 20560 df-dsmm 20666 df-frlm 20681 df-top 21763 df-topon 21780 df-topsp 21802 df-bases 21815 df-cld 21888 df-ntr 21889 df-cls 21890 df-xms 23190 df-ms 23191 df-nm 23452 df-ngp 23453 df-tng 23454 df-nrg 23455 df-nlm 23456 df-clm 23932 df-cph 24037 df-tcph 24038 df-rrx 24254 |
This theorem is referenced by: (None) |
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