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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 42567 | . . . . 5 ⊢ (𝐼 ∈ Fin → 𝐽 ∈ Top) |
4 | 1, 3 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐽 ∈ Top) |
5 | reex 10622 | . . . . . . 7 ⊢ ℝ ∈ V | |
6 | qssre 12352 | . . . . . . 7 ⊢ ℚ ⊆ ℝ | |
7 | mapss 8447 | . . . . . . 7 ⊢ ((ℝ ∈ V ∧ ℚ ⊆ ℝ) → (ℚ ↑m 𝐼) ⊆ (ℝ ↑m 𝐼)) | |
8 | 5, 6, 7 | mp2an 690 | . . . . . 6 ⊢ (ℚ ↑m 𝐼) ⊆ (ℝ ↑m 𝐼) |
9 | 8 | a1i 11 | . . . . 5 ⊢ (𝜑 → (ℚ ↑m 𝐼) ⊆ (ℝ ↑m 𝐼)) |
10 | eqid 2821 | . . . . . . . 8 ⊢ (ℝ^‘𝐼) = (ℝ^‘𝐼) | |
11 | eqid 2821 | . . . . . . . 8 ⊢ (Base‘(ℝ^‘𝐼)) = (Base‘(ℝ^‘𝐼)) | |
12 | 1, 10, 11 | rrxbasefi 24007 | . . . . . . 7 ⊢ (𝜑 → (Base‘(ℝ^‘𝐼)) = (ℝ ↑m 𝐼)) |
13 | 12 | eqcomd 2827 | . . . . . 6 ⊢ (𝜑 → (ℝ ↑m 𝐼) = (Base‘(ℝ^‘𝐼))) |
14 | rrxtps 42564 | . . . . . . 7 ⊢ (𝐼 ∈ Fin → (ℝ^‘𝐼) ∈ TopSp) | |
15 | eqid 2821 | . . . . . . . 8 ⊢ (TopOpen‘(ℝ^‘𝐼)) = (TopOpen‘(ℝ^‘𝐼)) | |
16 | 11, 15 | tpsuni 21538 | . . . . . . 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 4841 | . . . . . . . 8 ⊢ ∪ 𝐽 = ∪ (TopOpen‘(ℝ^‘𝐼)) |
19 | 18 | eqcomi 2830 | . . . . . . 7 ⊢ ∪ (TopOpen‘(ℝ^‘𝐼)) = ∪ 𝐽 |
20 | 19 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ∪ (TopOpen‘(ℝ^‘𝐼)) = ∪ 𝐽) |
21 | 13, 17, 20 | 3eqtrd 2860 | . . . . 5 ⊢ (𝜑 → (ℝ ↑m 𝐼) = ∪ 𝐽) |
22 | 9, 21 | sseqtrd 4007 | . . . 4 ⊢ (𝜑 → (ℚ ↑m 𝐼) ⊆ ∪ 𝐽) |
23 | eqid 2821 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
24 | 23 | clsss3 21661 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ (ℚ ↑m 𝐼) ⊆ ∪ 𝐽) → ((cls‘𝐽)‘(ℚ ↑m 𝐼)) ⊆ ∪ 𝐽) |
25 | 4, 22, 24 | syl2anc 586 | . . 3 ⊢ (𝜑 → ((cls‘𝐽)‘(ℚ ↑m 𝐼)) ⊆ ∪ 𝐽) |
26 | 21 | eqcomd 2827 | . . 3 ⊢ (𝜑 → ∪ 𝐽 = (ℝ ↑m 𝐼)) |
27 | 25, 26 | sseqtrd 4007 | . 2 ⊢ (𝜑 → ((cls‘𝐽)‘(ℚ ↑m 𝐼)) ⊆ (ℝ ↑m 𝐼)) |
28 | 1 | ad2antrr 724 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → 𝐼 ∈ Fin) |
29 | id 22 | . . . . . . . . . . . 12 ⊢ (𝑣 ∈ 𝐽 → 𝑣 ∈ 𝐽) | |
30 | 29, 2 | eleqtrdi 2923 | . . . . . . . . . . 11 ⊢ (𝑣 ∈ 𝐽 → 𝑣 ∈ (TopOpen‘(ℝ^‘𝐼))) |
31 | 30 | ad2antlr 725 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → 𝑣 ∈ (TopOpen‘(ℝ^‘𝐼))) |
32 | ne0i 4300 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑣 → 𝑣 ≠ ∅) | |
33 | 32 | adantl 484 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → 𝑣 ≠ ∅) |
34 | 28, 15, 31, 33 | qndenserrnopn 42576 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → ∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ 𝑣) |
35 | df-rex 3144 | . . . . . . . . 9 ⊢ (∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ 𝑣 ↔ ∃𝑦(𝑦 ∈ (ℚ ↑m 𝐼) ∧ 𝑦 ∈ 𝑣)) | |
36 | 34, 35 | sylib 220 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → ∃𝑦(𝑦 ∈ (ℚ ↑m 𝐼) ∧ 𝑦 ∈ 𝑣)) |
37 | simpr 487 | . . . . . . . . . . 11 ⊢ ((𝑦 ∈ (ℚ ↑m 𝐼) ∧ 𝑦 ∈ 𝑣) → 𝑦 ∈ 𝑣) | |
38 | simpl 485 | . . . . . . . . . . 11 ⊢ ((𝑦 ∈ (ℚ ↑m 𝐼) ∧ 𝑦 ∈ 𝑣) → 𝑦 ∈ (ℚ ↑m 𝐼)) | |
39 | 37, 38 | elind 4171 | . . . . . . . . . 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 1914 | . . . . . . . 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 4310 | . . . . . . 7 ⊢ ((𝑣 ∩ (ℚ ↑m 𝐼)) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝑣 ∩ (ℚ ↑m 𝐼))) | |
44 | 42, 43 | sylibr 236 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → (𝑣 ∩ (ℚ ↑m 𝐼)) ≠ ∅) |
45 | 44 | ex 415 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐽) → (𝑥 ∈ 𝑣 → (𝑣 ∩ (ℚ ↑m 𝐼)) ≠ ∅)) |
46 | 45 | adantlr 713 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℝ ↑m 𝐼)) ∧ 𝑣 ∈ 𝐽) → (𝑥 ∈ 𝑣 → (𝑣 ∩ (ℚ ↑m 𝐼)) ≠ ∅)) |
47 | 46 | ralrimiva 3182 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑m 𝐼)) → ∀𝑣 ∈ 𝐽 (𝑥 ∈ 𝑣 → (𝑣 ∩ (ℚ ↑m 𝐼)) ≠ ∅)) |
48 | 4 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑m 𝐼)) → 𝐽 ∈ Top) |
49 | 22 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑m 𝐼)) → (ℚ ↑m 𝐼) ⊆ ∪ 𝐽) |
50 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑m 𝐼)) → 𝑥 ∈ (ℝ ↑m 𝐼)) | |
51 | 21 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑m 𝐼)) → (ℝ ↑m 𝐼) = ∪ 𝐽) |
52 | 50, 51 | eleqtrd 2915 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑m 𝐼)) → 𝑥 ∈ ∪ 𝐽) |
53 | 23 | elcls 21675 | . . . 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 1367 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑m 𝐼)) → (𝑥 ∈ ((cls‘𝐽)‘(ℚ ↑m 𝐼)) ↔ ∀𝑣 ∈ 𝐽 (𝑥 ∈ 𝑣 → (𝑣 ∩ (ℚ ↑m 𝐼)) ≠ ∅))) |
55 | 47, 54 | mpbird 259 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑m 𝐼)) → 𝑥 ∈ ((cls‘𝐽)‘(ℚ ↑m 𝐼))) |
56 | 27, 55 | eqelssd 3988 | 1 ⊢ (𝜑 → ((cls‘𝐽)‘(ℚ ↑m 𝐼)) = (ℝ ↑m 𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 Vcvv 3495 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 ∪ cuni 4832 ‘cfv 6350 (class class class)co 7150 ↑m cmap 8400 Fincfn 8503 ℝcr 10530 ℚcq 12342 Basecbs 16477 TopOpenctopn 16689 Topctop 21495 TopSpctps 21534 clsccl 21620 ℝ^crrx 23980 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ico 12738 df-fz 12887 df-fzo 13028 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-sum 15037 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-rest 16690 df-topn 16691 df-0g 16709 df-gsum 16710 df-topgen 16711 df-prds 16715 df-pws 16717 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-subg 18270 df-ghm 18350 df-cntz 18441 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-dvr 19427 df-rnghom 19461 df-drng 19498 df-field 19499 df-subrg 19527 df-abv 19582 df-staf 19610 df-srng 19611 df-lmod 19630 df-lss 19698 df-lmhm 19788 df-lvec 19869 df-sra 19938 df-rgmod 19939 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-cnfld 20540 df-refld 20743 df-phl 20764 df-dsmm 20870 df-frlm 20885 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-cld 21621 df-ntr 21622 df-cls 21623 df-xms 22924 df-ms 22925 df-nm 23186 df-ngp 23187 df-tng 23188 df-nrg 23189 df-nlm 23190 df-clm 23661 df-cph 23766 df-tcph 23767 df-rrx 23982 |
This theorem is referenced by: (None) |
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