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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 41447 | . . . . 5 ⊢ (𝐼 ∈ Fin → 𝐽 ∈ Top) |
4 | 1, 3 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐽 ∈ Top) |
5 | reex 10365 | . . . . . . 7 ⊢ ℝ ∈ V | |
6 | qssre 12111 | . . . . . . 7 ⊢ ℚ ⊆ ℝ | |
7 | mapss 8188 | . . . . . . 7 ⊢ ((ℝ ∈ V ∧ ℚ ⊆ ℝ) → (ℚ ↑𝑚 𝐼) ⊆ (ℝ ↑𝑚 𝐼)) | |
8 | 5, 6, 7 | mp2an 682 | . . . . . 6 ⊢ (ℚ ↑𝑚 𝐼) ⊆ (ℝ ↑𝑚 𝐼) |
9 | 8 | a1i 11 | . . . . 5 ⊢ (𝜑 → (ℚ ↑𝑚 𝐼) ⊆ (ℝ ↑𝑚 𝐼)) |
10 | eqid 2778 | . . . . . . . 8 ⊢ (ℝ^‘𝐼) = (ℝ^‘𝐼) | |
11 | eqid 2778 | . . . . . . . 8 ⊢ (Base‘(ℝ^‘𝐼)) = (Base‘(ℝ^‘𝐼)) | |
12 | 1, 10, 11 | rrxbasefi 23627 | . . . . . . 7 ⊢ (𝜑 → (Base‘(ℝ^‘𝐼)) = (ℝ ↑𝑚 𝐼)) |
13 | 12 | eqcomd 2784 | . . . . . 6 ⊢ (𝜑 → (ℝ ↑𝑚 𝐼) = (Base‘(ℝ^‘𝐼))) |
14 | rrxtps 41444 | . . . . . . 7 ⊢ (𝐼 ∈ Fin → (ℝ^‘𝐼) ∈ TopSp) | |
15 | eqid 2778 | . . . . . . . 8 ⊢ (TopOpen‘(ℝ^‘𝐼)) = (TopOpen‘(ℝ^‘𝐼)) | |
16 | 11, 15 | tpsuni 21159 | . . . . . . 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 4682 | . . . . . . . 8 ⊢ ∪ 𝐽 = ∪ (TopOpen‘(ℝ^‘𝐼)) |
19 | 18 | eqcomi 2787 | . . . . . . 7 ⊢ ∪ (TopOpen‘(ℝ^‘𝐼)) = ∪ 𝐽 |
20 | 19 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ∪ (TopOpen‘(ℝ^‘𝐼)) = ∪ 𝐽) |
21 | 13, 17, 20 | 3eqtrd 2818 | . . . . 5 ⊢ (𝜑 → (ℝ ↑𝑚 𝐼) = ∪ 𝐽) |
22 | 9, 21 | sseqtrd 3860 | . . . 4 ⊢ (𝜑 → (ℚ ↑𝑚 𝐼) ⊆ ∪ 𝐽) |
23 | eqid 2778 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
24 | 23 | clsss3 21282 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ (ℚ ↑𝑚 𝐼) ⊆ ∪ 𝐽) → ((cls‘𝐽)‘(ℚ ↑𝑚 𝐼)) ⊆ ∪ 𝐽) |
25 | 4, 22, 24 | syl2anc 579 | . . 3 ⊢ (𝜑 → ((cls‘𝐽)‘(ℚ ↑𝑚 𝐼)) ⊆ ∪ 𝐽) |
26 | 21 | eqcomd 2784 | . . 3 ⊢ (𝜑 → ∪ 𝐽 = (ℝ ↑𝑚 𝐼)) |
27 | 25, 26 | sseqtrd 3860 | . 2 ⊢ (𝜑 → ((cls‘𝐽)‘(ℚ ↑𝑚 𝐼)) ⊆ (ℝ ↑𝑚 𝐼)) |
28 | 1 | ad2antrr 716 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → 𝐼 ∈ Fin) |
29 | id 22 | . . . . . . . . . . . 12 ⊢ (𝑣 ∈ 𝐽 → 𝑣 ∈ 𝐽) | |
30 | 29, 2 | syl6eleq 2869 | . . . . . . . . . . 11 ⊢ (𝑣 ∈ 𝐽 → 𝑣 ∈ (TopOpen‘(ℝ^‘𝐼))) |
31 | 30 | ad2antlr 717 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → 𝑣 ∈ (TopOpen‘(ℝ^‘𝐼))) |
32 | ne0i 4149 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑣 → 𝑣 ≠ ∅) | |
33 | 32 | adantl 475 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → 𝑣 ≠ ∅) |
34 | 28, 15, 31, 33 | qndenserrnopn 41456 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → ∃𝑦 ∈ (ℚ ↑𝑚 𝐼)𝑦 ∈ 𝑣) |
35 | df-rex 3096 | . . . . . . . . 9 ⊢ (∃𝑦 ∈ (ℚ ↑𝑚 𝐼)𝑦 ∈ 𝑣 ↔ ∃𝑦(𝑦 ∈ (ℚ ↑𝑚 𝐼) ∧ 𝑦 ∈ 𝑣)) | |
36 | 34, 35 | sylib 210 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → ∃𝑦(𝑦 ∈ (ℚ ↑𝑚 𝐼) ∧ 𝑦 ∈ 𝑣)) |
37 | simpr 479 | . . . . . . . . . . 11 ⊢ ((𝑦 ∈ (ℚ ↑𝑚 𝐼) ∧ 𝑦 ∈ 𝑣) → 𝑦 ∈ 𝑣) | |
38 | simpl 476 | . . . . . . . . . . 11 ⊢ ((𝑦 ∈ (ℚ ↑𝑚 𝐼) ∧ 𝑦 ∈ 𝑣) → 𝑦 ∈ (ℚ ↑𝑚 𝐼)) | |
39 | 37, 38 | elind 4021 | . . . . . . . . . 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 1960 | . . . . . . . 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 4159 | . . . . . . 7 ⊢ ((𝑣 ∩ (ℚ ↑𝑚 𝐼)) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝑣 ∩ (ℚ ↑𝑚 𝐼))) | |
44 | 42, 43 | sylibr 226 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → (𝑣 ∩ (ℚ ↑𝑚 𝐼)) ≠ ∅) |
45 | 44 | ex 403 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐽) → (𝑥 ∈ 𝑣 → (𝑣 ∩ (ℚ ↑𝑚 𝐼)) ≠ ∅)) |
46 | 45 | adantlr 705 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℝ ↑𝑚 𝐼)) ∧ 𝑣 ∈ 𝐽) → (𝑥 ∈ 𝑣 → (𝑣 ∩ (ℚ ↑𝑚 𝐼)) ≠ ∅)) |
47 | 46 | ralrimiva 3148 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑𝑚 𝐼)) → ∀𝑣 ∈ 𝐽 (𝑥 ∈ 𝑣 → (𝑣 ∩ (ℚ ↑𝑚 𝐼)) ≠ ∅)) |
48 | 4 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑𝑚 𝐼)) → 𝐽 ∈ Top) |
49 | 22 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑𝑚 𝐼)) → (ℚ ↑𝑚 𝐼) ⊆ ∪ 𝐽) |
50 | simpr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑𝑚 𝐼)) → 𝑥 ∈ (ℝ ↑𝑚 𝐼)) | |
51 | 21 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑𝑚 𝐼)) → (ℝ ↑𝑚 𝐼) = ∪ 𝐽) |
52 | 50, 51 | eleqtrd 2861 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑𝑚 𝐼)) → 𝑥 ∈ ∪ 𝐽) |
53 | 23 | elcls 21296 | . . . 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 1439 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑𝑚 𝐼)) → (𝑥 ∈ ((cls‘𝐽)‘(ℚ ↑𝑚 𝐼)) ↔ ∀𝑣 ∈ 𝐽 (𝑥 ∈ 𝑣 → (𝑣 ∩ (ℚ ↑𝑚 𝐼)) ≠ ∅))) |
55 | 47, 54 | mpbird 249 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑𝑚 𝐼)) → 𝑥 ∈ ((cls‘𝐽)‘(ℚ ↑𝑚 𝐼))) |
56 | 27, 55 | eqelssd 3841 | 1 ⊢ (𝜑 → ((cls‘𝐽)‘(ℚ ↑𝑚 𝐼)) = (ℝ ↑𝑚 𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∃wex 1823 ∈ wcel 2107 ≠ wne 2969 ∀wral 3090 ∃wrex 3091 Vcvv 3398 ∩ cin 3791 ⊆ wss 3792 ∅c0 4141 ∪ cuni 4673 ‘cfv 6137 (class class class)co 6924 ↑𝑚 cmap 8142 Fincfn 8243 ℝcr 10273 ℚcq 12100 Basecbs 16266 TopOpenctopn 16479 Topctop 21116 TopSpctps 21155 clsccl 21241 ℝ^crrx 23600 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 ax-addf 10353 ax-mulf 10354 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-of 7176 df-om 7346 df-1st 7447 df-2nd 7448 df-supp 7579 df-tpos 7636 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-map 8144 df-ixp 8197 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fsupp 8566 df-sup 8638 df-inf 8639 df-oi 8706 df-card 9100 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11036 df-nn 11380 df-2 11443 df-3 11444 df-4 11445 df-5 11446 df-6 11447 df-7 11448 df-8 11449 df-9 11450 df-n0 11648 df-z 11734 df-dec 11851 df-uz 11998 df-q 12101 df-rp 12143 df-xneg 12262 df-xadd 12263 df-xmul 12264 df-ioo 12496 df-ico 12498 df-fz 12649 df-fzo 12790 df-seq 13125 df-exp 13184 df-hash 13442 df-cj 14252 df-re 14253 df-im 14254 df-sqrt 14388 df-abs 14389 df-clim 14636 df-sum 14834 df-struct 16268 df-ndx 16269 df-slot 16270 df-base 16272 df-sets 16273 df-ress 16274 df-plusg 16362 df-mulr 16363 df-starv 16364 df-sca 16365 df-vsca 16366 df-ip 16367 df-tset 16368 df-ple 16369 df-ds 16371 df-unif 16372 df-hom 16373 df-cco 16374 df-rest 16480 df-topn 16481 df-0g 16499 df-gsum 16500 df-topgen 16501 df-prds 16505 df-pws 16507 df-mgm 17639 df-sgrp 17681 df-mnd 17692 df-mhm 17732 df-submnd 17733 df-grp 17823 df-minusg 17824 df-sbg 17825 df-subg 17986 df-ghm 18053 df-cntz 18144 df-cmn 18592 df-abl 18593 df-mgp 18888 df-ur 18900 df-ring 18947 df-cring 18948 df-oppr 19021 df-dvdsr 19039 df-unit 19040 df-invr 19070 df-dvr 19081 df-rnghom 19115 df-drng 19152 df-field 19153 df-subrg 19181 df-abv 19220 df-staf 19248 df-srng 19249 df-lmod 19268 df-lss 19336 df-lmhm 19428 df-lvec 19509 df-sra 19580 df-rgmod 19581 df-psmet 20145 df-xmet 20146 df-met 20147 df-bl 20148 df-mopn 20149 df-cnfld 20154 df-refld 20359 df-phl 20380 df-dsmm 20486 df-frlm 20501 df-top 21117 df-topon 21134 df-topsp 21156 df-bases 21169 df-cld 21242 df-ntr 21243 df-cls 21244 df-xms 22544 df-ms 22545 df-nm 22806 df-ngp 22807 df-tng 22808 df-nrg 22809 df-nlm 22810 df-clm 23281 df-cph 23386 df-tcph 23387 df-rrx 23602 |
This theorem is referenced by: (None) |
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