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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 46533 | . . . . 5 ⊢ (𝐼 ∈ Fin → 𝐽 ∈ Top) |
| 4 | 1, 3 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐽 ∈ Top) |
| 5 | reex 11117 | . . . . . . 7 ⊢ ℝ ∈ V | |
| 6 | qssre 12872 | . . . . . . 7 ⊢ ℚ ⊆ ℝ | |
| 7 | mapss 8827 | . . . . . . 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 2736 | . . . . . . . 8 ⊢ (ℝ^‘𝐼) = (ℝ^‘𝐼) | |
| 11 | eqid 2736 | . . . . . . . 8 ⊢ (Base‘(ℝ^‘𝐼)) = (Base‘(ℝ^‘𝐼)) | |
| 12 | 1, 10, 11 | rrxbasefi 25366 | . . . . . . 7 ⊢ (𝜑 → (Base‘(ℝ^‘𝐼)) = (ℝ ↑m 𝐼)) |
| 13 | 12 | eqcomd 2742 | . . . . . 6 ⊢ (𝜑 → (ℝ ↑m 𝐼) = (Base‘(ℝ^‘𝐼))) |
| 14 | rrxtps 46530 | . . . . . . 7 ⊢ (𝐼 ∈ Fin → (ℝ^‘𝐼) ∈ TopSp) | |
| 15 | eqid 2736 | . . . . . . . 8 ⊢ (TopOpen‘(ℝ^‘𝐼)) = (TopOpen‘(ℝ^‘𝐼)) | |
| 16 | 11, 15 | tpsuni 22880 | . . . . . . 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 4875 | . . . . . . . 8 ⊢ ∪ 𝐽 = ∪ (TopOpen‘(ℝ^‘𝐼)) |
| 19 | 18 | eqcomi 2745 | . . . . . . 7 ⊢ ∪ (TopOpen‘(ℝ^‘𝐼)) = ∪ 𝐽 |
| 20 | 19 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ∪ (TopOpen‘(ℝ^‘𝐼)) = ∪ 𝐽) |
| 21 | 13, 17, 20 | 3eqtrd 2775 | . . . . 5 ⊢ (𝜑 → (ℝ ↑m 𝐼) = ∪ 𝐽) |
| 22 | 9, 21 | sseqtrd 3970 | . . . 4 ⊢ (𝜑 → (ℚ ↑m 𝐼) ⊆ ∪ 𝐽) |
| 23 | eqid 2736 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
| 24 | 23 | clsss3 23003 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ (ℚ ↑m 𝐼) ⊆ ∪ 𝐽) → ((cls‘𝐽)‘(ℚ ↑m 𝐼)) ⊆ ∪ 𝐽) |
| 25 | 4, 22, 24 | syl2anc 584 | . . 3 ⊢ (𝜑 → ((cls‘𝐽)‘(ℚ ↑m 𝐼)) ⊆ ∪ 𝐽) |
| 26 | 21 | eqcomd 2742 | . . 3 ⊢ (𝜑 → ∪ 𝐽 = (ℝ ↑m 𝐼)) |
| 27 | 25, 26 | sseqtrd 3970 | . 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 2846 | . . . . . . . . . . 11 ⊢ (𝑣 ∈ 𝐽 → 𝑣 ∈ (TopOpen‘(ℝ^‘𝐼))) |
| 31 | 30 | ad2antlr 727 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → 𝑣 ∈ (TopOpen‘(ℝ^‘𝐼))) |
| 32 | ne0i 4293 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑣 → 𝑣 ≠ ∅) | |
| 33 | 32 | adantl 481 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → 𝑣 ≠ ∅) |
| 34 | 28, 15, 31, 33 | qndenserrnopn 46542 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → ∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ 𝑣) |
| 35 | df-rex 3061 | . . . . . . . . 9 ⊢ (∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ 𝑣 ↔ ∃𝑦(𝑦 ∈ (ℚ ↑m 𝐼) ∧ 𝑦 ∈ 𝑣)) | |
| 36 | 34, 35 | sylib 218 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → ∃𝑦(𝑦 ∈ (ℚ ↑m 𝐼) ∧ 𝑦 ∈ 𝑣)) |
| 37 | simpr 484 | . . . . . . . . . . 11 ⊢ ((𝑦 ∈ (ℚ ↑m 𝐼) ∧ 𝑦 ∈ 𝑣) → 𝑦 ∈ 𝑣) | |
| 38 | simpl 482 | . . . . . . . . . . 11 ⊢ ((𝑦 ∈ (ℚ ↑m 𝐼) ∧ 𝑦 ∈ 𝑣) → 𝑦 ∈ (ℚ ↑m 𝐼)) | |
| 39 | 37, 38 | elind 4152 | . . . . . . . . . 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 1918 | . . . . . . . 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 4305 | . . . . . . 7 ⊢ ((𝑣 ∩ (ℚ ↑m 𝐼)) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝑣 ∩ (ℚ ↑m 𝐼))) | |
| 44 | 42, 43 | sylibr 234 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → (𝑣 ∩ (ℚ ↑m 𝐼)) ≠ ∅) |
| 45 | 44 | ex 412 | . . . . 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 3128 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑m 𝐼)) → ∀𝑣 ∈ 𝐽 (𝑥 ∈ 𝑣 → (𝑣 ∩ (ℚ ↑m 𝐼)) ≠ ∅)) |
| 48 | 4 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑m 𝐼)) → 𝐽 ∈ Top) |
| 49 | 22 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑m 𝐼)) → (ℚ ↑m 𝐼) ⊆ ∪ 𝐽) |
| 50 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑m 𝐼)) → 𝑥 ∈ (ℝ ↑m 𝐼)) | |
| 51 | 21 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑m 𝐼)) → (ℝ ↑m 𝐼) = ∪ 𝐽) |
| 52 | 50, 51 | eleqtrd 2838 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑m 𝐼)) → 𝑥 ∈ ∪ 𝐽) |
| 53 | 23 | elcls 23017 | . . . 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 257 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑m 𝐼)) → 𝑥 ∈ ((cls‘𝐽)‘(ℚ ↑m 𝐼))) |
| 56 | 27, 55 | eqelssd 3955 | 1 ⊢ (𝜑 → ((cls‘𝐽)‘(ℚ ↑m 𝐼)) = (ℝ ↑m 𝐼)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∃wex 1780 ∈ wcel 2113 ≠ wne 2932 ∀wral 3051 ∃wrex 3060 Vcvv 3440 ∩ cin 3900 ⊆ wss 3901 ∅c0 4285 ∪ cuni 4863 ‘cfv 6492 (class class class)co 7358 ↑m cmap 8763 Fincfn 8883 ℝcr 11025 ℚcq 12861 Basecbs 17136 TopOpenctopn 17341 Topctop 22837 TopSpctps 22876 clsccl 22962 ℝ^crrx 25339 |
| 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 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105 ax-mulf 11106 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-tpos 8168 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-er 8635 df-map 8765 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-ioo 13265 df-ico 13267 df-fz 13424 df-fzo 13571 df-seq 13925 df-exp 13985 df-hash 14254 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-clim 15411 df-sum 15610 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-rest 17342 df-topn 17343 df-0g 17361 df-gsum 17362 df-topgen 17363 df-prds 17367 df-pws 17369 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18708 df-submnd 18709 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19053 df-ghm 19142 df-cntz 19246 df-cmn 19711 df-abl 19712 df-mgp 20076 df-rng 20088 df-ur 20117 df-ring 20170 df-cring 20171 df-oppr 20273 df-dvdsr 20293 df-unit 20294 df-invr 20324 df-dvr 20337 df-rhm 20408 df-subrng 20479 df-subrg 20503 df-drng 20664 df-field 20665 df-abv 20742 df-staf 20772 df-srng 20773 df-lmod 20813 df-lss 20883 df-lmhm 20974 df-lvec 21055 df-sra 21125 df-rgmod 21126 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-cnfld 21310 df-refld 21560 df-phl 21581 df-dsmm 21687 df-frlm 21702 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cld 22963 df-ntr 22964 df-cls 22965 df-xms 24264 df-ms 24265 df-nm 24526 df-ngp 24527 df-tng 24528 df-nrg 24529 df-nlm 24530 df-clm 25019 df-cph 25124 df-tcph 25125 df-rrx 25341 |
| This theorem is referenced by: (None) |
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