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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > qndenserrnopn | 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 |
---|---|
qndenserrnopn.i | β’ (π β πΌ β Fin) |
qndenserrnopn.j | β’ π½ = (TopOpenβ(β^βπΌ)) |
qndenserrnopn.v | β’ (π β π β π½) |
qndenserrnopn.n | β’ (π β π β β ) |
Ref | Expression |
---|---|
qndenserrnopn | β’ (π β βπ¦ β (β βm πΌ)π¦ β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qndenserrnopn.n | . . 3 β’ (π β π β β ) | |
2 | n0 4342 | . . 3 β’ (π β β β βπ₯ π₯ β π) | |
3 | 1, 2 | sylib 217 | . 2 β’ (π β βπ₯ π₯ β π) |
4 | qndenserrnopn.i | . . . . . 6 β’ (π β πΌ β Fin) | |
5 | 4 | adantr 480 | . . . . 5 β’ ((π β§ π₯ β π) β πΌ β Fin) |
6 | qndenserrnopn.j | . . . . 5 β’ π½ = (TopOpenβ(β^βπΌ)) | |
7 | qndenserrnopn.v | . . . . . 6 β’ (π β π β π½) | |
8 | 7 | adantr 480 | . . . . 5 β’ ((π β§ π₯ β π) β π β π½) |
9 | simpr 484 | . . . . 5 β’ ((π β§ π₯ β π) β π₯ β π) | |
10 | eqid 2727 | . . . . 5 β’ (distβ(β^βπΌ)) = (distβ(β^βπΌ)) | |
11 | 5, 6, 8, 9, 10 | qndenserrnopnlem 45608 | . . . 4 β’ ((π β§ π₯ β π) β βπ¦ β (β βm πΌ)π¦ β π) |
12 | 11 | ex 412 | . . 3 β’ (π β (π₯ β π β βπ¦ β (β βm πΌ)π¦ β π)) |
13 | 12 | exlimdv 1929 | . 2 β’ (π β (βπ₯ π₯ β π β βπ¦ β (β βm πΌ)π¦ β π)) |
14 | 3, 13 | mpd 15 | 1 β’ (π β βπ¦ β (β βm πΌ)π¦ β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 βwex 1774 β wcel 2099 β wne 2935 βwrex 3065 β c0 4318 βcfv 6542 (class class class)co 7414 βm cmap 8836 Fincfn 8955 βcq 12954 distcds 17233 TopOpenctopn 17394 β^crrx 25298 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9656 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208 ax-addf 11209 ax-mulf 11210 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8838 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9378 df-sup 9457 df-inf 9458 df-oi 9525 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-z 12581 df-dec 12700 df-uz 12845 df-q 12955 df-rp 12999 df-xneg 13116 df-xadd 13117 df-xmul 13118 df-ioo 13352 df-ico 13354 df-fz 13509 df-fzo 13652 df-seq 13991 df-exp 14051 df-hash 14314 df-cj 15070 df-re 15071 df-im 15072 df-sqrt 15206 df-abs 15207 df-clim 15456 df-sum 15657 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-starv 17239 df-sca 17240 df-vsca 17241 df-ip 17242 df-tset 17243 df-ple 17244 df-ds 17246 df-unif 17247 df-hom 17248 df-cco 17249 df-rest 17395 df-topn 17396 df-0g 17414 df-gsum 17415 df-topgen 17416 df-prds 17420 df-pws 17422 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-mhm 18731 df-submnd 18732 df-grp 18884 df-minusg 18885 df-sbg 18886 df-subg 19069 df-ghm 19159 df-cntz 19259 df-cmn 19728 df-abl 19729 df-mgp 20066 df-rng 20084 df-ur 20113 df-ring 20166 df-cring 20167 df-oppr 20262 df-dvdsr 20285 df-unit 20286 df-invr 20316 df-dvr 20329 df-rhm 20400 df-subrng 20472 df-subrg 20497 df-drng 20615 df-field 20616 df-abv 20686 df-staf 20714 df-srng 20715 df-lmod 20734 df-lss 20805 df-lmhm 20896 df-lvec 20977 df-sra 21047 df-rgmod 21048 df-psmet 21258 df-xmet 21259 df-met 21260 df-bl 21261 df-mopn 21262 df-cnfld 21267 df-refld 21524 df-phl 21545 df-dsmm 21653 df-frlm 21668 df-top 22783 df-topon 22800 df-topsp 22822 df-bases 22836 df-xms 24213 df-ms 24214 df-nm 24478 df-ngp 24479 df-tng 24480 df-nrg 24481 df-nlm 24482 df-clm 24977 df-cph 25083 df-tcph 25084 df-rrx 25300 |
This theorem is referenced by: qndenserrn 45610 |
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