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Mirrors > Home > MPE Home > Th. List > rrxmetfi | Structured version Visualization version GIF version |
Description: Euclidean space is a metric space. Finite dimensional version. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Ref | Expression |
---|---|
rrxmetfi.1 | β’ π· = (distβ(β^βπΌ)) |
Ref | Expression |
---|---|
rrxmetfi | β’ (πΌ β Fin β π· β (Metβ(β βm πΌ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . 3 β’ {β β (β βm πΌ) β£ β finSupp 0} = {β β (β βm πΌ) β£ β finSupp 0} | |
2 | rrxmetfi.1 | . . 3 β’ π· = (distβ(β^βπΌ)) | |
3 | 1, 2 | rrxmet 25291 | . 2 β’ (πΌ β Fin β π· β (Metβ{β β (β βm πΌ) β£ β finSupp 0})) |
4 | eqid 2726 | . . . . 5 β’ (β^βπΌ) = (β^βπΌ) | |
5 | eqid 2726 | . . . . 5 β’ (Baseβ(β^βπΌ)) = (Baseβ(β^βπΌ)) | |
6 | 4, 5 | rrxbase 25271 | . . . 4 β’ (πΌ β Fin β (Baseβ(β^βπΌ)) = {β β (β βm πΌ) β£ β finSupp 0}) |
7 | id 22 | . . . . 5 β’ (πΌ β Fin β πΌ β Fin) | |
8 | 7, 4, 5 | rrxbasefi 25293 | . . . 4 β’ (πΌ β Fin β (Baseβ(β^βπΌ)) = (β βm πΌ)) |
9 | 6, 8 | eqtr3d 2768 | . . 3 β’ (πΌ β Fin β {β β (β βm πΌ) β£ β finSupp 0} = (β βm πΌ)) |
10 | 9 | fveq2d 6889 | . 2 β’ (πΌ β Fin β (Metβ{β β (β βm πΌ) β£ β finSupp 0}) = (Metβ(β βm πΌ))) |
11 | 3, 10 | eleqtrd 2829 | 1 β’ (πΌ β Fin β π· β (Metβ(β βm πΌ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 {crab 3426 class class class wbr 5141 βcfv 6537 (class class class)co 7405 βm cmap 8822 Fincfn 8941 finSupp cfsupp 9363 βcr 11111 0cc0 11112 Basecbs 17153 distcds 17215 Metcmet 21226 β^crrx 25266 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-tpos 8212 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-rp 12981 df-ico 13336 df-fz 13491 df-fzo 13634 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 df-sum 15639 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-hom 17230 df-cco 17231 df-0g 17396 df-gsum 17397 df-prds 17402 df-pws 17404 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18713 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19050 df-ghm 19139 df-cntz 19233 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-cring 20141 df-oppr 20236 df-dvdsr 20259 df-unit 20260 df-invr 20290 df-dvr 20303 df-rhm 20374 df-subrng 20446 df-subrg 20471 df-drng 20589 df-field 20590 df-staf 20688 df-srng 20689 df-lmod 20708 df-lss 20779 df-sra 21021 df-rgmod 21022 df-met 21234 df-cnfld 21241 df-refld 21498 df-dsmm 21627 df-frlm 21642 df-nm 24446 df-tng 24448 df-tcph 25052 df-rrx 25268 |
This theorem is referenced by: qndenserrnbllem 45579 qndenserrnbl 45580 qndenserrnopnlem 45582 rrndsmet 45587 hoiqssbllem2 45908 hoiqssbl 45910 opnvonmbllem2 45918 rrxsphere 47706 |
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