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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 2728 | . . 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 25364 | . 2 β’ (πΌ β Fin β π· β (Metβ{β β (β βm πΌ) β£ β finSupp 0})) |
4 | eqid 2728 | . . . . 5 β’ (β^βπΌ) = (β^βπΌ) | |
5 | eqid 2728 | . . . . 5 β’ (Baseβ(β^βπΌ)) = (Baseβ(β^βπΌ)) | |
6 | 4, 5 | rrxbase 25344 | . . . 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 25366 | . . . 4 β’ (πΌ β Fin β (Baseβ(β^βπΌ)) = (β βm πΌ)) |
9 | 6, 8 | eqtr3d 2770 | . . 3 β’ (πΌ β Fin β {β β (β βm πΌ) β£ β finSupp 0} = (β βm πΌ)) |
10 | 9 | fveq2d 6906 | . 2 β’ (πΌ β Fin β (Metβ{β β (β βm πΌ) β£ β finSupp 0}) = (Metβ(β βm πΌ))) |
11 | 3, 10 | eleqtrd 2831 | 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 3430 class class class wbr 5152 βcfv 6553 (class class class)co 7426 βm cmap 8853 Fincfn 8972 finSupp cfsupp 9395 βcr 11147 0cc0 11148 Basecbs 17189 distcds 17251 Metcmet 21279 β^crrx 25339 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-inf2 9674 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226 ax-addf 11227 ax-mulf 11228 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-om 7879 df-1st 8001 df-2nd 8002 df-supp 8174 df-tpos 8240 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-map 8855 df-ixp 8925 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-fsupp 9396 df-sup 9475 df-oi 9543 df-card 9972 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-9 12322 df-n0 12513 df-z 12599 df-dec 12718 df-uz 12863 df-rp 13017 df-ico 13372 df-fz 13527 df-fzo 13670 df-seq 14009 df-exp 14069 df-hash 14332 df-cj 15088 df-re 15089 df-im 15090 df-sqrt 15224 df-abs 15225 df-clim 15474 df-sum 15675 df-struct 17125 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-mulr 17256 df-starv 17257 df-sca 17258 df-vsca 17259 df-ip 17260 df-tset 17261 df-ple 17262 df-ds 17264 df-unif 17265 df-hom 17266 df-cco 17267 df-0g 17432 df-gsum 17433 df-prds 17438 df-pws 17440 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-mhm 18749 df-grp 18907 df-minusg 18908 df-sbg 18909 df-subg 19092 df-ghm 19182 df-cntz 19282 df-cmn 19751 df-abl 19752 df-mgp 20089 df-rng 20107 df-ur 20136 df-ring 20189 df-cring 20190 df-oppr 20287 df-dvdsr 20310 df-unit 20311 df-invr 20341 df-dvr 20354 df-rhm 20425 df-subrng 20497 df-subrg 20522 df-drng 20640 df-field 20641 df-staf 20739 df-srng 20740 df-lmod 20759 df-lss 20830 df-sra 21072 df-rgmod 21073 df-met 21287 df-cnfld 21294 df-refld 21551 df-dsmm 21680 df-frlm 21695 df-nm 24519 df-tng 24521 df-tcph 25125 df-rrx 25341 |
This theorem is referenced by: qndenserrnbllem 45729 qndenserrnbl 45730 qndenserrnopnlem 45732 rrndsmet 45737 hoiqssbllem2 46058 hoiqssbl 46060 opnvonmbllem2 46068 rrxsphere 47917 |
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