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Mirrors > Home > MPE Home > Th. List > ehleudis | Structured version Visualization version GIF version |
Description: The Euclidean distance function in a real Euclidean space of finite dimension. (Contributed by AV, 15-Jan-2023.) |
Ref | Expression |
---|---|
ehleudis.i | β’ πΌ = (1...π) |
ehleudis.e | β’ πΈ = (πΌhilβπ) |
ehleudis.x | β’ π = (β βm πΌ) |
ehleudis.d | β’ π· = (distβπΈ) |
Ref | Expression |
---|---|
ehleudis | β’ (π β β0 β π· = (π β π, π β π β¦ (ββΞ£π β πΌ (((πβπ) β (πβπ))β2)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ehleudis.d | . . 3 β’ π· = (distβπΈ) | |
2 | ehleudis.e | . . . . 5 β’ πΈ = (πΌhilβπ) | |
3 | 2 | ehlval 25370 | . . . 4 β’ (π β β0 β πΈ = (β^β(1...π))) |
4 | 3 | fveq2d 6906 | . . 3 β’ (π β β0 β (distβπΈ) = (distβ(β^β(1...π)))) |
5 | 1, 4 | eqtrid 2780 | . 2 β’ (π β β0 β π· = (distβ(β^β(1...π)))) |
6 | ehleudis.i | . . . 4 β’ πΌ = (1...π) | |
7 | fzfi 13979 | . . . 4 β’ (1...π) β Fin | |
8 | 6, 7 | eqeltri 2825 | . . 3 β’ πΌ β Fin |
9 | 6 | eqcomi 2737 | . . . . . 6 β’ (1...π) = πΌ |
10 | 9 | fveq2i 6905 | . . . . 5 β’ (β^β(1...π)) = (β^βπΌ) |
11 | 10 | fveq2i 6905 | . . . 4 β’ (distβ(β^β(1...π))) = (distβ(β^βπΌ)) |
12 | eqid 2728 | . . . . 5 β’ (β^βπΌ) = (β^βπΌ) | |
13 | ehleudis.x | . . . . 5 β’ π = (β βm πΌ) | |
14 | 12, 13 | rrxdsfi 25367 | . . . 4 β’ (πΌ β Fin β (distβ(β^βπΌ)) = (π β π, π β π β¦ (ββΞ£π β πΌ (((πβπ) β (πβπ))β2)))) |
15 | 11, 14 | eqtrid 2780 | . . 3 β’ (πΌ β Fin β (distβ(β^β(1...π))) = (π β π, π β π β¦ (ββΞ£π β πΌ (((πβπ) β (πβπ))β2)))) |
16 | 8, 15 | mp1i 13 | . 2 β’ (π β β0 β (distβ(β^β(1...π))) = (π β π, π β π β¦ (ββΞ£π β πΌ (((πβπ) β (πβπ))β2)))) |
17 | 5, 16 | eqtrd 2768 | 1 β’ (π β β0 β π· = (π β π, π β π β¦ (ββΞ£π β πΌ (((πβπ) β (πβπ))β2)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βcfv 6553 (class class class)co 7426 β cmpo 7428 βm cmap 8853 Fincfn 8972 βcr 11147 1c1 11149 β cmin 11484 2c2 12307 β0cn0 12512 ...cfz 13526 βcexp 14068 βcsqrt 15222 Ξ£csu 15674 distcds 17251 β^crrx 25339 πΌhilcehl 25340 |
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-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-cnfld 21294 df-refld 21551 df-dsmm 21680 df-frlm 21695 df-nm 24519 df-tng 24521 df-tcph 25125 df-rrx 25341 df-ehl 25342 |
This theorem is referenced by: ehl1eudis 25376 ehl2eudis 25378 |
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