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Mirrors > Home > MPE Home > Th. List > ehleudisval | Structured version Visualization version GIF version |
Description: The value of 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 |
---|---|
ehleudisval | β’ ((π β β0 β§ πΉ β π β§ πΊ β π) β (πΉπ·πΊ) = (ββΞ£π β πΌ (((πΉβπ) β (πΊβπ))β2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ehleudis.d | . . . . 5 β’ π· = (distβπΈ) | |
2 | ehleudis.e | . . . . . . 7 β’ πΈ = (πΌhilβπ) | |
3 | 2 | ehlval 25316 | . . . . . 6 β’ (π β β0 β πΈ = (β^β(1...π))) |
4 | 3 | fveq2d 6895 | . . . . 5 β’ (π β β0 β (distβπΈ) = (distβ(β^β(1...π)))) |
5 | 1, 4 | eqtrid 2779 | . . . 4 β’ (π β β0 β π· = (distβ(β^β(1...π)))) |
6 | 5 | oveqd 7431 | . . 3 β’ (π β β0 β (πΉπ·πΊ) = (πΉ(distβ(β^β(1...π)))πΊ)) |
7 | 6 | 3ad2ant1 1131 | . 2 β’ ((π β β0 β§ πΉ β π β§ πΊ β π) β (πΉπ·πΊ) = (πΉ(distβ(β^β(1...π)))πΊ)) |
8 | ehleudis.i | . . . 4 β’ πΌ = (1...π) | |
9 | fzfi 13955 | . . . 4 β’ (1...π) β Fin | |
10 | 8, 9 | eqeltri 2824 | . . 3 β’ πΌ β Fin |
11 | ehleudis.x | . . . . . 6 β’ π = (β βm πΌ) | |
12 | 11 | eleq2i 2820 | . . . . 5 β’ (πΉ β π β πΉ β (β βm πΌ)) |
13 | 12 | biimpi 215 | . . . 4 β’ (πΉ β π β πΉ β (β βm πΌ)) |
14 | 13 | 3ad2ant2 1132 | . . 3 β’ ((π β β0 β§ πΉ β π β§ πΊ β π) β πΉ β (β βm πΌ)) |
15 | 11 | eleq2i 2820 | . . . . 5 β’ (πΊ β π β πΊ β (β βm πΌ)) |
16 | 15 | biimpi 215 | . . . 4 β’ (πΊ β π β πΊ β (β βm πΌ)) |
17 | 16 | 3ad2ant3 1133 | . . 3 β’ ((π β β0 β§ πΉ β π β§ πΊ β π) β πΊ β (β βm πΌ)) |
18 | eqid 2727 | . . . 4 β’ (β βm πΌ) = (β βm πΌ) | |
19 | 8 | eqcomi 2736 | . . . . . 6 β’ (1...π) = πΌ |
20 | 19 | fveq2i 6894 | . . . . 5 β’ (β^β(1...π)) = (β^βπΌ) |
21 | 20 | fveq2i 6894 | . . . 4 β’ (distβ(β^β(1...π))) = (distβ(β^βπΌ)) |
22 | 18, 21 | rrxdsfival 25315 | . . 3 β’ ((πΌ β Fin β§ πΉ β (β βm πΌ) β§ πΊ β (β βm πΌ)) β (πΉ(distβ(β^β(1...π)))πΊ) = (ββΞ£π β πΌ (((πΉβπ) β (πΊβπ))β2))) |
23 | 10, 14, 17, 22 | mp3an2i 1463 | . 2 β’ ((π β β0 β§ πΉ β π β§ πΊ β π) β (πΉ(distβ(β^β(1...π)))πΊ) = (ββΞ£π β πΌ (((πΉβπ) β (πΊβπ))β2))) |
24 | 7, 23 | eqtrd 2767 | 1 β’ ((π β β0 β§ πΉ β π β§ πΊ β π) β (πΉπ·πΊ) = (ββΞ£π β πΌ (((πΉβπ) β (πΊβπ))β2))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1085 = wceq 1534 β wcel 2099 βcfv 6542 (class class class)co 7414 βm cmap 8834 Fincfn 8953 βcr 11123 1c1 11125 β cmin 11460 2c2 12283 β0cn0 12488 ...cfz 13502 βcexp 14044 βcsqrt 15198 Ξ£csu 15650 distcds 17227 β^crrx 25285 πΌhilcehl 25286 |
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 7732 ax-inf2 9650 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 ax-addf 11203 ax-mulf 11204 |
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 7677 df-om 7863 df-1st 7985 df-2nd 7986 df-supp 8158 df-tpos 8223 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-map 8836 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9376 df-sup 9451 df-oi 9519 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-z 12575 df-dec 12694 df-uz 12839 df-rp 12993 df-fz 13503 df-fzo 13646 df-seq 13985 df-exp 14045 df-hash 14308 df-cj 15064 df-re 15065 df-im 15066 df-sqrt 15200 df-abs 15201 df-clim 15450 df-sum 15651 df-struct 17101 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ress 17195 df-plusg 17231 df-mulr 17232 df-starv 17233 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-unif 17241 df-hom 17242 df-cco 17243 df-0g 17408 df-gsum 17409 df-prds 17414 df-pws 17416 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-mhm 18725 df-grp 18878 df-minusg 18879 df-sbg 18880 df-subg 19062 df-ghm 19152 df-cntz 19252 df-cmn 19721 df-abl 19722 df-mgp 20059 df-rng 20077 df-ur 20106 df-ring 20159 df-cring 20160 df-oppr 20255 df-dvdsr 20278 df-unit 20279 df-invr 20309 df-dvr 20322 df-rhm 20393 df-subrng 20465 df-subrg 20490 df-drng 20608 df-field 20609 df-staf 20707 df-srng 20708 df-lmod 20727 df-lss 20798 df-sra 21040 df-rgmod 21041 df-cnfld 21260 df-refld 21517 df-dsmm 21646 df-frlm 21661 df-nm 24465 df-tng 24467 df-tcph 25071 df-rrx 25287 df-ehl 25288 |
This theorem is referenced by: (None) |
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