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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 | ⊢ 𝐼 = (1...𝑁) |
ehleudis.e | ⊢ 𝐸 = (𝔼hil‘𝑁) |
ehleudis.x | ⊢ 𝑋 = (ℝ ↑m 𝐼) |
ehleudis.d | ⊢ 𝐷 = (dist‘𝐸) |
Ref | Expression |
---|---|
ehleudis | ⊢ (𝑁 ∈ ℕ0 → 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ehleudis.d | . . 3 ⊢ 𝐷 = (dist‘𝐸) | |
2 | ehleudis.e | . . . . 5 ⊢ 𝐸 = (𝔼hil‘𝑁) | |
3 | 2 | ehlval 24576 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝐸 = (ℝ^‘(1...𝑁))) |
4 | 3 | fveq2d 6775 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (dist‘𝐸) = (dist‘(ℝ^‘(1...𝑁)))) |
5 | 1, 4 | eqtrid 2792 | . 2 ⊢ (𝑁 ∈ ℕ0 → 𝐷 = (dist‘(ℝ^‘(1...𝑁)))) |
6 | ehleudis.i | . . . 4 ⊢ 𝐼 = (1...𝑁) | |
7 | fzfi 13690 | . . . 4 ⊢ (1...𝑁) ∈ Fin | |
8 | 6, 7 | eqeltri 2837 | . . 3 ⊢ 𝐼 ∈ Fin |
9 | 6 | eqcomi 2749 | . . . . . 6 ⊢ (1...𝑁) = 𝐼 |
10 | 9 | fveq2i 6774 | . . . . 5 ⊢ (ℝ^‘(1...𝑁)) = (ℝ^‘𝐼) |
11 | 10 | fveq2i 6774 | . . . 4 ⊢ (dist‘(ℝ^‘(1...𝑁))) = (dist‘(ℝ^‘𝐼)) |
12 | eqid 2740 | . . . . 5 ⊢ (ℝ^‘𝐼) = (ℝ^‘𝐼) | |
13 | ehleudis.x | . . . . 5 ⊢ 𝑋 = (ℝ ↑m 𝐼) | |
14 | 12, 13 | rrxdsfi 24573 | . . . 4 ⊢ (𝐼 ∈ Fin → (dist‘(ℝ^‘𝐼)) = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) |
15 | 11, 14 | eqtrid 2792 | . . 3 ⊢ (𝐼 ∈ Fin → (dist‘(ℝ^‘(1...𝑁))) = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) |
16 | 8, 15 | mp1i 13 | . 2 ⊢ (𝑁 ∈ ℕ0 → (dist‘(ℝ^‘(1...𝑁))) = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) |
17 | 5, 16 | eqtrd 2780 | 1 ⊢ (𝑁 ∈ ℕ0 → 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 ‘cfv 6432 (class class class)co 7271 ∈ cmpo 7273 ↑m cmap 8598 Fincfn 8716 ℝcr 10871 1c1 10873 − cmin 11205 2c2 12028 ℕ0cn0 12233 ...cfz 13238 ↑cexp 13780 √csqrt 14942 Σcsu 15395 distcds 16969 ℝ^crrx 24545 𝔼hilcehl 24546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-inf2 9377 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950 ax-addf 10951 ax-mulf 10952 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-of 7527 df-om 7707 df-1st 7824 df-2nd 7825 df-supp 7969 df-tpos 8033 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-map 8600 df-ixp 8669 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-fsupp 9107 df-sup 9179 df-oi 9247 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12437 df-uz 12582 df-rp 12730 df-fz 13239 df-fzo 13382 df-seq 13720 df-exp 13781 df-hash 14043 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-clim 15195 df-sum 15396 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-starv 16975 df-sca 16976 df-vsca 16977 df-ip 16978 df-tset 16979 df-ple 16980 df-ds 16982 df-unif 16983 df-hom 16984 df-cco 16985 df-0g 17150 df-gsum 17151 df-prds 17156 df-pws 17158 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-mhm 18428 df-grp 18578 df-minusg 18579 df-sbg 18580 df-subg 18750 df-ghm 18830 df-cntz 18921 df-cmn 19386 df-abl 19387 df-mgp 19719 df-ur 19736 df-ring 19783 df-cring 19784 df-oppr 19860 df-dvdsr 19881 df-unit 19882 df-invr 19912 df-dvr 19923 df-rnghom 19957 df-drng 19991 df-field 19992 df-subrg 20020 df-staf 20103 df-srng 20104 df-lmod 20123 df-lss 20192 df-sra 20432 df-rgmod 20433 df-cnfld 20596 df-refld 20808 df-dsmm 20937 df-frlm 20952 df-nm 23736 df-tng 23738 df-tcph 24331 df-rrx 24547 df-ehl 24548 |
This theorem is referenced by: ehl1eudis 24582 ehl2eudis 24584 |
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