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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 | ⊢ 𝐼 = (1...𝑁) |
ehleudis.e | ⊢ 𝐸 = (𝔼hil‘𝑁) |
ehleudis.x | ⊢ 𝑋 = (ℝ ↑m 𝐼) |
ehleudis.d | ⊢ 𝐷 = (dist‘𝐸) |
Ref | Expression |
---|---|
ehleudisval | ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹𝐷𝐺) = (√‘Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ehleudis.d | . . . . 5 ⊢ 𝐷 = (dist‘𝐸) | |
2 | ehleudis.e | . . . . . . 7 ⊢ 𝐸 = (𝔼hil‘𝑁) | |
3 | 2 | ehlval 24900 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝐸 = (ℝ^‘(1...𝑁))) |
4 | 3 | fveq2d 6885 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (dist‘𝐸) = (dist‘(ℝ^‘(1...𝑁)))) |
5 | 1, 4 | eqtrid 2785 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝐷 = (dist‘(ℝ^‘(1...𝑁)))) |
6 | 5 | oveqd 7413 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝐹𝐷𝐺) = (𝐹(dist‘(ℝ^‘(1...𝑁)))𝐺)) |
7 | 6 | 3ad2ant1 1134 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹𝐷𝐺) = (𝐹(dist‘(ℝ^‘(1...𝑁)))𝐺)) |
8 | ehleudis.i | . . . 4 ⊢ 𝐼 = (1...𝑁) | |
9 | fzfi 13924 | . . . 4 ⊢ (1...𝑁) ∈ Fin | |
10 | 8, 9 | eqeltri 2830 | . . 3 ⊢ 𝐼 ∈ Fin |
11 | ehleudis.x | . . . . . 6 ⊢ 𝑋 = (ℝ ↑m 𝐼) | |
12 | 11 | eleq2i 2826 | . . . . 5 ⊢ (𝐹 ∈ 𝑋 ↔ 𝐹 ∈ (ℝ ↑m 𝐼)) |
13 | 12 | biimpi 215 | . . . 4 ⊢ (𝐹 ∈ 𝑋 → 𝐹 ∈ (ℝ ↑m 𝐼)) |
14 | 13 | 3ad2ant2 1135 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 𝐹 ∈ (ℝ ↑m 𝐼)) |
15 | 11 | eleq2i 2826 | . . . . 5 ⊢ (𝐺 ∈ 𝑋 ↔ 𝐺 ∈ (ℝ ↑m 𝐼)) |
16 | 15 | biimpi 215 | . . . 4 ⊢ (𝐺 ∈ 𝑋 → 𝐺 ∈ (ℝ ↑m 𝐼)) |
17 | 16 | 3ad2ant3 1136 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 𝐺 ∈ (ℝ ↑m 𝐼)) |
18 | eqid 2733 | . . . 4 ⊢ (ℝ ↑m 𝐼) = (ℝ ↑m 𝐼) | |
19 | 8 | eqcomi 2742 | . . . . . 6 ⊢ (1...𝑁) = 𝐼 |
20 | 19 | fveq2i 6884 | . . . . 5 ⊢ (ℝ^‘(1...𝑁)) = (ℝ^‘𝐼) |
21 | 20 | fveq2i 6884 | . . . 4 ⊢ (dist‘(ℝ^‘(1...𝑁))) = (dist‘(ℝ^‘𝐼)) |
22 | 18, 21 | rrxdsfival 24899 | . . 3 ⊢ ((𝐼 ∈ Fin ∧ 𝐹 ∈ (ℝ ↑m 𝐼) ∧ 𝐺 ∈ (ℝ ↑m 𝐼)) → (𝐹(dist‘(ℝ^‘(1...𝑁)))𝐺) = (√‘Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2))) |
23 | 10, 14, 17, 22 | mp3an2i 1467 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹(dist‘(ℝ^‘(1...𝑁)))𝐺) = (√‘Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2))) |
24 | 7, 23 | eqtrd 2773 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹𝐷𝐺) = (√‘Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ‘cfv 6535 (class class class)co 7396 ↑m cmap 8808 Fincfn 8927 ℝcr 11096 1c1 11098 − cmin 11431 2c2 12254 ℕ0cn0 12459 ...cfz 13471 ↑cexp 14014 √csqrt 15167 Σcsu 15619 distcds 17193 ℝ^crrx 24869 𝔼hilcehl 24870 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-inf2 9623 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 ax-pre-sup 11175 ax-addf 11176 ax-mulf 11177 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4905 df-int 4947 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 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 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-isom 6544 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7657 df-om 7843 df-1st 7962 df-2nd 7963 df-supp 8134 df-tpos 8198 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-er 8691 df-map 8810 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fsupp 9350 df-sup 9424 df-oi 9492 df-card 9921 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-div 11859 df-nn 12200 df-2 12262 df-3 12263 df-4 12264 df-5 12265 df-6 12266 df-7 12267 df-8 12268 df-9 12269 df-n0 12460 df-z 12546 df-dec 12665 df-uz 12810 df-rp 12962 df-fz 13472 df-fzo 13615 df-seq 13954 df-exp 14015 df-hash 14278 df-cj 15033 df-re 15034 df-im 15035 df-sqrt 15169 df-abs 15170 df-clim 15419 df-sum 15620 df-struct 17067 df-sets 17084 df-slot 17102 df-ndx 17114 df-base 17132 df-ress 17161 df-plusg 17197 df-mulr 17198 df-starv 17199 df-sca 17200 df-vsca 17201 df-ip 17202 df-tset 17203 df-ple 17204 df-ds 17206 df-unif 17207 df-hom 17208 df-cco 17209 df-0g 17374 df-gsum 17375 df-prds 17380 df-pws 17382 df-mgm 18548 df-sgrp 18597 df-mnd 18613 df-mhm 18658 df-grp 18809 df-minusg 18810 df-sbg 18811 df-subg 18988 df-ghm 19075 df-cntz 19166 df-cmn 19634 df-abl 19635 df-mgp 19971 df-ur 19988 df-ring 20040 df-cring 20041 df-oppr 20128 df-dvdsr 20149 df-unit 20150 df-invr 20180 df-dvr 20193 df-rnghom 20229 df-drng 20295 df-field 20296 df-subrg 20338 df-staf 20430 df-srng 20431 df-lmod 20450 df-lss 20520 df-sra 20762 df-rgmod 20763 df-cnfld 20919 df-refld 21131 df-dsmm 21260 df-frlm 21275 df-nm 24060 df-tng 24062 df-tcph 24655 df-rrx 24871 df-ehl 24872 |
This theorem is referenced by: (None) |
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