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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 24012 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝐸 = (ℝ^‘(1...𝑁))) |
4 | 3 | fveq2d 6667 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (dist‘𝐸) = (dist‘(ℝ^‘(1...𝑁)))) |
5 | 1, 4 | syl5eq 2867 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝐷 = (dist‘(ℝ^‘(1...𝑁)))) |
6 | 5 | oveqd 7166 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝐹𝐷𝐺) = (𝐹(dist‘(ℝ^‘(1...𝑁)))𝐺)) |
7 | 6 | 3ad2ant1 1128 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹𝐷𝐺) = (𝐹(dist‘(ℝ^‘(1...𝑁)))𝐺)) |
8 | ehleudis.i | . . . 4 ⊢ 𝐼 = (1...𝑁) | |
9 | fzfi 13337 | . . . 4 ⊢ (1...𝑁) ∈ Fin | |
10 | 8, 9 | eqeltri 2908 | . . 3 ⊢ 𝐼 ∈ Fin |
11 | ehleudis.x | . . . . . 6 ⊢ 𝑋 = (ℝ ↑m 𝐼) | |
12 | 11 | eleq2i 2903 | . . . . 5 ⊢ (𝐹 ∈ 𝑋 ↔ 𝐹 ∈ (ℝ ↑m 𝐼)) |
13 | 12 | biimpi 218 | . . . 4 ⊢ (𝐹 ∈ 𝑋 → 𝐹 ∈ (ℝ ↑m 𝐼)) |
14 | 13 | 3ad2ant2 1129 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 𝐹 ∈ (ℝ ↑m 𝐼)) |
15 | 11 | eleq2i 2903 | . . . . 5 ⊢ (𝐺 ∈ 𝑋 ↔ 𝐺 ∈ (ℝ ↑m 𝐼)) |
16 | 15 | biimpi 218 | . . . 4 ⊢ (𝐺 ∈ 𝑋 → 𝐺 ∈ (ℝ ↑m 𝐼)) |
17 | 16 | 3ad2ant3 1130 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 𝐺 ∈ (ℝ ↑m 𝐼)) |
18 | eqid 2820 | . . . 4 ⊢ (ℝ ↑m 𝐼) = (ℝ ↑m 𝐼) | |
19 | 8 | eqcomi 2829 | . . . . . 6 ⊢ (1...𝑁) = 𝐼 |
20 | 19 | fveq2i 6666 | . . . . 5 ⊢ (ℝ^‘(1...𝑁)) = (ℝ^‘𝐼) |
21 | 20 | fveq2i 6666 | . . . 4 ⊢ (dist‘(ℝ^‘(1...𝑁))) = (dist‘(ℝ^‘𝐼)) |
22 | 18, 21 | rrxdsfival 24011 | . . 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 1461 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹(dist‘(ℝ^‘(1...𝑁)))𝐺) = (√‘Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2))) |
24 | 7, 23 | eqtrd 2855 | 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 1082 = wceq 1536 ∈ wcel 2113 ‘cfv 6348 (class class class)co 7149 ↑m cmap 8399 Fincfn 8502 ℝcr 10529 1c1 10531 − cmin 10863 2c2 11686 ℕ0cn0 11891 ...cfz 12889 ↑cexp 13426 √csqrt 14587 Σcsu 15037 distcds 16569 ℝ^crrx 23981 𝔼hilcehl 23982 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-inf2 9097 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 ax-addf 10609 ax-mulf 10610 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-of 7402 df-om 7574 df-1st 7682 df-2nd 7683 df-supp 7824 df-tpos 7885 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-map 8401 df-ixp 8455 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-fsupp 8827 df-sup 8899 df-oi 8967 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-rp 12384 df-fz 12890 df-fzo 13031 df-seq 13367 df-exp 13427 df-hash 13688 df-cj 14453 df-re 14454 df-im 14455 df-sqrt 14589 df-abs 14590 df-clim 14840 df-sum 15038 df-struct 16480 df-ndx 16481 df-slot 16482 df-base 16484 df-sets 16485 df-ress 16486 df-plusg 16573 df-mulr 16574 df-starv 16575 df-sca 16576 df-vsca 16577 df-ip 16578 df-tset 16579 df-ple 16580 df-ds 16582 df-unif 16583 df-hom 16584 df-cco 16585 df-0g 16710 df-gsum 16711 df-prds 16716 df-pws 16718 df-mgm 17847 df-sgrp 17896 df-mnd 17907 df-mhm 17951 df-grp 18101 df-minusg 18102 df-sbg 18103 df-subg 18271 df-ghm 18351 df-cntz 18442 df-cmn 18903 df-abl 18904 df-mgp 19235 df-ur 19247 df-ring 19294 df-cring 19295 df-oppr 19368 df-dvdsr 19386 df-unit 19387 df-invr 19417 df-dvr 19428 df-rnghom 19462 df-drng 19499 df-field 19500 df-subrg 19528 df-staf 19611 df-srng 19612 df-lmod 19631 df-lss 19699 df-sra 19939 df-rgmod 19940 df-cnfld 20541 df-refld 20744 df-dsmm 20871 df-frlm 20886 df-nm 23187 df-tng 23189 df-tcph 23768 df-rrx 23983 df-ehl 23984 |
This theorem is referenced by: (None) |
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