![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > rrxdsfi | Structured version Visualization version GIF version |
Description: The distance over generalized Euclidean spaces. Finite dimensional case. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Ref | Expression |
---|---|
rrxdsfi.h | ⊢ 𝐻 = (ℝ^‘𝐼) |
rrxdsfi.b | ⊢ 𝐵 = (ℝ ↑𝑚 𝐼) |
Ref | Expression |
---|---|
rrxdsfi | ⊢ (𝐼 ∈ Fin → (dist‘𝐻) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . . 5 ⊢ (𝐼 ∈ Fin → 𝐼 ∈ Fin) | |
2 | rrxdsfi.h | . . . . 5 ⊢ 𝐻 = (ℝ^‘𝐼) | |
3 | eqid 2778 | . . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
4 | 1, 2, 3 | rrxbasefi 23719 | . . . 4 ⊢ (𝐼 ∈ Fin → (Base‘𝐻) = (ℝ ↑𝑚 𝐼)) |
5 | rrxdsfi.b | . . . 4 ⊢ 𝐵 = (ℝ ↑𝑚 𝐼) | |
6 | 4, 5 | syl6reqr 2833 | . . 3 ⊢ (𝐼 ∈ Fin → 𝐵 = (Base‘𝐻)) |
7 | 6 | adantr 473 | . . 3 ⊢ ((𝐼 ∈ Fin ∧ 𝑓 ∈ 𝐵) → 𝐵 = (Base‘𝐻)) |
8 | df-refld 20454 | . . . . . . 7 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
9 | 8 | oveq1i 6988 | . . . . . 6 ⊢ (ℝfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))) = ((ℂfld ↾s ℝ) Σg (𝑘 ∈ 𝐼 ↦ (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))) |
10 | simp1 1116 | . . . . . . 7 ⊢ ((𝐼 ∈ Fin ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝐼 ∈ Fin) | |
11 | simpr 477 | . . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ Fin ∧ 𝑓 ∈ 𝐵) → 𝑓 ∈ 𝐵) | |
12 | 11, 5 | syl6eleq 2876 | . . . . . . . . . . . 12 ⊢ ((𝐼 ∈ Fin ∧ 𝑓 ∈ 𝐵) → 𝑓 ∈ (ℝ ↑𝑚 𝐼)) |
13 | 12 | 3adant3 1112 | . . . . . . . . . . 11 ⊢ ((𝐼 ∈ Fin ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑓 ∈ (ℝ ↑𝑚 𝐼)) |
14 | elmapi 8230 | . . . . . . . . . . 11 ⊢ (𝑓 ∈ (ℝ ↑𝑚 𝐼) → 𝑓:𝐼⟶ℝ) | |
15 | 13, 14 | syl 17 | . . . . . . . . . 10 ⊢ ((𝐼 ∈ Fin ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑓:𝐼⟶ℝ) |
16 | 15 | ffvelrnda 6678 | . . . . . . . . 9 ⊢ (((𝐼 ∈ Fin ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑘 ∈ 𝐼) → (𝑓‘𝑘) ∈ ℝ) |
17 | simpr 477 | . . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ Fin ∧ 𝑔 ∈ 𝐵) → 𝑔 ∈ 𝐵) | |
18 | 17, 5 | syl6eleq 2876 | . . . . . . . . . . . 12 ⊢ ((𝐼 ∈ Fin ∧ 𝑔 ∈ 𝐵) → 𝑔 ∈ (ℝ ↑𝑚 𝐼)) |
19 | 18 | 3adant2 1111 | . . . . . . . . . . 11 ⊢ ((𝐼 ∈ Fin ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑔 ∈ (ℝ ↑𝑚 𝐼)) |
20 | elmapi 8230 | . . . . . . . . . . 11 ⊢ (𝑔 ∈ (ℝ ↑𝑚 𝐼) → 𝑔:𝐼⟶ℝ) | |
21 | 19, 20 | syl 17 | . . . . . . . . . 10 ⊢ ((𝐼 ∈ Fin ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑔:𝐼⟶ℝ) |
22 | 21 | ffvelrnda 6678 | . . . . . . . . 9 ⊢ (((𝐼 ∈ Fin ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑘 ∈ 𝐼) → (𝑔‘𝑘) ∈ ℝ) |
23 | 16, 22 | resubcld 10871 | . . . . . . . 8 ⊢ (((𝐼 ∈ Fin ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑘 ∈ 𝐼) → ((𝑓‘𝑘) − (𝑔‘𝑘)) ∈ ℝ) |
24 | 23 | resqcld 13429 | . . . . . . 7 ⊢ (((𝐼 ∈ Fin ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑘 ∈ 𝐼) → (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) ∈ ℝ) |
25 | 10, 24 | regsumfsum 20318 | . . . . . 6 ⊢ ((𝐼 ∈ Fin ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → ((ℂfld ↾s ℝ) Σg (𝑘 ∈ 𝐼 ↦ (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))) = Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)) |
26 | 9, 25 | syl5req 2827 | . . . . 5 ⊢ ((𝐼 ∈ Fin ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (ℝfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) |
27 | 26 | fveq2d 6505 | . . . 4 ⊢ ((𝐼 ∈ Fin ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)) = (√‘(ℝfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))))) |
28 | 27 | 3expb 1100 | . . 3 ⊢ ((𝐼 ∈ Fin ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)) = (√‘(ℝfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))))) |
29 | 6, 7, 28 | mpoeq123dva 7048 | . 2 ⊢ (𝐼 ∈ Fin → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))) = (𝑓 ∈ (Base‘𝐻), 𝑔 ∈ (Base‘𝐻) ↦ (√‘(ℝfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))))) |
30 | 2, 3 | rrxds 23702 | . 2 ⊢ (𝐼 ∈ Fin → (𝑓 ∈ (Base‘𝐻), 𝑔 ∈ (Base‘𝐻) ↦ (√‘(ℝfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))))) = (dist‘𝐻)) |
31 | 29, 30 | eqtr2d 2815 | 1 ⊢ (𝐼 ∈ Fin → (dist‘𝐻) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 ↦ cmpt 5009 ⟶wf 6186 ‘cfv 6190 (class class class)co 6978 ∈ cmpo 6980 ↑𝑚 cmap 8208 Fincfn 8308 ℝcr 10336 − cmin 10672 2c2 11498 ↑cexp 13247 √csqrt 14456 Σcsu 14906 Basecbs 16342 ↾s cress 16343 distcds 16433 Σg cgsu 16573 ℂfldccnfld 20250 ℝfldcrefld 20453 ℝ^crrx 23692 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-inf2 8900 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 ax-pre-sup 10415 ax-addf 10416 ax-mulf 10417 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-int 4751 df-iun 4795 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-se 5368 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-isom 6199 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-of 7229 df-om 7399 df-1st 7503 df-2nd 7504 df-supp 7636 df-tpos 7697 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-1o 7907 df-oadd 7911 df-er 8091 df-map 8210 df-ixp 8262 df-en 8309 df-dom 8310 df-sdom 8311 df-fin 8312 df-fsupp 8631 df-sup 8703 df-oi 8771 df-card 9164 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-div 11101 df-nn 11442 df-2 11506 df-3 11507 df-4 11508 df-5 11509 df-6 11510 df-7 11511 df-8 11512 df-9 11513 df-n0 11711 df-z 11797 df-dec 11915 df-uz 12062 df-rp 12208 df-fz 12712 df-fzo 12853 df-seq 13188 df-exp 13248 df-hash 13509 df-cj 14322 df-re 14323 df-im 14324 df-sqrt 14458 df-abs 14459 df-clim 14709 df-sum 14907 df-struct 16344 df-ndx 16345 df-slot 16346 df-base 16348 df-sets 16349 df-ress 16350 df-plusg 16437 df-mulr 16438 df-starv 16439 df-sca 16440 df-vsca 16441 df-ip 16442 df-tset 16443 df-ple 16444 df-ds 16446 df-unif 16447 df-hom 16448 df-cco 16449 df-0g 16574 df-gsum 16575 df-prds 16580 df-pws 16582 df-mgm 17713 df-sgrp 17755 df-mnd 17766 df-mhm 17806 df-grp 17897 df-minusg 17898 df-sbg 17899 df-subg 18063 df-ghm 18130 df-cntz 18221 df-cmn 18671 df-abl 18672 df-mgp 18966 df-ur 18978 df-ring 19025 df-cring 19026 df-oppr 19099 df-dvdsr 19117 df-unit 19118 df-invr 19148 df-dvr 19159 df-rnghom 19193 df-drng 19230 df-field 19231 df-subrg 19259 df-staf 19341 df-srng 19342 df-lmod 19361 df-lss 19429 df-sra 19669 df-rgmod 19670 df-cnfld 20251 df-refld 20454 df-dsmm 20581 df-frlm 20596 df-nm 22898 df-tng 22900 df-tcph 23479 df-rrx 23694 |
This theorem is referenced by: rrxdsfival 23722 ehleudis 23727 rrndistlt 42007 qndenserrnopnlem 42014 rrndsmet 42019 ioorrnopnlem 42021 hoiqssbllem2 42337 |
Copyright terms: Public domain | W3C validator |