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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 2771 | . . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
4 | 1, 2, 3 | rrxbasefi 41020 | . . . 4 ⊢ (𝐼 ∈ Fin → (Base‘𝐻) = (ℝ ↑𝑚 𝐼)) |
5 | rrxdsfi.b | . . . . . 6 ⊢ 𝐵 = (ℝ ↑𝑚 𝐼) | |
6 | 5 | eqcomi 2780 | . . . . 5 ⊢ (ℝ ↑𝑚 𝐼) = 𝐵 |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝐼 ∈ Fin → (ℝ ↑𝑚 𝐼) = 𝐵) |
8 | 4, 7 | eqtr2d 2806 | . . 3 ⊢ (𝐼 ∈ Fin → 𝐵 = (Base‘𝐻)) |
9 | 8 | adantr 466 | . . 3 ⊢ ((𝐼 ∈ Fin ∧ 𝑓 ∈ 𝐵) → 𝐵 = (Base‘𝐻)) |
10 | df-refld 20168 | . . . . . . . . 9 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
11 | 10 | oveq1i 6803 | . . . . . . . 8 ⊢ (ℝfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))) = ((ℂfld ↾s ℝ) Σg (𝑘 ∈ 𝐼 ↦ (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))) |
12 | 11 | a1i 11 | . . . . . . 7 ⊢ ((𝐼 ∈ Fin ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → (ℝfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))) = ((ℂfld ↾s ℝ) Σg (𝑘 ∈ 𝐼 ↦ (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) |
13 | simp1 1130 | . . . . . . . 8 ⊢ ((𝐼 ∈ Fin ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝐼 ∈ Fin) | |
14 | simpr 471 | . . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ Fin ∧ 𝑓 ∈ 𝐵) → 𝑓 ∈ 𝐵) | |
15 | 5 | a1i 11 | . . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ Fin ∧ 𝑓 ∈ 𝐵) → 𝐵 = (ℝ ↑𝑚 𝐼)) |
16 | 14, 15 | eleqtrd 2852 | . . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ Fin ∧ 𝑓 ∈ 𝐵) → 𝑓 ∈ (ℝ ↑𝑚 𝐼)) |
17 | 16 | 3adant3 1126 | . . . . . . . . . . . 12 ⊢ ((𝐼 ∈ Fin ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑓 ∈ (ℝ ↑𝑚 𝐼)) |
18 | elmapi 8031 | . . . . . . . . . . . 12 ⊢ (𝑓 ∈ (ℝ ↑𝑚 𝐼) → 𝑓:𝐼⟶ℝ) | |
19 | 17, 18 | syl 17 | . . . . . . . . . . 11 ⊢ ((𝐼 ∈ Fin ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑓:𝐼⟶ℝ) |
20 | 19 | ffvelrnda 6502 | . . . . . . . . . 10 ⊢ (((𝐼 ∈ Fin ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑘 ∈ 𝐼) → (𝑓‘𝑘) ∈ ℝ) |
21 | simpr 471 | . . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ Fin ∧ 𝑔 ∈ 𝐵) → 𝑔 ∈ 𝐵) | |
22 | 5 | a1i 11 | . . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ Fin ∧ 𝑔 ∈ 𝐵) → 𝐵 = (ℝ ↑𝑚 𝐼)) |
23 | 21, 22 | eleqtrd 2852 | . . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ Fin ∧ 𝑔 ∈ 𝐵) → 𝑔 ∈ (ℝ ↑𝑚 𝐼)) |
24 | 23 | 3adant2 1125 | . . . . . . . . . . . 12 ⊢ ((𝐼 ∈ Fin ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑔 ∈ (ℝ ↑𝑚 𝐼)) |
25 | elmapi 8031 | . . . . . . . . . . . 12 ⊢ (𝑔 ∈ (ℝ ↑𝑚 𝐼) → 𝑔:𝐼⟶ℝ) | |
26 | 24, 25 | syl 17 | . . . . . . . . . . 11 ⊢ ((𝐼 ∈ Fin ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → 𝑔:𝐼⟶ℝ) |
27 | 26 | ffvelrnda 6502 | . . . . . . . . . 10 ⊢ (((𝐼 ∈ Fin ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑘 ∈ 𝐼) → (𝑔‘𝑘) ∈ ℝ) |
28 | 20, 27 | resubcld 10660 | . . . . . . . . 9 ⊢ (((𝐼 ∈ Fin ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑘 ∈ 𝐼) → ((𝑓‘𝑘) − (𝑔‘𝑘)) ∈ ℝ) |
29 | 28 | resqcld 13242 | . . . . . . . 8 ⊢ (((𝐼 ∈ Fin ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ 𝑘 ∈ 𝐼) → (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) ∈ ℝ) |
30 | 13, 29 | regsumfsum 20029 | . . . . . . 7 ⊢ ((𝐼 ∈ Fin ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → ((ℂfld ↾s ℝ) Σg (𝑘 ∈ 𝐼 ↦ (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))) = Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)) |
31 | 12, 30 | eqtrd 2805 | . . . . . 6 ⊢ ((𝐼 ∈ Fin ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → (ℝfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))) = Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)) |
32 | 31 | eqcomd 2777 | . . . . 5 ⊢ ((𝐼 ∈ Fin ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (ℝfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) |
33 | 32 | fveq2d 6336 | . . . 4 ⊢ ((𝐼 ∈ Fin ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) → (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)) = (√‘(ℝfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))))) |
34 | 33 | 3expb 1113 | . . 3 ⊢ ((𝐼 ∈ Fin ∧ (𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)) = (√‘(ℝfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))))) |
35 | 8, 9, 34 | mpt2eq123dva 6863 | . 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)))))) |
36 | 2, 3 | rrxds 23400 | . 2 ⊢ (𝐼 ∈ Fin → (𝑓 ∈ (Base‘𝐻), 𝑔 ∈ (Base‘𝐻) ↦ (√‘(ℝfld Σg (𝑘 ∈ 𝐼 ↦ (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))))) = (dist‘𝐻)) |
37 | 35, 36 | eqtr2d 2806 | 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 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ↦ cmpt 4863 ⟶wf 6027 ‘cfv 6031 (class class class)co 6793 ↦ cmpt2 6795 ↑𝑚 cmap 8009 Fincfn 8109 ℝcr 10137 − cmin 10468 2c2 11272 ↑cexp 13067 √csqrt 14181 Σcsu 14624 Basecbs 16064 ↾s cress 16065 distcds 16158 Σg cgsu 16309 ℂfldccnfld 19961 ℝfldcrefld 20167 ℝ^crrx 23390 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-inf2 8702 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-pre-sup 10216 ax-addf 10217 ax-mulf 10218 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-isom 6040 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-of 7044 df-om 7213 df-1st 7315 df-2nd 7316 df-supp 7447 df-tpos 7504 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-oadd 7717 df-er 7896 df-map 8011 df-ixp 8063 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-fsupp 8432 df-sup 8504 df-oi 8571 df-card 8965 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-nn 11223 df-2 11281 df-3 11282 df-4 11283 df-5 11284 df-6 11285 df-7 11286 df-8 11287 df-9 11288 df-n0 11495 df-z 11580 df-dec 11696 df-uz 11889 df-rp 12036 df-fz 12534 df-fzo 12674 df-seq 13009 df-exp 13068 df-hash 13322 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-abs 14184 df-clim 14427 df-sum 14625 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-ress 16072 df-plusg 16162 df-mulr 16163 df-starv 16164 df-sca 16165 df-vsca 16166 df-ip 16167 df-tset 16168 df-ple 16169 df-ds 16172 df-unif 16173 df-hom 16174 df-cco 16175 df-0g 16310 df-gsum 16311 df-prds 16316 df-pws 16318 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-mhm 17543 df-grp 17633 df-minusg 17634 df-sbg 17635 df-subg 17799 df-ghm 17866 df-cntz 17957 df-cmn 18402 df-abl 18403 df-mgp 18698 df-ur 18710 df-ring 18757 df-cring 18758 df-oppr 18831 df-dvdsr 18849 df-unit 18850 df-invr 18880 df-dvr 18891 df-rnghom 18925 df-drng 18959 df-field 18960 df-subrg 18988 df-staf 19055 df-srng 19056 df-lmod 19075 df-lss 19143 df-sra 19387 df-rgmod 19388 df-cnfld 19962 df-refld 20168 df-dsmm 20293 df-frlm 20308 df-nm 22607 df-tng 22609 df-tch 23188 df-rrx 23392 |
This theorem is referenced by: rrndistlt 41027 qndenserrnopnlem 41034 rrndsmet 41039 ioorrnopnlem 41041 hoiqssbllem2 41357 |
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