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Mirrors > Home > MPE Home > Th. List > rrxmfval | Structured version Visualization version GIF version |
Description: The value of the Euclidean metric. Compare with rrnval 34583. (Contributed by Thierry Arnoux, 30-Jun-2019.) |
Ref | Expression |
---|---|
rrxmval.1 | ⊢ 𝑋 = {ℎ ∈ (ℝ ↑𝑚 𝐼) ∣ ℎ finSupp 0} |
rrxmval.d | ⊢ 𝐷 = (dist‘(ℝ^‘𝐼)) |
Ref | Expression |
---|---|
rrxmfval | ⊢ (𝐼 ∈ 𝑉 → 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ ((𝑓 supp 0) ∪ (𝑔 supp 0))(((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2793 | . . . . 5 ⊢ (𝑓 ∈ (Base‘(ℝ^‘𝐼)), 𝑔 ∈ (Base‘(ℝ^‘𝐼)) ↦ (√‘(ℝfld Σg (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))) = (𝑓 ∈ (Base‘(ℝ^‘𝐼)), 𝑔 ∈ (Base‘(ℝ^‘𝐼)) ↦ (√‘(ℝfld Σg (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))) | |
2 | fvex 6543 | . . . . 5 ⊢ (√‘(ℝfld Σg (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)))) ∈ V | |
3 | 1, 2 | fnmpoi 7615 | . . . 4 ⊢ (𝑓 ∈ (Base‘(ℝ^‘𝐼)), 𝑔 ∈ (Base‘(ℝ^‘𝐼)) ↦ (√‘(ℝfld Σg (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))) Fn ((Base‘(ℝ^‘𝐼)) × (Base‘(ℝ^‘𝐼))) |
4 | eqid 2793 | . . . . . . 7 ⊢ (ℝ^‘𝐼) = (ℝ^‘𝐼) | |
5 | eqid 2793 | . . . . . . 7 ⊢ (Base‘(ℝ^‘𝐼)) = (Base‘(ℝ^‘𝐼)) | |
6 | 4, 5 | rrxds 23667 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (𝑓 ∈ (Base‘(ℝ^‘𝐼)), 𝑔 ∈ (Base‘(ℝ^‘𝐼)) ↦ (√‘(ℝfld Σg (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))) = (dist‘(ℝ^‘𝐼))) |
7 | rrxmval.d | . . . . . 6 ⊢ 𝐷 = (dist‘(ℝ^‘𝐼)) | |
8 | 6, 7 | syl6reqr 2848 | . . . . 5 ⊢ (𝐼 ∈ 𝑉 → 𝐷 = (𝑓 ∈ (Base‘(ℝ^‘𝐼)), 𝑔 ∈ (Base‘(ℝ^‘𝐼)) ↦ (√‘(ℝfld Σg (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)))))) |
9 | 4, 5 | rrxbase 23662 | . . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → (Base‘(ℝ^‘𝐼)) = {ℎ ∈ (ℝ ↑𝑚 𝐼) ∣ ℎ finSupp 0}) |
10 | rrxmval.1 | . . . . . . 7 ⊢ 𝑋 = {ℎ ∈ (ℝ ↑𝑚 𝐼) ∣ ℎ finSupp 0} | |
11 | 9, 10 | syl6reqr 2848 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → 𝑋 = (Base‘(ℝ^‘𝐼))) |
12 | 11 | sqxpeqd 5467 | . . . . 5 ⊢ (𝐼 ∈ 𝑉 → (𝑋 × 𝑋) = ((Base‘(ℝ^‘𝐼)) × (Base‘(ℝ^‘𝐼)))) |
13 | 8, 12 | fneq12d 6310 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝐷 Fn (𝑋 × 𝑋) ↔ (𝑓 ∈ (Base‘(ℝ^‘𝐼)), 𝑔 ∈ (Base‘(ℝ^‘𝐼)) ↦ (√‘(ℝfld Σg (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))) Fn ((Base‘(ℝ^‘𝐼)) × (Base‘(ℝ^‘𝐼))))) |
14 | 3, 13 | mpbiri 259 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐷 Fn (𝑋 × 𝑋)) |
15 | fnov 7129 | . . 3 ⊢ (𝐷 Fn (𝑋 × 𝑋) ↔ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (𝑓𝐷𝑔))) | |
16 | 14, 15 | sylib 219 | . 2 ⊢ (𝐼 ∈ 𝑉 → 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (𝑓𝐷𝑔))) |
17 | 10, 7 | rrxmval 23679 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑓𝐷𝑔) = (√‘Σ𝑘 ∈ ((𝑓 supp 0) ∪ (𝑔 supp 0))(((𝑓‘𝑘) − (𝑔‘𝑘))↑2))) |
18 | 17 | mpoeq3dva 7080 | . 2 ⊢ (𝐼 ∈ 𝑉 → (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (𝑓𝐷𝑔)) = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ ((𝑓 supp 0) ∪ (𝑔 supp 0))(((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) |
19 | 16, 18 | eqtrd 2829 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ ((𝑓 supp 0) ∪ (𝑔 supp 0))(((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1520 ∈ wcel 2079 {crab 3107 ∪ cun 3852 class class class wbr 4956 ↦ cmpt 5035 × cxp 5433 Fn wfn 6212 ‘cfv 6217 (class class class)co 7007 ∈ cmpo 7009 supp csupp 7672 ↑𝑚 cmap 8247 finSupp cfsupp 8669 ℝcr 10371 0cc0 10372 − cmin 10706 2c2 11529 ↑cexp 13267 √csqrt 14414 Σcsu 14864 Basecbs 16300 distcds 16391 Σg cgsu 16531 ℝfldcrefld 20418 ℝ^crrx 23657 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-inf2 8939 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 ax-pre-sup 10450 ax-addf 10451 ax-mulf 10452 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-fal 1533 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-int 4777 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-se 5395 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-isom 6226 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-of 7258 df-om 7428 df-1st 7536 df-2nd 7537 df-supp 7673 df-tpos 7734 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-1o 7944 df-oadd 7948 df-er 8130 df-map 8249 df-ixp 8301 df-en 8348 df-dom 8349 df-sdom 8350 df-fin 8351 df-fsupp 8670 df-sup 8742 df-oi 8810 df-card 9203 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-div 11135 df-nn 11476 df-2 11537 df-3 11538 df-4 11539 df-5 11540 df-6 11541 df-7 11542 df-8 11543 df-9 11544 df-n0 11735 df-z 11819 df-dec 11937 df-uz 12083 df-rp 12229 df-fz 12732 df-fzo 12873 df-seq 13208 df-exp 13268 df-hash 13529 df-cj 14280 df-re 14281 df-im 14282 df-sqrt 14416 df-abs 14417 df-clim 14667 df-sum 14865 df-struct 16302 df-ndx 16303 df-slot 16304 df-base 16306 df-sets 16307 df-ress 16308 df-plusg 16395 df-mulr 16396 df-starv 16397 df-sca 16398 df-vsca 16399 df-ip 16400 df-tset 16401 df-ple 16402 df-ds 16404 df-unif 16405 df-hom 16406 df-cco 16407 df-0g 16532 df-gsum 16533 df-prds 16538 df-pws 16540 df-mgm 17669 df-sgrp 17711 df-mnd 17722 df-mhm 17762 df-grp 17852 df-minusg 17853 df-sbg 17854 df-subg 18018 df-ghm 18085 df-cntz 18176 df-cmn 18623 df-abl 18624 df-mgp 18918 df-ur 18930 df-ring 18977 df-cring 18978 df-oppr 19051 df-dvdsr 19069 df-unit 19070 df-invr 19100 df-dvr 19111 df-rnghom 19145 df-drng 19182 df-field 19183 df-subrg 19211 df-staf 19294 df-srng 19295 df-lmod 19314 df-lss 19382 df-sra 19622 df-rgmod 19623 df-cnfld 20216 df-refld 20419 df-dsmm 20546 df-frlm 20561 df-nm 22863 df-tng 22865 df-tcph 23444 df-rrx 23659 |
This theorem is referenced by: rrxmet 23682 |
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