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Mirrors > Home > MPE Home > Th. List > ehl1eudis | Structured version Visualization version GIF version |
Description: The Euclidean distance function in a real Euclidean space of dimension 1. (Contributed by AV, 16-Jan-2023.) |
Ref | Expression |
---|---|
ehl1eudis.e | ⊢ 𝐸 = (𝔼hil‘1) |
ehl1eudis.x | ⊢ 𝑋 = (ℝ ↑𝑚 {1}) |
ehl1eudis.d | ⊢ 𝐷 = (dist‘𝐸) |
Ref | Expression |
---|---|
ehl1eudis | ⊢ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (abs‘((𝑓‘1) − (𝑔‘1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn0 11725 | . . 3 ⊢ 1 ∈ ℕ0 | |
2 | 1z 11825 | . . . . . 6 ⊢ 1 ∈ ℤ | |
3 | fzsn 12765 | . . . . . 6 ⊢ (1 ∈ ℤ → (1...1) = {1}) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ (1...1) = {1} |
5 | 4 | eqcomi 2788 | . . . 4 ⊢ {1} = (1...1) |
6 | ehl1eudis.e | . . . 4 ⊢ 𝐸 = (𝔼hil‘1) | |
7 | ehl1eudis.x | . . . 4 ⊢ 𝑋 = (ℝ ↑𝑚 {1}) | |
8 | ehl1eudis.d | . . . 4 ⊢ 𝐷 = (dist‘𝐸) | |
9 | 5, 6, 7, 8 | ehleudis 23724 | . . 3 ⊢ (1 ∈ ℕ0 → 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ {1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) |
10 | 1, 9 | ax-mp 5 | . 2 ⊢ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ {1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))) |
11 | 7 | eleq2i 2858 | . . . . . . . . . . . 12 ⊢ (𝑓 ∈ 𝑋 ↔ 𝑓 ∈ (ℝ ↑𝑚 {1})) |
12 | reex 10426 | . . . . . . . . . . . . 13 ⊢ ℝ ∈ V | |
13 | snex 5188 | . . . . . . . . . . . . 13 ⊢ {1} ∈ V | |
14 | 12, 13 | elmap 8235 | . . . . . . . . . . . 12 ⊢ (𝑓 ∈ (ℝ ↑𝑚 {1}) ↔ 𝑓:{1}⟶ℝ) |
15 | 11, 14 | bitri 267 | . . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝑋 ↔ 𝑓:{1}⟶ℝ) |
16 | id 22 | . . . . . . . . . . . 12 ⊢ (𝑓:{1}⟶ℝ → 𝑓:{1}⟶ℝ) | |
17 | 1ex 10435 | . . . . . . . . . . . . . 14 ⊢ 1 ∈ V | |
18 | 17 | snid 4473 | . . . . . . . . . . . . 13 ⊢ 1 ∈ {1} |
19 | 18 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑓:{1}⟶ℝ → 1 ∈ {1}) |
20 | 16, 19 | ffvelrnd 6677 | . . . . . . . . . . 11 ⊢ (𝑓:{1}⟶ℝ → (𝑓‘1) ∈ ℝ) |
21 | 15, 20 | sylbi 209 | . . . . . . . . . 10 ⊢ (𝑓 ∈ 𝑋 → (𝑓‘1) ∈ ℝ) |
22 | 21 | adantr 473 | . . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑓‘1) ∈ ℝ) |
23 | 7 | eleq2i 2858 | . . . . . . . . . . . 12 ⊢ (𝑔 ∈ 𝑋 ↔ 𝑔 ∈ (ℝ ↑𝑚 {1})) |
24 | 12, 13 | elmap 8235 | . . . . . . . . . . . 12 ⊢ (𝑔 ∈ (ℝ ↑𝑚 {1}) ↔ 𝑔:{1}⟶ℝ) |
25 | 23, 24 | bitri 267 | . . . . . . . . . . 11 ⊢ (𝑔 ∈ 𝑋 ↔ 𝑔:{1}⟶ℝ) |
26 | id 22 | . . . . . . . . . . . 12 ⊢ (𝑔:{1}⟶ℝ → 𝑔:{1}⟶ℝ) | |
27 | 18 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑔:{1}⟶ℝ → 1 ∈ {1}) |
28 | 26, 27 | ffvelrnd 6677 | . . . . . . . . . . 11 ⊢ (𝑔:{1}⟶ℝ → (𝑔‘1) ∈ ℝ) |
29 | 25, 28 | sylbi 209 | . . . . . . . . . 10 ⊢ (𝑔 ∈ 𝑋 → (𝑔‘1) ∈ ℝ) |
30 | 29 | adantl 474 | . . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑔‘1) ∈ ℝ) |
31 | 22, 30 | resubcld 10869 | . . . . . . . 8 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → ((𝑓‘1) − (𝑔‘1)) ∈ ℝ) |
32 | 31 | resqcld 13426 | . . . . . . 7 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘1) − (𝑔‘1))↑2) ∈ ℝ) |
33 | 32 | recnd 10468 | . . . . . 6 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘1) − (𝑔‘1))↑2) ∈ ℂ) |
34 | fveq2 6499 | . . . . . . . . 9 ⊢ (𝑘 = 1 → (𝑓‘𝑘) = (𝑓‘1)) | |
35 | fveq2 6499 | . . . . . . . . 9 ⊢ (𝑘 = 1 → (𝑔‘𝑘) = (𝑔‘1)) | |
36 | 34, 35 | oveq12d 6994 | . . . . . . . 8 ⊢ (𝑘 = 1 → ((𝑓‘𝑘) − (𝑔‘𝑘)) = ((𝑓‘1) − (𝑔‘1))) |
37 | 36 | oveq1d 6991 | . . . . . . 7 ⊢ (𝑘 = 1 → (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (((𝑓‘1) − (𝑔‘1))↑2)) |
38 | 37 | sumsn 14961 | . . . . . 6 ⊢ ((1 ∈ ℤ ∧ (((𝑓‘1) − (𝑔‘1))↑2) ∈ ℂ) → Σ𝑘 ∈ {1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (((𝑓‘1) − (𝑔‘1))↑2)) |
39 | 2, 33, 38 | sylancr 578 | . . . . 5 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → Σ𝑘 ∈ {1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (((𝑓‘1) − (𝑔‘1))↑2)) |
40 | 39 | fveq2d 6503 | . . . 4 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (√‘Σ𝑘 ∈ {1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)) = (√‘(((𝑓‘1) − (𝑔‘1))↑2))) |
41 | 31 | absred 14637 | . . . 4 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (abs‘((𝑓‘1) − (𝑔‘1))) = (√‘(((𝑓‘1) − (𝑔‘1))↑2))) |
42 | 40, 41 | eqtr4d 2818 | . . 3 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (√‘Σ𝑘 ∈ {1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)) = (abs‘((𝑓‘1) − (𝑔‘1)))) |
43 | 42 | mpoeq3ia 7050 | . 2 ⊢ (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ {1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))) = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (abs‘((𝑓‘1) − (𝑔‘1)))) |
44 | 10, 43 | eqtri 2803 | 1 ⊢ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (abs‘((𝑓‘1) − (𝑔‘1)))) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 387 = wceq 1507 ∈ wcel 2050 {csn 4441 ⟶wf 6184 ‘cfv 6188 (class class class)co 6976 ∈ cmpo 6978 ↑𝑚 cmap 8206 ℂcc 10333 ℝcr 10334 1c1 10336 − cmin 10670 2c2 11495 ℕ0cn0 11707 ℤcz 11793 ...cfz 12708 ↑cexp 13244 √csqrt 14453 abscabs 14454 Σcsu 14903 distcds 16430 𝔼hilcehl 23690 |
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 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-inf2 8898 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 ax-pre-sup 10413 ax-addf 10414 ax-mulf 10415 |
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 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-se 5367 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-isom 6197 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-of 7227 df-om 7397 df-1st 7501 df-2nd 7502 df-supp 7634 df-tpos 7695 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-oadd 7909 df-er 8089 df-map 8208 df-ixp 8260 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-fsupp 8629 df-sup 8701 df-oi 8769 df-card 9162 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-nn 11440 df-2 11503 df-3 11504 df-4 11505 df-5 11506 df-6 11507 df-7 11508 df-8 11509 df-9 11510 df-n0 11708 df-z 11794 df-dec 11912 df-uz 12059 df-rp 12205 df-fz 12709 df-fzo 12850 df-seq 13185 df-exp 13245 df-hash 13506 df-cj 14319 df-re 14320 df-im 14321 df-sqrt 14455 df-abs 14456 df-clim 14706 df-sum 14904 df-struct 16341 df-ndx 16342 df-slot 16343 df-base 16345 df-sets 16346 df-ress 16347 df-plusg 16434 df-mulr 16435 df-starv 16436 df-sca 16437 df-vsca 16438 df-ip 16439 df-tset 16440 df-ple 16441 df-ds 16443 df-unif 16444 df-hom 16445 df-cco 16446 df-0g 16571 df-gsum 16572 df-prds 16577 df-pws 16579 df-mgm 17710 df-sgrp 17752 df-mnd 17763 df-mhm 17803 df-grp 17894 df-minusg 17895 df-sbg 17896 df-subg 18060 df-ghm 18127 df-cntz 18218 df-cmn 18668 df-abl 18669 df-mgp 18963 df-ur 18975 df-ring 19022 df-cring 19023 df-oppr 19096 df-dvdsr 19114 df-unit 19115 df-invr 19145 df-dvr 19156 df-rnghom 19190 df-drng 19227 df-field 19228 df-subrg 19256 df-staf 19338 df-srng 19339 df-lmod 19358 df-lss 19426 df-sra 19666 df-rgmod 19667 df-cnfld 20248 df-refld 20451 df-dsmm 20578 df-frlm 20593 df-nm 22895 df-tng 22897 df-tcph 23476 df-rrx 23691 df-ehl 23692 |
This theorem is referenced by: ehl1eudisval 23727 |
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