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Mirrors > Home > MPE Home > Th. List > ehl2eudis | Structured version Visualization version GIF version |
Description: The Euclidean distance function in a real Euclidean space of dimension 2. (Contributed by AV, 16-Jan-2023.) |
Ref | Expression |
---|---|
ehl2eudis.e | ⊢ 𝐸 = (𝔼hil‘2) |
ehl2eudis.x | ⊢ 𝑋 = (ℝ ↑𝑚 {1, 2}) |
ehl2eudis.d | ⊢ 𝐷 = (dist‘𝐸) |
Ref | Expression |
---|---|
ehl2eudis | ⊢ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘((((𝑓‘1) − (𝑔‘1))↑2) + (((𝑓‘2) − (𝑔‘2))↑2)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nn0 11666 | . . 3 ⊢ 2 ∈ ℕ0 | |
2 | fz12pr 12720 | . . . . 5 ⊢ (1...2) = {1, 2} | |
3 | 2 | eqcomi 2787 | . . . 4 ⊢ {1, 2} = (1...2) |
4 | ehl2eudis.e | . . . 4 ⊢ 𝐸 = (𝔼hil‘2) | |
5 | ehl2eudis.x | . . . 4 ⊢ 𝑋 = (ℝ ↑𝑚 {1, 2}) | |
6 | ehl2eudis.d | . . . 4 ⊢ 𝐷 = (dist‘𝐸) | |
7 | 3, 4, 5, 6 | ehleudis 23635 | . . 3 ⊢ (2 ∈ ℕ0 → 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ {1, 2} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) |
8 | 1, 7 | ax-mp 5 | . 2 ⊢ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ {1, 2} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))) |
9 | fveq2 6448 | . . . . . . 7 ⊢ (𝑘 = 1 → (𝑓‘𝑘) = (𝑓‘1)) | |
10 | fveq2 6448 | . . . . . . 7 ⊢ (𝑘 = 1 → (𝑔‘𝑘) = (𝑔‘1)) | |
11 | 9, 10 | oveq12d 6942 | . . . . . 6 ⊢ (𝑘 = 1 → ((𝑓‘𝑘) − (𝑔‘𝑘)) = ((𝑓‘1) − (𝑔‘1))) |
12 | 11 | oveq1d 6939 | . . . . 5 ⊢ (𝑘 = 1 → (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (((𝑓‘1) − (𝑔‘1))↑2)) |
13 | fveq2 6448 | . . . . . . 7 ⊢ (𝑘 = 2 → (𝑓‘𝑘) = (𝑓‘2)) | |
14 | fveq2 6448 | . . . . . . 7 ⊢ (𝑘 = 2 → (𝑔‘𝑘) = (𝑔‘2)) | |
15 | 13, 14 | oveq12d 6942 | . . . . . 6 ⊢ (𝑘 = 2 → ((𝑓‘𝑘) − (𝑔‘𝑘)) = ((𝑓‘2) − (𝑔‘2))) |
16 | 15 | oveq1d 6939 | . . . . 5 ⊢ (𝑘 = 2 → (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (((𝑓‘2) − (𝑔‘2))↑2)) |
17 | 5 | eleq2i 2851 | . . . . . . . . . . . 12 ⊢ (𝑓 ∈ 𝑋 ↔ 𝑓 ∈ (ℝ ↑𝑚 {1, 2})) |
18 | reex 10365 | . . . . . . . . . . . . 13 ⊢ ℝ ∈ V | |
19 | prex 5143 | . . . . . . . . . . . . 13 ⊢ {1, 2} ∈ V | |
20 | 18, 19 | elmap 8171 | . . . . . . . . . . . 12 ⊢ (𝑓 ∈ (ℝ ↑𝑚 {1, 2}) ↔ 𝑓:{1, 2}⟶ℝ) |
21 | 17, 20 | bitri 267 | . . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝑋 ↔ 𝑓:{1, 2}⟶ℝ) |
22 | id 22 | . . . . . . . . . . . 12 ⊢ (𝑓:{1, 2}⟶ℝ → 𝑓:{1, 2}⟶ℝ) | |
23 | 1ex 10374 | . . . . . . . . . . . . . 14 ⊢ 1 ∈ V | |
24 | 23 | prid1 4529 | . . . . . . . . . . . . 13 ⊢ 1 ∈ {1, 2} |
25 | 24 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑓:{1, 2}⟶ℝ → 1 ∈ {1, 2}) |
26 | 22, 25 | ffvelrnd 6626 | . . . . . . . . . . 11 ⊢ (𝑓:{1, 2}⟶ℝ → (𝑓‘1) ∈ ℝ) |
27 | 21, 26 | sylbi 209 | . . . . . . . . . 10 ⊢ (𝑓 ∈ 𝑋 → (𝑓‘1) ∈ ℝ) |
28 | 27 | adantr 474 | . . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑓‘1) ∈ ℝ) |
29 | 5 | eleq2i 2851 | . . . . . . . . . . . 12 ⊢ (𝑔 ∈ 𝑋 ↔ 𝑔 ∈ (ℝ ↑𝑚 {1, 2})) |
30 | 18, 19 | elmap 8171 | . . . . . . . . . . . 12 ⊢ (𝑔 ∈ (ℝ ↑𝑚 {1, 2}) ↔ 𝑔:{1, 2}⟶ℝ) |
31 | 29, 30 | bitri 267 | . . . . . . . . . . 11 ⊢ (𝑔 ∈ 𝑋 ↔ 𝑔:{1, 2}⟶ℝ) |
32 | id 22 | . . . . . . . . . . . 12 ⊢ (𝑔:{1, 2}⟶ℝ → 𝑔:{1, 2}⟶ℝ) | |
33 | 24 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑔:{1, 2}⟶ℝ → 1 ∈ {1, 2}) |
34 | 32, 33 | ffvelrnd 6626 | . . . . . . . . . . 11 ⊢ (𝑔:{1, 2}⟶ℝ → (𝑔‘1) ∈ ℝ) |
35 | 31, 34 | sylbi 209 | . . . . . . . . . 10 ⊢ (𝑔 ∈ 𝑋 → (𝑔‘1) ∈ ℝ) |
36 | 35 | adantl 475 | . . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑔‘1) ∈ ℝ) |
37 | 28, 36 | resubcld 10806 | . . . . . . . 8 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → ((𝑓‘1) − (𝑔‘1)) ∈ ℝ) |
38 | 37 | resqcld 13362 | . . . . . . 7 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘1) − (𝑔‘1))↑2) ∈ ℝ) |
39 | 38 | recnd 10407 | . . . . . 6 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘1) − (𝑔‘1))↑2) ∈ ℂ) |
40 | 2ex 11457 | . . . . . . . . . . . . . 14 ⊢ 2 ∈ V | |
41 | 40 | prid2 4530 | . . . . . . . . . . . . 13 ⊢ 2 ∈ {1, 2} |
42 | 41 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑓:{1, 2}⟶ℝ → 2 ∈ {1, 2}) |
43 | 22, 42 | ffvelrnd 6626 | . . . . . . . . . . 11 ⊢ (𝑓:{1, 2}⟶ℝ → (𝑓‘2) ∈ ℝ) |
44 | 21, 43 | sylbi 209 | . . . . . . . . . 10 ⊢ (𝑓 ∈ 𝑋 → (𝑓‘2) ∈ ℝ) |
45 | 44 | adantr 474 | . . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑓‘2) ∈ ℝ) |
46 | 41 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑔:{1, 2}⟶ℝ → 2 ∈ {1, 2}) |
47 | 32, 46 | ffvelrnd 6626 | . . . . . . . . . . 11 ⊢ (𝑔:{1, 2}⟶ℝ → (𝑔‘2) ∈ ℝ) |
48 | 31, 47 | sylbi 209 | . . . . . . . . . 10 ⊢ (𝑔 ∈ 𝑋 → (𝑔‘2) ∈ ℝ) |
49 | 48 | adantl 475 | . . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑔‘2) ∈ ℝ) |
50 | 45, 49 | resubcld 10806 | . . . . . . . 8 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → ((𝑓‘2) − (𝑔‘2)) ∈ ℝ) |
51 | 50 | resqcld 13362 | . . . . . . 7 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘2) − (𝑔‘2))↑2) ∈ ℝ) |
52 | 51 | recnd 10407 | . . . . . 6 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘2) − (𝑔‘2))↑2) ∈ ℂ) |
53 | 39, 52 | jca 507 | . . . . 5 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → ((((𝑓‘1) − (𝑔‘1))↑2) ∈ ℂ ∧ (((𝑓‘2) − (𝑔‘2))↑2) ∈ ℂ)) |
54 | 23, 40 | pm3.2i 464 | . . . . . 6 ⊢ (1 ∈ V ∧ 2 ∈ V) |
55 | 54 | a1i 11 | . . . . 5 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (1 ∈ V ∧ 2 ∈ V)) |
56 | 1ne2 11595 | . . . . . 6 ⊢ 1 ≠ 2 | |
57 | 56 | a1i 11 | . . . . 5 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → 1 ≠ 2) |
58 | 12, 16, 53, 55, 57 | sumpr 14893 | . . . 4 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → Σ𝑘 ∈ {1, 2} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = ((((𝑓‘1) − (𝑔‘1))↑2) + (((𝑓‘2) − (𝑔‘2))↑2))) |
59 | 58 | fveq2d 6452 | . . 3 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (√‘Σ𝑘 ∈ {1, 2} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)) = (√‘((((𝑓‘1) − (𝑔‘1))↑2) + (((𝑓‘2) − (𝑔‘2))↑2)))) |
60 | 59 | mpt2eq3ia 6999 | . 2 ⊢ (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ {1, 2} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))) = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘((((𝑓‘1) − (𝑔‘1))↑2) + (((𝑓‘2) − (𝑔‘2))↑2)))) |
61 | 8, 60 | eqtri 2802 | 1 ⊢ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘((((𝑓‘1) − (𝑔‘1))↑2) + (((𝑓‘2) − (𝑔‘2))↑2)))) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 386 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 Vcvv 3398 {cpr 4400 ⟶wf 6133 ‘cfv 6137 (class class class)co 6924 ↦ cmpt2 6926 ↑𝑚 cmap 8142 ℂcc 10272 ℝcr 10273 1c1 10275 + caddc 10277 − cmin 10608 2c2 11435 ℕ0cn0 11647 ...cfz 12648 ↑cexp 13183 √csqrt 14386 Σcsu 14833 distcds 16358 𝔼hilcehl 23601 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 ax-addf 10353 ax-mulf 10354 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-of 7176 df-om 7346 df-1st 7447 df-2nd 7448 df-supp 7579 df-tpos 7636 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-map 8144 df-ixp 8197 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fsupp 8566 df-sup 8638 df-oi 8706 df-card 9100 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11036 df-nn 11380 df-2 11443 df-3 11444 df-4 11445 df-5 11446 df-6 11447 df-7 11448 df-8 11449 df-9 11450 df-n0 11648 df-z 11734 df-dec 11851 df-uz 11998 df-rp 12143 df-fz 12649 df-fzo 12790 df-seq 13125 df-exp 13184 df-hash 13442 df-cj 14252 df-re 14253 df-im 14254 df-sqrt 14388 df-abs 14389 df-clim 14636 df-sum 14834 df-struct 16268 df-ndx 16269 df-slot 16270 df-base 16272 df-sets 16273 df-ress 16274 df-plusg 16362 df-mulr 16363 df-starv 16364 df-sca 16365 df-vsca 16366 df-ip 16367 df-tset 16368 df-ple 16369 df-ds 16371 df-unif 16372 df-hom 16373 df-cco 16374 df-0g 16499 df-gsum 16500 df-prds 16505 df-pws 16507 df-mgm 17639 df-sgrp 17681 df-mnd 17692 df-mhm 17732 df-grp 17823 df-minusg 17824 df-sbg 17825 df-subg 17986 df-ghm 18053 df-cntz 18144 df-cmn 18592 df-abl 18593 df-mgp 18888 df-ur 18900 df-ring 18947 df-cring 18948 df-oppr 19021 df-dvdsr 19039 df-unit 19040 df-invr 19070 df-dvr 19081 df-rnghom 19115 df-drng 19152 df-field 19153 df-subrg 19181 df-staf 19248 df-srng 19249 df-lmod 19268 df-lss 19336 df-sra 19580 df-rgmod 19581 df-cnfld 20154 df-refld 20359 df-dsmm 20486 df-frlm 20501 df-nm 22806 df-tng 22808 df-tcph 23387 df-rrx 23602 df-ehl 23603 |
This theorem is referenced by: ehl2eudisval 23640 |
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