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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 11902 | . . 3 ⊢ 2 ∈ ℕ0 | |
2 | fz12pr 12952 | . . . . 5 ⊢ (1...2) = {1, 2} | |
3 | 2 | eqcomi 2827 | . . . 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 23948 | . . 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 6663 | . . . . . . 7 ⊢ (𝑘 = 1 → (𝑓‘𝑘) = (𝑓‘1)) | |
10 | fveq2 6663 | . . . . . . 7 ⊢ (𝑘 = 1 → (𝑔‘𝑘) = (𝑔‘1)) | |
11 | 9, 10 | oveq12d 7163 | . . . . . 6 ⊢ (𝑘 = 1 → ((𝑓‘𝑘) − (𝑔‘𝑘)) = ((𝑓‘1) − (𝑔‘1))) |
12 | 11 | oveq1d 7160 | . . . . 5 ⊢ (𝑘 = 1 → (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (((𝑓‘1) − (𝑔‘1))↑2)) |
13 | fveq2 6663 | . . . . . . 7 ⊢ (𝑘 = 2 → (𝑓‘𝑘) = (𝑓‘2)) | |
14 | fveq2 6663 | . . . . . . 7 ⊢ (𝑘 = 2 → (𝑔‘𝑘) = (𝑔‘2)) | |
15 | 13, 14 | oveq12d 7163 | . . . . . 6 ⊢ (𝑘 = 2 → ((𝑓‘𝑘) − (𝑔‘𝑘)) = ((𝑓‘2) − (𝑔‘2))) |
16 | 15 | oveq1d 7160 | . . . . 5 ⊢ (𝑘 = 2 → (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (((𝑓‘2) − (𝑔‘2))↑2)) |
17 | 5 | eleq2i 2901 | . . . . . . . . . . . 12 ⊢ (𝑓 ∈ 𝑋 ↔ 𝑓 ∈ (ℝ ↑m {1, 2})) |
18 | reex 10616 | . . . . . . . . . . . . 13 ⊢ ℝ ∈ V | |
19 | prex 5323 | . . . . . . . . . . . . 13 ⊢ {1, 2} ∈ V | |
20 | 18, 19 | elmap 8424 | . . . . . . . . . . . 12 ⊢ (𝑓 ∈ (ℝ ↑m {1, 2}) ↔ 𝑓:{1, 2}⟶ℝ) |
21 | 17, 20 | bitri 276 | . . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝑋 ↔ 𝑓:{1, 2}⟶ℝ) |
22 | id 22 | . . . . . . . . . . . 12 ⊢ (𝑓:{1, 2}⟶ℝ → 𝑓:{1, 2}⟶ℝ) | |
23 | 1ex 10625 | . . . . . . . . . . . . . 14 ⊢ 1 ∈ V | |
24 | 23 | prid1 4690 | . . . . . . . . . . . . 13 ⊢ 1 ∈ {1, 2} |
25 | 24 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑓:{1, 2}⟶ℝ → 1 ∈ {1, 2}) |
26 | 22, 25 | ffvelrnd 6844 | . . . . . . . . . . 11 ⊢ (𝑓:{1, 2}⟶ℝ → (𝑓‘1) ∈ ℝ) |
27 | 21, 26 | sylbi 218 | . . . . . . . . . 10 ⊢ (𝑓 ∈ 𝑋 → (𝑓‘1) ∈ ℝ) |
28 | 27 | adantr 481 | . . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑓‘1) ∈ ℝ) |
29 | 5 | eleq2i 2901 | . . . . . . . . . . . 12 ⊢ (𝑔 ∈ 𝑋 ↔ 𝑔 ∈ (ℝ ↑m {1, 2})) |
30 | 18, 19 | elmap 8424 | . . . . . . . . . . . 12 ⊢ (𝑔 ∈ (ℝ ↑m {1, 2}) ↔ 𝑔:{1, 2}⟶ℝ) |
31 | 29, 30 | bitri 276 | . . . . . . . . . . 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 6844 | . . . . . . . . . . 11 ⊢ (𝑔:{1, 2}⟶ℝ → (𝑔‘1) ∈ ℝ) |
35 | 31, 34 | sylbi 218 | . . . . . . . . . 10 ⊢ (𝑔 ∈ 𝑋 → (𝑔‘1) ∈ ℝ) |
36 | 35 | adantl 482 | . . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑔‘1) ∈ ℝ) |
37 | 28, 36 | resubcld 11056 | . . . . . . . 8 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → ((𝑓‘1) − (𝑔‘1)) ∈ ℝ) |
38 | 37 | resqcld 13599 | . . . . . . 7 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘1) − (𝑔‘1))↑2) ∈ ℝ) |
39 | 38 | recnd 10657 | . . . . . 6 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘1) − (𝑔‘1))↑2) ∈ ℂ) |
40 | 2ex 11702 | . . . . . . . . . . . . . 14 ⊢ 2 ∈ V | |
41 | 40 | prid2 4691 | . . . . . . . . . . . . 13 ⊢ 2 ∈ {1, 2} |
42 | 41 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑓:{1, 2}⟶ℝ → 2 ∈ {1, 2}) |
43 | 22, 42 | ffvelrnd 6844 | . . . . . . . . . . 11 ⊢ (𝑓:{1, 2}⟶ℝ → (𝑓‘2) ∈ ℝ) |
44 | 21, 43 | sylbi 218 | . . . . . . . . . 10 ⊢ (𝑓 ∈ 𝑋 → (𝑓‘2) ∈ ℝ) |
45 | 44 | adantr 481 | . . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑓‘2) ∈ ℝ) |
46 | 41 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑔:{1, 2}⟶ℝ → 2 ∈ {1, 2}) |
47 | 32, 46 | ffvelrnd 6844 | . . . . . . . . . . 11 ⊢ (𝑔:{1, 2}⟶ℝ → (𝑔‘2) ∈ ℝ) |
48 | 31, 47 | sylbi 218 | . . . . . . . . . 10 ⊢ (𝑔 ∈ 𝑋 → (𝑔‘2) ∈ ℝ) |
49 | 48 | adantl 482 | . . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑔‘2) ∈ ℝ) |
50 | 45, 49 | resubcld 11056 | . . . . . . . 8 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → ((𝑓‘2) − (𝑔‘2)) ∈ ℝ) |
51 | 50 | resqcld 13599 | . . . . . . 7 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘2) − (𝑔‘2))↑2) ∈ ℝ) |
52 | 51 | recnd 10657 | . . . . . 6 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘2) − (𝑔‘2))↑2) ∈ ℂ) |
53 | 39, 52 | jca 512 | . . . . 5 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → ((((𝑓‘1) − (𝑔‘1))↑2) ∈ ℂ ∧ (((𝑓‘2) − (𝑔‘2))↑2) ∈ ℂ)) |
54 | 23, 40 | pm3.2i 471 | . . . . . 6 ⊢ (1 ∈ V ∧ 2 ∈ V) |
55 | 54 | a1i 11 | . . . . 5 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (1 ∈ V ∧ 2 ∈ V)) |
56 | 1ne2 11833 | . . . . . 6 ⊢ 1 ≠ 2 | |
57 | 56 | a1i 11 | . . . . 5 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → 1 ≠ 2) |
58 | 12, 16, 53, 55, 57 | sumpr 15091 | . . . 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 6667 | . . 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 | mpoeq3ia 7221 | . 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 2841 | 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 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 Vcvv 3492 {cpr 4559 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 ↑m cmap 8395 ℂcc 10523 ℝcr 10524 1c1 10526 + caddc 10528 − cmin 10858 2c2 11680 ℕ0cn0 11885 ...cfz 12880 ↑cexp 13417 √csqrt 14580 Σcsu 15030 distcds 16562 𝔼hilcehl 23914 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-sup 8894 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-fz 12881 df-fzo 13022 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 df-sum 15031 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-0g 16703 df-gsum 16704 df-prds 16709 df-pws 16711 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-grp 18044 df-minusg 18045 df-sbg 18046 df-subg 18214 df-ghm 18294 df-cntz 18385 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-rnghom 19396 df-drng 19433 df-field 19434 df-subrg 19462 df-staf 19545 df-srng 19546 df-lmod 19565 df-lss 19633 df-sra 19873 df-rgmod 19874 df-cnfld 20474 df-refld 20677 df-dsmm 20804 df-frlm 20819 df-nm 23119 df-tng 23121 df-tcph 23700 df-rrx 23915 df-ehl 23916 |
This theorem is referenced by: ehl2eudisval 23953 |
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