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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 12249 | . . 3 ⊢ 1 ∈ ℕ0 | |
2 | 1z 12350 | . . . . . 6 ⊢ 1 ∈ ℤ | |
3 | fzsn 13297 | . . . . . 6 ⊢ (1 ∈ ℤ → (1...1) = {1}) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ (1...1) = {1} |
5 | 4 | eqcomi 2749 | . . . 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 24580 | . . 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 2832 | . . . . . . . . . . . 12 ⊢ (𝑓 ∈ 𝑋 ↔ 𝑓 ∈ (ℝ ↑m {1})) |
12 | reex 10963 | . . . . . . . . . . . . 13 ⊢ ℝ ∈ V | |
13 | snex 5358 | . . . . . . . . . . . . 13 ⊢ {1} ∈ V | |
14 | 12, 13 | elmap 8642 | . . . . . . . . . . . 12 ⊢ (𝑓 ∈ (ℝ ↑m {1}) ↔ 𝑓:{1}⟶ℝ) |
15 | 11, 14 | bitri 274 | . . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝑋 ↔ 𝑓:{1}⟶ℝ) |
16 | id 22 | . . . . . . . . . . . 12 ⊢ (𝑓:{1}⟶ℝ → 𝑓:{1}⟶ℝ) | |
17 | 1ex 10972 | . . . . . . . . . . . . . 14 ⊢ 1 ∈ V | |
18 | 17 | snid 4603 | . . . . . . . . . . . . 13 ⊢ 1 ∈ {1} |
19 | 18 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑓:{1}⟶ℝ → 1 ∈ {1}) |
20 | 16, 19 | ffvelrnd 6959 | . . . . . . . . . . 11 ⊢ (𝑓:{1}⟶ℝ → (𝑓‘1) ∈ ℝ) |
21 | 15, 20 | sylbi 216 | . . . . . . . . . 10 ⊢ (𝑓 ∈ 𝑋 → (𝑓‘1) ∈ ℝ) |
22 | 21 | adantr 481 | . . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑓‘1) ∈ ℝ) |
23 | 7 | eleq2i 2832 | . . . . . . . . . . . 12 ⊢ (𝑔 ∈ 𝑋 ↔ 𝑔 ∈ (ℝ ↑m {1})) |
24 | 12, 13 | elmap 8642 | . . . . . . . . . . . 12 ⊢ (𝑔 ∈ (ℝ ↑m {1}) ↔ 𝑔:{1}⟶ℝ) |
25 | 23, 24 | bitri 274 | . . . . . . . . . . 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 6959 | . . . . . . . . . . 11 ⊢ (𝑔:{1}⟶ℝ → (𝑔‘1) ∈ ℝ) |
29 | 25, 28 | sylbi 216 | . . . . . . . . . 10 ⊢ (𝑔 ∈ 𝑋 → (𝑔‘1) ∈ ℝ) |
30 | 29 | adantl 482 | . . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑔‘1) ∈ ℝ) |
31 | 22, 30 | resubcld 11403 | . . . . . . . 8 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → ((𝑓‘1) − (𝑔‘1)) ∈ ℝ) |
32 | 31 | resqcld 13963 | . . . . . . 7 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘1) − (𝑔‘1))↑2) ∈ ℝ) |
33 | 32 | recnd 11004 | . . . . . 6 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘1) − (𝑔‘1))↑2) ∈ ℂ) |
34 | fveq2 6771 | . . . . . . . . 9 ⊢ (𝑘 = 1 → (𝑓‘𝑘) = (𝑓‘1)) | |
35 | fveq2 6771 | . . . . . . . . 9 ⊢ (𝑘 = 1 → (𝑔‘𝑘) = (𝑔‘1)) | |
36 | 34, 35 | oveq12d 7289 | . . . . . . . 8 ⊢ (𝑘 = 1 → ((𝑓‘𝑘) − (𝑔‘𝑘)) = ((𝑓‘1) − (𝑔‘1))) |
37 | 36 | oveq1d 7286 | . . . . . . 7 ⊢ (𝑘 = 1 → (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (((𝑓‘1) − (𝑔‘1))↑2)) |
38 | 37 | sumsn 15456 | . . . . . 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 587 | . . . . 5 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → Σ𝑘 ∈ {1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (((𝑓‘1) − (𝑔‘1))↑2)) |
40 | 39 | fveq2d 6775 | . . . 4 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (√‘Σ𝑘 ∈ {1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)) = (√‘(((𝑓‘1) − (𝑔‘1))↑2))) |
41 | 31 | absred 15126 | . . . 4 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (abs‘((𝑓‘1) − (𝑔‘1))) = (√‘(((𝑓‘1) − (𝑔‘1))↑2))) |
42 | 40, 41 | eqtr4d 2783 | . . 3 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (√‘Σ𝑘 ∈ {1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)) = (abs‘((𝑓‘1) − (𝑔‘1)))) |
43 | 42 | mpoeq3ia 7347 | . 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 2768 | 1 ⊢ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (abs‘((𝑓‘1) − (𝑔‘1)))) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1542 ∈ wcel 2110 {csn 4567 ⟶wf 6428 ‘cfv 6432 (class class class)co 7271 ∈ cmpo 7273 ↑m cmap 8598 ℂcc 10870 ℝcr 10871 1c1 10873 − cmin 11205 2c2 12028 ℕ0cn0 12233 ℤcz 12319 ...cfz 13238 ↑cexp 13780 √csqrt 14942 abscabs 14943 Σcsu 15395 distcds 16969 𝔼hilcehl 24546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-inf2 9377 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950 ax-addf 10951 ax-mulf 10952 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-of 7527 df-om 7707 df-1st 7824 df-2nd 7825 df-supp 7969 df-tpos 8033 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-map 8600 df-ixp 8669 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-fsupp 9107 df-sup 9179 df-oi 9247 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12437 df-uz 12582 df-rp 12730 df-fz 13239 df-fzo 13382 df-seq 13720 df-exp 13781 df-hash 14043 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-clim 15195 df-sum 15396 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-starv 16975 df-sca 16976 df-vsca 16977 df-ip 16978 df-tset 16979 df-ple 16980 df-ds 16982 df-unif 16983 df-hom 16984 df-cco 16985 df-0g 17150 df-gsum 17151 df-prds 17156 df-pws 17158 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-mhm 18428 df-grp 18578 df-minusg 18579 df-sbg 18580 df-subg 18750 df-ghm 18830 df-cntz 18921 df-cmn 19386 df-abl 19387 df-mgp 19719 df-ur 19736 df-ring 19783 df-cring 19784 df-oppr 19860 df-dvdsr 19881 df-unit 19882 df-invr 19912 df-dvr 19923 df-rnghom 19957 df-drng 19991 df-field 19992 df-subrg 20020 df-staf 20103 df-srng 20104 df-lmod 20123 df-lss 20192 df-sra 20432 df-rgmod 20433 df-cnfld 20596 df-refld 20808 df-dsmm 20937 df-frlm 20952 df-nm 23736 df-tng 23738 df-tcph 24331 df-rrx 24547 df-ehl 24548 |
This theorem is referenced by: ehl1eudisval 24583 |
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