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Mirrors > Home > MPE Home > Th. List > ehl2eudisval | Structured version Visualization version GIF version |
Description: The value of 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 |
---|---|
ehl2eudisval | ⊢ ((𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹𝐷𝐺) = (√‘((((𝐹‘1) − (𝐺‘1))↑2) + (((𝐹‘2) − (𝐺‘2))↑2)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ehl2eudis.e | . . . 4 ⊢ 𝐸 = (𝔼hil‘2) | |
2 | ehl2eudis.x | . . . 4 ⊢ 𝑋 = (ℝ ↑m {1, 2}) | |
3 | ehl2eudis.d | . . . 4 ⊢ 𝐷 = (dist‘𝐸) | |
4 | 1, 2, 3 | ehl2eudis 24582 | . . 3 ⊢ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘((((𝑓‘1) − (𝑔‘1))↑2) + (((𝑓‘2) − (𝑔‘2))↑2)))) |
5 | 4 | oveqi 7282 | . 2 ⊢ (𝐹𝐷𝐺) = (𝐹(𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘((((𝑓‘1) − (𝑔‘1))↑2) + (((𝑓‘2) − (𝑔‘2))↑2))))𝐺) |
6 | eqidd 2741 | . . 3 ⊢ ((𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘((((𝑓‘1) − (𝑔‘1))↑2) + (((𝑓‘2) − (𝑔‘2))↑2)))) = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘((((𝑓‘1) − (𝑔‘1))↑2) + (((𝑓‘2) − (𝑔‘2))↑2))))) | |
7 | fveq1 6768 | . . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘1) = (𝐹‘1)) | |
8 | fveq1 6768 | . . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑔‘1) = (𝐺‘1)) | |
9 | 7, 8 | oveqan12d 7288 | . . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓‘1) − (𝑔‘1)) = ((𝐹‘1) − (𝐺‘1))) |
10 | 9 | oveq1d 7284 | . . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (((𝑓‘1) − (𝑔‘1))↑2) = (((𝐹‘1) − (𝐺‘1))↑2)) |
11 | fveq1 6768 | . . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘2) = (𝐹‘2)) | |
12 | fveq1 6768 | . . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑔‘2) = (𝐺‘2)) | |
13 | 11, 12 | oveqan12d 7288 | . . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓‘2) − (𝑔‘2)) = ((𝐹‘2) − (𝐺‘2))) |
14 | 13 | oveq1d 7284 | . . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (((𝑓‘2) − (𝑔‘2))↑2) = (((𝐹‘2) − (𝐺‘2))↑2)) |
15 | 10, 14 | oveq12d 7287 | . . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((((𝑓‘1) − (𝑔‘1))↑2) + (((𝑓‘2) − (𝑔‘2))↑2)) = ((((𝐹‘1) − (𝐺‘1))↑2) + (((𝐹‘2) − (𝐺‘2))↑2))) |
16 | 15 | fveq2d 6773 | . . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (√‘((((𝑓‘1) − (𝑔‘1))↑2) + (((𝑓‘2) − (𝑔‘2))↑2))) = (√‘((((𝐹‘1) − (𝐺‘1))↑2) + (((𝐹‘2) − (𝐺‘2))↑2)))) |
17 | 16 | adantl 482 | . . 3 ⊢ (((𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (√‘((((𝑓‘1) − (𝑔‘1))↑2) + (((𝑓‘2) − (𝑔‘2))↑2))) = (√‘((((𝐹‘1) − (𝐺‘1))↑2) + (((𝐹‘2) − (𝐺‘2))↑2)))) |
18 | simpl 483 | . . 3 ⊢ ((𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 𝐹 ∈ 𝑋) | |
19 | simpr 485 | . . 3 ⊢ ((𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 𝐺 ∈ 𝑋) | |
20 | fvexd 6784 | . . 3 ⊢ ((𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (√‘((((𝐹‘1) − (𝐺‘1))↑2) + (((𝐹‘2) − (𝐺‘2))↑2))) ∈ V) | |
21 | 6, 17, 18, 19, 20 | ovmpod 7417 | . 2 ⊢ ((𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹(𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘((((𝑓‘1) − (𝑔‘1))↑2) + (((𝑓‘2) − (𝑔‘2))↑2))))𝐺) = (√‘((((𝐹‘1) − (𝐺‘1))↑2) + (((𝐹‘2) − (𝐺‘2))↑2)))) |
22 | 5, 21 | eqtrid 2792 | 1 ⊢ ((𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹𝐷𝐺) = (√‘((((𝐹‘1) − (𝐺‘1))↑2) + (((𝐹‘2) − (𝐺‘2))↑2)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 Vcvv 3431 {cpr 4569 ‘cfv 6431 (class class class)co 7269 ∈ cmpo 7271 ↑m cmap 8596 ℝcr 10869 1c1 10871 + caddc 10873 − cmin 11203 2c2 12026 ↑cexp 13778 √csqrt 14940 distcds 16967 𝔼hilcehl 24544 |
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 7580 ax-inf2 9375 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 ax-pre-sup 10948 ax-addf 10949 ax-mulf 10950 |
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 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-isom 6440 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-of 7525 df-om 7705 df-1st 7822 df-2nd 7823 df-supp 7967 df-tpos 8031 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-er 8479 df-map 8598 df-ixp 8667 df-en 8715 df-dom 8716 df-sdom 8717 df-fin 8718 df-fsupp 9105 df-sup 9177 df-oi 9245 df-card 9696 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-div 11631 df-nn 11972 df-2 12034 df-3 12035 df-4 12036 df-5 12037 df-6 12038 df-7 12039 df-8 12040 df-9 12041 df-n0 12232 df-z 12318 df-dec 12435 df-uz 12580 df-rp 12728 df-fz 13237 df-fzo 13380 df-seq 13718 df-exp 13779 df-hash 14041 df-cj 14806 df-re 14807 df-im 14808 df-sqrt 14942 df-abs 14943 df-clim 15193 df-sum 15394 df-struct 16844 df-sets 16861 df-slot 16879 df-ndx 16891 df-base 16909 df-ress 16938 df-plusg 16971 df-mulr 16972 df-starv 16973 df-sca 16974 df-vsca 16975 df-ip 16976 df-tset 16977 df-ple 16978 df-ds 16980 df-unif 16981 df-hom 16982 df-cco 16983 df-0g 17148 df-gsum 17149 df-prds 17154 df-pws 17156 df-mgm 18322 df-sgrp 18371 df-mnd 18382 df-mhm 18426 df-grp 18576 df-minusg 18577 df-sbg 18578 df-subg 18748 df-ghm 18828 df-cntz 18919 df-cmn 19384 df-abl 19385 df-mgp 19717 df-ur 19734 df-ring 19781 df-cring 19782 df-oppr 19858 df-dvdsr 19879 df-unit 19880 df-invr 19910 df-dvr 19921 df-rnghom 19955 df-drng 19989 df-field 19990 df-subrg 20018 df-staf 20101 df-srng 20102 df-lmod 20121 df-lss 20190 df-sra 20430 df-rgmod 20431 df-cnfld 20594 df-refld 20806 df-dsmm 20935 df-frlm 20950 df-nm 23734 df-tng 23736 df-tcph 24329 df-rrx 24545 df-ehl 24546 |
This theorem is referenced by: ehl2eudisval0 46038 2sphere 46062 |
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