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Mirrors > Home > MPE Home > Th. List > Mathboxes > ehl2eudisval0 | Structured version Visualization version GIF version |
Description: The Euclidean distance of a point to the origin in a real Euclidean space of dimension 2. (Contributed by AV, 26-Feb-2023.) |
Ref | Expression |
---|---|
ehl2eudisval0.e | ⊢ 𝐸 = (𝔼hil‘2) |
ehl2eudisval0.x | ⊢ 𝑋 = (ℝ ↑m {1, 2}) |
ehl2eudisval0.d | ⊢ 𝐷 = (dist‘𝐸) |
ehl2eudisval0.0 | ⊢ 0 = ({1, 2} × {0}) |
Ref | Expression |
---|---|
ehl2eudisval0 | ⊢ (𝐹 ∈ 𝑋 → (𝐹𝐷 0 ) = (√‘(((𝐹‘1)↑2) + ((𝐹‘2)↑2)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prex 5434 | . . . 4 ⊢ {1, 2} ∈ V | |
2 | ehl2eudisval0.0 | . . . . 5 ⊢ 0 = ({1, 2} × {0}) | |
3 | ehl2eudisval0.x | . . . . 5 ⊢ 𝑋 = (ℝ ↑m {1, 2}) | |
4 | 2, 3 | rrx0el 25370 | . . . 4 ⊢ ({1, 2} ∈ V → 0 ∈ 𝑋) |
5 | 1, 4 | mp1i 13 | . . 3 ⊢ (𝐹 ∈ 𝑋 → 0 ∈ 𝑋) |
6 | ehl2eudisval0.e | . . . 4 ⊢ 𝐸 = (𝔼hil‘2) | |
7 | ehl2eudisval0.d | . . . 4 ⊢ 𝐷 = (dist‘𝐸) | |
8 | 6, 3, 7 | ehl2eudisval 25395 | . . 3 ⊢ ((𝐹 ∈ 𝑋 ∧ 0 ∈ 𝑋) → (𝐹𝐷 0 ) = (√‘((((𝐹‘1) − ( 0 ‘1))↑2) + (((𝐹‘2) − ( 0 ‘2))↑2)))) |
9 | 5, 8 | mpdan 685 | . 2 ⊢ (𝐹 ∈ 𝑋 → (𝐹𝐷 0 ) = (√‘((((𝐹‘1) − ( 0 ‘1))↑2) + (((𝐹‘2) − ( 0 ‘2))↑2)))) |
10 | 1ex 11242 | . . . . . . . . . . . 12 ⊢ 1 ∈ V | |
11 | 2ex 12322 | . . . . . . . . . . . 12 ⊢ 2 ∈ V | |
12 | c0ex 11240 | . . . . . . . . . . . 12 ⊢ 0 ∈ V | |
13 | xpprsng 7149 | . . . . . . . . . . . 12 ⊢ ((1 ∈ V ∧ 2 ∈ V ∧ 0 ∈ V) → ({1, 2} × {0}) = {〈1, 0〉, 〈2, 0〉}) | |
14 | 10, 11, 12, 13 | mp3an 1457 | . . . . . . . . . . 11 ⊢ ({1, 2} × {0}) = {〈1, 0〉, 〈2, 0〉} |
15 | 2, 14 | eqtri 2753 | . . . . . . . . . 10 ⊢ 0 = {〈1, 0〉, 〈2, 0〉} |
16 | 15 | fveq1i 6897 | . . . . . . . . 9 ⊢ ( 0 ‘1) = ({〈1, 0〉, 〈2, 0〉}‘1) |
17 | 1ne2 12453 | . . . . . . . . . 10 ⊢ 1 ≠ 2 | |
18 | 10, 12 | fvpr1 7202 | . . . . . . . . . 10 ⊢ (1 ≠ 2 → ({〈1, 0〉, 〈2, 0〉}‘1) = 0) |
19 | 17, 18 | ax-mp 5 | . . . . . . . . 9 ⊢ ({〈1, 0〉, 〈2, 0〉}‘1) = 0 |
20 | 16, 19 | eqtri 2753 | . . . . . . . 8 ⊢ ( 0 ‘1) = 0 |
21 | 20 | a1i 11 | . . . . . . 7 ⊢ (𝐹 ∈ 𝑋 → ( 0 ‘1) = 0) |
22 | 21 | oveq2d 7435 | . . . . . 6 ⊢ (𝐹 ∈ 𝑋 → ((𝐹‘1) − ( 0 ‘1)) = ((𝐹‘1) − 0)) |
23 | eqid 2725 | . . . . . . . . 9 ⊢ {1, 2} = {1, 2} | |
24 | 23, 3 | rrx2pxel 47970 | . . . . . . . 8 ⊢ (𝐹 ∈ 𝑋 → (𝐹‘1) ∈ ℝ) |
25 | 24 | recnd 11274 | . . . . . . 7 ⊢ (𝐹 ∈ 𝑋 → (𝐹‘1) ∈ ℂ) |
26 | 25 | subid1d 11592 | . . . . . 6 ⊢ (𝐹 ∈ 𝑋 → ((𝐹‘1) − 0) = (𝐹‘1)) |
27 | 22, 26 | eqtrd 2765 | . . . . 5 ⊢ (𝐹 ∈ 𝑋 → ((𝐹‘1) − ( 0 ‘1)) = (𝐹‘1)) |
28 | 27 | oveq1d 7434 | . . . 4 ⊢ (𝐹 ∈ 𝑋 → (((𝐹‘1) − ( 0 ‘1))↑2) = ((𝐹‘1)↑2)) |
29 | 15 | fveq1i 6897 | . . . . . . . 8 ⊢ ( 0 ‘2) = ({〈1, 0〉, 〈2, 0〉}‘2) |
30 | 11, 12 | fvpr2 7204 | . . . . . . . . 9 ⊢ (1 ≠ 2 → ({〈1, 0〉, 〈2, 0〉}‘2) = 0) |
31 | 17, 30 | mp1i 13 | . . . . . . . 8 ⊢ (𝐹 ∈ 𝑋 → ({〈1, 0〉, 〈2, 0〉}‘2) = 0) |
32 | 29, 31 | eqtrid 2777 | . . . . . . 7 ⊢ (𝐹 ∈ 𝑋 → ( 0 ‘2) = 0) |
33 | 32 | oveq2d 7435 | . . . . . 6 ⊢ (𝐹 ∈ 𝑋 → ((𝐹‘2) − ( 0 ‘2)) = ((𝐹‘2) − 0)) |
34 | 23, 3 | rrx2pyel 47971 | . . . . . . . 8 ⊢ (𝐹 ∈ 𝑋 → (𝐹‘2) ∈ ℝ) |
35 | 34 | recnd 11274 | . . . . . . 7 ⊢ (𝐹 ∈ 𝑋 → (𝐹‘2) ∈ ℂ) |
36 | 35 | subid1d 11592 | . . . . . 6 ⊢ (𝐹 ∈ 𝑋 → ((𝐹‘2) − 0) = (𝐹‘2)) |
37 | 33, 36 | eqtrd 2765 | . . . . 5 ⊢ (𝐹 ∈ 𝑋 → ((𝐹‘2) − ( 0 ‘2)) = (𝐹‘2)) |
38 | 37 | oveq1d 7434 | . . . 4 ⊢ (𝐹 ∈ 𝑋 → (((𝐹‘2) − ( 0 ‘2))↑2) = ((𝐹‘2)↑2)) |
39 | 28, 38 | oveq12d 7437 | . . 3 ⊢ (𝐹 ∈ 𝑋 → ((((𝐹‘1) − ( 0 ‘1))↑2) + (((𝐹‘2) − ( 0 ‘2))↑2)) = (((𝐹‘1)↑2) + ((𝐹‘2)↑2))) |
40 | 39 | fveq2d 6900 | . 2 ⊢ (𝐹 ∈ 𝑋 → (√‘((((𝐹‘1) − ( 0 ‘1))↑2) + (((𝐹‘2) − ( 0 ‘2))↑2))) = (√‘(((𝐹‘1)↑2) + ((𝐹‘2)↑2)))) |
41 | 9, 40 | eqtrd 2765 | 1 ⊢ (𝐹 ∈ 𝑋 → (𝐹𝐷 0 ) = (√‘(((𝐹‘1)↑2) + ((𝐹‘2)↑2)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 Vcvv 3461 {csn 4630 {cpr 4632 〈cop 4636 × cxp 5676 ‘cfv 6549 (class class class)co 7419 ↑m cmap 8845 ℝcr 11139 0cc0 11140 1c1 11141 + caddc 11143 − cmin 11476 2c2 12300 ↑cexp 14062 √csqrt 15216 distcds 17245 𝔼hilcehl 25356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9666 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 ax-addf 11219 ax-mulf 11220 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9388 df-sup 9467 df-oi 9535 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-rp 13010 df-fz 13520 df-fzo 13663 df-seq 14003 df-exp 14063 df-hash 14326 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-clim 15468 df-sum 15669 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-starv 17251 df-sca 17252 df-vsca 17253 df-ip 17254 df-tset 17255 df-ple 17256 df-ds 17258 df-unif 17259 df-hom 17260 df-cco 17261 df-0g 17426 df-gsum 17427 df-prds 17432 df-pws 17434 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18743 df-grp 18901 df-minusg 18902 df-sbg 18903 df-subg 19086 df-ghm 19176 df-cntz 19280 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-cring 20188 df-oppr 20285 df-dvdsr 20308 df-unit 20309 df-invr 20339 df-dvr 20352 df-rhm 20423 df-subrng 20495 df-subrg 20520 df-drng 20638 df-field 20639 df-staf 20737 df-srng 20738 df-lmod 20757 df-lss 20828 df-sra 21070 df-rgmod 21071 df-cnfld 21297 df-refld 21554 df-dsmm 21683 df-frlm 21698 df-nm 24535 df-tng 24537 df-tcph 25141 df-rrx 25357 df-ehl 25358 |
This theorem is referenced by: ehl2eudis0lt 47985 itscnhlinecirc02plem3 48043 |
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