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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 5437 | . . . 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 25432 | . . . 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 25457 | . . 3 ⊢ ((𝐹 ∈ 𝑋 ∧ 0 ∈ 𝑋) → (𝐹𝐷 0 ) = (√‘((((𝐹‘1) − ( 0 ‘1))↑2) + (((𝐹‘2) − ( 0 ‘2))↑2)))) |
| 9 | 5, 8 | mpdan 687 | . 2 ⊢ (𝐹 ∈ 𝑋 → (𝐹𝐷 0 ) = (√‘((((𝐹‘1) − ( 0 ‘1))↑2) + (((𝐹‘2) − ( 0 ‘2))↑2)))) |
| 10 | 1ex 11257 | . . . . . . . . . . . 12 ⊢ 1 ∈ V | |
| 11 | 2ex 12343 | . . . . . . . . . . . 12 ⊢ 2 ∈ V | |
| 12 | c0ex 11255 | . . . . . . . . . . . 12 ⊢ 0 ∈ V | |
| 13 | xpprsng 7160 | . . . . . . . . . . . 12 ⊢ ((1 ∈ V ∧ 2 ∈ V ∧ 0 ∈ V) → ({1, 2} × {0}) = {〈1, 0〉, 〈2, 0〉}) | |
| 14 | 10, 11, 12, 13 | mp3an 1463 | . . . . . . . . . . 11 ⊢ ({1, 2} × {0}) = {〈1, 0〉, 〈2, 0〉} |
| 15 | 2, 14 | eqtri 2765 | . . . . . . . . . 10 ⊢ 0 = {〈1, 0〉, 〈2, 0〉} |
| 16 | 15 | fveq1i 6907 | . . . . . . . . 9 ⊢ ( 0 ‘1) = ({〈1, 0〉, 〈2, 0〉}‘1) |
| 17 | 1ne2 12474 | . . . . . . . . . 10 ⊢ 1 ≠ 2 | |
| 18 | 10, 12 | fvpr1 7212 | . . . . . . . . . 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 2765 | . . . . . . . 8 ⊢ ( 0 ‘1) = 0 |
| 21 | 20 | a1i 11 | . . . . . . 7 ⊢ (𝐹 ∈ 𝑋 → ( 0 ‘1) = 0) |
| 22 | 21 | oveq2d 7447 | . . . . . 6 ⊢ (𝐹 ∈ 𝑋 → ((𝐹‘1) − ( 0 ‘1)) = ((𝐹‘1) − 0)) |
| 23 | eqid 2737 | . . . . . . . . 9 ⊢ {1, 2} = {1, 2} | |
| 24 | 23, 3 | rrx2pxel 48632 | . . . . . . . 8 ⊢ (𝐹 ∈ 𝑋 → (𝐹‘1) ∈ ℝ) |
| 25 | 24 | recnd 11289 | . . . . . . 7 ⊢ (𝐹 ∈ 𝑋 → (𝐹‘1) ∈ ℂ) |
| 26 | 25 | subid1d 11609 | . . . . . 6 ⊢ (𝐹 ∈ 𝑋 → ((𝐹‘1) − 0) = (𝐹‘1)) |
| 27 | 22, 26 | eqtrd 2777 | . . . . 5 ⊢ (𝐹 ∈ 𝑋 → ((𝐹‘1) − ( 0 ‘1)) = (𝐹‘1)) |
| 28 | 27 | oveq1d 7446 | . . . 4 ⊢ (𝐹 ∈ 𝑋 → (((𝐹‘1) − ( 0 ‘1))↑2) = ((𝐹‘1)↑2)) |
| 29 | 15 | fveq1i 6907 | . . . . . . . 8 ⊢ ( 0 ‘2) = ({〈1, 0〉, 〈2, 0〉}‘2) |
| 30 | 11, 12 | fvpr2 7213 | . . . . . . . . 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 2789 | . . . . . . 7 ⊢ (𝐹 ∈ 𝑋 → ( 0 ‘2) = 0) |
| 33 | 32 | oveq2d 7447 | . . . . . 6 ⊢ (𝐹 ∈ 𝑋 → ((𝐹‘2) − ( 0 ‘2)) = ((𝐹‘2) − 0)) |
| 34 | 23, 3 | rrx2pyel 48633 | . . . . . . . 8 ⊢ (𝐹 ∈ 𝑋 → (𝐹‘2) ∈ ℝ) |
| 35 | 34 | recnd 11289 | . . . . . . 7 ⊢ (𝐹 ∈ 𝑋 → (𝐹‘2) ∈ ℂ) |
| 36 | 35 | subid1d 11609 | . . . . . 6 ⊢ (𝐹 ∈ 𝑋 → ((𝐹‘2) − 0) = (𝐹‘2)) |
| 37 | 33, 36 | eqtrd 2777 | . . . . 5 ⊢ (𝐹 ∈ 𝑋 → ((𝐹‘2) − ( 0 ‘2)) = (𝐹‘2)) |
| 38 | 37 | oveq1d 7446 | . . . 4 ⊢ (𝐹 ∈ 𝑋 → (((𝐹‘2) − ( 0 ‘2))↑2) = ((𝐹‘2)↑2)) |
| 39 | 28, 38 | oveq12d 7449 | . . 3 ⊢ (𝐹 ∈ 𝑋 → ((((𝐹‘1) − ( 0 ‘1))↑2) + (((𝐹‘2) − ( 0 ‘2))↑2)) = (((𝐹‘1)↑2) + ((𝐹‘2)↑2))) |
| 40 | 39 | fveq2d 6910 | . 2 ⊢ (𝐹 ∈ 𝑋 → (√‘((((𝐹‘1) − ( 0 ‘1))↑2) + (((𝐹‘2) − ( 0 ‘2))↑2))) = (√‘(((𝐹‘1)↑2) + ((𝐹‘2)↑2)))) |
| 41 | 9, 40 | eqtrd 2777 | 1 ⊢ (𝐹 ∈ 𝑋 → (𝐹𝐷 0 ) = (√‘(((𝐹‘1)↑2) + ((𝐹‘2)↑2)))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 {csn 4626 {cpr 4628 〈cop 4632 × cxp 5683 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 − cmin 11492 2c2 12321 ↑cexp 14102 √csqrt 15272 distcds 17306 𝔼hilcehl 25418 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-ghm 19231 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-rhm 20472 df-subrng 20546 df-subrg 20570 df-drng 20731 df-field 20732 df-staf 20840 df-srng 20841 df-lmod 20860 df-lss 20930 df-sra 21172 df-rgmod 21173 df-cnfld 21365 df-refld 21623 df-dsmm 21752 df-frlm 21767 df-nm 24595 df-tng 24597 df-tcph 25203 df-rrx 25419 df-ehl 25420 |
| This theorem is referenced by: ehl2eudis0lt 48647 itscnhlinecirc02plem3 48705 |
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