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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 5358 | . . . 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 24543 | . . . 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 24568 | . . 3 ⊢ ((𝐹 ∈ 𝑋 ∧ 0 ∈ 𝑋) → (𝐹𝐷 0 ) = (√‘((((𝐹‘1) − ( 0 ‘1))↑2) + (((𝐹‘2) − ( 0 ‘2))↑2)))) |
9 | 5, 8 | mpdan 683 | . 2 ⊢ (𝐹 ∈ 𝑋 → (𝐹𝐷 0 ) = (√‘((((𝐹‘1) − ( 0 ‘1))↑2) + (((𝐹‘2) − ( 0 ‘2))↑2)))) |
10 | 1ex 10955 | . . . . . . . . . . . 12 ⊢ 1 ∈ V | |
11 | 2ex 12033 | . . . . . . . . . . . 12 ⊢ 2 ∈ V | |
12 | c0ex 10953 | . . . . . . . . . . . 12 ⊢ 0 ∈ V | |
13 | xpprsng 7006 | . . . . . . . . . . . 12 ⊢ ((1 ∈ V ∧ 2 ∈ V ∧ 0 ∈ V) → ({1, 2} × {0}) = {〈1, 0〉, 〈2, 0〉}) | |
14 | 10, 11, 12, 13 | mp3an 1459 | . . . . . . . . . . 11 ⊢ ({1, 2} × {0}) = {〈1, 0〉, 〈2, 0〉} |
15 | 2, 14 | eqtri 2767 | . . . . . . . . . 10 ⊢ 0 = {〈1, 0〉, 〈2, 0〉} |
16 | 15 | fveq1i 6769 | . . . . . . . . 9 ⊢ ( 0 ‘1) = ({〈1, 0〉, 〈2, 0〉}‘1) |
17 | 1ne2 12164 | . . . . . . . . . 10 ⊢ 1 ≠ 2 | |
18 | 10, 12 | fvpr1 7059 | . . . . . . . . . 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 2767 | . . . . . . . 8 ⊢ ( 0 ‘1) = 0 |
21 | 20 | a1i 11 | . . . . . . 7 ⊢ (𝐹 ∈ 𝑋 → ( 0 ‘1) = 0) |
22 | 21 | oveq2d 7284 | . . . . . 6 ⊢ (𝐹 ∈ 𝑋 → ((𝐹‘1) − ( 0 ‘1)) = ((𝐹‘1) − 0)) |
23 | eqid 2739 | . . . . . . . . 9 ⊢ {1, 2} = {1, 2} | |
24 | 23, 3 | rrx2pxel 46009 | . . . . . . . 8 ⊢ (𝐹 ∈ 𝑋 → (𝐹‘1) ∈ ℝ) |
25 | 24 | recnd 10987 | . . . . . . 7 ⊢ (𝐹 ∈ 𝑋 → (𝐹‘1) ∈ ℂ) |
26 | 25 | subid1d 11304 | . . . . . 6 ⊢ (𝐹 ∈ 𝑋 → ((𝐹‘1) − 0) = (𝐹‘1)) |
27 | 22, 26 | eqtrd 2779 | . . . . 5 ⊢ (𝐹 ∈ 𝑋 → ((𝐹‘1) − ( 0 ‘1)) = (𝐹‘1)) |
28 | 27 | oveq1d 7283 | . . . 4 ⊢ (𝐹 ∈ 𝑋 → (((𝐹‘1) − ( 0 ‘1))↑2) = ((𝐹‘1)↑2)) |
29 | 15 | fveq1i 6769 | . . . . . . . 8 ⊢ ( 0 ‘2) = ({〈1, 0〉, 〈2, 0〉}‘2) |
30 | 11, 12 | fvpr2 7061 | . . . . . . . . 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 2791 | . . . . . . 7 ⊢ (𝐹 ∈ 𝑋 → ( 0 ‘2) = 0) |
33 | 32 | oveq2d 7284 | . . . . . 6 ⊢ (𝐹 ∈ 𝑋 → ((𝐹‘2) − ( 0 ‘2)) = ((𝐹‘2) − 0)) |
34 | 23, 3 | rrx2pyel 46010 | . . . . . . . 8 ⊢ (𝐹 ∈ 𝑋 → (𝐹‘2) ∈ ℝ) |
35 | 34 | recnd 10987 | . . . . . . 7 ⊢ (𝐹 ∈ 𝑋 → (𝐹‘2) ∈ ℂ) |
36 | 35 | subid1d 11304 | . . . . . 6 ⊢ (𝐹 ∈ 𝑋 → ((𝐹‘2) − 0) = (𝐹‘2)) |
37 | 33, 36 | eqtrd 2779 | . . . . 5 ⊢ (𝐹 ∈ 𝑋 → ((𝐹‘2) − ( 0 ‘2)) = (𝐹‘2)) |
38 | 37 | oveq1d 7283 | . . . 4 ⊢ (𝐹 ∈ 𝑋 → (((𝐹‘2) − ( 0 ‘2))↑2) = ((𝐹‘2)↑2)) |
39 | 28, 38 | oveq12d 7286 | . . 3 ⊢ (𝐹 ∈ 𝑋 → ((((𝐹‘1) − ( 0 ‘1))↑2) + (((𝐹‘2) − ( 0 ‘2))↑2)) = (((𝐹‘1)↑2) + ((𝐹‘2)↑2))) |
40 | 39 | fveq2d 6772 | . 2 ⊢ (𝐹 ∈ 𝑋 → (√‘((((𝐹‘1) − ( 0 ‘1))↑2) + (((𝐹‘2) − ( 0 ‘2))↑2))) = (√‘(((𝐹‘1)↑2) + ((𝐹‘2)↑2)))) |
41 | 9, 40 | eqtrd 2779 | 1 ⊢ (𝐹 ∈ 𝑋 → (𝐹𝐷 0 ) = (√‘(((𝐹‘1)↑2) + ((𝐹‘2)↑2)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2109 ≠ wne 2944 Vcvv 3430 {csn 4566 {cpr 4568 〈cop 4572 × cxp 5586 ‘cfv 6430 (class class class)co 7268 ↑m cmap 8589 ℝcr 10854 0cc0 10855 1c1 10856 + caddc 10858 − cmin 11188 2c2 12011 ↑cexp 13763 √csqrt 14925 distcds 16952 𝔼hilcehl 24529 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-inf2 9360 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-pre-sup 10933 ax-addf 10934 ax-mulf 10935 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-se 5544 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-isom 6439 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-of 7524 df-om 7701 df-1st 7817 df-2nd 7818 df-supp 7962 df-tpos 8026 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-er 8472 df-map 8591 df-ixp 8660 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-fsupp 9090 df-sup 9162 df-oi 9230 df-card 9681 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-div 11616 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-5 12022 df-6 12023 df-7 12024 df-8 12025 df-9 12026 df-n0 12217 df-z 12303 df-dec 12420 df-uz 12565 df-rp 12713 df-fz 13222 df-fzo 13365 df-seq 13703 df-exp 13764 df-hash 14026 df-cj 14791 df-re 14792 df-im 14793 df-sqrt 14927 df-abs 14928 df-clim 15178 df-sum 15379 df-struct 16829 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-ress 16923 df-plusg 16956 df-mulr 16957 df-starv 16958 df-sca 16959 df-vsca 16960 df-ip 16961 df-tset 16962 df-ple 16963 df-ds 16965 df-unif 16966 df-hom 16967 df-cco 16968 df-0g 17133 df-gsum 17134 df-prds 17139 df-pws 17141 df-mgm 18307 df-sgrp 18356 df-mnd 18367 df-mhm 18411 df-grp 18561 df-minusg 18562 df-sbg 18563 df-subg 18733 df-ghm 18813 df-cntz 18904 df-cmn 19369 df-abl 19370 df-mgp 19702 df-ur 19719 df-ring 19766 df-cring 19767 df-oppr 19843 df-dvdsr 19864 df-unit 19865 df-invr 19895 df-dvr 19906 df-rnghom 19940 df-drng 19974 df-field 19975 df-subrg 20003 df-staf 20086 df-srng 20087 df-lmod 20106 df-lss 20175 df-sra 20415 df-rgmod 20416 df-cnfld 20579 df-refld 20791 df-dsmm 20920 df-frlm 20935 df-nm 23719 df-tng 23721 df-tcph 24314 df-rrx 24530 df-ehl 24531 |
This theorem is referenced by: ehl2eudis0lt 46024 itscnhlinecirc02plem3 46082 |
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