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Mirrors > Home > MPE Home > Th. List > Mathboxes > ehl2eudis0lt | Structured version Visualization version GIF version |
Description: An upper bound of the Euclidean distance of a point to the origin in a real Euclidean space of dimension 2. (Contributed by AV, 9-May-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 |
---|---|
ehl2eudis0lt | ⊢ ((𝐹 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+) → ((𝐹𝐷 0 ) < 𝑅 ↔ (((𝐹‘1)↑2) + ((𝐹‘2)↑2)) < (𝑅↑2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ehl2eudisval0.e | . . . . 5 ⊢ 𝐸 = (𝔼hil‘2) | |
2 | ehl2eudisval0.x | . . . . 5 ⊢ 𝑋 = (ℝ ↑m {1, 2}) | |
3 | ehl2eudisval0.d | . . . . 5 ⊢ 𝐷 = (dist‘𝐸) | |
4 | ehl2eudisval0.0 | . . . . 5 ⊢ 0 = ({1, 2} × {0}) | |
5 | 1, 2, 3, 4 | ehl2eudisval0 46801 | . . . 4 ⊢ (𝐹 ∈ 𝑋 → (𝐹𝐷 0 ) = (√‘(((𝐹‘1)↑2) + ((𝐹‘2)↑2)))) |
6 | 5 | adantr 481 | . . 3 ⊢ ((𝐹 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+) → (𝐹𝐷 0 ) = (√‘(((𝐹‘1)↑2) + ((𝐹‘2)↑2)))) |
7 | 6 | breq1d 5115 | . 2 ⊢ ((𝐹 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+) → ((𝐹𝐷 0 ) < 𝑅 ↔ (√‘(((𝐹‘1)↑2) + ((𝐹‘2)↑2))) < 𝑅)) |
8 | eqid 2736 | . . . . . . 7 ⊢ {1, 2} = {1, 2} | |
9 | 8, 2 | rrx2pxel 46787 | . . . . . 6 ⊢ (𝐹 ∈ 𝑋 → (𝐹‘1) ∈ ℝ) |
10 | 8, 2 | rrx2pyel 46788 | . . . . . 6 ⊢ (𝐹 ∈ 𝑋 → (𝐹‘2) ∈ ℝ) |
11 | eqid 2736 | . . . . . . 7 ⊢ (((𝐹‘1)↑2) + ((𝐹‘2)↑2)) = (((𝐹‘1)↑2) + ((𝐹‘2)↑2)) | |
12 | 11 | resum2sqcl 46782 | . . . . . 6 ⊢ (((𝐹‘1) ∈ ℝ ∧ (𝐹‘2) ∈ ℝ) → (((𝐹‘1)↑2) + ((𝐹‘2)↑2)) ∈ ℝ) |
13 | 9, 10, 12 | syl2anc 584 | . . . . 5 ⊢ (𝐹 ∈ 𝑋 → (((𝐹‘1)↑2) + ((𝐹‘2)↑2)) ∈ ℝ) |
14 | resqcl 14029 | . . . . . . . 8 ⊢ ((𝐹‘1) ∈ ℝ → ((𝐹‘1)↑2) ∈ ℝ) | |
15 | resqcl 14029 | . . . . . . . 8 ⊢ ((𝐹‘2) ∈ ℝ → ((𝐹‘2)↑2) ∈ ℝ) | |
16 | 14, 15 | anim12i 613 | . . . . . . 7 ⊢ (((𝐹‘1) ∈ ℝ ∧ (𝐹‘2) ∈ ℝ) → (((𝐹‘1)↑2) ∈ ℝ ∧ ((𝐹‘2)↑2) ∈ ℝ)) |
17 | sqge0 14041 | . . . . . . . 8 ⊢ ((𝐹‘1) ∈ ℝ → 0 ≤ ((𝐹‘1)↑2)) | |
18 | sqge0 14041 | . . . . . . . 8 ⊢ ((𝐹‘2) ∈ ℝ → 0 ≤ ((𝐹‘2)↑2)) | |
19 | 17, 18 | anim12i 613 | . . . . . . 7 ⊢ (((𝐹‘1) ∈ ℝ ∧ (𝐹‘2) ∈ ℝ) → (0 ≤ ((𝐹‘1)↑2) ∧ 0 ≤ ((𝐹‘2)↑2))) |
20 | addge0 11644 | . . . . . . 7 ⊢ (((((𝐹‘1)↑2) ∈ ℝ ∧ ((𝐹‘2)↑2) ∈ ℝ) ∧ (0 ≤ ((𝐹‘1)↑2) ∧ 0 ≤ ((𝐹‘2)↑2))) → 0 ≤ (((𝐹‘1)↑2) + ((𝐹‘2)↑2))) | |
21 | 16, 19, 20 | syl2anc 584 | . . . . . 6 ⊢ (((𝐹‘1) ∈ ℝ ∧ (𝐹‘2) ∈ ℝ) → 0 ≤ (((𝐹‘1)↑2) + ((𝐹‘2)↑2))) |
22 | 9, 10, 21 | syl2anc 584 | . . . . 5 ⊢ (𝐹 ∈ 𝑋 → 0 ≤ (((𝐹‘1)↑2) + ((𝐹‘2)↑2))) |
23 | 13, 22 | resqrtcld 15302 | . . . 4 ⊢ (𝐹 ∈ 𝑋 → (√‘(((𝐹‘1)↑2) + ((𝐹‘2)↑2))) ∈ ℝ) |
24 | 13, 22 | sqrtge0d 15305 | . . . 4 ⊢ (𝐹 ∈ 𝑋 → 0 ≤ (√‘(((𝐹‘1)↑2) + ((𝐹‘2)↑2)))) |
25 | 23, 24 | jca 512 | . . 3 ⊢ (𝐹 ∈ 𝑋 → ((√‘(((𝐹‘1)↑2) + ((𝐹‘2)↑2))) ∈ ℝ ∧ 0 ≤ (√‘(((𝐹‘1)↑2) + ((𝐹‘2)↑2))))) |
26 | rprege0 12930 | . . 3 ⊢ (𝑅 ∈ ℝ+ → (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅)) | |
27 | lt2sq 14038 | . . 3 ⊢ ((((√‘(((𝐹‘1)↑2) + ((𝐹‘2)↑2))) ∈ ℝ ∧ 0 ≤ (√‘(((𝐹‘1)↑2) + ((𝐹‘2)↑2)))) ∧ (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅)) → ((√‘(((𝐹‘1)↑2) + ((𝐹‘2)↑2))) < 𝑅 ↔ ((√‘(((𝐹‘1)↑2) + ((𝐹‘2)↑2)))↑2) < (𝑅↑2))) | |
28 | 25, 26, 27 | syl2an 596 | . 2 ⊢ ((𝐹 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+) → ((√‘(((𝐹‘1)↑2) + ((𝐹‘2)↑2))) < 𝑅 ↔ ((√‘(((𝐹‘1)↑2) + ((𝐹‘2)↑2)))↑2) < (𝑅↑2))) |
29 | 13, 22 | jca 512 | . . . . 5 ⊢ (𝐹 ∈ 𝑋 → ((((𝐹‘1)↑2) + ((𝐹‘2)↑2)) ∈ ℝ ∧ 0 ≤ (((𝐹‘1)↑2) + ((𝐹‘2)↑2)))) |
30 | 29 | adantr 481 | . . . 4 ⊢ ((𝐹 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+) → ((((𝐹‘1)↑2) + ((𝐹‘2)↑2)) ∈ ℝ ∧ 0 ≤ (((𝐹‘1)↑2) + ((𝐹‘2)↑2)))) |
31 | resqrtth 15140 | . . . 4 ⊢ (((((𝐹‘1)↑2) + ((𝐹‘2)↑2)) ∈ ℝ ∧ 0 ≤ (((𝐹‘1)↑2) + ((𝐹‘2)↑2))) → ((√‘(((𝐹‘1)↑2) + ((𝐹‘2)↑2)))↑2) = (((𝐹‘1)↑2) + ((𝐹‘2)↑2))) | |
32 | 30, 31 | syl 17 | . . 3 ⊢ ((𝐹 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+) → ((√‘(((𝐹‘1)↑2) + ((𝐹‘2)↑2)))↑2) = (((𝐹‘1)↑2) + ((𝐹‘2)↑2))) |
33 | 32 | breq1d 5115 | . 2 ⊢ ((𝐹 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+) → (((√‘(((𝐹‘1)↑2) + ((𝐹‘2)↑2)))↑2) < (𝑅↑2) ↔ (((𝐹‘1)↑2) + ((𝐹‘2)↑2)) < (𝑅↑2))) |
34 | 7, 28, 33 | 3bitrd 304 | 1 ⊢ ((𝐹 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+) → ((𝐹𝐷 0 ) < 𝑅 ↔ (((𝐹‘1)↑2) + ((𝐹‘2)↑2)) < (𝑅↑2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {csn 4586 {cpr 4588 class class class wbr 5105 × cxp 5631 ‘cfv 6496 (class class class)co 7357 ↑m cmap 8765 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 < clt 11189 ≤ cle 11190 2c2 12208 ℝ+crp 12915 ↑cexp 13967 √csqrt 15118 distcds 17142 𝔼hilcehl 24748 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-tpos 8157 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-sup 9378 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-rp 12916 df-fz 13425 df-fzo 13568 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-sum 15571 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-0g 17323 df-gsum 17324 df-prds 17329 df-pws 17331 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-mhm 18601 df-grp 18751 df-minusg 18752 df-sbg 18753 df-subg 18925 df-ghm 19006 df-cntz 19097 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-cring 19967 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-invr 20101 df-dvr 20112 df-rnghom 20146 df-drng 20187 df-field 20188 df-subrg 20220 df-staf 20304 df-srng 20305 df-lmod 20324 df-lss 20393 df-sra 20633 df-rgmod 20634 df-cnfld 20797 df-refld 21009 df-dsmm 21138 df-frlm 21153 df-nm 23938 df-tng 23940 df-tcph 24533 df-rrx 24749 df-ehl 24750 |
This theorem is referenced by: inlinecirc02p 46863 |
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