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Mathbox for Alexander van der Vekens |
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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 | β’ πΈ = (πΌhilβ2) |
ehl2eudisval0.x | β’ π = (β βm {1, 2}) |
ehl2eudisval0.d | β’ π· = (distβπΈ) |
ehl2eudisval0.0 | β’ 0 = ({1, 2} Γ {0}) |
Ref | Expression |
---|---|
ehl2eudis0lt | β’ ((πΉ β π β§ π β β+) β ((πΉπ· 0 ) < π β (((πΉβ1)β2) + ((πΉβ2)β2)) < (π β2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ehl2eudisval0.e | . . . . 5 β’ πΈ = (πΌhilβ2) | |
2 | ehl2eudisval0.x | . . . . 5 β’ π = (β βm {1, 2}) | |
3 | ehl2eudisval0.d | . . . . 5 β’ π· = (distβπΈ) | |
4 | ehl2eudisval0.0 | . . . . 5 β’ 0 = ({1, 2} Γ {0}) | |
5 | 1, 2, 3, 4 | ehl2eudisval0 46901 | . . . 4 β’ (πΉ β π β (πΉπ· 0 ) = (ββ(((πΉβ1)β2) + ((πΉβ2)β2)))) |
6 | 5 | adantr 482 | . . 3 β’ ((πΉ β π β§ π β β+) β (πΉπ· 0 ) = (ββ(((πΉβ1)β2) + ((πΉβ2)β2)))) |
7 | 6 | breq1d 5119 | . 2 β’ ((πΉ β π β§ π β β+) β ((πΉπ· 0 ) < π β (ββ(((πΉβ1)β2) + ((πΉβ2)β2))) < π )) |
8 | eqid 2733 | . . . . . . 7 β’ {1, 2} = {1, 2} | |
9 | 8, 2 | rrx2pxel 46887 | . . . . . 6 β’ (πΉ β π β (πΉβ1) β β) |
10 | 8, 2 | rrx2pyel 46888 | . . . . . 6 β’ (πΉ β π β (πΉβ2) β β) |
11 | eqid 2733 | . . . . . . 7 β’ (((πΉβ1)β2) + ((πΉβ2)β2)) = (((πΉβ1)β2) + ((πΉβ2)β2)) | |
12 | 11 | resum2sqcl 46882 | . . . . . 6 β’ (((πΉβ1) β β β§ (πΉβ2) β β) β (((πΉβ1)β2) + ((πΉβ2)β2)) β β) |
13 | 9, 10, 12 | syl2anc 585 | . . . . 5 β’ (πΉ β π β (((πΉβ1)β2) + ((πΉβ2)β2)) β β) |
14 | resqcl 14038 | . . . . . . . 8 β’ ((πΉβ1) β β β ((πΉβ1)β2) β β) | |
15 | resqcl 14038 | . . . . . . . 8 β’ ((πΉβ2) β β β ((πΉβ2)β2) β β) | |
16 | 14, 15 | anim12i 614 | . . . . . . 7 β’ (((πΉβ1) β β β§ (πΉβ2) β β) β (((πΉβ1)β2) β β β§ ((πΉβ2)β2) β β)) |
17 | sqge0 14050 | . . . . . . . 8 β’ ((πΉβ1) β β β 0 β€ ((πΉβ1)β2)) | |
18 | sqge0 14050 | . . . . . . . 8 β’ ((πΉβ2) β β β 0 β€ ((πΉβ2)β2)) | |
19 | 17, 18 | anim12i 614 | . . . . . . 7 β’ (((πΉβ1) β β β§ (πΉβ2) β β) β (0 β€ ((πΉβ1)β2) β§ 0 β€ ((πΉβ2)β2))) |
20 | addge0 11652 | . . . . . . 7 β’ (((((πΉβ1)β2) β β β§ ((πΉβ2)β2) β β) β§ (0 β€ ((πΉβ1)β2) β§ 0 β€ ((πΉβ2)β2))) β 0 β€ (((πΉβ1)β2) + ((πΉβ2)β2))) | |
21 | 16, 19, 20 | syl2anc 585 | . . . . . 6 β’ (((πΉβ1) β β β§ (πΉβ2) β β) β 0 β€ (((πΉβ1)β2) + ((πΉβ2)β2))) |
22 | 9, 10, 21 | syl2anc 585 | . . . . 5 β’ (πΉ β π β 0 β€ (((πΉβ1)β2) + ((πΉβ2)β2))) |
23 | 13, 22 | resqrtcld 15311 | . . . 4 β’ (πΉ β π β (ββ(((πΉβ1)β2) + ((πΉβ2)β2))) β β) |
24 | 13, 22 | sqrtge0d 15314 | . . . 4 β’ (πΉ β π β 0 β€ (ββ(((πΉβ1)β2) + ((πΉβ2)β2)))) |
25 | 23, 24 | jca 513 | . . 3 β’ (πΉ β π β ((ββ(((πΉβ1)β2) + ((πΉβ2)β2))) β β β§ 0 β€ (ββ(((πΉβ1)β2) + ((πΉβ2)β2))))) |
26 | rprege0 12938 | . . 3 β’ (π β β+ β (π β β β§ 0 β€ π )) | |
27 | lt2sq 14047 | . . 3 β’ ((((ββ(((πΉβ1)β2) + ((πΉβ2)β2))) β β β§ 0 β€ (ββ(((πΉβ1)β2) + ((πΉβ2)β2)))) β§ (π β β β§ 0 β€ π )) β ((ββ(((πΉβ1)β2) + ((πΉβ2)β2))) < π β ((ββ(((πΉβ1)β2) + ((πΉβ2)β2)))β2) < (π β2))) | |
28 | 25, 26, 27 | syl2an 597 | . 2 β’ ((πΉ β π β§ π β β+) β ((ββ(((πΉβ1)β2) + ((πΉβ2)β2))) < π β ((ββ(((πΉβ1)β2) + ((πΉβ2)β2)))β2) < (π β2))) |
29 | 13, 22 | jca 513 | . . . . 5 β’ (πΉ β π β ((((πΉβ1)β2) + ((πΉβ2)β2)) β β β§ 0 β€ (((πΉβ1)β2) + ((πΉβ2)β2)))) |
30 | 29 | adantr 482 | . . . 4 β’ ((πΉ β π β§ π β β+) β ((((πΉβ1)β2) + ((πΉβ2)β2)) β β β§ 0 β€ (((πΉβ1)β2) + ((πΉβ2)β2)))) |
31 | resqrtth 15149 | . . . 4 β’ (((((πΉβ1)β2) + ((πΉβ2)β2)) β β β§ 0 β€ (((πΉβ1)β2) + ((πΉβ2)β2))) β ((ββ(((πΉβ1)β2) + ((πΉβ2)β2)))β2) = (((πΉβ1)β2) + ((πΉβ2)β2))) | |
32 | 30, 31 | syl 17 | . . 3 β’ ((πΉ β π β§ π β β+) β ((ββ(((πΉβ1)β2) + ((πΉβ2)β2)))β2) = (((πΉβ1)β2) + ((πΉβ2)β2))) |
33 | 32 | breq1d 5119 | . 2 β’ ((πΉ β π β§ π β β+) β (((ββ(((πΉβ1)β2) + ((πΉβ2)β2)))β2) < (π β2) β (((πΉβ1)β2) + ((πΉβ2)β2)) < (π β2))) |
34 | 7, 28, 33 | 3bitrd 305 | 1 β’ ((πΉ β π β§ π β β+) β ((πΉπ· 0 ) < π β (((πΉβ1)β2) + ((πΉβ2)β2)) < (π β2))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 {csn 4590 {cpr 4592 class class class wbr 5109 Γ cxp 5635 βcfv 6500 (class class class)co 7361 βm cmap 8771 βcr 11058 0cc0 11059 1c1 11060 + caddc 11062 < clt 11197 β€ cle 11198 2c2 12216 β+crp 12923 βcexp 13976 βcsqrt 15127 distcds 17150 πΌhilcehl 24771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-inf2 9585 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 ax-addf 11138 ax-mulf 11139 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-om 7807 df-1st 7925 df-2nd 7926 df-supp 8097 df-tpos 8161 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-map 8773 df-ixp 8842 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fsupp 9312 df-sup 9386 df-oi 9454 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-uz 12772 df-rp 12924 df-fz 13434 df-fzo 13577 df-seq 13916 df-exp 13977 df-hash 14240 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-clim 15379 df-sum 15580 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-starv 17156 df-sca 17157 df-vsca 17158 df-ip 17159 df-tset 17160 df-ple 17161 df-ds 17163 df-unif 17164 df-hom 17165 df-cco 17166 df-0g 17331 df-gsum 17332 df-prds 17337 df-pws 17339 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-mhm 18609 df-grp 18759 df-minusg 18760 df-sbg 18761 df-subg 18933 df-ghm 19014 df-cntz 19105 df-cmn 19572 df-abl 19573 df-mgp 19905 df-ur 19922 df-ring 19974 df-cring 19975 df-oppr 20057 df-dvdsr 20078 df-unit 20079 df-invr 20109 df-dvr 20120 df-rnghom 20156 df-drng 20221 df-field 20222 df-subrg 20262 df-staf 20347 df-srng 20348 df-lmod 20367 df-lss 20437 df-sra 20678 df-rgmod 20679 df-cnfld 20820 df-refld 21032 df-dsmm 21161 df-frlm 21176 df-nm 23961 df-tng 23963 df-tcph 24556 df-rrx 24772 df-ehl 24773 |
This theorem is referenced by: inlinecirc02p 46963 |
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