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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 47910 | . . . 4 β’ (πΉ β π β (πΉπ· 0 ) = (ββ(((πΉβ1)β2) + ((πΉβ2)β2)))) |
6 | 5 | adantr 479 | . . 3 β’ ((πΉ β π β§ π β β+) β (πΉπ· 0 ) = (ββ(((πΉβ1)β2) + ((πΉβ2)β2)))) |
7 | 6 | breq1d 5153 | . 2 β’ ((πΉ β π β§ π β β+) β ((πΉπ· 0 ) < π β (ββ(((πΉβ1)β2) + ((πΉβ2)β2))) < π )) |
8 | eqid 2725 | . . . . . . 7 β’ {1, 2} = {1, 2} | |
9 | 8, 2 | rrx2pxel 47896 | . . . . . 6 β’ (πΉ β π β (πΉβ1) β β) |
10 | 8, 2 | rrx2pyel 47897 | . . . . . 6 β’ (πΉ β π β (πΉβ2) β β) |
11 | eqid 2725 | . . . . . . 7 β’ (((πΉβ1)β2) + ((πΉβ2)β2)) = (((πΉβ1)β2) + ((πΉβ2)β2)) | |
12 | 11 | resum2sqcl 47891 | . . . . . 6 β’ (((πΉβ1) β β β§ (πΉβ2) β β) β (((πΉβ1)β2) + ((πΉβ2)β2)) β β) |
13 | 9, 10, 12 | syl2anc 582 | . . . . 5 β’ (πΉ β π β (((πΉβ1)β2) + ((πΉβ2)β2)) β β) |
14 | resqcl 14120 | . . . . . . . 8 β’ ((πΉβ1) β β β ((πΉβ1)β2) β β) | |
15 | resqcl 14120 | . . . . . . . 8 β’ ((πΉβ2) β β β ((πΉβ2)β2) β β) | |
16 | 14, 15 | anim12i 611 | . . . . . . 7 β’ (((πΉβ1) β β β§ (πΉβ2) β β) β (((πΉβ1)β2) β β β§ ((πΉβ2)β2) β β)) |
17 | sqge0 14132 | . . . . . . . 8 β’ ((πΉβ1) β β β 0 β€ ((πΉβ1)β2)) | |
18 | sqge0 14132 | . . . . . . . 8 β’ ((πΉβ2) β β β 0 β€ ((πΉβ2)β2)) | |
19 | 17, 18 | anim12i 611 | . . . . . . 7 β’ (((πΉβ1) β β β§ (πΉβ2) β β) β (0 β€ ((πΉβ1)β2) β§ 0 β€ ((πΉβ2)β2))) |
20 | addge0 11733 | . . . . . . 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 582 | . . . . . 6 β’ (((πΉβ1) β β β§ (πΉβ2) β β) β 0 β€ (((πΉβ1)β2) + ((πΉβ2)β2))) |
22 | 9, 10, 21 | syl2anc 582 | . . . . 5 β’ (πΉ β π β 0 β€ (((πΉβ1)β2) + ((πΉβ2)β2))) |
23 | 13, 22 | resqrtcld 15396 | . . . 4 β’ (πΉ β π β (ββ(((πΉβ1)β2) + ((πΉβ2)β2))) β β) |
24 | 13, 22 | sqrtge0d 15399 | . . . 4 β’ (πΉ β π β 0 β€ (ββ(((πΉβ1)β2) + ((πΉβ2)β2)))) |
25 | 23, 24 | jca 510 | . . 3 β’ (πΉ β π β ((ββ(((πΉβ1)β2) + ((πΉβ2)β2))) β β β§ 0 β€ (ββ(((πΉβ1)β2) + ((πΉβ2)β2))))) |
26 | rprege0 13021 | . . 3 β’ (π β β+ β (π β β β§ 0 β€ π )) | |
27 | lt2sq 14129 | . . 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 594 | . 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 510 | . . . . 5 β’ (πΉ β π β ((((πΉβ1)β2) + ((πΉβ2)β2)) β β β§ 0 β€ (((πΉβ1)β2) + ((πΉβ2)β2)))) |
30 | 29 | adantr 479 | . . . 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 15234 | . . . 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 5153 | . 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 304 | 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 394 = wceq 1533 β wcel 2098 {csn 4624 {cpr 4626 class class class wbr 5143 Γ cxp 5670 βcfv 6543 (class class class)co 7416 βm cmap 8843 βcr 11137 0cc0 11138 1c1 11139 + caddc 11141 < clt 11278 β€ cle 11279 2c2 12297 β+crp 13006 βcexp 14058 βcsqrt 15212 distcds 17241 πΌhilcehl 25330 |
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 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217 ax-mulf 11218 |
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 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7991 df-2nd 7992 df-supp 8164 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-sup 9465 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-rp 13007 df-fz 13517 df-fzo 13660 df-seq 13999 df-exp 14059 df-hash 14322 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-clim 15464 df-sum 15665 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-hom 17256 df-cco 17257 df-0g 17422 df-gsum 17423 df-prds 17428 df-pws 17430 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-grp 18897 df-minusg 18898 df-sbg 18899 df-subg 19082 df-ghm 19172 df-cntz 19272 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-ring 20179 df-cring 20180 df-oppr 20277 df-dvdsr 20300 df-unit 20301 df-invr 20331 df-dvr 20344 df-rhm 20415 df-subrng 20487 df-subrg 20512 df-drng 20630 df-field 20631 df-staf 20729 df-srng 20730 df-lmod 20749 df-lss 20820 df-sra 21062 df-rgmod 21063 df-cnfld 21284 df-refld 21541 df-dsmm 21670 df-frlm 21685 df-nm 24509 df-tng 24511 df-tcph 25115 df-rrx 25331 df-ehl 25332 |
This theorem is referenced by: inlinecirc02p 47972 |
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