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Mathbox for Alexander van der Vekens |
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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 | 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 |
---|---|
ehl2eudisval0 | β’ (πΉ β π β (πΉπ· 0 ) = (ββ(((πΉβ1)β2) + ((πΉβ2)β2)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prex 5428 | . . . 4 β’ {1, 2} β V | |
2 | ehl2eudisval0.0 | . . . . 5 β’ 0 = ({1, 2} Γ {0}) | |
3 | ehl2eudisval0.x | . . . . 5 β’ π = (β βm {1, 2}) | |
4 | 2, 3 | rrx0el 25313 | . . . 4 β’ ({1, 2} β V β 0 β π) |
5 | 1, 4 | mp1i 13 | . . 3 β’ (πΉ β π β 0 β π) |
6 | ehl2eudisval0.e | . . . 4 β’ πΈ = (πΌhilβ2) | |
7 | ehl2eudisval0.d | . . . 4 β’ π· = (distβπΈ) | |
8 | 6, 3, 7 | ehl2eudisval 25338 | . . 3 β’ ((πΉ β π β§ 0 β π) β (πΉπ· 0 ) = (ββ((((πΉβ1) β ( 0 β1))β2) + (((πΉβ2) β ( 0 β2))β2)))) |
9 | 5, 8 | mpdan 686 | . 2 β’ (πΉ β π β (πΉπ· 0 ) = (ββ((((πΉβ1) β ( 0 β1))β2) + (((πΉβ2) β ( 0 β2))β2)))) |
10 | 1ex 11232 | . . . . . . . . . . . 12 β’ 1 β V | |
11 | 2ex 12311 | . . . . . . . . . . . 12 β’ 2 β V | |
12 | c0ex 11230 | . . . . . . . . . . . 12 β’ 0 β V | |
13 | xpprsng 7143 | . . . . . . . . . . . 12 β’ ((1 β V β§ 2 β V β§ 0 β V) β ({1, 2} Γ {0}) = {β¨1, 0β©, β¨2, 0β©}) | |
14 | 10, 11, 12, 13 | mp3an 1458 | . . . . . . . . . . 11 β’ ({1, 2} Γ {0}) = {β¨1, 0β©, β¨2, 0β©} |
15 | 2, 14 | eqtri 2755 | . . . . . . . . . 10 β’ 0 = {β¨1, 0β©, β¨2, 0β©} |
16 | 15 | fveq1i 6892 | . . . . . . . . 9 β’ ( 0 β1) = ({β¨1, 0β©, β¨2, 0β©}β1) |
17 | 1ne2 12442 | . . . . . . . . . 10 β’ 1 β 2 | |
18 | 10, 12 | fvpr1 7196 | . . . . . . . . . 10 β’ (1 β 2 β ({β¨1, 0β©, β¨2, 0β©}β1) = 0) |
19 | 17, 18 | ax-mp 5 | . . . . . . . . 9 β’ ({β¨1, 0β©, β¨2, 0β©}β1) = 0 |
20 | 16, 19 | eqtri 2755 | . . . . . . . 8 β’ ( 0 β1) = 0 |
21 | 20 | a1i 11 | . . . . . . 7 β’ (πΉ β π β ( 0 β1) = 0) |
22 | 21 | oveq2d 7430 | . . . . . 6 β’ (πΉ β π β ((πΉβ1) β ( 0 β1)) = ((πΉβ1) β 0)) |
23 | eqid 2727 | . . . . . . . . 9 β’ {1, 2} = {1, 2} | |
24 | 23, 3 | rrx2pxel 47707 | . . . . . . . 8 β’ (πΉ β π β (πΉβ1) β β) |
25 | 24 | recnd 11264 | . . . . . . 7 β’ (πΉ β π β (πΉβ1) β β) |
26 | 25 | subid1d 11582 | . . . . . 6 β’ (πΉ β π β ((πΉβ1) β 0) = (πΉβ1)) |
27 | 22, 26 | eqtrd 2767 | . . . . 5 β’ (πΉ β π β ((πΉβ1) β ( 0 β1)) = (πΉβ1)) |
28 | 27 | oveq1d 7429 | . . . 4 β’ (πΉ β π β (((πΉβ1) β ( 0 β1))β2) = ((πΉβ1)β2)) |
29 | 15 | fveq1i 6892 | . . . . . . . 8 β’ ( 0 β2) = ({β¨1, 0β©, β¨2, 0β©}β2) |
30 | 11, 12 | fvpr2 7198 | . . . . . . . . 9 β’ (1 β 2 β ({β¨1, 0β©, β¨2, 0β©}β2) = 0) |
31 | 17, 30 | mp1i 13 | . . . . . . . 8 β’ (πΉ β π β ({β¨1, 0β©, β¨2, 0β©}β2) = 0) |
32 | 29, 31 | eqtrid 2779 | . . . . . . 7 β’ (πΉ β π β ( 0 β2) = 0) |
33 | 32 | oveq2d 7430 | . . . . . 6 β’ (πΉ β π β ((πΉβ2) β ( 0 β2)) = ((πΉβ2) β 0)) |
34 | 23, 3 | rrx2pyel 47708 | . . . . . . . 8 β’ (πΉ β π β (πΉβ2) β β) |
35 | 34 | recnd 11264 | . . . . . . 7 β’ (πΉ β π β (πΉβ2) β β) |
36 | 35 | subid1d 11582 | . . . . . 6 β’ (πΉ β π β ((πΉβ2) β 0) = (πΉβ2)) |
37 | 33, 36 | eqtrd 2767 | . . . . 5 β’ (πΉ β π β ((πΉβ2) β ( 0 β2)) = (πΉβ2)) |
38 | 37 | oveq1d 7429 | . . . 4 β’ (πΉ β π β (((πΉβ2) β ( 0 β2))β2) = ((πΉβ2)β2)) |
39 | 28, 38 | oveq12d 7432 | . . 3 β’ (πΉ β π β ((((πΉβ1) β ( 0 β1))β2) + (((πΉβ2) β ( 0 β2))β2)) = (((πΉβ1)β2) + ((πΉβ2)β2))) |
40 | 39 | fveq2d 6895 | . 2 β’ (πΉ β π β (ββ((((πΉβ1) β ( 0 β1))β2) + (((πΉβ2) β ( 0 β2))β2))) = (ββ(((πΉβ1)β2) + ((πΉβ2)β2)))) |
41 | 9, 40 | eqtrd 2767 | 1 β’ (πΉ β π β (πΉπ· 0 ) = (ββ(((πΉβ1)β2) + ((πΉβ2)β2)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 β wne 2935 Vcvv 3469 {csn 4624 {cpr 4626 β¨cop 4630 Γ cxp 5670 βcfv 6542 (class class class)co 7414 βm cmap 8836 βcr 11129 0cc0 11130 1c1 11131 + caddc 11133 β cmin 11466 2c2 12289 βcexp 14050 βcsqrt 15204 distcds 17233 πΌhilcehl 25299 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9656 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208 ax-addf 11209 ax-mulf 11210 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 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 5143 df-opab 5205 df-mpt 5226 df-tr 5260 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8838 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9378 df-sup 9457 df-oi 9525 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-z 12581 df-dec 12700 df-uz 12845 df-rp 12999 df-fz 13509 df-fzo 13652 df-seq 13991 df-exp 14051 df-hash 14314 df-cj 15070 df-re 15071 df-im 15072 df-sqrt 15206 df-abs 15207 df-clim 15456 df-sum 15657 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-starv 17239 df-sca 17240 df-vsca 17241 df-ip 17242 df-tset 17243 df-ple 17244 df-ds 17246 df-unif 17247 df-hom 17248 df-cco 17249 df-0g 17414 df-gsum 17415 df-prds 17420 df-pws 17422 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-mhm 18731 df-grp 18884 df-minusg 18885 df-sbg 18886 df-subg 19069 df-ghm 19159 df-cntz 19259 df-cmn 19728 df-abl 19729 df-mgp 20066 df-rng 20084 df-ur 20113 df-ring 20166 df-cring 20167 df-oppr 20262 df-dvdsr 20285 df-unit 20286 df-invr 20316 df-dvr 20329 df-rhm 20400 df-subrng 20472 df-subrg 20497 df-drng 20615 df-field 20616 df-staf 20714 df-srng 20715 df-lmod 20734 df-lss 20805 df-sra 21047 df-rgmod 21048 df-cnfld 21267 df-refld 21524 df-dsmm 21653 df-frlm 21668 df-nm 24478 df-tng 24480 df-tcph 25084 df-rrx 25300 df-ehl 25301 |
This theorem is referenced by: ehl2eudis0lt 47722 itscnhlinecirc02plem3 47780 |
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