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| Mirrors > Home > MPE Home > Th. List > ehl2eudisval | Structured version Visualization version GIF version | ||
| Description: The value of the Euclidean distance function in a real Euclidean space of dimension 2. (Contributed by AV, 16-Jan-2023.) |
| Ref | Expression |
|---|---|
| ehl2eudis.e | β’ πΈ = (πΌhilβ2) |
| ehl2eudis.x | β’ π = (β βm {1, 2}) |
| ehl2eudis.d | β’ π· = (distβπΈ) |
| Ref | Expression |
|---|---|
| ehl2eudisval | β’ ((πΉ β π β§ πΊ β π) β (πΉπ·πΊ) = (ββ((((πΉβ1) β (πΊβ1))β2) + (((πΉβ2) β (πΊβ2))β2)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ehl2eudis.e | . . . 4 β’ πΈ = (πΌhilβ2) | |
| 2 | ehl2eudis.x | . . . 4 β’ π = (β βm {1, 2}) | |
| 3 | ehl2eudis.d | . . . 4 β’ π· = (distβπΈ) | |
| 4 | 1, 2, 3 | ehl2eudis 25171 | . . 3 β’ π· = (π β π, π β π β¦ (ββ((((πβ1) β (πβ1))β2) + (((πβ2) β (πβ2))β2)))) |
| 5 | 4 | oveqi 7425 | . 2 β’ (πΉπ·πΊ) = (πΉ(π β π, π β π β¦ (ββ((((πβ1) β (πβ1))β2) + (((πβ2) β (πβ2))β2))))πΊ) |
| 6 | eqidd 2732 | . . 3 β’ ((πΉ β π β§ πΊ β π) β (π β π, π β π β¦ (ββ((((πβ1) β (πβ1))β2) + (((πβ2) β (πβ2))β2)))) = (π β π, π β π β¦ (ββ((((πβ1) β (πβ1))β2) + (((πβ2) β (πβ2))β2))))) | |
| 7 | fveq1 6890 | . . . . . . . 8 β’ (π = πΉ β (πβ1) = (πΉβ1)) | |
| 8 | fveq1 6890 | . . . . . . . 8 β’ (π = πΊ β (πβ1) = (πΊβ1)) | |
| 9 | 7, 8 | oveqan12d 7431 | . . . . . . 7 β’ ((π = πΉ β§ π = πΊ) β ((πβ1) β (πβ1)) = ((πΉβ1) β (πΊβ1))) |
| 10 | 9 | oveq1d 7427 | . . . . . 6 β’ ((π = πΉ β§ π = πΊ) β (((πβ1) β (πβ1))β2) = (((πΉβ1) β (πΊβ1))β2)) |
| 11 | fveq1 6890 | . . . . . . . 8 β’ (π = πΉ β (πβ2) = (πΉβ2)) | |
| 12 | fveq1 6890 | . . . . . . . 8 β’ (π = πΊ β (πβ2) = (πΊβ2)) | |
| 13 | 11, 12 | oveqan12d 7431 | . . . . . . 7 β’ ((π = πΉ β§ π = πΊ) β ((πβ2) β (πβ2)) = ((πΉβ2) β (πΊβ2))) |
| 14 | 13 | oveq1d 7427 | . . . . . 6 β’ ((π = πΉ β§ π = πΊ) β (((πβ2) β (πβ2))β2) = (((πΉβ2) β (πΊβ2))β2)) |
| 15 | 10, 14 | oveq12d 7430 | . . . . 5 β’ ((π = πΉ β§ π = πΊ) β ((((πβ1) β (πβ1))β2) + (((πβ2) β (πβ2))β2)) = ((((πΉβ1) β (πΊβ1))β2) + (((πΉβ2) β (πΊβ2))β2))) |
| 16 | 15 | fveq2d 6895 | . . . 4 β’ ((π = πΉ β§ π = πΊ) β (ββ((((πβ1) β (πβ1))β2) + (((πβ2) β (πβ2))β2))) = (ββ((((πΉβ1) β (πΊβ1))β2) + (((πΉβ2) β (πΊβ2))β2)))) |
| 17 | 16 | adantl 481 | . . 3 β’ (((πΉ β π β§ πΊ β π) β§ (π = πΉ β§ π = πΊ)) β (ββ((((πβ1) β (πβ1))β2) + (((πβ2) β (πβ2))β2))) = (ββ((((πΉβ1) β (πΊβ1))β2) + (((πΉβ2) β (πΊβ2))β2)))) |
| 18 | simpl 482 | . . 3 β’ ((πΉ β π β§ πΊ β π) β πΉ β π) | |
| 19 | simpr 484 | . . 3 β’ ((πΉ β π β§ πΊ β π) β πΊ β π) | |
| 20 | fvexd 6906 | . . 3 β’ ((πΉ β π β§ πΊ β π) β (ββ((((πΉβ1) β (πΊβ1))β2) + (((πΉβ2) β (πΊβ2))β2))) β V) | |
| 21 | 6, 17, 18, 19, 20 | ovmpod 7563 | . 2 β’ ((πΉ β π β§ πΊ β π) β (πΉ(π β π, π β π β¦ (ββ((((πβ1) β (πβ1))β2) + (((πβ2) β (πβ2))β2))))πΊ) = (ββ((((πΉβ1) β (πΊβ1))β2) + (((πΉβ2) β (πΊβ2))β2)))) |
| 22 | 5, 21 | eqtrid 2783 | 1 β’ ((πΉ β π β§ πΊ β π) β (πΉπ·πΊ) = (ββ((((πΉβ1) β (πΊβ1))β2) + (((πΉβ2) β (πΊβ2))β2)))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 = wceq 1540 β wcel 2105 Vcvv 3473 {cpr 4630 βcfv 6543 (class class class)co 7412 β cmpo 7414 βm cmap 8824 βcr 11113 1c1 11115 + caddc 11117 β cmin 11449 2c2 12272 βcexp 14032 βcsqrt 15185 distcds 17211 πΌhilcehl 25133 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9640 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 ax-addf 11193 ax-mulf 11194 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 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 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8151 df-tpos 8215 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-map 8826 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9366 df-sup 9441 df-oi 9509 df-card 9938 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-rp 12980 df-fz 13490 df-fzo 13633 df-seq 13972 df-exp 14033 df-hash 14296 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-clim 15437 df-sum 15638 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-0g 17392 df-gsum 17393 df-prds 17398 df-pws 17400 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-mhm 18706 df-grp 18859 df-minusg 18860 df-sbg 18861 df-subg 19040 df-ghm 19129 df-cntz 19223 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-cring 20131 df-oppr 20226 df-dvdsr 20249 df-unit 20250 df-invr 20280 df-dvr 20293 df-rhm 20364 df-subrng 20435 df-subrg 20460 df-drng 20503 df-field 20504 df-staf 20597 df-srng 20598 df-lmod 20617 df-lss 20688 df-sra 20931 df-rgmod 20932 df-cnfld 21146 df-refld 21378 df-dsmm 21507 df-frlm 21522 df-nm 24312 df-tng 24314 df-tcph 24918 df-rrx 25134 df-ehl 25135 |
| This theorem is referenced by: ehl2eudisval0 47499 2sphere 47523 |
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