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Mirrors > Home > MPE Home > Th. List > rrxdsfi | Structured version Visualization version GIF version |
Description: The distance over generalized Euclidean spaces. Finite dimensional case. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Ref | Expression |
---|---|
rrxdsfi.h | β’ π» = (β^βπΌ) |
rrxdsfi.b | β’ π΅ = (β βm πΌ) |
Ref | Expression |
---|---|
rrxdsfi | β’ (πΌ β Fin β (distβπ») = (π β π΅, π β π΅ β¦ (ββΞ£π β πΌ (((πβπ) β (πβπ))β2)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrxdsfi.b | . . . 4 β’ π΅ = (β βm πΌ) | |
2 | id 22 | . . . . 5 β’ (πΌ β Fin β πΌ β Fin) | |
3 | rrxdsfi.h | . . . . 5 β’ π» = (β^βπΌ) | |
4 | eqid 2737 | . . . . 5 β’ (Baseβπ») = (Baseβπ») | |
5 | 2, 3, 4 | rrxbasefi 24777 | . . . 4 β’ (πΌ β Fin β (Baseβπ») = (β βm πΌ)) |
6 | 1, 5 | eqtr4id 2796 | . . 3 β’ (πΌ β Fin β π΅ = (Baseβπ»)) |
7 | 6 | adantr 482 | . . 3 β’ ((πΌ β Fin β§ π β π΅) β π΅ = (Baseβπ»)) |
8 | df-refld 21012 | . . . . . . 7 β’ βfld = (βfld βΎs β) | |
9 | 8 | oveq1i 7368 | . . . . . 6 β’ (βfld Ξ£g (π β πΌ β¦ (((πβπ) β (πβπ))β2))) = ((βfld βΎs β) Ξ£g (π β πΌ β¦ (((πβπ) β (πβπ))β2))) |
10 | simp1 1137 | . . . . . . 7 β’ ((πΌ β Fin β§ π β π΅ β§ π β π΅) β πΌ β Fin) | |
11 | simpr 486 | . . . . . . . . . . . . 13 β’ ((πΌ β Fin β§ π β π΅) β π β π΅) | |
12 | 11, 1 | eleqtrdi 2848 | . . . . . . . . . . . 12 β’ ((πΌ β Fin β§ π β π΅) β π β (β βm πΌ)) |
13 | 12 | 3adant3 1133 | . . . . . . . . . . 11 β’ ((πΌ β Fin β§ π β π΅ β§ π β π΅) β π β (β βm πΌ)) |
14 | elmapi 8788 | . . . . . . . . . . 11 β’ (π β (β βm πΌ) β π:πΌβΆβ) | |
15 | 13, 14 | syl 17 | . . . . . . . . . 10 β’ ((πΌ β Fin β§ π β π΅ β§ π β π΅) β π:πΌβΆβ) |
16 | 15 | ffvelcdmda 7036 | . . . . . . . . 9 β’ (((πΌ β Fin β§ π β π΅ β§ π β π΅) β§ π β πΌ) β (πβπ) β β) |
17 | simpr 486 | . . . . . . . . . . . . 13 β’ ((πΌ β Fin β§ π β π΅) β π β π΅) | |
18 | 17, 1 | eleqtrdi 2848 | . . . . . . . . . . . 12 β’ ((πΌ β Fin β§ π β π΅) β π β (β βm πΌ)) |
19 | 18 | 3adant2 1132 | . . . . . . . . . . 11 β’ ((πΌ β Fin β§ π β π΅ β§ π β π΅) β π β (β βm πΌ)) |
20 | elmapi 8788 | . . . . . . . . . . 11 β’ (π β (β βm πΌ) β π:πΌβΆβ) | |
21 | 19, 20 | syl 17 | . . . . . . . . . 10 β’ ((πΌ β Fin β§ π β π΅ β§ π β π΅) β π:πΌβΆβ) |
22 | 21 | ffvelcdmda 7036 | . . . . . . . . 9 β’ (((πΌ β Fin β§ π β π΅ β§ π β π΅) β§ π β πΌ) β (πβπ) β β) |
23 | 16, 22 | resubcld 11584 | . . . . . . . 8 β’ (((πΌ β Fin β§ π β π΅ β§ π β π΅) β§ π β πΌ) β ((πβπ) β (πβπ)) β β) |
24 | 23 | resqcld 14031 | . . . . . . 7 β’ (((πΌ β Fin β§ π β π΅ β§ π β π΅) β§ π β πΌ) β (((πβπ) β (πβπ))β2) β β) |
25 | 10, 24 | regsumfsum 20868 | . . . . . 6 β’ ((πΌ β Fin β§ π β π΅ β§ π β π΅) β ((βfld βΎs β) Ξ£g (π β πΌ β¦ (((πβπ) β (πβπ))β2))) = Ξ£π β πΌ (((πβπ) β (πβπ))β2)) |
26 | 9, 25 | eqtr2id 2790 | . . . . 5 β’ ((πΌ β Fin β§ π β π΅ β§ π β π΅) β Ξ£π β πΌ (((πβπ) β (πβπ))β2) = (βfld Ξ£g (π β πΌ β¦ (((πβπ) β (πβπ))β2)))) |
27 | 26 | fveq2d 6847 | . . . 4 β’ ((πΌ β Fin β§ π β π΅ β§ π β π΅) β (ββΞ£π β πΌ (((πβπ) β (πβπ))β2)) = (ββ(βfld Ξ£g (π β πΌ β¦ (((πβπ) β (πβπ))β2))))) |
28 | 27 | 3expb 1121 | . . 3 β’ ((πΌ β Fin β§ (π β π΅ β§ π β π΅)) β (ββΞ£π β πΌ (((πβπ) β (πβπ))β2)) = (ββ(βfld Ξ£g (π β πΌ β¦ (((πβπ) β (πβπ))β2))))) |
29 | 6, 7, 28 | mpoeq123dva 7432 | . 2 β’ (πΌ β Fin β (π β π΅, π β π΅ β¦ (ββΞ£π β πΌ (((πβπ) β (πβπ))β2))) = (π β (Baseβπ»), π β (Baseβπ») β¦ (ββ(βfld Ξ£g (π β πΌ β¦ (((πβπ) β (πβπ))β2)))))) |
30 | 3, 4 | rrxds 24760 | . 2 β’ (πΌ β Fin β (π β (Baseβπ»), π β (Baseβπ») β¦ (ββ(βfld Ξ£g (π β πΌ β¦ (((πβπ) β (πβπ))β2))))) = (distβπ»)) |
31 | 29, 30 | eqtr2d 2778 | 1 β’ (πΌ β Fin β (distβπ») = (π β π΅, π β π΅ β¦ (ββΞ£π β πΌ (((πβπ) β (πβπ))β2)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β¦ cmpt 5189 βΆwf 6493 βcfv 6497 (class class class)co 7358 β cmpo 7360 βm cmap 8766 Fincfn 8884 βcr 11051 β cmin 11386 2c2 12209 βcexp 13968 βcsqrt 15119 Ξ£csu 15571 Basecbs 17084 βΎs cress 17113 distcds 17143 Ξ£g cgsu 17323 βfldccnfld 20799 βfldcrefld 21011 β^crrx 24750 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9578 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 ax-pre-sup 11130 ax-addf 11131 ax-mulf 11132 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-tpos 8158 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8649 df-map 8768 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9307 df-sup 9379 df-oi 9447 df-card 9876 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-div 11814 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-5 12220 df-6 12221 df-7 12222 df-8 12223 df-9 12224 df-n0 12415 df-z 12501 df-dec 12620 df-uz 12765 df-rp 12917 df-fz 13426 df-fzo 13569 df-seq 13908 df-exp 13969 df-hash 14232 df-cj 14985 df-re 14986 df-im 14987 df-sqrt 15121 df-abs 15122 df-clim 15371 df-sum 15572 df-struct 17020 df-sets 17037 df-slot 17055 df-ndx 17067 df-base 17085 df-ress 17114 df-plusg 17147 df-mulr 17148 df-starv 17149 df-sca 17150 df-vsca 17151 df-ip 17152 df-tset 17153 df-ple 17154 df-ds 17156 df-unif 17157 df-hom 17158 df-cco 17159 df-0g 17324 df-gsum 17325 df-prds 17330 df-pws 17332 df-mgm 18498 df-sgrp 18547 df-mnd 18558 df-mhm 18602 df-grp 18752 df-minusg 18753 df-sbg 18754 df-subg 18926 df-ghm 19007 df-cntz 19098 df-cmn 19565 df-abl 19566 df-mgp 19898 df-ur 19915 df-ring 19967 df-cring 19968 df-oppr 20050 df-dvdsr 20071 df-unit 20072 df-invr 20102 df-dvr 20113 df-rnghom 20147 df-drng 20188 df-field 20189 df-subrg 20223 df-staf 20307 df-srng 20308 df-lmod 20327 df-lss 20396 df-sra 20636 df-rgmod 20637 df-cnfld 20800 df-refld 21012 df-dsmm 21141 df-frlm 21156 df-nm 23941 df-tng 23943 df-tcph 24536 df-rrx 24752 |
This theorem is referenced by: rrxdsfival 24780 ehleudis 24785 rrndistlt 44538 qndenserrnopnlem 44545 rrndsmet 44550 ioorrnopnlem 44552 hoiqssbllem2 44871 |
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