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Mirrors > Home > MPE Home > Th. List > rrxmfval | Structured version Visualization version GIF version |
Description: The value of the Euclidean metric. Compare with rrnval 36289. (Contributed by Thierry Arnoux, 30-Jun-2019.) |
Ref | Expression |
---|---|
rrxmval.1 | β’ π = {β β (β βm πΌ) β£ β finSupp 0} |
rrxmval.d | β’ π· = (distβ(β^βπΌ)) |
Ref | Expression |
---|---|
rrxmfval | β’ (πΌ β π β π· = (π β π, π β π β¦ (ββΞ£π β ((π supp 0) βͺ (π supp 0))(((πβπ) β (πβπ))β2)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . . . 5 β’ (π β (Baseβ(β^βπΌ)), π β (Baseβ(β^βπΌ)) β¦ (ββ(βfld Ξ£g (π₯ β πΌ β¦ (((πβπ₯) β (πβπ₯))β2))))) = (π β (Baseβ(β^βπΌ)), π β (Baseβ(β^βπΌ)) β¦ (ββ(βfld Ξ£g (π₯ β πΌ β¦ (((πβπ₯) β (πβπ₯))β2))))) | |
2 | fvex 6856 | . . . . 5 β’ (ββ(βfld Ξ£g (π₯ β πΌ β¦ (((πβπ₯) β (πβπ₯))β2)))) β V | |
3 | 1, 2 | fnmpoi 8003 | . . . 4 β’ (π β (Baseβ(β^βπΌ)), π β (Baseβ(β^βπΌ)) β¦ (ββ(βfld Ξ£g (π₯ β πΌ β¦ (((πβπ₯) β (πβπ₯))β2))))) Fn ((Baseβ(β^βπΌ)) Γ (Baseβ(β^βπΌ))) |
4 | rrxmval.d | . . . . . 6 β’ π· = (distβ(β^βπΌ)) | |
5 | eqid 2737 | . . . . . . 7 β’ (β^βπΌ) = (β^βπΌ) | |
6 | eqid 2737 | . . . . . . 7 β’ (Baseβ(β^βπΌ)) = (Baseβ(β^βπΌ)) | |
7 | 5, 6 | rrxds 24760 | . . . . . 6 β’ (πΌ β π β (π β (Baseβ(β^βπΌ)), π β (Baseβ(β^βπΌ)) β¦ (ββ(βfld Ξ£g (π₯ β πΌ β¦ (((πβπ₯) β (πβπ₯))β2))))) = (distβ(β^βπΌ))) |
8 | 4, 7 | eqtr4id 2796 | . . . . 5 β’ (πΌ β π β π· = (π β (Baseβ(β^βπΌ)), π β (Baseβ(β^βπΌ)) β¦ (ββ(βfld Ξ£g (π₯ β πΌ β¦ (((πβπ₯) β (πβπ₯))β2)))))) |
9 | rrxmval.1 | . . . . . . 7 β’ π = {β β (β βm πΌ) β£ β finSupp 0} | |
10 | 5, 6 | rrxbase 24755 | . . . . . . 7 β’ (πΌ β π β (Baseβ(β^βπΌ)) = {β β (β βm πΌ) β£ β finSupp 0}) |
11 | 9, 10 | eqtr4id 2796 | . . . . . 6 β’ (πΌ β π β π = (Baseβ(β^βπΌ))) |
12 | 11 | sqxpeqd 5666 | . . . . 5 β’ (πΌ β π β (π Γ π) = ((Baseβ(β^βπΌ)) Γ (Baseβ(β^βπΌ)))) |
13 | 8, 12 | fneq12d 6598 | . . . 4 β’ (πΌ β π β (π· Fn (π Γ π) β (π β (Baseβ(β^βπΌ)), π β (Baseβ(β^βπΌ)) β¦ (ββ(βfld Ξ£g (π₯ β πΌ β¦ (((πβπ₯) β (πβπ₯))β2))))) Fn ((Baseβ(β^βπΌ)) Γ (Baseβ(β^βπΌ))))) |
14 | 3, 13 | mpbiri 258 | . . 3 β’ (πΌ β π β π· Fn (π Γ π)) |
15 | fnov 7488 | . . 3 β’ (π· Fn (π Γ π) β π· = (π β π, π β π β¦ (ππ·π))) | |
16 | 14, 15 | sylib 217 | . 2 β’ (πΌ β π β π· = (π β π, π β π β¦ (ππ·π))) |
17 | 9, 4 | rrxmval 24772 | . . 3 β’ ((πΌ β π β§ π β π β§ π β π) β (ππ·π) = (ββΞ£π β ((π supp 0) βͺ (π supp 0))(((πβπ) β (πβπ))β2))) |
18 | 17 | mpoeq3dva 7435 | . 2 β’ (πΌ β π β (π β π, π β π β¦ (ππ·π)) = (π β π, π β π β¦ (ββΞ£π β ((π supp 0) βͺ (π supp 0))(((πβπ) β (πβπ))β2)))) |
19 | 16, 18 | eqtrd 2777 | 1 β’ (πΌ β π β π· = (π β π, π β π β¦ (ββΞ£π β ((π supp 0) βͺ (π supp 0))(((πβπ) β (πβπ))β2)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 {crab 3408 βͺ cun 3909 class class class wbr 5106 β¦ cmpt 5189 Γ cxp 5632 Fn wfn 6492 βcfv 6497 (class class class)co 7358 β cmpo 7360 supp csupp 8093 βm cmap 8766 finSupp cfsupp 9306 βcr 11051 0cc0 11052 β cmin 11386 2c2 12209 βcexp 13968 βcsqrt 15119 Ξ£csu 15571 Basecbs 17084 distcds 17143 Ξ£g cgsu 17323 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: rrxmet 24775 |
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