Theorem rrndistlt 40510

Description: Given two points in the space of n-dimensional real numbers, if every component is closer than E then the distance between the two points is less then ( ( sqr ` n ) x. E ) (Contributed by Glauco Siliprandi, 24-Dec-2020.)

Hypotheses

Ref	Expression
rrndistlt.i	|- ( ph -> I e. Fin )
rrndistlt.z	|- ( ph -> I =/= (/) )
rrndistlt.n	|- N = ( # ` I )
rrndistlt.x	|- ( ph -> X e. ( RR ^m I ) )
rrndistlt.y	|- ( ph -> Y e. ( RR ^m I ) )
rrndistlt.l	|- ( ( ph /\ i e. I ) -> ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) < E )
rrndistlt.e	|- ( ph -> E e. RR+ )
rrndistlt.d	|- D = ( dist ` (ℝ^ ` I ) )

Assertion

Ref	Expression
rrndistlt	|- ( ph -> ( X D Y ) < ( ( sqr ` N ) x. E ) )

Distinct variable groups:   i, E   i, I   i, X   i, Y   ph, i

Allowed substitution hints:   D( i)   N( i)

Proof of Theorem rrndistlt

Dummy variables f g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rrndistlt.i	. . . . 5 |- ( ph -> I e. Fin )
2		rrndistlt.z	. . . . 5 |- ( ph -> I =/= (/) )
3		rrndistlt.x	. . . . . . . . . . 11 |- ( ph -> X e. ( RR ^m I ) )
4		elmapi 7879	. . . . . . . . . . 11 |- ( X e. ( RR ^m I ) -> X : I --> RR )
5	3, 4	syl 17	. . . . . . . . . 10 |- ( ph -> X : I --> RR )
6		ax-resscn 9993	. . . . . . . . . . 11 |- RR C_ CC
7	6	a1i 11	. . . . . . . . . 10 |- ( ph -> RR C_ CC )
8	5, 7	fssd 6057	. . . . . . . . 9 |- ( ph -> X : I --> CC )
9	8	ffvelrnda 6359	. . . . . . . 8 |- ( ( ph /\ i e. I ) -> ( X ` i ) e. CC )
10		rrndistlt.y	. . . . . . . . . . 11 |- ( ph -> Y e. ( RR ^m I ) )
11		elmapi 7879	. . . . . . . . . . 11 |- ( Y e. ( RR ^m I ) -> Y : I --> RR )
12	10, 11	syl 17	. . . . . . . . . 10 |- ( ph -> Y : I --> RR )
13	12, 7	fssd 6057	. . . . . . . . 9 |- ( ph -> Y : I --> CC )
14	13	ffvelrnda 6359	. . . . . . . 8 |- ( ( ph /\ i e. I ) -> ( Y ` i ) e. CC )
15	9, 14	subcld 10392	. . . . . . 7 |- ( ( ph /\ i e. I ) -> ( ( X ` i ) - ( Y ` i ) ) e. CC )
16	15	abscld 14175	. . . . . 6 |- ( ( ph /\ i e. I ) -> ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) e. RR )
17	16	resqcld 13035	. . . . 5 |- ( ( ph /\ i e. I ) -> ( ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) ^ 2 ) e. RR )
18		rrndistlt.e	. . . . . . . 8 |- ( ph -> E e. RR+ )
19	18	rpred 11872	. . . . . . 7 |- ( ph -> E e. RR )
20	19	resqcld 13035	. . . . . 6 |- ( ph -> ( E ^ 2 ) e. RR )
21	20	adantr 481	. . . . 5 |- ( ( ph /\ i e. I ) -> ( E ^ 2 ) e. RR )
22		rrndistlt.l	. . . . . 6 |- ( ( ph /\ i e. I ) -> ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) < E )
23	15	absge0d 14183	. . . . . . 7 |- ( ( ph /\ i e. I ) -> 0 <_ ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) )
24	19	adantr 481	. . . . . . 7 |- ( ( ph /\ i e. I ) -> E e. RR )
25	18	adantr 481	. . . . . . . 8 |- ( ( ph /\ i e. I ) -> E e. RR+ )
26	25	rpge0d 11876	. . . . . . 7 |- ( ( ph /\ i e. I ) -> 0 <_ E )
27		lt2sq 12937	. . . . . . 7 |- ( ( ( ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) e. RR /\ 0 <_ ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) ) /\ ( E e. RR /\ 0 <_ E ) ) -> ( ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) < E <-> ( ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) ^ 2 ) < ( E ^ 2 ) ) )
28	16, 23, 24, 26, 27	syl22anc 1327	. . . . . 6 |- ( ( ph /\ i e. I ) -> ( ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) < E <-> ( ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) ^ 2 ) < ( E ^ 2 ) ) )
29	22, 28	mpbid 222	. . . . 5 |- ( ( ph /\ i e. I ) -> ( ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) ^ 2 ) < ( E ^ 2 ) )
30	1, 2, 17, 21, 29	fsumlt 14532	. . . 4 |- ( ph -> sum_ i e. I ( ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) ^ 2 ) < sum_ i e. I ( E ^ 2 ) )
31	5	ffvelrnda 6359	. . . . . . . . 9 |- ( ( ph /\ i e. I ) -> ( X ` i ) e. RR )
32	12	ffvelrnda 6359	. . . . . . . . 9 |- ( ( ph /\ i e. I ) -> ( Y ` i ) e. RR )
33	31, 32	resubcld 10458	. . . . . . . 8 |- ( ( ph /\ i e. I ) -> ( ( X ` i ) - ( Y ` i ) ) e. RR )
34		absresq 14042	. . . . . . . 8 |- ( ( ( X ` i ) - ( Y ` i ) ) e. RR -> ( ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) ^ 2 ) = ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) )
35	33, 34	syl 17	. . . . . . 7 |- ( ( ph /\ i e. I ) -> ( ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) ^ 2 ) = ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) )
36	35	eqcomd 2628	. . . . . 6 |- ( ( ph /\ i e. I ) -> ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) = ( ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) ^ 2 ) )
37	36	sumeq2dv 14433	. . . . 5 |- ( ph -> sum_ i e. I ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) = sum_ i e. I ( ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) ^ 2 ) )
38	6, 20	sseldi 3601	. . . . . . 7 |- ( ph -> ( E ^ 2 ) e. CC )
39		fsumconst 14522	. . . . . . 7 |- ( ( I e. Fin /\ ( E ^ 2 ) e. CC ) -> sum_ i e. I ( E ^ 2 ) = ( ( # ` I ) x. ( E ^ 2 ) ) )
40	1, 38, 39	syl2anc 693	. . . . . 6 |- ( ph -> sum_ i e. I ( E ^ 2 ) = ( ( # ` I ) x. ( E ^ 2 ) ) )
41		rrndistlt.n	. . . . . . . . 9 |- N = ( # ` I )
42		eqcom 2629	. . . . . . . . 9 |- ( N = ( # ` I ) <-> ( # ` I ) = N )
43	41, 42	mpbi 220	. . . . . . . 8 |- ( # ` I ) = N
44	43	oveq1i 6660	. . . . . . 7 |- ( ( # ` I ) x. ( E ^ 2 ) ) = ( N x. ( E ^ 2 ) )
45	44	a1i 11	. . . . . 6 |- ( ph -> ( ( # ` I ) x. ( E ^ 2 ) ) = ( N x. ( E ^ 2 ) ) )
46	40, 45	eqtr2d 2657	. . . . 5 |- ( ph -> ( N x. ( E ^ 2 ) ) = sum_ i e. I ( E ^ 2 ) )
47	37, 46	breq12d 4666	. . . 4 |- ( ph -> ( sum_ i e. I ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) < ( N x. ( E ^ 2 ) ) <-> sum_ i e. I ( ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) ^ 2 ) < sum_ i e. I ( E ^ 2 ) ) )
48	30, 47	mpbird 247	. . 3 |- ( ph -> sum_ i e. I ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) < ( N x. ( E ^ 2 ) ) )
49		nfv 1843	. . . . 5 |- F/ i ph
50	33	resqcld 13035	. . . . 5 |- ( ( ph /\ i e. I ) -> ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) e. RR )
51	49, 1, 50	fsumreclf 39808	. . . 4 |- ( ph -> sum_ i e. I ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) e. RR )
52	33	sqge0d 13036	. . . . 5 |- ( ( ph /\ i e. I ) -> 0 <_ ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) )
53	1, 50, 52	fsumge0 14527	. . . 4 |- ( ph -> 0 <_ sum_ i e. I ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) )
54		hashcl 13147	. . . . . . . 8 |- ( I e. Fin -> ( # ` I ) e. NN0 )
55	1, 54	syl 17	. . . . . . 7 |- ( ph -> ( # ` I ) e. NN0 )
56	41, 55	syl5eqel 2705	. . . . . 6 |- ( ph -> N e. NN0 )
57	56	nn0red 11352	. . . . 5 |- ( ph -> N e. RR )
58	57, 20	remulcld 10070	. . . 4 |- ( ph -> ( N x. ( E ^ 2 ) ) e. RR )
59	56	nn0ge0d 11354	. . . . 5 |- ( ph -> 0 <_ N )
60	19	sqge0d 13036	. . . . 5 |- ( ph -> 0 <_ ( E ^ 2 ) )
61	57, 20, 59, 60	mulge0d 10604	. . . 4 |- ( ph -> 0 <_ ( N x. ( E ^ 2 ) ) )
62	51, 53, 58, 61	sqrtltd 14166	. . 3 |- ( ph -> ( sum_ i e. I ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) < ( N x. ( E ^ 2 ) ) <-> ( sqr ` sum_ i e. I ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) ) < ( sqr ` ( N x. ( E ^ 2 ) ) ) ) )
63	48, 62	mpbid 222	. 2 |- ( ph -> ( sqr ` sum_ i e. I ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) ) < ( sqr ` ( N x. ( E ^ 2 ) ) ) )
64		rrndistlt.d	. . . . . 6 |- D = ( dist ` (ℝ^ ` I ) )
65	64	a1i 11	. . . . 5 |- ( ph -> D = ( dist ` (ℝ^ ` I ) ) )
66		eqid 2622	. . . . . . 7 |- (ℝ^ ` I ) = (ℝ^ ` I )
67		eqid 2622	. . . . . . 7 |- ( RR ^m I ) = ( RR ^m I )
68	66, 67	rrxdsfi 40505	. . . . . 6 |- ( I e. Fin -> ( dist ` (ℝ^ ` I ) ) = ( f e. ( RR ^m I ) , g e. ( RR ^m I ) |-> ( sqr ` sum_ i e. I ( ( ( f ` i ) - ( g ` i ) ) ^ 2 ) ) ) )
69	1, 68	syl 17	. . . . 5 |- ( ph -> ( dist ` (ℝ^ ` I ) ) = ( f e. ( RR ^m I ) , g e. ( RR ^m I ) |-> ( sqr ` sum_ i e. I ( ( ( f ` i ) - ( g ` i ) ) ^ 2 ) ) ) )
70	65, 69	eqtrd 2656	. . . 4 |- ( ph -> D = ( f e. ( RR ^m I ) , g e. ( RR ^m I ) |-> ( sqr ` sum_ i e. I ( ( ( f ` i ) - ( g ` i ) ) ^ 2 ) ) ) )
71		fveq1 6190	. . . . . . . . . 10 |- ( f = X -> ( f ` i ) = ( X ` i ) )
72	71	adantr 481	. . . . . . . . 9 |- ( ( f = X /\ g = Y ) -> ( f ` i ) = ( X ` i ) )
73		fveq1 6190	. . . . . . . . . 10 |- ( g = Y -> ( g ` i ) = ( Y ` i ) )
74	73	adantl 482	. . . . . . . . 9 |- ( ( f = X /\ g = Y ) -> ( g ` i ) = ( Y ` i ) )
75	72, 74	oveq12d 6668	. . . . . . . 8 |- ( ( f = X /\ g = Y ) -> ( ( f ` i ) - ( g ` i ) ) = ( ( X ` i ) - ( Y ` i ) ) )
76	75	oveq1d 6665	. . . . . . 7 |- ( ( f = X /\ g = Y ) -> ( ( ( f ` i ) - ( g ` i ) ) ^ 2 ) = ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) )
77	76	sumeq2ad 14434	. . . . . 6 |- ( ( f = X /\ g = Y ) -> sum_ i e. I ( ( ( f ` i ) - ( g ` i ) ) ^ 2 ) = sum_ i e. I ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) )
78	77	fveq2d 6195	. . . . 5 |- ( ( f = X /\ g = Y ) -> ( sqr ` sum_ i e. I ( ( ( f ` i ) - ( g ` i ) ) ^ 2 ) ) = ( sqr ` sum_ i e. I ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) ) )
79	78	adantl 482	. . . 4 |- ( ( ph /\ ( f = X /\ g = Y ) ) -> ( sqr ` sum_ i e. I ( ( ( f ` i ) - ( g ` i ) ) ^ 2 ) ) = ( sqr ` sum_ i e. I ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) ) )
80	51, 53	resqrtcld 14156	. . . 4 |- ( ph -> ( sqr ` sum_ i e. I ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) ) e. RR )
81	70, 79, 3, 10, 80	ovmpt2d 6788	. . 3 |- ( ph -> ( X D Y ) = ( sqr ` sum_ i e. I ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) ) )
82		sqrtmul 14000	. . . . 5 |- ( ( ( N e. RR /\ 0 <_ N ) /\ ( ( E ^ 2 ) e. RR /\ 0 <_ ( E ^ 2 ) ) ) -> ( sqr ` ( N x. ( E ^ 2 ) ) ) = ( ( sqr ` N ) x. ( sqr ` ( E ^ 2 ) ) ) )
83	57, 59, 20, 60, 82	syl22anc 1327	. . . 4 |- ( ph -> ( sqr ` ( N x. ( E ^ 2 ) ) ) = ( ( sqr ` N ) x. ( sqr ` ( E ^ 2 ) ) ) )
84	18	rpge0d 11876	. . . . . 6 |- ( ph -> 0 <_ E )
85	19, 84	sqrtsqd 14158	. . . . 5 |- ( ph -> ( sqr ` ( E ^ 2 ) ) = E )
86	85	oveq2d 6666	. . . 4 |- ( ph -> ( ( sqr ` N ) x. ( sqr ` ( E ^ 2 ) ) ) = ( ( sqr ` N ) x. E ) )
87	83, 86	eqtr2d 2657	. . 3 |- ( ph -> ( ( sqr ` N ) x. E ) = ( sqr ` ( N x. ( E ^ 2 ) ) ) )
88	81, 87	breq12d 4666	. 2 |- ( ph -> ( ( X D Y ) < ( ( sqr ` N ) x. E ) <-> ( sqr ` sum_ i e. I ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) ) < ( sqr ` ( N x. ( E ^ 2 ) ) ) ) )
89	63, 88	mpbird 247	1 |- ( ph -> ( X D Y ) < ( ( sqr ` N ) x. E ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   (/)c0 3915   class class class wbr 4653   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ^m cmap 7857   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   2c2 11070   NN0cn0 11292   RR+crp 11832   ^cexp 12860   #chash 13117   sqrcsqrt 13973   abscabs 13974   sum_csu 14416   distcds 15950  ℝ^crrx 23171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-rp 11833  df-ico 12181  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-0g 16102  df-gsum 16103  df-prds 16108  df-pws 16110  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-rnghom 18715  df-drng 18749  df-field 18750  df-subrg 18778  df-staf 18845  df-srng 18846  df-lmod 18865  df-lss 18933  df-sra 19172  df-rgmod 19173  df-cnfld 19747  df-refld 19951  df-dsmm 20076  df-frlm 20091  df-nm 22387  df-tng 22389  df-tch 22969  df-rrx 23173

This theorem is referenced by:  qndenserrnbllem  40514