rrndstprj2 - Mathbox for Jeff Madsen

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Theorem rrndstprj2 33630

Description: Bound on the distance between two points in Euclidean space given bounds on the distances in each coordinate. This theorem and rrndstprj1 33629 can be used to show that the supremum norm and Euclidean norm are equivalent. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.)

**Hypotheses**
Ref	Expression
rrnval.1
rrndstprj1.1

**Assertion**
Ref	Expression
rrndstprj2	$\$ $/\$ $/\$ $/\$ $/\$

Distinct variable groups:

Allowed substitution hint:

(

)

Proof of Theorem rrndstprj2 Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1064	. . . 4 $\$ $/\$ $/\$ $/\$ $/\$ $\$
2	1	eldifad 3586	. . 3 $\$ $/\$ $/\$ $/\$ $/\$
3		simpl2 1065	. . 3 $\$ $/\$ $/\$ $/\$ $/\$
4		simpl3 1066	. . 3 $\$ $/\$ $/\$ $/\$ $/\$
5		rrnval.1	. . . 4
6	5	rrnmval 33627	. . 3 $/\$ $/\$
7	2, 3, 4, 6	syl3anc 1326	. 2 $\$ $/\$ $/\$ $/\$ $/\$
8		eldifsni 4320	. . . . . 6 $\$
9	1, 8	syl 17	. . . . 5 $\$ $/\$ $/\$ $/\$ $/\$
10	3, 5	syl6eleq 2711	. . . . . . . . 9 $\$ $/\$ $/\$ $/\$ $/\$
11		elmapi 7879	. . . . . . . . 9
12	10, 11	syl 17	. . . . . . . 8 $\$ $/\$ $/\$ $/\$ $/\$
13	12	ffvelrnda 6359	. . . . . . 7 $\$ $/\$ $/\$ $/\$ $/\$ $/\$
14	4, 5	syl6eleq 2711	. . . . . . . . 9 $\$ $/\$ $/\$ $/\$ $/\$
15		elmapi 7879	. . . . . . . . 9
16	14, 15	syl 17	. . . . . . . 8 $\$ $/\$ $/\$ $/\$ $/\$
17	16	ffvelrnda 6359	. . . . . . 7 $\$ $/\$ $/\$ $/\$ $/\$ $/\$
18	13, 17	resubcld 10458	. . . . . 6 $\$ $/\$ $/\$ $/\$ $/\$ $/\$
19	18	resqcld 13035	. . . . 5 $\$ $/\$ $/\$ $/\$ $/\$ $/\$
20		simprl 794	. . . . . . . 8 $\$ $/\$ $/\$ $/\$ $/\$
21	20	rpred 11872	. . . . . . 7 $\$ $/\$ $/\$ $/\$ $/\$
22	21	resqcld 13035	. . . . . 6 $\$ $/\$ $/\$ $/\$ $/\$
23	22	adantr 481	. . . . 5 $\$ $/\$ $/\$ $/\$ $/\$ $/\$
24		absresq 14042	. . . . . . 7
25	18, 24	syl 17	. . . . . 6 $\$ $/\$ $/\$ $/\$ $/\$ $/\$
26		rrndstprj1.1	. . . . . . . . . 10
27	26	remetdval 22592	. . . . . . . . 9 $/\$
28	13, 17, 27	syl2anc 693	. . . . . . . 8 $\$ $/\$ $/\$ $/\$ $/\$ $/\$
29		simprr 796	. . . . . . . . 9 $\$ $/\$ $/\$ $/\$ $/\$
30		fveq2 6191	. . . . . . . . . . . 12
31		fveq2 6191	. . . . . . . . . . . 12
32	30, 31	oveq12d 6668	. . . . . . . . . . 11
33	32	breq1d 4663	. . . . . . . . . 10
34	33	rspccva 3308	. . . . . . . . 9 $/\$
35	29, 34	sylan 488	. . . . . . . 8 $\$ $/\$ $/\$ $/\$ $/\$ $/\$
36	28, 35	eqbrtrrd 4677	. . . . . . 7 $\$ $/\$ $/\$ $/\$ $/\$ $/\$
37	18	recnd 10068	. . . . . . . . 9 $\$ $/\$ $/\$ $/\$ $/\$ $/\$
38	37	abscld 14175	. . . . . . . 8 $\$ $/\$ $/\$ $/\$ $/\$ $/\$
39	21	adantr 481	. . . . . . . 8 $\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	37	absge0d 14183	. . . . . . . 8 $\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	20	rpge0d 11876	. . . . . . . . 9 $\$ $/\$ $/\$ $/\$ $/\$
42	41	adantr 481	. . . . . . . 8 $\$ $/\$ $/\$ $/\$ $/\$ $/\$
43	38, 39, 40, 42	lt2sqd 13043	. . . . . . 7 $\$ $/\$ $/\$ $/\$ $/\$ $/\$
44	36, 43	mpbid 222	. . . . . 6 $\$ $/\$ $/\$ $/\$ $/\$ $/\$
45	25, 44	eqbrtrrd 4677	. . . . 5 $\$ $/\$ $/\$ $/\$ $/\$ $/\$
46	2, 9, 19, 23, 45	fsumlt 14532	. . . 4 $\$ $/\$ $/\$ $/\$ $/\$
47	2, 19	fsumrecl 14465	. . . . 5 $\$ $/\$ $/\$ $/\$ $/\$
48	18	sqge0d 13036	. . . . . 6 $\$ $/\$ $/\$ $/\$ $/\$ $/\$
49	2, 19, 48	fsumge0 14527	. . . . 5 $\$ $/\$ $/\$ $/\$ $/\$
50		resqrtth 13996	. . . . 5 $/\$
51	47, 49, 50	syl2anc 693	. . . 4 $\$ $/\$ $/\$ $/\$ $/\$
52		hashnncl 13157	. . . . . . . . . . . 12
53	2, 52	syl 17	. . . . . . . . . . 11 $\$ $/\$ $/\$ $/\$ $/\$
54	9, 53	mpbird 247	. . . . . . . . . 10 $\$ $/\$ $/\$ $/\$ $/\$
55	54	nnrpd 11870	. . . . . . . . 9 $\$ $/\$ $/\$ $/\$ $/\$
56	55	rpred 11872	. . . . . . . 8 $\$ $/\$ $/\$ $/\$ $/\$
57	55	rpge0d 11876	. . . . . . . 8 $\$ $/\$ $/\$ $/\$ $/\$
58		resqrtth 13996	. . . . . . . 8 $/\$
59	56, 57, 58	syl2anc 693	. . . . . . 7 $\$ $/\$ $/\$ $/\$ $/\$
60	59	oveq2d 6666	. . . . . 6 $\$ $/\$ $/\$ $/\$ $/\$
61	22	recnd 10068	. . . . . . 7 $\$ $/\$ $/\$ $/\$ $/\$
62	55	rpcnd 11874	. . . . . . 7 $\$ $/\$ $/\$ $/\$ $/\$
63	61, 62	mulcomd 10061	. . . . . 6 $\$ $/\$ $/\$ $/\$ $/\$
64	60, 63	eqtrd 2656	. . . . 5 $\$ $/\$ $/\$ $/\$ $/\$
65	20	rpcnd 11874	. . . . . 6 $\$ $/\$ $/\$ $/\$ $/\$
66	55	rpsqrtcld 14150	. . . . . . 7 $\$ $/\$ $/\$ $/\$ $/\$
67	66	rpcnd 11874	. . . . . 6 $\$ $/\$ $/\$ $/\$ $/\$
68	65, 67	sqmuld 13020	. . . . 5 $\$ $/\$ $/\$ $/\$ $/\$
69		fsumconst 14522	. . . . . 6 $/\$
70	2, 61, 69	syl2anc 693	. . . . 5 $\$ $/\$ $/\$ $/\$ $/\$
71	64, 68, 70	3eqtr4d 2666	. . . 4 $\$ $/\$ $/\$ $/\$ $/\$
72	46, 51, 71	3brtr4d 4685	. . 3 $\$ $/\$ $/\$ $/\$ $/\$
73	47, 49	resqrtcld 14156	. . . 4 $\$ $/\$ $/\$ $/\$ $/\$
74	20, 66	rpmulcld 11888	. . . . 5 $\$ $/\$ $/\$ $/\$ $/\$
75	74	rpred 11872	. . . 4 $\$ $/\$ $/\$ $/\$ $/\$
76	47, 49	sqrtge0d 14159	. . . 4 $\$ $/\$ $/\$ $/\$ $/\$
77	74	rpge0d 11876	. . . 4 $\$ $/\$ $/\$ $/\$ $/\$
78	73, 75, 76, 77	lt2sqd 13043	. . 3 $\$ $/\$ $/\$ $/\$ $/\$
79	72, 78	mpbird 247	. 2 $\$ $/\$ $/\$ $/\$ $/\$
80	7, 79	eqbrtrd 4675	1 $\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wne 2794

wral 2912 $\$ cdif 3571

c0 3915

csn 4177 class class class wbr 4653

cxp 5112

cres 5116

ccom 5118

wf 5884

cfv 5888 (class class class)co 6650

cmap 7857

cfn 7955

cc 9934

cr 9935

cc0 9936

cmul 9941

clt 10074

cle 10075

cmin 10266

cn 11020

c2 11070

crp 11832

cexp 12860

chash 13117

csqrt 13973

cabs 13974

csu 14416

crrn 33624

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

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-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 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-n0 11293 df-z 11378 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-rrn 33625

This theorem is referenced by: rrncmslem 33631 rrnequiv 33634

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