Theorem rrntotbnd 33635

Description: A set in Euclidean space is totally bounded iff its is bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 16-Sep-2015.)

Hypotheses

Ref	Expression
rrntotbnd.1	|- X = ( RR ^m I )
rrntotbnd.2	|- M = ( ( Rn ` I ) |` ( Y X. Y ) )

Assertion

Ref	Expression
rrntotbnd	|- ( I e. Fin -> ( M e. ( TotBnd ` Y ) <-> M e. ( Bnd ` Y ) ) )

Proof of Theorem rrntotbnd

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( (ℂfld ↾s RR ) ^s I ) = ( (ℂfld ↾s RR ) ^s I )
2		eqid 2622	. . 3 |- ( dist ` ( (ℂfld ↾s RR ) ^s I ) ) = ( dist ` ( (ℂfld ↾s RR ) ^s I ) )
3		rrntotbnd.1	. . 3 |- X = ( RR ^m I )
4	1, 2, 3	repwsmet 33633	. 2 |- ( I e. Fin -> ( dist ` ( (ℂfld ↾s RR ) ^s I ) ) e. ( Met ` X ) )
5	3	rrnmet 33628	. 2 |- ( I e. Fin -> ( Rn ` I ) e. ( Met ` X ) )
6		hashcl 13147	. . . 4 |- ( I e. Fin -> ( # ` I ) e. NN0 )
7		nn0re 11301	. . . . 5 |- ( ( # ` I ) e. NN0 -> ( # ` I ) e. RR )
8		nn0ge0 11318	. . . . 5 |- ( ( # ` I ) e. NN0 -> 0 <_ ( # ` I ) )
9	7, 8	resqrtcld 14156	. . . 4 |- ( ( # ` I ) e. NN0 -> ( sqr ` ( # ` I ) ) e. RR )
10	6, 9	syl 17	. . 3 |- ( I e. Fin -> ( sqr ` ( # ` I ) ) e. RR )
11	7, 8	sqrtge0d 14159	. . . 4 |- ( ( # ` I ) e. NN0 -> 0 <_ ( sqr ` ( # ` I ) ) )
12	6, 11	syl 17	. . 3 |- ( I e. Fin -> 0 <_ ( sqr ` ( # ` I ) ) )
13	10, 12	ge0p1rpd 11902	. 2 |- ( I e. Fin -> ( ( sqr ` ( # ` I ) ) + 1 ) e. RR+ )
14		1rp 11836	. . 3 |- 1 e. RR+
15	14	a1i 11	. 2 |- ( I e. Fin -> 1 e. RR+ )
16		metcl 22137	. . . . 5 |- ( ( ( Rn ` I ) e. ( Met ` X ) /\ x e. X /\ y e. X ) -> ( x ( Rn ` I ) y ) e. RR )
17	16	3expb 1266	. . . 4 |- ( ( ( Rn ` I ) e. ( Met ` X ) /\ ( x e. X /\ y e. X ) ) -> ( x ( Rn ` I ) y ) e. RR )
18	5, 17	sylan 488	. . 3 |- ( ( I e. Fin /\ ( x e. X /\ y e. X ) ) -> ( x ( Rn ` I ) y ) e. RR )
19	10	adantr 481	. . . 4 |- ( ( I e. Fin /\ ( x e. X /\ y e. X ) ) -> ( sqr ` ( # ` I ) ) e. RR )
20	4	adantr 481	. . . . . 6 |- ( ( I e. Fin /\ ( x e. X /\ y e. X ) ) -> ( dist ` ( (ℂfld ↾s RR ) ^s I ) ) e. ( Met ` X ) )
21		simprl 794	. . . . . 6 |- ( ( I e. Fin /\ ( x e. X /\ y e. X ) ) -> x e. X )
22		simprr 796	. . . . . 6 |- ( ( I e. Fin /\ ( x e. X /\ y e. X ) ) -> y e. X )
23		metcl 22137	. . . . . . 7 |- ( ( ( dist ` ( (ℂfld ↾s RR ) ^s I ) ) e. ( Met ` X ) /\ x e. X /\ y e. X ) -> ( x ( dist ` ( (ℂfld ↾s RR ) ^s I ) ) y ) e. RR )
24		metge0 22150	. . . . . . 7 |- ( ( ( dist ` ( (ℂfld ↾s RR ) ^s I ) ) e. ( Met ` X ) /\ x e. X /\ y e. X ) -> 0 <_ ( x ( dist ` ( (ℂfld ↾s RR ) ^s I ) ) y ) )
25	23, 24	jca 554	. . . . . 6 |- ( ( ( dist ` ( (ℂfld ↾s RR ) ^s I ) ) e. ( Met ` X ) /\ x e. X /\ y e. X ) -> ( ( x ( dist ` ( (ℂfld ↾s RR ) ^s I ) ) y ) e. RR /\ 0 <_ ( x ( dist ` ( (ℂfld ↾s RR ) ^s I ) ) y ) ) )
26	20, 21, 22, 25	syl3anc 1326	. . . . 5 |- ( ( I e. Fin /\ ( x e. X /\ y e. X ) ) -> ( ( x ( dist ` ( (ℂfld ↾s RR ) ^s I ) ) y ) e. RR /\ 0 <_ ( x ( dist ` ( (ℂfld ↾s RR ) ^s I ) ) y ) ) )
27	26	simpld 475	. . . 4 |- ( ( I e. Fin /\ ( x e. X /\ y e. X ) ) -> ( x ( dist ` ( (ℂfld ↾s RR ) ^s I ) ) y ) e. RR )
28	19, 27	remulcld 10070	. . 3 |- ( ( I e. Fin /\ ( x e. X /\ y e. X ) ) -> ( ( sqr ` ( # ` I ) ) x. ( x ( dist ` ( (ℂfld ↾s RR ) ^s I ) ) y ) ) e. RR )
29		peano2re 10209	. . . . . 6 |- ( ( sqr ` ( # ` I ) ) e. RR -> ( ( sqr ` ( # ` I ) ) + 1 ) e. RR )
30	10, 29	syl 17	. . . . 5 |- ( I e. Fin -> ( ( sqr ` ( # ` I ) ) + 1 ) e. RR )
31	30	adantr 481	. . . 4 |- ( ( I e. Fin /\ ( x e. X /\ y e. X ) ) -> ( ( sqr ` ( # ` I ) ) + 1 ) e. RR )
32	31, 27	remulcld 10070	. . 3 |- ( ( I e. Fin /\ ( x e. X /\ y e. X ) ) -> ( ( ( sqr ` ( # ` I ) ) + 1 ) x. ( x ( dist ` ( (ℂfld ↾s RR ) ^s I ) ) y ) ) e. RR )
33		id 22	. . . . 5 |- ( I e. Fin -> I e. Fin )
34	1, 2, 3, 33	rrnequiv 33634	. . . 4 |- ( ( I e. Fin /\ ( x e. X /\ y e. X ) ) -> ( ( x ( dist ` ( (ℂfld ↾s RR ) ^s I ) ) y ) <_ ( x ( Rn ` I ) y ) /\ ( x ( Rn ` I ) y ) <_ ( ( sqr ` ( # ` I ) ) x. ( x ( dist ` ( (ℂfld ↾s RR ) ^s I ) ) y ) ) ) )
35	34	simprd 479	. . 3 |- ( ( I e. Fin /\ ( x e. X /\ y e. X ) ) -> ( x ( Rn ` I ) y ) <_ ( ( sqr ` ( # ` I ) ) x. ( x ( dist ` ( (ℂfld ↾s RR ) ^s I ) ) y ) ) )
36	19	lep1d 10955	. . . 4 |- ( ( I e. Fin /\ ( x e. X /\ y e. X ) ) -> ( sqr ` ( # ` I ) ) <_ ( ( sqr ` ( # ` I ) ) + 1 ) )
37		lemul1a 10877	. . . 4 |- ( ( ( ( sqr ` ( # ` I ) ) e. RR /\ ( ( sqr ` ( # ` I ) ) + 1 ) e. RR /\ ( ( x ( dist ` ( (ℂfld ↾s RR ) ^s I ) ) y ) e. RR /\ 0 <_ ( x ( dist ` ( (ℂfld ↾s RR ) ^s I ) ) y ) ) ) /\ ( sqr ` ( # ` I ) ) <_ ( ( sqr ` ( # ` I ) ) + 1 ) ) -> ( ( sqr ` ( # ` I ) ) x. ( x ( dist ` ( (ℂfld ↾s RR ) ^s I ) ) y ) ) <_ ( ( ( sqr ` ( # ` I ) ) + 1 ) x. ( x ( dist ` ( (ℂfld ↾s RR ) ^s I ) ) y ) ) )
38	19, 31, 26, 36, 37	syl31anc 1329	. . 3 |- ( ( I e. Fin /\ ( x e. X /\ y e. X ) ) -> ( ( sqr ` ( # ` I ) ) x. ( x ( dist ` ( (ℂfld ↾s RR ) ^s I ) ) y ) ) <_ ( ( ( sqr ` ( # ` I ) ) + 1 ) x. ( x ( dist ` ( (ℂfld ↾s RR ) ^s I ) ) y ) ) )
39	18, 28, 32, 35, 38	letrd 10194	. 2 |- ( ( I e. Fin /\ ( x e. X /\ y e. X ) ) -> ( x ( Rn ` I ) y ) <_ ( ( ( sqr ` ( # ` I ) ) + 1 ) x. ( x ( dist ` ( (ℂfld ↾s RR ) ^s I ) ) y ) ) )
40	34	simpld 475	. . 3 |- ( ( I e. Fin /\ ( x e. X /\ y e. X ) ) -> ( x ( dist ` ( (ℂfld ↾s RR ) ^s I ) ) y ) <_ ( x ( Rn ` I ) y ) )
41	18	recnd 10068	. . . 4 |- ( ( I e. Fin /\ ( x e. X /\ y e. X ) ) -> ( x ( Rn ` I ) y ) e. CC )
42	41	mulid2d 10058	. . 3 |- ( ( I e. Fin /\ ( x e. X /\ y e. X ) ) -> ( 1 x. ( x ( Rn ` I ) y ) ) = ( x ( Rn ` I ) y ) )
43	40, 42	breqtrrd 4681	. 2 |- ( ( I e. Fin /\ ( x e. X /\ y e. X ) ) -> ( x ( dist ` ( (ℂfld ↾s RR ) ^s I ) ) y ) <_ ( 1 x. ( x ( Rn ` I ) y ) ) )
44		eqid 2622	. 2 |- ( ( dist ` ( (ℂfld ↾s RR ) ^s I ) ) |` ( Y X. Y ) ) = ( ( dist ` ( (ℂfld ↾s RR ) ^s I ) ) |` ( Y X. Y ) )
45		rrntotbnd.2	. 2 |- M = ( ( Rn ` I ) |` ( Y X. Y ) )
46		ax-resscn 9993	. . 3 |- RR C_ CC
47	1, 44	cnpwstotbnd 33596	. . 3 |- ( ( RR C_ CC /\ I e. Fin ) -> ( ( ( dist ` ( (ℂfld ↾s RR ) ^s I ) ) |` ( Y X. Y ) ) e. ( TotBnd ` Y ) <-> ( ( dist ` ( (ℂfld ↾s RR ) ^s I ) ) |` ( Y X. Y ) ) e. ( Bnd ` Y ) ) )
48	46, 47	mpan 706	. 2 |- ( I e. Fin -> ( ( ( dist ` ( (ℂfld ↾s RR ) ^s I ) ) |` ( Y X. Y ) ) e. ( TotBnd ` Y ) <-> ( ( dist ` ( (ℂfld ↾s RR ) ^s I ) ) |` ( Y X. Y ) ) e. ( Bnd ` Y ) ) )
49	4, 5, 13, 15, 39, 43, 44, 45, 48	equivbnd2 33591	1 |- ( I e. Fin -> ( M e. ( TotBnd ` Y ) <-> M e. ( Bnd ` Y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   class class class wbr 4653   X. cxp 5112   |` cres 5116   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   <_ cle 10075   NN0cn0 11292   RR+crp 11832   #chash 13117   sqrcsqrt 13973   ↾_s cress 15858   distcds 15950   ^_s cpws 16107   Metcme 19732  ℂ_fldccnfld 19746   TotBndctotbnd 33565   Bndcbnd 33566   Rncrrn 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-2o 7561  df-oadd 7564  df-er 7742  df-ec 7744  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  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-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  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-gz 15634  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-rest 16083  df-topn 16084  df-topgen 16104  df-prds 16108  df-pws 16110  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-xms 22125  df-ms 22126  df-totbnd 33567  df-bnd 33578  df-rrn 33625

This theorem is referenced by:  rrnheibor  33636