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Theorem rrntotbnd 29933
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.
StepHypRef Expression
1 eqid 2467 . . 3  |-  ( (flds  RR )  ^s  I )  =  ( (flds  RR )  ^s  I )
2 eqid 2467 . . 3  |-  ( dist `  ( (flds  RR )  ^s  I ) )  =  ( dist `  (
(flds  RR )  ^s  I ) )
3 rrntotbnd.1 . . 3  |-  X  =  ( RR  ^m  I
)
41, 2, 3repwsmet 29931 . 2  |-  ( I  e.  Fin  ->  ( dist `  ( (flds  RR )  ^s  I ) )  e.  ( Met `  X ) )
53rrnmet 29926 . 2  |-  ( I  e.  Fin  ->  ( Rn `  I )  e.  ( Met `  X
) )
6 hashcl 12390 . . . 4  |-  ( I  e.  Fin  ->  ( # `
 I )  e. 
NN0 )
7 nn0re 10800 . . . . 5  |-  ( (
# `  I )  e.  NN0  ->  ( # `  I
)  e.  RR )
8 nn0ge0 10817 . . . . 5  |-  ( (
# `  I )  e.  NN0  ->  0  <_  (
# `  I )
)
97, 8resqrtcld 13205 . . . 4  |-  ( (
# `  I )  e.  NN0  ->  ( sqr `  ( # `  I
) )  e.  RR )
106, 9syl 16 . . 3  |-  ( I  e.  Fin  ->  ( sqr `  ( # `  I
) )  e.  RR )
117, 8sqrtge0d 13208 . . . 4  |-  ( (
# `  I )  e.  NN0  ->  0  <_  ( sqr `  ( # `  I ) ) )
126, 11syl 16 . . 3  |-  ( I  e.  Fin  ->  0  <_  ( sqr `  ( # `
 I ) ) )
1310, 12ge0p1rpd 11278 . 2  |-  ( I  e.  Fin  ->  (
( sqr `  ( # `
 I ) )  +  1 )  e.  RR+ )
14 1rp 11220 . . 3  |-  1  e.  RR+
1514a1i 11 . 2  |-  ( I  e.  Fin  ->  1  e.  RR+ )
16 metcl 20567 . . . . 5  |-  ( ( ( Rn `  I
)  e.  ( Met `  X )  /\  x  e.  X  /\  y  e.  X )  ->  (
x ( Rn `  I ) y )  e.  RR )
17163expb 1197 . . . 4  |-  ( ( ( Rn `  I
)  e.  ( Met `  X )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( x
( Rn `  I
) y )  e.  RR )
185, 17sylan 471 . . 3  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
x ( Rn `  I ) y )  e.  RR )
1910adantr 465 . . . 4  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  ( sqr `  ( # `  I
) )  e.  RR )
204adantr 465 . . . . . 6  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  ( dist `  ( (flds  RR )  ^s  I ) )  e.  ( Met `  X ) )
21 simprl 755 . . . . . 6  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  x  e.  X )
22 simprr 756 . . . . . 6  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  y  e.  X )
23 metcl 20567 . . . . . . 7  |-  ( ( ( dist `  (
(flds  RR )  ^s  I ) )  e.  ( Met `  X
)  /\  x  e.  X  /\  y  e.  X
)  ->  ( x
( dist `  ( (flds  RR )  ^s  I ) ) y )  e.  RR )
24 metge0 20580 . . . . . . 7  |-  ( ( ( dist `  (
(flds  RR )  ^s  I ) )  e.  ( Met `  X
)  /\  x  e.  X  /\  y  e.  X
)  ->  0  <_  ( x ( dist `  (
(flds  RR )  ^s  I ) ) y ) )
2523, 24jca 532 . . . . . 6  |-  ( ( ( dist `  (
(flds  RR )  ^s  I ) )  e.  ( Met `  X
)  /\  x  e.  X  /\  y  e.  X
)  ->  ( (
x ( dist `  (
(flds  RR )  ^s  I ) ) y )  e.  RR  /\  0  <_  ( x (
dist `  ( (flds  RR )  ^s  I ) ) y ) ) )
2620, 21, 22, 25syl3anc 1228 . . . . 5  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
( x ( dist `  ( (flds  RR )  ^s  I ) ) y )  e.  RR  /\  0  <_  ( x (
dist `  ( (flds  RR )  ^s  I ) ) y ) ) )
2726simpld 459 . . . 4  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
x ( dist `  (
(flds  RR )  ^s  I ) ) y )  e.  RR )
2819, 27remulcld 9620 . . 3  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
( sqr `  ( # `
 I ) )  x.  ( x (
dist `  ( (flds  RR )  ^s  I ) ) y ) )  e.  RR )
29 peano2re 9748 . . . . . 6  |-  ( ( sqr `  ( # `  I ) )  e.  RR  ->  ( ( sqr `  ( # `  I
) )  +  1 )  e.  RR )
3010, 29syl 16 . . . . 5  |-  ( I  e.  Fin  ->  (
( sqr `  ( # `
 I ) )  +  1 )  e.  RR )
3130adantr 465 . . . 4  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
( sqr `  ( # `
 I ) )  +  1 )  e.  RR )
3231, 27remulcld 9620 . . 3  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
( ( sqr `  ( # `
 I ) )  +  1 )  x.  ( x ( dist `  ( (flds  RR )  ^s  I ) ) y ) )  e.  RR )
33 id 22 . . . . 5  |-  ( I  e.  Fin  ->  I  e.  Fin )
341, 2, 3, 33rrnequiv 29932 . . . 4  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
( x ( dist `  ( (flds  RR )  ^s  I ) ) y )  <_  ( x
( Rn `  I
) y )  /\  ( x ( Rn
`  I ) y )  <_  ( ( sqr `  ( # `  I
) )  x.  (
x ( dist `  (
(flds  RR )  ^s  I ) ) y ) ) ) )
3534simprd 463 . . 3  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
x ( Rn `  I ) y )  <_  ( ( sqr `  ( # `  I
) )  x.  (
x ( dist `  (
(flds  RR )  ^s  I ) ) y ) ) )
3619lep1d 10473 . . . 4  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  ( sqr `  ( # `  I
) )  <_  (
( sqr `  ( # `
 I ) )  +  1 ) )
37 lemul1a 10392 . . . 4  |-  ( ( ( ( sqr `  ( # `
 I ) )  e.  RR  /\  (
( sqr `  ( # `
 I ) )  +  1 )  e.  RR  /\  ( ( x ( dist `  (
(flds  RR )  ^s  I ) ) y )  e.  RR  /\  0  <_  ( x (
dist `  ( (flds  RR )  ^s  I ) ) y ) ) )  /\  ( sqr `  ( # `  I ) )  <_ 
( ( sqr `  ( # `
 I ) )  +  1 ) )  ->  ( ( sqr `  ( # `  I
) )  x.  (
x ( dist `  (
(flds  RR )  ^s  I ) ) y ) )  <_  (
( ( sqr `  ( # `
 I ) )  +  1 )  x.  ( x ( dist `  ( (flds  RR )  ^s  I ) ) y ) ) )
3819, 31, 26, 36, 37syl31anc 1231 . . 3  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
( sqr `  ( # `
 I ) )  x.  ( x (
dist `  ( (flds  RR )  ^s  I ) ) y ) )  <_  (
( ( sqr `  ( # `
 I ) )  +  1 )  x.  ( x ( dist `  ( (flds  RR )  ^s  I ) ) y ) ) )
3918, 28, 32, 35, 38letrd 9734 . 2  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
x ( Rn `  I ) y )  <_  ( ( ( sqr `  ( # `  I ) )  +  1 )  x.  (
x ( dist `  (
(flds  RR )  ^s  I ) ) y ) ) )
4034simpld 459 . . 3  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
x ( dist `  (
(flds  RR )  ^s  I ) ) y )  <_  ( x
( Rn `  I
) y ) )
4118recnd 9618 . . . 4  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
x ( Rn `  I ) y )  e.  CC )
4241mulid2d 9610 . . 3  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
1  x.  ( x ( Rn `  I
) y ) )  =  ( x ( Rn `  I ) y ) )
4340, 42breqtrrd 4473 . 2  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
x ( dist `  (
(flds  RR )  ^s  I ) ) y )  <_  ( 1  x.  ( x ( Rn `  I ) y ) ) )
44 eqid 2467 . 2  |-  ( (
dist `  ( (flds  RR )  ^s  I ) )  |`  ( Y  X.  Y
) )  =  ( ( dist `  (
(flds  RR )  ^s  I ) )  |`  ( Y  X.  Y
) )
45 rrntotbnd.2 . 2  |-  M  =  ( ( Rn `  I )  |`  ( Y  X.  Y ) )
46 ax-resscn 9545 . . 3  |-  RR  C_  CC
471, 44cnpwstotbnd 29894 . . 3  |-  ( ( RR  C_  CC  /\  I  e.  Fin )  ->  (
( ( dist `  (
(flds  RR )  ^s  I ) )  |`  ( Y  X.  Y
) )  e.  (
TotBnd `  Y )  <->  ( ( dist `  ( (flds  RR )  ^s  I ) )  |`  ( Y  X.  Y ) )  e.  ( Bnd `  Y
) ) )
4846, 47mpan 670 . 2  |-  ( I  e.  Fin  ->  (
( ( dist `  (
(flds  RR )  ^s  I ) )  |`  ( Y  X.  Y
) )  e.  (
TotBnd `  Y )  <->  ( ( dist `  ( (flds  RR )  ^s  I ) )  |`  ( Y  X.  Y ) )  e.  ( Bnd `  Y
) ) )
494, 5, 13, 15, 39, 43, 44, 45, 48equivbnd2 29889 1  |-  ( I  e.  Fin  ->  ( M  e.  ( TotBnd `  Y )  <->  M  e.  ( Bnd `  Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    C_ wss 3476   class class class wbr 4447    X. cxp 4997    |` cres 5001   ` cfv 5586  (class class class)co 6282    ^m cmap 7417   Fincfn 7513   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489    + caddc 9491    x. cmul 9493    <_ cle 9625   NN0cn0 10791   RR+crp 11216   #chash 12367   sqrcsqrt 13023   ↾s cress 14484   distcds 14557    ^s cpws 14695   Metcme 18172  ℂfldccnfld 18188   TotBndctotbnd 29863   Bndcbnd 29864   Rncrrn 29922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-ec 7310  df-map 7419  df-pm 7420  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ico 11531  df-icc 11532  df-fz 11669  df-fzo 11789  df-fl 11893  df-seq 12071  df-exp 12130  df-hash 12368  df-cj 12889  df-re 12890  df-im 12891  df-sqrt 13025  df-abs 13026  df-clim 13267  df-sum 13465  df-gz 14300  df-struct 14485  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-plusg 14561  df-mulr 14562  df-starv 14563  df-sca 14564  df-vsca 14565  df-ip 14566  df-tset 14567  df-ple 14568  df-ds 14570  df-unif 14571  df-hom 14572  df-cco 14573  df-rest 14671  df-topn 14672  df-topgen 14692  df-prds 14696  df-pws 14698  df-psmet 18179  df-xmet 18180  df-met 18181  df-bl 18182  df-mopn 18183  df-cnfld 18189  df-top 19163  df-bases 19165  df-topon 19166  df-topsp 19167  df-xms 20555  df-ms 20556  df-totbnd 29865  df-bnd 29876  df-rrn 29923
This theorem is referenced by:  rrnheibor  29934
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