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Theorem rrntotbnd 26545
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 2436 . . 3  |-  ( (flds  RR )  ^s  I )  =  ( (flds  RR )  ^s  I )
2 eqid 2436 . . 3  |-  ( dist `  ( (flds  RR )  ^s  I ) )  =  ( dist `  (
(flds  RR )  ^s  I ) )
3 rrntotbnd.1 . . 3  |-  X  =  ( RR  ^m  I
)
41, 2, 3repwsmet 26543 . 2  |-  ( I  e.  Fin  ->  ( dist `  ( (flds  RR )  ^s  I ) )  e.  ( Met `  X ) )
53rrnmet 26538 . 2  |-  ( I  e.  Fin  ->  ( Rn `  I )  e.  ( Met `  X
) )
6 hashcl 11639 . . . 4  |-  ( I  e.  Fin  ->  ( # `
 I )  e. 
NN0 )
7 nn0re 10230 . . . . 5  |-  ( (
# `  I )  e.  NN0  ->  ( # `  I
)  e.  RR )
8 nn0ge0 10247 . . . . 5  |-  ( (
# `  I )  e.  NN0  ->  0  <_  (
# `  I )
)
97, 8resqrcld 12220 . . . 4  |-  ( (
# `  I )  e.  NN0  ->  ( sqr `  ( # `  I
) )  e.  RR )
106, 9syl 16 . . 3  |-  ( I  e.  Fin  ->  ( sqr `  ( # `  I
) )  e.  RR )
117, 8sqrge0d 12223 . . . 4  |-  ( (
# `  I )  e.  NN0  ->  0  <_  ( sqr `  ( # `  I ) ) )
126, 11syl 16 . . 3  |-  ( I  e.  Fin  ->  0  <_  ( sqr `  ( # `
 I ) ) )
1310, 12ge0p1rpd 10674 . 2  |-  ( I  e.  Fin  ->  (
( sqr `  ( # `
 I ) )  +  1 )  e.  RR+ )
14 1rp 10616 . . 3  |-  1  e.  RR+
1514a1i 11 . 2  |-  ( I  e.  Fin  ->  1  e.  RR+ )
16 metcl 18362 . . . . 5  |-  ( ( ( Rn `  I
)  e.  ( Met `  X )  /\  x  e.  X  /\  y  e.  X )  ->  (
x ( Rn `  I ) y )  e.  RR )
17163expb 1154 . . . 4  |-  ( ( ( Rn `  I
)  e.  ( Met `  X )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( x
( Rn `  I
) y )  e.  RR )
185, 17sylan 458 . . 3  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
x ( Rn `  I ) y )  e.  RR )
1910adantr 452 . . . 4  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  ( sqr `  ( # `  I
) )  e.  RR )
204adantr 452 . . . . . 6  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  ( dist `  ( (flds  RR )  ^s  I ) )  e.  ( Met `  X ) )
21 simprl 733 . . . . . 6  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  x  e.  X )
22 simprr 734 . . . . . 6  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  y  e.  X )
23 metcl 18362 . . . . . . 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 18375 . . . . . . 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 519 . . . . . 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 1184 . . . . 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 446 . . . 4  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
x ( dist `  (
(flds  RR )  ^s  I ) ) y )  e.  RR )
2819, 27remulcld 9116 . . 3  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
( sqr `  ( # `
 I ) )  x.  ( x (
dist `  ( (flds  RR )  ^s  I ) ) y ) )  e.  RR )
29 peano2re 9239 . . . . . 6  |-  ( ( sqr `  ( # `  I ) )  e.  RR  ->  ( ( sqr `  ( # `  I
) )  +  1 )  e.  RR )
3010, 29syl 16 . . . . 5  |-  ( I  e.  Fin  ->  (
( sqr `  ( # `
 I ) )  +  1 )  e.  RR )
3130adantr 452 . . . 4  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
( sqr `  ( # `
 I ) )  +  1 )  e.  RR )
3231, 27remulcld 9116 . . 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 20 . . . . 5  |-  ( I  e.  Fin  ->  I  e.  Fin )
341, 2, 3, 33rrnequiv 26544 . . . 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 450 . . 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 9942 . . . 4  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  ( sqr `  ( # `  I
) )  <_  (
( sqr `  ( # `
 I ) )  +  1 ) )
37 lemul1a 9864 . . . 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 1187 . . 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 9227 . 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 446 . . 3  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
x ( dist `  (
(flds  RR )  ^s  I ) ) y )  <_  ( x
( Rn `  I
) y ) )
4118recnd 9114 . . . 4  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
x ( Rn `  I ) y )  e.  CC )
4241mulid2d 9106 . . 3  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
1  x.  ( x ( Rn `  I
) y ) )  =  ( x ( Rn `  I ) y ) )
4340, 42breqtrrd 4238 . 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 2436 . 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 9047 . . 3  |-  RR  C_  CC
471, 44cnpwstotbnd 26506 . . 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 652 . 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 26501 1  |-  ( I  e.  Fin  ->  ( M  e.  ( TotBnd `  Y )  <->  M  e.  ( Bnd `  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    C_ wss 3320   class class class wbr 4212    X. cxp 4876    |` cres 4880   ` cfv 5454  (class class class)co 6081    ^m cmap 7018   Fincfn 7109   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995    <_ cle 9121   NN0cn0 10221   RR+crp 10612   #chash 11618   sqrcsqr 12038   ↾s cress 13470   distcds 13538    ^s cpws 13670   Metcme 16687  ℂfldccnfld 16703   TotBndctotbnd 26475   Bndcbnd 26476   Rncrrn 26534
This theorem is referenced by:  rrnheibor  26546
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-ec 6907  df-map 7020  df-pm 7021  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-ico 10922  df-icc 10923  df-fz 11044  df-fzo 11136  df-fl 11202  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-sum 12480  df-gz 13298  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-starv 13544  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-hom 13553  df-cco 13554  df-rest 13650  df-topn 13651  df-topgen 13667  df-prds 13671  df-pws 13673  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-mopn 16698  df-cnfld 16704  df-top 16963  df-bases 16965  df-topon 16966  df-topsp 16967  df-xms 18350  df-ms 18351  df-totbnd 26477  df-bnd 26488  df-rrn 26535
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