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Theorem rrntotbnd 26663
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 2296 . . 3  |-  ( (flds  RR )  ^s  I )  =  ( (flds  RR )  ^s  I )
2 eqid 2296 . . 3  |-  ( dist `  ( (flds  RR )  ^s  I ) )  =  ( dist `  (
(flds  RR )  ^s  I ) )
3 rrntotbnd.1 . . 3  |-  X  =  ( RR  ^m  I
)
41, 2, 3repwsmet 26661 . 2  |-  ( I  e.  Fin  ->  ( dist `  ( (flds  RR )  ^s  I ) )  e.  ( Met `  X ) )
53rrnmet 26656 . 2  |-  ( I  e.  Fin  ->  ( Rn `  I )  e.  ( Met `  X
) )
6 hashcl 11366 . . . 4  |-  ( I  e.  Fin  ->  ( # `
 I )  e. 
NN0 )
7 nn0re 9990 . . . . 5  |-  ( (
# `  I )  e.  NN0  ->  ( # `  I
)  e.  RR )
8 nn0ge0 10007 . . . . 5  |-  ( (
# `  I )  e.  NN0  ->  0  <_  (
# `  I )
)
97, 8resqrcld 11916 . . . 4  |-  ( (
# `  I )  e.  NN0  ->  ( sqr `  ( # `  I
) )  e.  RR )
106, 9syl 15 . . 3  |-  ( I  e.  Fin  ->  ( sqr `  ( # `  I
) )  e.  RR )
117, 8sqrge0d 11919 . . . 4  |-  ( (
# `  I )  e.  NN0  ->  0  <_  ( sqr `  ( # `  I ) ) )
126, 11syl 15 . . 3  |-  ( I  e.  Fin  ->  0  <_  ( sqr `  ( # `
 I ) ) )
1310, 12ge0p1rpd 10432 . 2  |-  ( I  e.  Fin  ->  (
( sqr `  ( # `
 I ) )  +  1 )  e.  RR+ )
14 1rp 10374 . . 3  |-  1  e.  RR+
1514a1i 10 . 2  |-  ( I  e.  Fin  ->  1  e.  RR+ )
16 metcl 17913 . . . . 5  |-  ( ( ( Rn `  I
)  e.  ( Met `  X )  /\  x  e.  X  /\  y  e.  X )  ->  (
x ( Rn `  I ) y )  e.  RR )
17163expb 1152 . . . 4  |-  ( ( ( Rn `  I
)  e.  ( Met `  X )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( x
( Rn `  I
) y )  e.  RR )
185, 17sylan 457 . . 3  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
x ( Rn `  I ) y )  e.  RR )
1910adantr 451 . . . 4  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  ( sqr `  ( # `  I
) )  e.  RR )
204adantr 451 . . . . . 6  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  ( dist `  ( (flds  RR )  ^s  I ) )  e.  ( Met `  X ) )
21 simprl 732 . . . . . 6  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  x  e.  X )
22 simprr 733 . . . . . 6  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  y  e.  X )
23 metcl 17913 . . . . . . 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 17926 . . . . . . 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 518 . . . . . 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 1182 . . . . 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 445 . . . 4  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
x ( dist `  (
(flds  RR )  ^s  I ) ) y )  e.  RR )
2819, 27remulcld 8879 . . 3  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
( sqr `  ( # `
 I ) )  x.  ( x (
dist `  ( (flds  RR )  ^s  I ) ) y ) )  e.  RR )
29 peano2re 9001 . . . . . 6  |-  ( ( sqr `  ( # `  I ) )  e.  RR  ->  ( ( sqr `  ( # `  I
) )  +  1 )  e.  RR )
3010, 29syl 15 . . . . 5  |-  ( I  e.  Fin  ->  (
( sqr `  ( # `
 I ) )  +  1 )  e.  RR )
3130adantr 451 . . . 4  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
( sqr `  ( # `
 I ) )  +  1 )  e.  RR )
3231, 27remulcld 8879 . . 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 19 . . . . 5  |-  ( I  e.  Fin  ->  I  e.  Fin )
341, 2, 3, 33rrnequiv 26662 . . . 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 449 . . 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 9704 . . . 4  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  ( sqr `  ( # `  I
) )  <_  (
( sqr `  ( # `
 I ) )  +  1 ) )
37 lemul1a 9626 . . . 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 1185 . . 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 8989 . 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 445 . . 3  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
x ( dist `  (
(flds  RR )  ^s  I ) ) y )  <_  ( x
( Rn `  I
) y ) )
4118recnd 8877 . . . 4  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
x ( Rn `  I ) y )  e.  CC )
4241mulid2d 8869 . . 3  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
1  x.  ( x ( Rn `  I
) y ) )  =  ( x ( Rn `  I ) y ) )
4340, 42breqtrrd 4065 . 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 2296 . 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 8810 . . 3  |-  RR  C_  CC
471, 44cnpwstotbnd 26624 . . 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 651 . 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 26619 1  |-  ( I  e.  Fin  ->  ( M  e.  ( TotBnd `  Y )  <->  M  e.  ( Bnd `  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    C_ wss 3165   class class class wbr 4039    X. cxp 4703    |` cres 4707   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   Fincfn 6879   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    <_ cle 8884   NN0cn0 9981   RR+crp 10370   #chash 11353   sqrcsqr 11734   ↾s cress 13165   distcds 13233    ^s cpws 13363   Metcme 16386  ℂfldccnfld 16393   TotBndctotbnd 26593   Bndcbnd 26594   Rncrrn 26652
This theorem is referenced by:  rrnheibor  26664
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-ec 6678  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-gz 12993  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-prds 13364  df-pws 13366  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-xms 17901  df-ms 17902  df-totbnd 26595  df-bnd 26606  df-rrn 26653
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