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Theorem rrntotbnd 28740
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 2443 . . 3  |-  ( (flds  RR )  ^s  I )  =  ( (flds  RR )  ^s  I )
2 eqid 2443 . . 3  |-  ( dist `  ( (flds  RR )  ^s  I ) )  =  ( dist `  (
(flds  RR )  ^s  I ) )
3 rrntotbnd.1 . . 3  |-  X  =  ( RR  ^m  I
)
41, 2, 3repwsmet 28738 . 2  |-  ( I  e.  Fin  ->  ( dist `  ( (flds  RR )  ^s  I ) )  e.  ( Met `  X ) )
53rrnmet 28733 . 2  |-  ( I  e.  Fin  ->  ( Rn `  I )  e.  ( Met `  X
) )
6 hashcl 12131 . . . 4  |-  ( I  e.  Fin  ->  ( # `
 I )  e. 
NN0 )
7 nn0re 10593 . . . . 5  |-  ( (
# `  I )  e.  NN0  ->  ( # `  I
)  e.  RR )
8 nn0ge0 10610 . . . . 5  |-  ( (
# `  I )  e.  NN0  ->  0  <_  (
# `  I )
)
97, 8resqrcld 12909 . . . 4  |-  ( (
# `  I )  e.  NN0  ->  ( sqr `  ( # `  I
) )  e.  RR )
106, 9syl 16 . . 3  |-  ( I  e.  Fin  ->  ( sqr `  ( # `  I
) )  e.  RR )
117, 8sqrge0d 12912 . . . 4  |-  ( (
# `  I )  e.  NN0  ->  0  <_  ( sqr `  ( # `  I ) ) )
126, 11syl 16 . . 3  |-  ( I  e.  Fin  ->  0  <_  ( sqr `  ( # `
 I ) ) )
1310, 12ge0p1rpd 11058 . 2  |-  ( I  e.  Fin  ->  (
( sqr `  ( # `
 I ) )  +  1 )  e.  RR+ )
14 1rp 11000 . . 3  |-  1  e.  RR+
1514a1i 11 . 2  |-  ( I  e.  Fin  ->  1  e.  RR+ )
16 metcl 19912 . . . . 5  |-  ( ( ( Rn `  I
)  e.  ( Met `  X )  /\  x  e.  X  /\  y  e.  X )  ->  (
x ( Rn `  I ) y )  e.  RR )
17163expb 1188 . . . 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 19912 . . . . . . 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 19925 . . . . . . 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 1218 . . . . 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 9419 . . 3  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
( sqr `  ( # `
 I ) )  x.  ( x (
dist `  ( (flds  RR )  ^s  I ) ) y ) )  e.  RR )
29 peano2re 9547 . . . . . 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 9419 . . 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 28739 . . . 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 10269 . . . 4  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  ( sqr `  ( # `  I
) )  <_  (
( sqr `  ( # `
 I ) )  +  1 ) )
37 lemul1a 10188 . . . 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 1221 . . 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 9533 . 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 9417 . . . 4  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
x ( Rn `  I ) y )  e.  CC )
4241mulid2d 9409 . . 3  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
1  x.  ( x ( Rn `  I
) y ) )  =  ( x ( Rn `  I ) y ) )
4340, 42breqtrrd 4323 . 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 2443 . 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 9344 . . 3  |-  RR  C_  CC
471, 44cnpwstotbnd 28701 . . 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 28696 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 965    = wceq 1369    e. wcel 1756    C_ wss 3333   class class class wbr 4297    X. cxp 4843    |` cres 4847   ` cfv 5423  (class class class)co 6096    ^m cmap 7219   Fincfn 7315   CCcc 9285   RRcr 9286   0cc0 9287   1c1 9288    + caddc 9290    x. cmul 9292    <_ cle 9424   NN0cn0 10584   RR+crp 10996   #chash 12108   sqrcsqr 12727   ↾s cress 14180   distcds 14252    ^s cpws 14390   Metcme 17807  ℂfldccnfld 17823   TotBndctotbnd 28670   Bndcbnd 28671   Rncrrn 28729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-ec 7108  df-map 7221  df-pm 7222  df-ixp 7269  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-oi 7729  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-q 10959  df-rp 10997  df-xneg 11094  df-xadd 11095  df-xmul 11096  df-ico 11311  df-icc 11312  df-fz 11443  df-fzo 11554  df-fl 11647  df-seq 11812  df-exp 11871  df-hash 12109  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-clim 12971  df-sum 13169  df-gz 13996  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-starv 14258  df-sca 14259  df-vsca 14260  df-ip 14261  df-tset 14262  df-ple 14263  df-ds 14265  df-unif 14266  df-hom 14267  df-cco 14268  df-rest 14366  df-topn 14367  df-topgen 14387  df-prds 14391  df-pws 14393  df-psmet 17814  df-xmet 17815  df-met 17816  df-bl 17817  df-mopn 17818  df-cnfld 17824  df-top 18508  df-bases 18510  df-topon 18511  df-topsp 18512  df-xms 19900  df-ms 19901  df-totbnd 28672  df-bnd 28683  df-rrn 28730
This theorem is referenced by:  rrnheibor  28741
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