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Theorem rrntotbnd 30572
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 2454 . . 3  |-  ( (flds  RR )  ^s  I )  =  ( (flds  RR )  ^s  I )
2 eqid 2454 . . 3  |-  ( dist `  ( (flds  RR )  ^s  I ) )  =  ( dist `  (
(flds  RR )  ^s  I ) )
3 rrntotbnd.1 . . 3  |-  X  =  ( RR  ^m  I
)
41, 2, 3repwsmet 30570 . 2  |-  ( I  e.  Fin  ->  ( dist `  ( (flds  RR )  ^s  I ) )  e.  ( Met `  X ) )
53rrnmet 30565 . 2  |-  ( I  e.  Fin  ->  ( Rn `  I )  e.  ( Met `  X
) )
6 hashcl 12410 . . . 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 13331 . . . 4  |-  ( (
# `  I )  e.  NN0  ->  ( sqr `  ( # `  I
) )  e.  RR )
106, 9syl 16 . . 3  |-  ( I  e.  Fin  ->  ( sqr `  ( # `  I
) )  e.  RR )
117, 8sqrtge0d 13334 . . . 4  |-  ( (
# `  I )  e.  NN0  ->  0  <_  ( sqr `  ( # `  I ) ) )
126, 11syl 16 . . 3  |-  ( I  e.  Fin  ->  0  <_  ( sqr `  ( # `
 I ) ) )
1310, 12ge0p1rpd 11285 . 2  |-  ( I  e.  Fin  ->  (
( sqr `  ( # `
 I ) )  +  1 )  e.  RR+ )
14 1rp 11225 . . 3  |-  1  e.  RR+
1514a1i 11 . 2  |-  ( I  e.  Fin  ->  1  e.  RR+ )
16 metcl 21001 . . . . 5  |-  ( ( ( Rn `  I
)  e.  ( Met `  X )  /\  x  e.  X  /\  y  e.  X )  ->  (
x ( Rn `  I ) y )  e.  RR )
17163expb 1195 . . . 4  |-  ( ( ( Rn `  I
)  e.  ( Met `  X )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( x
( Rn `  I
) y )  e.  RR )
185, 17sylan 469 . . 3  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
x ( Rn `  I ) y )  e.  RR )
1910adantr 463 . . . 4  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  ( sqr `  ( # `  I
) )  e.  RR )
204adantr 463 . . . . . 6  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  ( dist `  ( (flds  RR )  ^s  I ) )  e.  ( Met `  X ) )
21 simprl 754 . . . . . 6  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  x  e.  X )
22 simprr 755 . . . . . 6  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  y  e.  X )
23 metcl 21001 . . . . . . 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 21014 . . . . . . 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 530 . . . . . 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 1226 . . . . 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 457 . . . 4  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
x ( dist `  (
(flds  RR )  ^s  I ) ) y )  e.  RR )
2819, 27remulcld 9613 . . 3  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
( sqr `  ( # `
 I ) )  x.  ( x (
dist `  ( (flds  RR )  ^s  I ) ) y ) )  e.  RR )
29 peano2re 9742 . . . . . 6  |-  ( ( sqr `  ( # `  I ) )  e.  RR  ->  ( ( sqr `  ( # `  I
) )  +  1 )  e.  RR )
3010, 29syl 16 . . . . 5  |-  ( I  e.  Fin  ->  (
( sqr `  ( # `
 I ) )  +  1 )  e.  RR )
3130adantr 463 . . . 4  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
( sqr `  ( # `
 I ) )  +  1 )  e.  RR )
3231, 27remulcld 9613 . . 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 30571 . . . 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 461 . . 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 10472 . . . 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 1229 . . 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 9728 . 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 457 . . 3  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
x ( dist `  (
(flds  RR )  ^s  I ) ) y )  <_  ( x
( Rn `  I
) y ) )
4118recnd 9611 . . . 4  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
x ( Rn `  I ) y )  e.  CC )
4241mulid2d 9603 . . 3  |-  ( ( I  e.  Fin  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
1  x.  ( x ( Rn `  I
) y ) )  =  ( x ( Rn `  I ) y ) )
4340, 42breqtrrd 4465 . 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 2454 . 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 9538 . . 3  |-  RR  C_  CC
471, 44cnpwstotbnd 30533 . . 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 668 . 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 30528 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    C_ wss 3461   class class class wbr 4439    X. cxp 4986    |` cres 4990   ` cfv 5570  (class class class)co 6270    ^m cmap 7412   Fincfn 7509   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486    <_ cle 9618   NN0cn0 10791   RR+crp 11221   #chash 12387   sqrcsqrt 13148   ↾s cress 14717   distcds 14793    ^s cpws 14936   Metcme 18599  ℂfldccnfld 18615   TotBndctotbnd 30502   Bndcbnd 30503   Rncrrn 30561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-ec 7305  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  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 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-fl 11910  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-clim 13393  df-sum 13591  df-gz 14532  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-starv 14799  df-sca 14800  df-vsca 14801  df-ip 14802  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-hom 14808  df-cco 14809  df-rest 14912  df-topn 14913  df-topgen 14933  df-prds 14937  df-pws 14939  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-cnfld 18616  df-top 19566  df-bases 19568  df-topon 19569  df-topsp 19570  df-xms 20989  df-ms 20990  df-totbnd 30504  df-bnd 30515  df-rrn 30562
This theorem is referenced by:  rrnheibor  30573
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