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Theorem cnpwstotbnd 26396
Description: A subset of  A ^
I, where  A  C_  CC, is totally bounded iff it is bounded. (Contributed by Mario Carneiro, 14-Sep-2015.)
Hypotheses
Ref Expression
cnpwstotbnd.y  |-  Y  =  ( (flds  A )  ^s  I )
cnpwstotbnd.d  |-  D  =  ( ( dist `  Y
)  |`  ( X  X.  X ) )
Assertion
Ref Expression
cnpwstotbnd  |-  ( ( A  C_  CC  /\  I  e.  Fin )  ->  ( D  e.  ( TotBnd `  X )  <->  D  e.  ( Bnd `  X ) ) )

Proof of Theorem cnpwstotbnd
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2404 . . 3  |-  ( (Scalar `  (flds  A ) ) X_s ( I  X.  {
(flds  A
) } ) )  =  ( (Scalar `  (flds  A
) ) X_s ( I  X.  {
(flds  A
) } ) )
2 eqid 2404 . . 3  |-  ( Base `  ( (Scalar `  (flds  A )
) X_s ( I  X.  {
(flds  A
) } ) ) )  =  ( Base `  ( (Scalar `  (flds  A )
) X_s ( I  X.  {
(flds  A
) } ) ) )
3 eqid 2404 . . 3  |-  ( Base `  ( ( I  X.  { (flds  A ) } ) `  x ) )  =  ( Base `  (
( I  X.  {
(flds  A
) } ) `  x ) )
4 eqid 2404 . . 3  |-  ( (
dist `  ( (
I  X.  { (flds  A ) } ) `  x
) )  |`  (
( Base `  ( (
I  X.  { (flds  A ) } ) `  x
) )  X.  ( Base `  ( ( I  X.  { (flds  A ) } ) `
 x ) ) ) )  =  ( ( dist `  (
( I  X.  {
(flds  A
) } ) `  x ) )  |`  ( ( Base `  (
( I  X.  {
(flds  A
) } ) `  x ) )  X.  ( Base `  (
( I  X.  {
(flds  A
) } ) `  x ) ) ) )
5 eqid 2404 . . 3  |-  ( dist `  ( (Scalar `  (flds  A )
) X_s ( I  X.  {
(flds  A
) } ) ) )  =  ( dist `  ( (Scalar `  (flds  A )
) X_s ( I  X.  {
(flds  A
) } ) ) )
6 fvex 5701 . . . 4  |-  (Scalar `  (flds  A
) )  e.  _V
76a1i 11 . . 3  |-  ( ( A  C_  CC  /\  I  e.  Fin )  ->  (Scalar `  (flds  A ) )  e.  _V )
8 simpr 448 . . 3  |-  ( ( A  C_  CC  /\  I  e.  Fin )  ->  I  e.  Fin )
9 ovex 6065 . . . 4  |-  (flds  A )  e.  _V
10 fnconstg 5590 . . . 4  |-  ( (flds  A )  e.  _V  ->  (
I  X.  { (flds  A ) } )  Fn  I
)
119, 10mp1i 12 . . 3  |-  ( ( A  C_  CC  /\  I  e.  Fin )  ->  (
I  X.  { (flds  A ) } )  Fn  I
)
12 eqid 2404 . . 3  |-  ( (
dist `  ( (Scalar `  (flds  A ) ) X_s ( I  X.  {
(flds  A
) } ) ) )  |`  ( X  X.  X ) )  =  ( ( dist `  (
(Scalar `  (flds  A ) ) X_s (
I  X.  { (flds  A ) } ) ) )  |`  ( X  X.  X
) )
13 cnfldms 18763 . . . . . 6  |-fld  e.  MetSp
14 cnex 9027 . . . . . . . 8  |-  CC  e.  _V
1514ssex 4307 . . . . . . 7  |-  ( A 
C_  CC  ->  A  e. 
_V )
1615ad2antrr 707 . . . . . 6  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  A  e.  _V )
17 ressms 18509 . . . . . 6  |-  ( (fld  e. 
MetSp  /\  A  e.  _V )  ->  (flds  A )  e.  MetSp )
1813, 16, 17sylancr 645 . . . . 5  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  (flds  A )  e.  MetSp )
19 eqid 2404 . . . . . 6  |-  ( Base `  (flds  A ) )  =  (
Base `  (flds  A ) )
20 eqid 2404 . . . . . 6  |-  ( (
dist `  (flds  A ) )  |`  ( ( Base `  (flds  A )
)  X.  ( Base `  (flds  A ) ) ) )  =  ( ( dist `  (flds  A ) )  |`  (
( Base `  (flds  A ) )  X.  ( Base `  (flds  A )
) ) )
2119, 20msmet 18440 . . . . 5  |-  ( (flds  A )  e.  MetSp  ->  ( ( dist `  (flds  A ) )  |`  (
( Base `  (flds  A ) )  X.  ( Base `  (flds  A )
) ) )  e.  ( Met `  ( Base `  (flds  A ) ) ) )
2218, 21syl 16 . . . 4  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( ( dist `  (flds  A ) )  |`  (
( Base `  (flds  A ) )  X.  ( Base `  (flds  A )
) ) )  e.  ( Met `  ( Base `  (flds  A ) ) ) )
239fvconst2 5906 . . . . . . 7  |-  ( x  e.  I  ->  (
( I  X.  {
(flds  A
) } ) `  x )  =  (flds  A ) )
2423adantl 453 . . . . . 6  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( (
I  X.  { (flds  A ) } ) `  x
)  =  (flds  A ) )
2524fveq2d 5691 . . . . 5  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( dist `  ( ( I  X.  { (flds  A ) } ) `  x ) )  =  ( dist `  (flds  A )
) )
2624fveq2d 5691 . . . . . 6  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( Base `  ( ( I  X.  { (flds  A ) } ) `  x ) )  =  ( Base `  (flds  A )
) )
2726, 26xpeq12d 4862 . . . . 5  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( ( Base `  ( ( I  X.  { (flds  A ) } ) `
 x ) )  X.  ( Base `  (
( I  X.  {
(flds  A
) } ) `  x ) ) )  =  ( ( Base `  (flds  A ) )  X.  ( Base `  (flds  A ) ) ) )
2825, 27reseq12d 5106 . . . 4  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( ( dist `  ( ( I  X.  { (flds  A ) } ) `
 x ) )  |`  ( ( Base `  (
( I  X.  {
(flds  A
) } ) `  x ) )  X.  ( Base `  (
( I  X.  {
(flds  A
) } ) `  x ) ) ) )  =  ( (
dist `  (flds  A ) )  |`  ( ( Base `  (flds  A )
)  X.  ( Base `  (flds  A ) ) ) ) )
2926fveq2d 5691 . . . 4  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( Met `  ( Base `  (
( I  X.  {
(flds  A
) } ) `  x ) ) )  =  ( Met `  ( Base `  (flds  A ) ) ) )
3022, 28, 293eltr4d 2485 . . 3  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( ( dist `  ( ( I  X.  { (flds  A ) } ) `
 x ) )  |`  ( ( Base `  (
( I  X.  {
(flds  A
) } ) `  x ) )  X.  ( Base `  (
( I  X.  {
(flds  A
) } ) `  x ) ) ) )  e.  ( Met `  ( Base `  (
( I  X.  {
(flds  A
) } ) `  x ) ) ) )
31 totbndbnd 26388 . . . . . 6  |-  ( ( ( ( dist `  (flds  A )
)  |`  ( ( Base `  (flds  A ) )  X.  ( Base `  (flds  A ) ) ) )  |`  ( y  X.  y
) )  e.  (
TotBnd `  y )  -> 
( ( ( dist `  (flds  A ) )  |`  (
( Base `  (flds  A ) )  X.  ( Base `  (flds  A )
) ) )  |`  ( y  X.  y
) )  e.  ( Bnd `  y ) )
32 eqid 2404 . . . . . . . . . . 11  |-  (flds  A )  =  (flds  A )
33 cnfldbas 16662 . . . . . . . . . . 11  |-  CC  =  ( Base ` fld )
3432, 33ressbas2 13475 . . . . . . . . . 10  |-  ( A 
C_  CC  ->  A  =  ( Base `  (flds  A )
) )
3534ad2antrr 707 . . . . . . . . 9  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  A  =  ( Base `  (flds  A ) ) )
3635fveq2d 5691 . . . . . . . 8  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( Met `  A )  =  ( Met `  ( Base `  (flds  A ) ) ) )
3722, 36eleqtrrd 2481 . . . . . . 7  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( ( dist `  (flds  A ) )  |`  (
( Base `  (flds  A ) )  X.  ( Base `  (flds  A )
) ) )  e.  ( Met `  A
) )
38 eqid 2404 . . . . . . . . 9  |-  ( ( ( dist `  (flds  A )
)  |`  ( ( Base `  (flds  A ) )  X.  ( Base `  (flds  A ) ) ) )  |`  ( y  X.  y
) )  =  ( ( ( dist `  (flds  A )
)  |`  ( ( Base `  (flds  A ) )  X.  ( Base `  (flds  A ) ) ) )  |`  ( y  X.  y
) )
3938bnd2lem 26390 . . . . . . . 8  |-  ( ( ( ( dist `  (flds  A )
)  |`  ( ( Base `  (flds  A ) )  X.  ( Base `  (flds  A ) ) ) )  e.  ( Met `  A
)  /\  ( (
( dist `  (flds  A ) )  |`  ( ( Base `  (flds  A )
)  X.  ( Base `  (flds  A ) ) ) )  |`  ( y  X.  y
) )  e.  ( Bnd `  y ) )  ->  y  C_  A )
4039ex 424 . . . . . . 7  |-  ( ( ( dist `  (flds  A )
)  |`  ( ( Base `  (flds  A ) )  X.  ( Base `  (flds  A ) ) ) )  e.  ( Met `  A
)  ->  ( (
( ( dist `  (flds  A )
)  |`  ( ( Base `  (flds  A ) )  X.  ( Base `  (flds  A ) ) ) )  |`  ( y  X.  y
) )  e.  ( Bnd `  y )  ->  y  C_  A
) )
4137, 40syl 16 . . . . . 6  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( (
( ( dist `  (flds  A )
)  |`  ( ( Base `  (flds  A ) )  X.  ( Base `  (flds  A ) ) ) )  |`  ( y  X.  y
) )  e.  ( Bnd `  y )  ->  y  C_  A
) )
4231, 41syl5 30 . . . . 5  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( (
( ( dist `  (flds  A )
)  |`  ( ( Base `  (flds  A ) )  X.  ( Base `  (flds  A ) ) ) )  |`  ( y  X.  y
) )  e.  (
TotBnd `  y )  -> 
y  C_  A )
)
43 eqid 2404 . . . . . . . . 9  |-  ( ( abs  o.  -  )  |`  ( y  X.  y
) )  =  ( ( abs  o.  -  )  |`  ( y  X.  y ) )
4443cntotbnd 26395 . . . . . . . 8  |-  ( ( ( abs  o.  -  )  |`  ( y  X.  y ) )  e.  ( TotBnd `  y )  <->  ( ( abs  o.  -  )  |`  ( y  X.  y ) )  e.  ( Bnd `  y
) )
4544a1i 11 . . . . . . 7  |-  ( ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  /\  y  C_  A )  ->  (
( ( abs  o.  -  )  |`  ( y  X.  y ) )  e.  ( TotBnd `  y
)  <->  ( ( abs 
o.  -  )  |`  (
y  X.  y ) )  e.  ( Bnd `  y ) ) )
4635sseq2d 3336 . . . . . . . . . . . 12  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( y  C_  A  <->  y  C_  ( Base `  (flds  A ) ) ) )
4746biimpa 471 . . . . . . . . . . 11  |-  ( ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  /\  y  C_  A )  ->  y  C_  ( Base `  (flds  A )
) )
48 xpss12 4940 . . . . . . . . . . 11  |-  ( ( y  C_  ( Base `  (flds  A ) )  /\  y  C_  ( Base `  (flds  A )
) )  ->  (
y  X.  y ) 
C_  ( ( Base `  (flds  A ) )  X.  ( Base `  (flds  A ) ) ) )
4947, 47, 48syl2anc 643 . . . . . . . . . 10  |-  ( ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  /\  y  C_  A )  ->  (
y  X.  y ) 
C_  ( ( Base `  (flds  A ) )  X.  ( Base `  (flds  A ) ) ) )
50 resabs1 5134 . . . . . . . . . 10  |-  ( ( y  X.  y ) 
C_  ( ( Base `  (flds  A ) )  X.  ( Base `  (flds  A ) ) )  -> 
( ( ( dist `  (flds  A ) )  |`  (
( Base `  (flds  A ) )  X.  ( Base `  (flds  A )
) ) )  |`  ( y  X.  y
) )  =  ( ( dist `  (flds  A )
)  |`  ( y  X.  y ) ) )
5149, 50syl 16 . . . . . . . . 9  |-  ( ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  /\  y  C_  A )  ->  (
( ( dist `  (flds  A )
)  |`  ( ( Base `  (flds  A ) )  X.  ( Base `  (flds  A ) ) ) )  |`  ( y  X.  y
) )  =  ( ( dist `  (flds  A )
)  |`  ( y  X.  y ) ) )
5216adantr 452 . . . . . . . . . . 11  |-  ( ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  /\  y  C_  A )  ->  A  e.  _V )
53 cnfldds 16668 . . . . . . . . . . . 12  |-  ( abs 
o.  -  )  =  ( dist ` fld )
5432, 53ressds 13596 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  ( abs  o.  -  )  =  ( dist `  (flds  A )
) )
5552, 54syl 16 . . . . . . . . . 10  |-  ( ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  /\  y  C_  A )  ->  ( abs  o.  -  )  =  ( dist `  (flds  A )
) )
5655reseq1d 5104 . . . . . . . . 9  |-  ( ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  /\  y  C_  A )  ->  (
( abs  o.  -  )  |`  ( y  X.  y
) )  =  ( ( dist `  (flds  A )
)  |`  ( y  X.  y ) ) )
5751, 56eqtr4d 2439 . . . . . . . 8  |-  ( ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  /\  y  C_  A )  ->  (
( ( dist `  (flds  A )
)  |`  ( ( Base `  (flds  A ) )  X.  ( Base `  (flds  A ) ) ) )  |`  ( y  X.  y
) )  =  ( ( abs  o.  -  )  |`  ( y  X.  y ) ) )
5857eleq1d 2470 . . . . . . 7  |-  ( ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  /\  y  C_  A )  ->  (
( ( ( dist `  (flds  A ) )  |`  (
( Base `  (flds  A ) )  X.  ( Base `  (flds  A )
) ) )  |`  ( y  X.  y
) )  e.  (
TotBnd `  y )  <->  ( ( abs  o.  -  )  |`  ( y  X.  y
) )  e.  (
TotBnd `  y ) ) )
5957eleq1d 2470 . . . . . . 7  |-  ( ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  /\  y  C_  A )  ->  (
( ( ( dist `  (flds  A ) )  |`  (
( Base `  (flds  A ) )  X.  ( Base `  (flds  A )
) ) )  |`  ( y  X.  y
) )  e.  ( Bnd `  y )  <-> 
( ( abs  o.  -  )  |`  ( y  X.  y ) )  e.  ( Bnd `  y
) ) )
6045, 58, 593bitr4d 277 . . . . . 6  |-  ( ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  /\  y  C_  A )  ->  (
( ( ( dist `  (flds  A ) )  |`  (
( Base `  (flds  A ) )  X.  ( Base `  (flds  A )
) ) )  |`  ( y  X.  y
) )  e.  (
TotBnd `  y )  <->  ( (
( dist `  (flds  A ) )  |`  ( ( Base `  (flds  A )
)  X.  ( Base `  (flds  A ) ) ) )  |`  ( y  X.  y
) )  e.  ( Bnd `  y ) ) )
6160ex 424 . . . . 5  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( y  C_  A  ->  ( (
( ( dist `  (flds  A )
)  |`  ( ( Base `  (flds  A ) )  X.  ( Base `  (flds  A ) ) ) )  |`  ( y  X.  y
) )  e.  (
TotBnd `  y )  <->  ( (
( dist `  (flds  A ) )  |`  ( ( Base `  (flds  A )
)  X.  ( Base `  (flds  A ) ) ) )  |`  ( y  X.  y
) )  e.  ( Bnd `  y ) ) ) )
6242, 41, 61pm5.21ndd 344 . . . 4  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( (
( ( dist `  (flds  A )
)  |`  ( ( Base `  (flds  A ) )  X.  ( Base `  (flds  A ) ) ) )  |`  ( y  X.  y
) )  e.  (
TotBnd `  y )  <->  ( (
( dist `  (flds  A ) )  |`  ( ( Base `  (flds  A )
)  X.  ( Base `  (flds  A ) ) ) )  |`  ( y  X.  y
) )  e.  ( Bnd `  y ) ) )
6328reseq1d 5104 . . . . 5  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( (
( dist `  ( (
I  X.  { (flds  A ) } ) `  x
) )  |`  (
( Base `  ( (
I  X.  { (flds  A ) } ) `  x
) )  X.  ( Base `  ( ( I  X.  { (flds  A ) } ) `
 x ) ) ) )  |`  (
y  X.  y ) )  =  ( ( ( dist `  (flds  A )
)  |`  ( ( Base `  (flds  A ) )  X.  ( Base `  (flds  A ) ) ) )  |`  ( y  X.  y
) ) )
6463eleq1d 2470 . . . 4  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( (
( ( dist `  (
( I  X.  {
(flds  A
) } ) `  x ) )  |`  ( ( Base `  (
( I  X.  {
(flds  A
) } ) `  x ) )  X.  ( Base `  (
( I  X.  {
(flds  A
) } ) `  x ) ) ) )  |`  ( y  X.  y ) )  e.  ( TotBnd `  y )  <->  ( ( ( dist `  (flds  A )
)  |`  ( ( Base `  (flds  A ) )  X.  ( Base `  (flds  A ) ) ) )  |`  ( y  X.  y
) )  e.  (
TotBnd `  y ) ) )
6563eleq1d 2470 . . . 4  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( (
( ( dist `  (
( I  X.  {
(flds  A
) } ) `  x ) )  |`  ( ( Base `  (
( I  X.  {
(flds  A
) } ) `  x ) )  X.  ( Base `  (
( I  X.  {
(flds  A
) } ) `  x ) ) ) )  |`  ( y  X.  y ) )  e.  ( Bnd `  y
)  <->  ( ( (
dist `  (flds  A ) )  |`  ( ( Base `  (flds  A )
)  X.  ( Base `  (flds  A ) ) ) )  |`  ( y  X.  y
) )  e.  ( Bnd `  y ) ) )
6662, 64, 653bitr4d 277 . . 3  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( (
( ( dist `  (
( I  X.  {
(flds  A
) } ) `  x ) )  |`  ( ( Base `  (
( I  X.  {
(flds  A
) } ) `  x ) )  X.  ( Base `  (
( I  X.  {
(flds  A
) } ) `  x ) ) ) )  |`  ( y  X.  y ) )  e.  ( TotBnd `  y )  <->  ( ( ( dist `  (
( I  X.  {
(flds  A
) } ) `  x ) )  |`  ( ( Base `  (
( I  X.  {
(flds  A
) } ) `  x ) )  X.  ( Base `  (
( I  X.  {
(flds  A
) } ) `  x ) ) ) )  |`  ( y  X.  y ) )  e.  ( Bnd `  y
) ) )
671, 2, 3, 4, 5, 7, 8, 11, 12, 30, 66prdsbnd2 26394 . 2  |-  ( ( A  C_  CC  /\  I  e.  Fin )  ->  (
( ( dist `  (
(Scalar `  (flds  A ) ) X_s (
I  X.  { (flds  A ) } ) ) )  |`  ( X  X.  X
) )  e.  (
TotBnd `  X )  <->  ( ( dist `  ( (Scalar `  (flds  A
) ) X_s ( I  X.  {
(flds  A
) } ) ) )  |`  ( X  X.  X ) )  e.  ( Bnd `  X
) ) )
68 cnpwstotbnd.d . . . 4  |-  D  =  ( ( dist `  Y
)  |`  ( X  X.  X ) )
69 cnpwstotbnd.y . . . . . . . 8  |-  Y  =  ( (flds  A )  ^s  I )
70 eqid 2404 . . . . . . . 8  |-  (Scalar `  (flds  A
) )  =  (Scalar `  (flds  A ) )
7169, 70pwsval 13663 . . . . . . 7  |-  ( ( (flds  A )  e.  _V  /\  I  e.  Fin )  ->  Y  =  ( (Scalar `  (flds  A ) ) X_s ( I  X.  {
(flds  A
) } ) ) )
729, 8, 71sylancr 645 . . . . . 6  |-  ( ( A  C_  CC  /\  I  e.  Fin )  ->  Y  =  ( (Scalar `  (flds  A
) ) X_s ( I  X.  {
(flds  A
) } ) ) )
7372fveq2d 5691 . . . . 5  |-  ( ( A  C_  CC  /\  I  e.  Fin )  ->  ( dist `  Y )  =  ( dist `  (
(Scalar `  (flds  A ) ) X_s (
I  X.  { (flds  A ) } ) ) ) )
7473reseq1d 5104 . . . 4  |-  ( ( A  C_  CC  /\  I  e.  Fin )  ->  (
( dist `  Y )  |`  ( X  X.  X
) )  =  ( ( dist `  (
(Scalar `  (flds  A ) ) X_s (
I  X.  { (flds  A ) } ) ) )  |`  ( X  X.  X
) ) )
7568, 74syl5eq 2448 . . 3  |-  ( ( A  C_  CC  /\  I  e.  Fin )  ->  D  =  ( ( dist `  ( (Scalar `  (flds  A )
) X_s ( I  X.  {
(flds  A
) } ) ) )  |`  ( X  X.  X ) ) )
7675eleq1d 2470 . 2  |-  ( ( A  C_  CC  /\  I  e.  Fin )  ->  ( D  e.  ( TotBnd `  X )  <->  ( ( dist `  ( (Scalar `  (flds  A
) ) X_s ( I  X.  {
(flds  A
) } ) ) )  |`  ( X  X.  X ) )  e.  ( TotBnd `  X )
) )
7775eleq1d 2470 . 2  |-  ( ( A  C_  CC  /\  I  e.  Fin )  ->  ( D  e.  ( Bnd `  X )  <->  ( ( dist `  ( (Scalar `  (flds  A
) ) X_s ( I  X.  {
(flds  A
) } ) ) )  |`  ( X  X.  X ) )  e.  ( Bnd `  X
) ) )
7867, 76, 773bitr4d 277 1  |-  ( ( A  C_  CC  /\  I  e.  Fin )  ->  ( D  e.  ( TotBnd `  X )  <->  D  e.  ( Bnd `  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916    C_ wss 3280   {csn 3774    X. cxp 4835    |` cres 4839    o. ccom 4841    Fn wfn 5408   ` cfv 5413  (class class class)co 6040   Fincfn 7068   CCcc 8944    - cmin 9247   abscabs 11994   Basecbs 13424   ↾s cress 13425  Scalarcsca 13487   distcds 13493   X_scprds 13624    ^s cpws 13625   Metcme 16642  ℂfldccnfld 16658   MetSpcmt 18301   TotBndctotbnd 26365   Bndcbnd 26366
This theorem is referenced by:  rrntotbnd  26435
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-ec 6866  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-icc 10879  df-fz 11000  df-fl 11157  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-gz 13253  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-prds 13626  df-pws 13628  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-xms 18303  df-ms 18304  df-totbnd 26367  df-bnd 26378
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