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Theorem cnpwstotbnd 28649
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 2438 . . 3  |-  ( (Scalar `  (flds  A ) ) X_s ( I  X.  {
(flds  A
) } ) )  =  ( (Scalar `  (flds  A
) ) X_s ( I  X.  {
(flds  A
) } ) )
2 eqid 2438 . . 3  |-  ( Base `  ( (Scalar `  (flds  A )
) X_s ( I  X.  {
(flds  A
) } ) ) )  =  ( Base `  ( (Scalar `  (flds  A )
) X_s ( I  X.  {
(flds  A
) } ) ) )
3 eqid 2438 . . 3  |-  ( Base `  ( ( I  X.  { (flds  A ) } ) `  x ) )  =  ( Base `  (
( I  X.  {
(flds  A
) } ) `  x ) )
4 eqid 2438 . . 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 2438 . . 3  |-  ( dist `  ( (Scalar `  (flds  A )
) X_s ( I  X.  {
(flds  A
) } ) ) )  =  ( dist `  ( (Scalar `  (flds  A )
) X_s ( I  X.  {
(flds  A
) } ) ) )
6 fvex 5696 . . . 4  |-  (Scalar `  (flds  A
) )  e.  _V
76a1i 11 . . 3  |-  ( ( A  C_  CC  /\  I  e.  Fin )  ->  (Scalar `  (flds  A ) )  e.  _V )
8 simpr 461 . . 3  |-  ( ( A  C_  CC  /\  I  e.  Fin )  ->  I  e.  Fin )
9 ovex 6111 . . . 4  |-  (flds  A )  e.  _V
10 fnconstg 5593 . . . 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 2438 . . 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 20330 . . . . . 6  |-fld  e.  MetSp
14 cnex 9355 . . . . . . . 8  |-  CC  e.  _V
1514ssex 4431 . . . . . . 7  |-  ( A 
C_  CC  ->  A  e. 
_V )
1615ad2antrr 725 . . . . . 6  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  A  e.  _V )
17 ressms 20076 . . . . . 6  |-  ( (fld  e. 
MetSp  /\  A  e.  _V )  ->  (flds  A )  e.  MetSp )
1813, 16, 17sylancr 663 . . . . 5  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  (flds  A )  e.  MetSp )
19 eqid 2438 . . . . . 6  |-  ( Base `  (flds  A ) )  =  (
Base `  (flds  A ) )
20 eqid 2438 . . . . . 6  |-  ( (
dist `  (flds  A ) )  |`  ( ( Base `  (flds  A )
)  X.  ( Base `  (flds  A ) ) ) )  =  ( ( dist `  (flds  A ) )  |`  (
( Base `  (flds  A ) )  X.  ( Base `  (flds  A )
) ) )
2119, 20msmet 20007 . . . . 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 5928 . . . . . . 7  |-  ( x  e.  I  ->  (
( I  X.  {
(flds  A
) } ) `  x )  =  (flds  A ) )
2423adantl 466 . . . . . 6  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( (
I  X.  { (flds  A ) } ) `  x
)  =  (flds  A ) )
2524fveq2d 5690 . . . . 5  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( dist `  ( ( I  X.  { (flds  A ) } ) `  x ) )  =  ( dist `  (flds  A )
) )
2624fveq2d 5690 . . . . . 6  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( Base `  ( ( I  X.  { (flds  A ) } ) `  x ) )  =  ( Base `  (flds  A )
) )
2726, 26xpeq12d 4860 . . . . 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 5690 . . . 4  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( Met `  ( Base `  (
( I  X.  {
(flds  A
) } ) `  x ) ) )  =  ( Met `  ( Base `  (flds  A ) ) ) )
3022, 28, 293eltr4d 2519 . . 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 28641 . . . . . 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 2438 . . . . . . . . . . 11  |-  (flds  A )  =  (flds  A )
33 cnfldbas 17797 . . . . . . . . . . 11  |-  CC  =  ( Base ` fld )
3432, 33ressbas2 14221 . . . . . . . . . 10  |-  ( A 
C_  CC  ->  A  =  ( Base `  (flds  A )
) )
3534ad2antrr 725 . . . . . . . . 9  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  A  =  ( Base `  (flds  A ) ) )
3635fveq2d 5690 . . . . . . . 8  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( Met `  A )  =  ( Met `  ( Base `  (flds  A ) ) ) )
3722, 36eleqtrrd 2515 . . . . . . 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 2438 . . . . . . . . 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 28643 . . . . . . . 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 434 . . . . . . 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 32 . . . . 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 2438 . . . . . . . . 9  |-  ( ( abs  o.  -  )  |`  ( y  X.  y
) )  =  ( ( abs  o.  -  )  |`  ( y  X.  y ) )
4443cntotbnd 28648 . . . . . . . 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 3379 . . . . . . . . . . . 12  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( y  C_  A  <->  y  C_  ( Base `  (flds  A ) ) ) )
4746biimpa 484 . . . . . . . . . . 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 661 . . . . . . . . . 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 465 . . . . . . . . . . 11  |-  ( ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  /\  y  C_  A )  ->  A  e.  _V )
53 cnfldds 17803 . . . . . . . . . . . 12  |-  ( abs 
o.  -  )  =  ( dist ` fld )
5432, 53ressds 14344 . . . . . . . . . . 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 2473 . . . . . . . 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 2504 . . . . . . 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 2504 . . . . . . 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 285 . . . . . 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 434 . . . . 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 354 . . . 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 2504 . . . 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 2504 . . . 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 285 . . 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 28647 . 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 2438 . . . . . . . 8  |-  (Scalar `  (flds  A
) )  =  (Scalar `  (flds  A ) )
7169, 70pwsval 14416 . . . . . . 7  |-  ( ( (flds  A )  e.  _V  /\  I  e.  Fin )  ->  Y  =  ( (Scalar `  (flds  A ) ) X_s ( I  X.  {
(flds  A
) } ) ) )
729, 8, 71sylancr 663 . . . . . 6  |-  ( ( A  C_  CC  /\  I  e.  Fin )  ->  Y  =  ( (Scalar `  (flds  A
) ) X_s ( I  X.  {
(flds  A
) } ) ) )
7372fveq2d 5690 . . . . 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 2482 . . 3  |-  ( ( A  C_  CC  /\  I  e.  Fin )  ->  D  =  ( ( dist `  ( (Scalar `  (flds  A )
) X_s ( I  X.  {
(flds  A
) } ) ) )  |`  ( X  X.  X ) ) )
7675eleq1d 2504 . 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 2504 . 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 285 1  |-  ( ( A  C_  CC  /\  I  e.  Fin )  ->  ( D  e.  ( TotBnd `  X )  <->  D  e.  ( Bnd `  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2967    C_ wss 3323   {csn 3872    X. cxp 4833    |` cres 4837    o. ccom 4839    Fn wfn 5408   ` cfv 5413  (class class class)co 6086   Fincfn 7302   CCcc 9272    - cmin 9587   abscabs 12715   Basecbs 14166   ↾s cress 14167  Scalarcsca 14233   distcds 14239   X_scprds 14376    ^s cpws 14377   Metcme 17777  ℂfldccnfld 17793   MetSpcmt 19868   TotBndctotbnd 28618   Bndcbnd 28619
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-ec 7095  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-icc 11299  df-fz 11430  df-fl 11634  df-seq 11799  df-exp 11858  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-gz 13983  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-topgen 14374  df-prds 14378  df-pws 14380  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-cnfld 17794  df-top 18478  df-bases 18480  df-topon 18481  df-topsp 18482  df-xms 19870  df-ms 19871  df-totbnd 28620  df-bnd 28631
This theorem is referenced by:  rrntotbnd  28688
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