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Theorem cnpwstotbnd 28843
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 2454 . . 3  |-  ( (Scalar `  (flds  A ) ) X_s ( I  X.  {
(flds  A
) } ) )  =  ( (Scalar `  (flds  A
) ) X_s ( I  X.  {
(flds  A
) } ) )
2 eqid 2454 . . 3  |-  ( Base `  ( (Scalar `  (flds  A )
) X_s ( I  X.  {
(flds  A
) } ) ) )  =  ( Base `  ( (Scalar `  (flds  A )
) X_s ( I  X.  {
(flds  A
) } ) ) )
3 eqid 2454 . . 3  |-  ( Base `  ( ( I  X.  { (flds  A ) } ) `  x ) )  =  ( Base `  (
( I  X.  {
(flds  A
) } ) `  x ) )
4 eqid 2454 . . 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 2454 . . 3  |-  ( dist `  ( (Scalar `  (flds  A )
) X_s ( I  X.  {
(flds  A
) } ) ) )  =  ( dist `  ( (Scalar `  (flds  A )
) X_s ( I  X.  {
(flds  A
) } ) ) )
6 fvex 5808 . . . 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 6224 . . . 4  |-  (flds  A )  e.  _V
10 fnconstg 5705 . . . 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 2454 . . 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 20486 . . . . . 6  |-fld  e.  MetSp
14 cnex 9473 . . . . . . . 8  |-  CC  e.  _V
1514ssex 4543 . . . . . . 7  |-  ( A 
C_  CC  ->  A  e. 
_V )
1615ad2antrr 725 . . . . . 6  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  A  e.  _V )
17 ressms 20232 . . . . . 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 2454 . . . . . 6  |-  ( Base `  (flds  A ) )  =  (
Base `  (flds  A ) )
20 eqid 2454 . . . . . 6  |-  ( (
dist `  (flds  A ) )  |`  ( ( Base `  (flds  A )
)  X.  ( Base `  (flds  A ) ) ) )  =  ( ( dist `  (flds  A ) )  |`  (
( Base `  (flds  A ) )  X.  ( Base `  (flds  A )
) ) )
2119, 20msmet 20163 . . . . 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 6041 . . . . . . 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 5802 . . . . 5  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( dist `  ( ( I  X.  { (flds  A ) } ) `  x ) )  =  ( dist `  (flds  A )
) )
2624fveq2d 5802 . . . . . 6  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( Base `  ( ( I  X.  { (flds  A ) } ) `  x ) )  =  ( Base `  (flds  A )
) )
2726, 26xpeq12d 4972 . . . . 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 5218 . . . 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 5802 . . . 4  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( Met `  ( Base `  (
( I  X.  {
(flds  A
) } ) `  x ) ) )  =  ( Met `  ( Base `  (flds  A ) ) ) )
3022, 28, 293eltr4d 2557 . . 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 28835 . . . . . 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 2454 . . . . . . . . . . 11  |-  (flds  A )  =  (flds  A )
33 cnfldbas 17946 . . . . . . . . . . 11  |-  CC  =  ( Base ` fld )
3432, 33ressbas2 14347 . . . . . . . . . 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 5802 . . . . . . . 8  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( Met `  A )  =  ( Met `  ( Base `  (flds  A ) ) ) )
3722, 36eleqtrrd 2545 . . . . . . 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 2454 . . . . . . . . 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 28837 . . . . . . . 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 2454 . . . . . . . . 9  |-  ( ( abs  o.  -  )  |`  ( y  X.  y
) )  =  ( ( abs  o.  -  )  |`  ( y  X.  y ) )
4443cntotbnd 28842 . . . . . . . 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 3491 . . . . . . . . . . . 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 5052 . . . . . . . . . . 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 5246 . . . . . . . . . 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 17952 . . . . . . . . . . . 12  |-  ( abs 
o.  -  )  =  ( dist ` fld )
5432, 53ressds 14470 . . . . . . . . . . 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 5216 . . . . . . . . 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 2498 . . . . . . . 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 2523 . . . . . . 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 2523 . . . . . . 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 5216 . . . . 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 2523 . . . 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 2523 . . . 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 28841 . 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 2454 . . . . . . . 8  |-  (Scalar `  (flds  A
) )  =  (Scalar `  (flds  A ) )
7169, 70pwsval 14542 . . . . . . 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 5802 . . . . 5  |-  ( ( A  C_  CC  /\  I  e.  Fin )  ->  ( dist `  Y )  =  ( dist `  (
(Scalar `  (flds  A ) ) X_s (
I  X.  { (flds  A ) } ) ) ) )
7473reseq1d 5216 . . . 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 2507 . . 3  |-  ( ( A  C_  CC  /\  I  e.  Fin )  ->  D  =  ( ( dist `  ( (Scalar `  (flds  A )
) X_s ( I  X.  {
(flds  A
) } ) ) )  |`  ( X  X.  X ) ) )
7675eleq1d 2523 . 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 2523 . 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 1370    e. wcel 1758   _Vcvv 3076    C_ wss 3435   {csn 3984    X. cxp 4945    |` cres 4949    o. ccom 4951    Fn wfn 5520   ` cfv 5525  (class class class)co 6199   Fincfn 7419   CCcc 9390    - cmin 9705   abscabs 12840   Basecbs 14291   ↾s cress 14292  Scalarcsca 14359   distcds 14365   X_scprds 14502    ^s cpws 14503   Metcme 17926  ℂfldccnfld 17942   MetSpcmt 20024   TotBndctotbnd 28812   Bndcbnd 28813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-1o 7029  df-2o 7030  df-oadd 7033  df-er 7210  df-ec 7212  df-map 7325  df-pm 7326  df-ixp 7373  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-sup 7801  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-4 10492  df-5 10493  df-6 10494  df-7 10495  df-8 10496  df-9 10497  df-10 10498  df-n0 10690  df-z 10757  df-dec 10866  df-uz 10972  df-q 11064  df-rp 11102  df-xneg 11199  df-xadd 11200  df-xmul 11201  df-icc 11417  df-fz 11554  df-fl 11758  df-seq 11923  df-exp 11982  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-gz 14108  df-struct 14293  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-plusg 14369  df-mulr 14370  df-starv 14371  df-sca 14372  df-vsca 14373  df-ip 14374  df-tset 14375  df-ple 14376  df-ds 14378  df-unif 14379  df-hom 14380  df-cco 14381  df-rest 14479  df-topn 14480  df-topgen 14500  df-prds 14504  df-pws 14506  df-psmet 17933  df-xmet 17934  df-met 17935  df-bl 17936  df-mopn 17937  df-cnfld 17943  df-top 18634  df-bases 18636  df-topon 18637  df-topsp 18638  df-xms 20026  df-ms 20027  df-totbnd 28814  df-bnd 28825
This theorem is referenced by:  rrntotbnd  28882
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