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Theorem cnpwstotbnd 26521
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 2283 . . 3  |-  ( (Scalar `  (flds  A ) ) X_s ( I  X.  {
(flds  A
) } ) )  =  ( (Scalar `  (flds  A
) ) X_s ( I  X.  {
(flds  A
) } ) )
2 eqid 2283 . . 3  |-  ( Base `  ( (Scalar `  (flds  A )
) X_s ( I  X.  {
(flds  A
) } ) ) )  =  ( Base `  ( (Scalar `  (flds  A )
) X_s ( I  X.  {
(flds  A
) } ) ) )
3 eqid 2283 . . 3  |-  ( Base `  ( ( I  X.  { (flds  A ) } ) `  x ) )  =  ( Base `  (
( I  X.  {
(flds  A
) } ) `  x ) )
4 eqid 2283 . . 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 2283 . . 3  |-  ( dist `  ( (Scalar `  (flds  A )
) X_s ( I  X.  {
(flds  A
) } ) ) )  =  ( dist `  ( (Scalar `  (flds  A )
) X_s ( I  X.  {
(flds  A
) } ) ) )
6 fvex 5539 . . . 4  |-  (Scalar `  (flds  A
) )  e.  _V
76a1i 10 . . 3  |-  ( ( A  C_  CC  /\  I  e.  Fin )  ->  (Scalar `  (flds  A ) )  e.  _V )
8 simpr 447 . . 3  |-  ( ( A  C_  CC  /\  I  e.  Fin )  ->  I  e.  Fin )
9 ovex 5883 . . . 4  |-  (flds  A )  e.  _V
10 fnconstg 5429 . . . 4  |-  ( (flds  A )  e.  _V  ->  (
I  X.  { (flds  A ) } )  Fn  I
)
119, 10mp1i 11 . . 3  |-  ( ( A  C_  CC  /\  I  e.  Fin )  ->  (
I  X.  { (flds  A ) } )  Fn  I
)
12 eqid 2283 . . 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 18285 . . . . . 6  |-fld  e.  MetSp
14 cnex 8818 . . . . . . . 8  |-  CC  e.  _V
1514ssex 4158 . . . . . . 7  |-  ( A 
C_  CC  ->  A  e. 
_V )
1615ad2antrr 706 . . . . . 6  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  A  e.  _V )
17 ressms 18072 . . . . . 6  |-  ( (fld  e. 
MetSp  /\  A  e.  _V )  ->  (flds  A )  e.  MetSp )
1813, 16, 17sylancr 644 . . . . 5  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  (flds  A )  e.  MetSp )
19 eqid 2283 . . . . . 6  |-  ( Base `  (flds  A ) )  =  (
Base `  (flds  A ) )
20 eqid 2283 . . . . . 6  |-  ( (
dist `  (flds  A ) )  |`  ( ( Base `  (flds  A )
)  X.  ( Base `  (flds  A ) ) ) )  =  ( ( dist `  (flds  A ) )  |`  (
( Base `  (flds  A ) )  X.  ( Base `  (flds  A )
) ) )
2119, 20msmet 18003 . . . . 5  |-  ( (flds  A )  e.  MetSp  ->  ( ( dist `  (flds  A ) )  |`  (
( Base `  (flds  A ) )  X.  ( Base `  (flds  A )
) ) )  e.  ( Met `  ( Base `  (flds  A ) ) ) )
2218, 21syl 15 . . . 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 5729 . . . . . . . 8  |-  ( x  e.  I  ->  (
( I  X.  {
(flds  A
) } ) `  x )  =  (flds  A ) )
2423adantl 452 . . . . . . 7  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( (
I  X.  { (flds  A ) } ) `  x
)  =  (flds  A ) )
2524fveq2d 5529 . . . . . 6  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( dist `  ( ( I  X.  { (flds  A ) } ) `  x ) )  =  ( dist `  (flds  A )
) )
2624fveq2d 5529 . . . . . . 7  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( Base `  ( ( I  X.  { (flds  A ) } ) `  x ) )  =  ( Base `  (flds  A )
) )
2726, 26xpeq12d 4714 . . . . . 6  |-  ( ( ( 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 4956 . . . . 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 ) ) ) )  =  ( (
dist `  (flds  A ) )  |`  ( ( Base `  (flds  A )
)  X.  ( Base `  (flds  A ) ) ) ) )
2926fveq2d 5529 . . . . 5  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( Met `  ( Base `  (
( I  X.  {
(flds  A
) } ) `  x ) ) )  =  ( Met `  ( Base `  (flds  A ) ) ) )
3028, 29eleq12d 2351 . . . 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 ) ) ) )  e.  ( Met `  ( Base `  ( ( I  X.  { (flds  A ) } ) `  x ) ) )  <-> 
( ( dist `  (flds  A )
)  |`  ( ( Base `  (flds  A ) )  X.  ( Base `  (flds  A ) ) ) )  e.  ( Met `  ( Base `  (flds  A ) ) ) ) )
3122, 30mpbird 223 . . 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 ) ) ) )
32 totbndbnd 26513 . . . . . 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 ) )
33 eqid 2283 . . . . . . . . . . 11  |-  (flds  A )  =  (flds  A )
34 cnfldbas 16383 . . . . . . . . . . 11  |-  CC  =  ( Base ` fld )
3533, 34ressbas2 13199 . . . . . . . . . 10  |-  ( A 
C_  CC  ->  A  =  ( Base `  (flds  A )
) )
3635ad2antrr 706 . . . . . . . . 9  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  A  =  ( Base `  (flds  A ) ) )
3736fveq2d 5529 . . . . . . . 8  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( Met `  A )  =  ( Met `  ( Base `  (flds  A ) ) ) )
3822, 37eleqtrrd 2360 . . . . . . 7  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( ( dist `  (flds  A ) )  |`  (
( Base `  (flds  A ) )  X.  ( Base `  (flds  A )
) ) )  e.  ( Met `  A
) )
39 eqid 2283 . . . . . . . . 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
) )
4039bnd2lem 26515 . . . . . . . 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 )
4140ex 423 . . . . . . 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
) )
4238, 41syl 15 . . . . . 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
) )
4332, 42syl5 28 . . . . 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 )
)
44 eqid 2283 . . . . . . . . 9  |-  ( ( abs  o.  -  )  |`  ( y  X.  y
) )  =  ( ( abs  o.  -  )  |`  ( y  X.  y ) )
4544cntotbnd 26520 . . . . . . . 8  |-  ( ( ( abs  o.  -  )  |`  ( y  X.  y ) )  e.  ( TotBnd `  y )  <->  ( ( abs  o.  -  )  |`  ( y  X.  y ) )  e.  ( Bnd `  y
) )
4645a1i 10 . . . . . . 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 ) ) )
4736sseq2d 3206 . . . . . . . . . . . 12  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( y  C_  A  <->  y  C_  ( Base `  (flds  A ) ) ) )
4847biimpa 470 . . . . . . . . . . 11  |-  ( ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  /\  y  C_  A )  ->  y  C_  ( Base `  (flds  A )
) )
49 xpss12 4792 . . . . . . . . . . 11  |-  ( ( y  C_  ( Base `  (flds  A ) )  /\  y  C_  ( Base `  (flds  A )
) )  ->  (
y  X.  y ) 
C_  ( ( Base `  (flds  A ) )  X.  ( Base `  (flds  A ) ) ) )
5048, 48, 49syl2anc 642 . . . . . . . . . 10  |-  ( ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  /\  y  C_  A )  ->  (
y  X.  y ) 
C_  ( ( Base `  (flds  A ) )  X.  ( Base `  (flds  A ) ) ) )
51 resabs1 4984 . . . . . . . . . 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 ) ) )
5250, 51syl 15 . . . . . . . . 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 ) ) )
5316adantr 451 . . . . . . . . . . 11  |-  ( ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  /\  y  C_  A )  ->  A  e.  _V )
54 cnfldds 16389 . . . . . . . . . . . 12  |-  ( abs 
o.  -  )  =  ( dist ` fld )
5533, 54ressds 13318 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  ( abs  o.  -  )  =  ( dist `  (flds  A )
) )
5653, 55syl 15 . . . . . . . . . 10  |-  ( ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  /\  y  C_  A )  ->  ( abs  o.  -  )  =  ( dist `  (flds  A )
) )
5756reseq1d 4954 . . . . . . . . 9  |-  ( ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  /\  y  C_  A )  ->  (
( abs  o.  -  )  |`  ( y  X.  y
) )  =  ( ( dist `  (flds  A )
)  |`  ( y  X.  y ) ) )
5852, 57eqtr4d 2318 . . . . . . . 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 ) ) )
5958eleq1d 2349 . . . . . . 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 ) ) )
6058eleq1d 2349 . . . . . . 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
) ) )
6146, 59, 603bitr4d 276 . . . . . 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 ) ) )
6261ex 423 . . . . 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 ) ) ) )
6343, 42, 62pm5.21ndd 343 . . . 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 ) ) )
6428reseq1d 4954 . . . . 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
) ) )
6564eleq1d 2349 . . . 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 ) ) )
6664eleq1d 2349 . . . 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 ) ) )
6763, 65, 663bitr4d 276 . . 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
) ) )
681, 2, 3, 4, 5, 7, 8, 11, 12, 31, 67prdsbnd2 26519 . 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
) ) )
69 cnpwstotbnd.d . . . 4  |-  D  =  ( ( dist `  Y
)  |`  ( X  X.  X ) )
70 cnpwstotbnd.y . . . . . . . 8  |-  Y  =  ( (flds  A )  ^s  I )
71 eqid 2283 . . . . . . . 8  |-  (Scalar `  (flds  A
) )  =  (Scalar `  (flds  A ) )
7270, 71pwsval 13385 . . . . . . 7  |-  ( ( (flds  A )  e.  _V  /\  I  e.  Fin )  ->  Y  =  ( (Scalar `  (flds  A ) ) X_s ( I  X.  {
(flds  A
) } ) ) )
739, 8, 72sylancr 644 . . . . . 6  |-  ( ( A  C_  CC  /\  I  e.  Fin )  ->  Y  =  ( (Scalar `  (flds  A
) ) X_s ( I  X.  {
(flds  A
) } ) ) )
7473fveq2d 5529 . . . . 5  |-  ( ( A  C_  CC  /\  I  e.  Fin )  ->  ( dist `  Y )  =  ( dist `  (
(Scalar `  (flds  A ) ) X_s (
I  X.  { (flds  A ) } ) ) ) )
7574reseq1d 4954 . . . 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
) ) )
7669, 75syl5eq 2327 . . 3  |-  ( ( A  C_  CC  /\  I  e.  Fin )  ->  D  =  ( ( dist `  ( (Scalar `  (flds  A )
) X_s ( I  X.  {
(flds  A
) } ) ) )  |`  ( X  X.  X ) ) )
7776eleq1d 2349 . 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 )
) )
7876eleq1d 2349 . 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
) ) )
7968, 77, 783bitr4d 276 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   {csn 3640    X. cxp 4687    |` cres 4691    o. ccom 4693    Fn wfn 5250   ` cfv 5255  (class class class)co 5858   Fincfn 6863   CCcc 8735    - cmin 9037   abscabs 11719   Basecbs 13148   ↾s cress 13149  Scalarcsca 13211   distcds 13217   X_scprds 13346    ^s cpws 13347   Metcme 16370  ℂfldccnfld 16377   MetSpcmt 17883   TotBndctotbnd 26490   Bndcbnd 26491
This theorem is referenced by:  rrntotbnd  26560
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-ec 6662  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-icc 10663  df-fz 10783  df-fl 10925  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-gz 12977  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-prds 13348  df-pws 13350  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-xms 17885  df-ms 17886  df-totbnd 26492  df-bnd 26503
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