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Theorem cnpwstotbnd 30455
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 2457 . . 3  |-  ( (Scalar `  (flds  A ) ) X_s ( I  X.  {
(flds  A
) } ) )  =  ( (Scalar `  (flds  A
) ) X_s ( I  X.  {
(flds  A
) } ) )
2 eqid 2457 . . 3  |-  ( Base `  ( (Scalar `  (flds  A )
) X_s ( I  X.  {
(flds  A
) } ) ) )  =  ( Base `  ( (Scalar `  (flds  A )
) X_s ( I  X.  {
(flds  A
) } ) ) )
3 eqid 2457 . . 3  |-  ( Base `  ( ( I  X.  { (flds  A ) } ) `  x ) )  =  ( Base `  (
( I  X.  {
(flds  A
) } ) `  x ) )
4 eqid 2457 . . 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 2457 . . 3  |-  ( dist `  ( (Scalar `  (flds  A )
) X_s ( I  X.  {
(flds  A
) } ) ) )  =  ( dist `  ( (Scalar `  (flds  A )
) X_s ( I  X.  {
(flds  A
) } ) ) )
6 fvex 5882 . . . 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 6324 . . . 4  |-  (flds  A )  e.  _V
10 fnconstg 5779 . . . 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 2457 . . 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 21408 . . . . . 6  |-fld  e.  MetSp
14 cnex 9590 . . . . . . . 8  |-  CC  e.  _V
1514ssex 4600 . . . . . . 7  |-  ( A 
C_  CC  ->  A  e. 
_V )
1615ad2antrr 725 . . . . . 6  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  A  e.  _V )
17 ressms 21154 . . . . . 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 2457 . . . . . 6  |-  ( Base `  (flds  A ) )  =  (
Base `  (flds  A ) )
20 eqid 2457 . . . . . 6  |-  ( (
dist `  (flds  A ) )  |`  ( ( Base `  (flds  A )
)  X.  ( Base `  (flds  A ) ) ) )  =  ( ( dist `  (flds  A ) )  |`  (
( Base `  (flds  A ) )  X.  ( Base `  (flds  A )
) ) )
2119, 20msmet 21085 . . . . 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 6128 . . . . . . 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 5876 . . . . 5  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( dist `  ( ( I  X.  { (flds  A ) } ) `  x ) )  =  ( dist `  (flds  A )
) )
2624fveq2d 5876 . . . . . 6  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( Base `  ( ( I  X.  { (flds  A ) } ) `  x ) )  =  ( Base `  (flds  A )
) )
2726sqxpeqd 5034 . . . . 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 5284 . . . 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 5876 . . . 4  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( Met `  ( Base `  (
( I  X.  {
(flds  A
) } ) `  x ) ) )  =  ( Met `  ( Base `  (flds  A ) ) ) )
3022, 28, 293eltr4d 2560 . . 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 30447 . . . . . 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 2457 . . . . . . . . . . 11  |-  (flds  A )  =  (flds  A )
33 cnfldbas 18550 . . . . . . . . . . 11  |-  CC  =  ( Base ` fld )
3432, 33ressbas2 14701 . . . . . . . . . 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 5876 . . . . . . . 8  |-  ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  ->  ( Met `  A )  =  ( Met `  ( Base `  (flds  A ) ) ) )
3722, 36eleqtrrd 2548 . . . . . . 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 2457 . . . . . . . . 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 30449 . . . . . . . 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 2457 . . . . . . . . 9  |-  ( ( abs  o.  -  )  |`  ( y  X.  y
) )  =  ( ( abs  o.  -  )  |`  ( y  X.  y ) )
4443cntotbnd 30454 . . . . . . . 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 3527 . . . . . . . . . . . 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 5117 . . . . . . . . . . 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 ) ) ) )
5049resabs1d 5313 . . . . . . . . 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 ) ) )
5116adantr 465 . . . . . . . . . . 11  |-  ( ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  /\  y  C_  A )  ->  A  e.  _V )
52 cnfldds 18556 . . . . . . . . . . . 12  |-  ( abs 
o.  -  )  =  ( dist ` fld )
5332, 52ressds 14829 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  ( abs  o.  -  )  =  ( dist `  (flds  A )
) )
5451, 53syl 16 . . . . . . . . . 10  |-  ( ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  /\  y  C_  A )  ->  ( abs  o.  -  )  =  ( dist `  (flds  A )
) )
5554reseq1d 5282 . . . . . . . . 9  |-  ( ( ( ( A  C_  CC  /\  I  e.  Fin )  /\  x  e.  I
)  /\  y  C_  A )  ->  (
( abs  o.  -  )  |`  ( y  X.  y
) )  =  ( ( dist `  (flds  A )
)  |`  ( y  X.  y ) ) )
5650, 55eqtr4d 2501 . . . . . . . 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 ) ) )
5756eleq1d 2526 . . . . . . 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 ) ) )
5856eleq1d 2526 . . . . . . 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
) ) )
5945, 57, 583bitr4d 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 ) ) )
6059ex 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 ) ) ) )
6142, 41, 60pm5.21ndd 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 ) ) )
6228reseq1d 5282 . . . . 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
) ) )
6362eleq1d 2526 . . . 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 ) ) )
6462eleq1d 2526 . . . 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 ) ) )
6561, 63, 643bitr4d 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
) ) )
661, 2, 3, 4, 5, 7, 8, 11, 12, 30, 65prdsbnd2 30453 . 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
) ) )
67 cnpwstotbnd.d . . . 4  |-  D  =  ( ( dist `  Y
)  |`  ( X  X.  X ) )
68 cnpwstotbnd.y . . . . . . . 8  |-  Y  =  ( (flds  A )  ^s  I )
69 eqid 2457 . . . . . . . 8  |-  (Scalar `  (flds  A
) )  =  (Scalar `  (flds  A ) )
7068, 69pwsval 14902 . . . . . . 7  |-  ( ( (flds  A )  e.  _V  /\  I  e.  Fin )  ->  Y  =  ( (Scalar `  (flds  A ) ) X_s ( I  X.  {
(flds  A
) } ) ) )
719, 8, 70sylancr 663 . . . . . 6  |-  ( ( A  C_  CC  /\  I  e.  Fin )  ->  Y  =  ( (Scalar `  (flds  A
) ) X_s ( I  X.  {
(flds  A
) } ) ) )
7271fveq2d 5876 . . . . 5  |-  ( ( A  C_  CC  /\  I  e.  Fin )  ->  ( dist `  Y )  =  ( dist `  (
(Scalar `  (flds  A ) ) X_s (
I  X.  { (flds  A ) } ) ) ) )
7372reseq1d 5282 . . . 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
) ) )
7467, 73syl5eq 2510 . . 3  |-  ( ( A  C_  CC  /\  I  e.  Fin )  ->  D  =  ( ( dist `  ( (Scalar `  (flds  A )
) X_s ( I  X.  {
(flds  A
) } ) ) )  |`  ( X  X.  X ) ) )
7574eleq1d 2526 . 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 )
) )
7674eleq1d 2526 . 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
) ) )
7766, 75, 763bitr4d 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 1395    e. wcel 1819   _Vcvv 3109    C_ wss 3471   {csn 4032    X. cxp 5006    |` cres 5010    o. ccom 5012    Fn wfn 5589   ` cfv 5594  (class class class)co 6296   Fincfn 7535   CCcc 9507    - cmin 9824   abscabs 13078   Basecbs 14643   ↾s cress 14644  Scalarcsca 14714   distcds 14720   X_scprds 14862    ^s cpws 14863   Metcme 18530  ℂfldccnfld 18546   MetSpcmt 20946   TotBndctotbnd 30424   Bndcbnd 30425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-ec 7331  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-icc 11561  df-fz 11698  df-fl 11931  df-seq 12110  df-exp 12169  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-gz 14459  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-mulr 14725  df-starv 14726  df-sca 14727  df-vsca 14728  df-ip 14729  df-tset 14730  df-ple 14731  df-ds 14733  df-unif 14734  df-hom 14735  df-cco 14736  df-rest 14839  df-topn 14840  df-topgen 14860  df-prds 14864  df-pws 14866  df-psmet 18537  df-xmet 18538  df-met 18539  df-bl 18540  df-mopn 18541  df-cnfld 18547  df-top 19525  df-bases 19527  df-topon 19528  df-topsp 19529  df-xms 20948  df-ms 20949  df-totbnd 30426  df-bnd 30437
This theorem is referenced by:  rrntotbnd  30494
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