HomeHome Metamath Proof Explorer
Theorem List (p. 187 of 315)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21459)
  Hilbert Space Explorer  Hilbert Space Explorer
(21460-22982)
  Users' Mathboxes  Users' Mathboxes
(22983-31404)
 

Theorem List for Metamath Proof Explorer - 18601-18700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcphip0r 18601 Inner product with a zero second argument. Complex version of ip0r 16504. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   =>    |-  ( ( W  e.  CPreHil  /\  A  e.  V ) 
 ->  ( A  .,  .0.  )  =  0 )
 
Theoremcphipeq0 18602 The inner product of a vector with itself is zero iff the vector is zero. Part of Definition 3.1-1 of [Kreyszig] p. 129. Complex version of ipeq0 16505. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   =>    |-  ( ( W  e.  CPreHil  /\  A  e.  V ) 
 ->  ( ( A  .,  A )  =  0  <->  A  =  .0.  ) )
 
Theoremcphdir 18603 Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. Complex version of ipdir 16506. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   =>    |-  (
 ( W  e.  CPreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )  ->  ( ( A  .+  B )  .,  C )  =  (
 ( A  .,  C )  +  ( B  .,  C ) ) )
 
Theoremcphdi 18604 Distributive law for inner product. Complex version of ipdi 16507. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   =>    |-  (
 ( W  e.  CPreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )  ->  ( A  .,  ( B  .+  C ) )  =  (
 ( A  .,  B )  +  ( A  .,  C ) ) )
 
Theoremcph2di 18605 Distributive law for inner product. Complex version of ip2di 16508. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  ( ph  ->  W  e.  CPreHil )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  D  e.  V )   =>    |-  ( ph  ->  (
 ( A  .+  B )  .,  ( C  .+  D ) )  =  ( ( ( A 
 .,  C )  +  ( B  .,  D ) )  +  ( ( A  .,  D )  +  ( B  .,  C ) ) ) )
 
Theoremcphsubdir 18606 Distributive law for inner product subtraction. Complex version of ipsubdir 16509. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  .-  =  ( -g `  W )   =>    |-  ( ( W  e.  CPreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )  ->  ( ( A  .-  B )  .,  C )  =  (
 ( A  .,  C )  -  ( B  .,  C ) ) )
 
Theoremcphsubdi 18607 Distributive law for inner product subtraction. Complex version of ipsubdi 16510. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  .-  =  ( -g `  W )   =>    |-  ( ( W  e.  CPreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )  ->  ( A  .,  ( B  .-  C ) )  =  (
 ( A  .,  B )  -  ( A  .,  C ) ) )
 
Theoremcph2subdi 18608 Distributive law for inner product subtraction. Complex version of ip2subdi 16511. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  .-  =  ( -g `  W )   &    |-  ( ph  ->  W  e.  CPreHil )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  D  e.  V )   =>    |-  ( ph  ->  (
 ( A  .-  B )  .,  ( C  .-  D ) )  =  ( ( ( A 
 .,  C )  +  ( B  .,  D ) )  -  ( ( A  .,  D )  +  ( B  .,  C ) ) ) )
 
Theoremcphass 18609 Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. See ipass 16512, his5 21626. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  .x.  =  ( .s `  W )   =>    |-  ( ( W  e.  CPreHil  /\  ( A  e.  K  /\  B  e.  V  /\  C  e.  V )
 )  ->  ( ( A  .x.  B )  .,  C )  =  ( A  x.  ( B  .,  C ) ) )
 
Theoremcphassr 18610 "Associative" law for second argument of inner product (compare cphass 18609). See ipassr 16513, his52 . (Contributed by Mario Carneiro, 16-Oct-2015.)
 |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  .x.  =  ( .s `  W )   =>    |-  ( ( W  e.  CPreHil  /\  ( A  e.  K  /\  B  e.  V  /\  C  e.  V )
 )  ->  ( B  .,  ( A  .x.  C ) )  =  (
 ( * `  A )  x.  ( B  .,  C ) ) )
 
Theoremcph2ass 18611 Move scalar multiplication to outside of inner product. See his35 21628. (Contributed by Mario Carneiro, 17-Oct-2015.)
 |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  .x.  =  ( .s `  W )   =>    |-  ( ( W  e.  CPreHil  /\  ( A  e.  K  /\  B  e.  K ) 
 /\  ( C  e.  V  /\  D  e.  V ) )  ->  ( ( A  .x.  C )  .,  ( B  .x.  D ) )  =  (
 ( A  x.  ( * `  B ) )  x.  ( C  .,  D ) ) )
 
Theoremtchex 18612* Lemma for tchbas 18614 and similar theorems. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  V  =  ( Base `  W )   =>    |-  ( x  e.  V  |->  ( sqr `  ( x  .,  x ) ) )  e.  _V
 
Theoremtchval 18613* Define a function to augment a pre-Hilbert space with norm. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  G  =  (toCHil `  W )   &    |-  V  =  ( Base `  W )   &    |-  .,  =  ( .i `  W )   =>    |-  G  =  ( W toNrmGrp  ( x  e.  V  |->  ( sqr `  ( x  .,  x ) ) ) )
 
Theoremtchbas 18614 The base set of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
 |-  G  =  (toCHil `  W )   &    |-  V  =  ( Base `  W )   =>    |-  V  =  ( Base `  G )
 
Theoremtchplusg 18615 The addition operation of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
 |-  G  =  (toCHil `  W )   &    |- 
 .+  =  ( +g  `  W )   =>    |- 
 .+  =  ( +g  `  G )
 
Theoremtchmulr 18616 The ring operation of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
 |-  G  =  (toCHil `  W )   &    |- 
 .x.  =  ( .r `  W )   =>    |- 
 .x.  =  ( .r `  G )
 
Theoremtchsca 18617 The scalar field of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
 |-  G  =  (toCHil `  W )   &    |-  F  =  (Scalar `  W )   =>    |-  F  =  (Scalar `  G )
 
Theoremtchvsca 18618 The scalar multiplication of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
 |-  G  =  (toCHil `  W )   &    |- 
 .x.  =  ( .s `  W )   =>    |- 
 .x.  =  ( .s `  G )
 
Theoremtchip 18619 The inner product of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
 |-  G  =  (toCHil `  W )   &    |- 
 .x.  =  ( .i `  W )   =>    |- 
 .x.  =  ( .i `  G )
 
Theoremtchtopn 18620 The topology of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
 |-  G  =  (toCHil `  W )   &    |-  D  =  ( dist `  G )   &    |-  J  =  (
 TopOpen `  G )   =>    |-  ( W  e.  V  ->  J  =  (
 MetOpen `  D ) )
 
Theoremtchphl 18621 Augmentation of a pre-Hilbert space with a norm does not affect whether it is still a pre-Hilbert space because all the orginal components are the same. (Contributed by Mario Carneiro, 8-Oct-2015.)
 |-  G  =  (toCHil `  W )   =>    |-  ( W  e.  PreHil  <->  G  e.  PreHil )
 
Theoremtchnmfval 18622* The norm of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
 |-  G  =  (toCHil `  W )   &    |-  N  =  ( norm `  G )   &    |-  V  =  (
 Base `  W )   &    |-  .,  =  ( .i `  W )   =>    |-  ( W  e.  Grp  ->  N  =  ( x  e.  V  |->  ( sqr `  ( x  .,  x ) ) ) )
 
Theoremtchnmval 18623 The norm of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
 |-  G  =  (toCHil `  W )   &    |-  N  =  ( norm `  G )   &    |-  V  =  (
 Base `  W )   &    |-  .,  =  ( .i `  W )   =>    |-  ( ( W  e.  Grp  /\  X  e.  V ) 
 ->  ( N `  X )  =  ( sqr `  ( X  .,  X ) ) )
 
Theoremcphtchnm 18624 The norm of a norm-augmented complex pre-Hilbert space is the same as the original norm on it. (Contributed by Mario Carneiro, 11-Oct-2015.)
 |-  G  =  (toCHil `  W )   &    |-  N  =  ( norm `  W )   =>    |-  ( W  e.  CPreHil  ->  N  =  ( norm `  G ) )
 
Theoremtchclm 18625 Lemma for tchcph 18630. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  G  =  (toCHil `  W )   &    |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  ( ph  ->  W  e.  PreHil )   &    |-  ( ph  ->  F  =  (flds  K ) )   =>    |-  ( ph  ->  W  e. CMod )
 
Theoremtchcphlem3 18626 Lemma for tchcph 18630: real closure of an inner product of a vector with itself. (Contributed by Mario Carneiro, 10-Oct-2015.)
 |-  G  =  (toCHil `  W )   &    |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  ( ph  ->  W  e.  PreHil )   &    |-  ( ph  ->  F  =  (flds  K ) )   &    |-  .,  =  ( .i `  W )   =>    |-  ( ( ph  /\  X  e.  V ) 
 ->  ( X  .,  X )  e.  RR )
 
Theoremipcau2 18627* The Cauchy-Schwarz inequality for a complex pre-Hilbert space. (Contributed by Mario Carneiro, 11-Oct-2015.)
 |-  G  =  (toCHil `  W )   &    |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  ( ph  ->  W  e.  PreHil )   &    |-  ( ph  ->  F  =  (flds  K ) )   &    |-  .,  =  ( .i `  W )   &    |-  (
 ( ph  /\  ( x  e.  K  /\  x  e.  RR  /\  0  <_  x ) )  ->  ( sqr `  x )  e.  K )   &    |-  ( ( ph  /\  x  e.  V ) 
 ->  0  <_  ( x 
 .,  x ) )   &    |-  K  =  ( Base `  F )   &    |-  N  =  (
 norm `  G )   &    |-  C  =  ( ( Y  .,  X )  /  ( Y  .,  Y ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( abs `  ( X  .,  Y ) )  <_  ( ( N `  X )  x.  ( N `  Y ) ) )
 
Theoremtchcphlem1 18628* Lemma for tchcph 18630: the triangle inequality. (Contributed by Mario Carneiro, 8-Oct-2015.)
 |-  G  =  (toCHil `  W )   &    |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  ( ph  ->  W  e.  PreHil )   &    |-  ( ph  ->  F  =  (flds  K ) )   &    |-  .,  =  ( .i `  W )   &    |-  (
 ( ph  /\  ( x  e.  K  /\  x  e.  RR  /\  0  <_  x ) )  ->  ( sqr `  x )  e.  K )   &    |-  ( ( ph  /\  x  e.  V ) 
 ->  0  <_  ( x 
 .,  x ) )   &    |-  K  =  ( Base `  F )   &    |-  .-  =  ( -g `  W )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( sqr `  ( ( X 
 .-  Y )  .,  ( X  .-  Y ) ) )  <_  (
 ( sqr `  ( X  .,  X ) )  +  ( sqr `  ( Y  .,  Y ) ) ) )
 
Theoremtchcphlem2 18629* Lemma for tchcph 18630: homogeneity. (Contributed by Mario Carneiro, 8-Oct-2015.)
 |-  G  =  (toCHil `  W )   &    |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  ( ph  ->  W  e.  PreHil )   &    |-  ( ph  ->  F  =  (flds  K ) )   &    |-  .,  =  ( .i `  W )   &    |-  (
 ( ph  /\  ( x  e.  K  /\  x  e.  RR  /\  0  <_  x ) )  ->  ( sqr `  x )  e.  K )   &    |-  ( ( ph  /\  x  e.  V ) 
 ->  0  <_  ( x 
 .,  x ) )   &    |-  K  =  ( Base `  F )   &    |-  .x.  =  ( .s `  W )   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( sqr `  ( ( X 
 .x.  Y )  .,  ( X  .x.  Y ) ) )  =  ( ( abs `  X )  x.  ( sqr `  ( Y  .,  Y ) ) ) )
 
Theoremtchcph 18630* The standard definition of a norm turns any pre-Hilbert space over a quadratically closed subfield of  CC into a complex pre-Hilbert space (which allows access to a norm, metric, and topology). (Contributed by Mario Carneiro, 11-Oct-2015.)
 |-  G  =  (toCHil `  W )   &    |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  ( ph  ->  W  e.  PreHil )   &    |-  ( ph  ->  F  =  (flds  K ) )   &    |-  .,  =  ( .i `  W )   &    |-  (
 ( ph  /\  ( x  e.  K  /\  x  e.  RR  /\  0  <_  x ) )  ->  ( sqr `  x )  e.  K )   &    |-  ( ( ph  /\  x  e.  V ) 
 ->  0  <_  ( x 
 .,  x ) )   =>    |-  ( ph  ->  G  e.  CPreHil )
 
Theoremipcau 18631 The Cauchy-Schwarz inequality for a complex pre-Hilbert space. (Contributed by Mario Carneiro, 11-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  .,  =  ( .i `  W )   &    |-  N  =  ( norm `  W )   =>    |-  (
 ( W  e.  CPreHil  /\  X  e.  V  /\  Y  e.  V )  ->  ( abs `  ( X  .,  Y ) ) 
 <_  ( ( N `  X )  x.  ( N `  Y ) ) )
 
Theoremnmparlem 18632 Lemma for nmpar 18633. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .-  =  ( -g `  W )   &    |-  N  =  ( norm `  W )   &    |-  .,  =  ( .i `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  ( ph  ->  W  e.  CPreHil )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  V )   =>    |-  ( ph  ->  ( ( ( N `  ( A  .+  B ) ) ^ 2 )  +  ( ( N `
  ( A  .-  B ) ) ^
 2 ) )  =  ( 2  x.  (
 ( ( N `  A ) ^ 2
 )  +  ( ( N `  B ) ^ 2 ) ) ) )
 
Theoremnmpar 18633 A complex pre-Hilbert space satisfies the parallelogram law. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .-  =  ( -g `  W )   &    |-  N  =  ( norm `  W )   =>    |-  ( ( W  e.  CPreHil  /\  A  e.  V  /\  B  e.  V )  ->  ( ( ( N `
  ( A  .+  B ) ) ^
 2 )  +  (
 ( N `  ( A  .-  B ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `
  A ) ^
 2 )  +  (
 ( N `  B ) ^ 2 ) ) ) )
 
Theoremipcnlem2 18634 The inner product operation of a complex pre-Hilbert space is continuous. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  .,  =  ( .i `  W )   &    |-  D  =  ( dist `  W )   &    |-  N  =  ( norm `  W )   &    |-  T  =  ( ( R  / 
 2 )  /  (
 ( N `  A )  +  1 )
 )   &    |-  U  =  ( ( R  /  2 ) 
 /  ( ( N `
  B )  +  T ) )   &    |-  ( ph  ->  W  e.  CPreHil )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  ( A D X )  <  U )   &    |-  ( ph  ->  ( B D Y )  <  T )   =>    |-  ( ph  ->  ( abs `  ( ( A 
 .,  B )  -  ( X  .,  Y ) ) )  <  R )
 
Theoremipcnlem1 18635* The inner product operation of a complex pre-Hilbert space is continuous. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  .,  =  ( .i `  W )   &    |-  D  =  ( dist `  W )   &    |-  N  =  ( norm `  W )   &    |-  T  =  ( ( R  / 
 2 )  /  (
 ( N `  A )  +  1 )
 )   &    |-  U  =  ( ( R  /  2 ) 
 /  ( ( N `
  B )  +  T ) )   &    |-  ( ph  ->  W  e.  CPreHil )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  R  e.  RR+ )   =>    |-  ( ph  ->  E. r  e.  RR+  A. x  e.  V  A. y  e.  V  ( ( ( A D x )  <  r  /\  ( B D y )  <  r )  ->  ( abs `  ( ( A  .,  B )  -  ( x  .,  y ) ) )  <  R ) )
 
Theoremipcn 18636 The inner product operation of a complex pre-Hilbert space is continuous. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |- 
 .,  =  ( .i f `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  K  =  (
 TopOpen ` fld )   =>    |-  ( W  e.  CPreHil  ->  .,  e.  ( ( J 
 tX  J )  Cn  K ) )
 
Theoremcnmpt1ip 18637* Continuity of inner product; analogue of cnmpt12f 17323 which cannot be used directly because  .i is not a function. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  J  =  ( TopOpen `  W )   &    |-  C  =  (
 TopOpen ` fld )   &    |-  .,  =  ( .i `  W )   &    |-  ( ph  ->  W  e.  CPreHil )   &    |-  ( ph  ->  K  e.  (TopOn `  X ) )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( K  Cn  J ) )   &    |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( K  Cn  J ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( A  .,  B ) )  e.  ( K  Cn  C ) )
 
Theoremcnmpt2ip 18638* Continuity of inner product; analogue of cnmpt22f 17332 which cannot be used directly because  .i is not a function. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  J  =  ( TopOpen `  W )   &    |-  C  =  (
 TopOpen ` fld )   &    |-  .,  =  ( .i `  W )   &    |-  ( ph  ->  W  e.  CPreHil )   &    |-  ( ph  ->  K  e.  (TopOn `  X ) )   &    |-  ( ph  ->  L  e.  (TopOn `  Y ) )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( K  tX  L )  Cn  J ) )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  B )  e.  ( ( K 
 tX  L )  Cn  J ) )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  ( A 
 .,  B ) )  e.  ( ( K 
 tX  L )  Cn  C ) )
 
Theoremcsscld 18639 A "closed subspace" in a complex pre-Hilbert space is actually closed in the topology induced by the norm, thus justifying the terminology "closed subspace". (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  C  =  ( CSubSp `  W )   &    |-  J  =  (
 TopOpen `  W )   =>    |-  ( ( W  e.  CPreHil  /\  S  e.  C )  ->  S  e.  ( Clsd `  J )
 )
 
Theoremclsocv 18640 The orthogonal complement of the closure of a subset is the same as the orthogonal complement of the subset itself. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  O  =  ( ocv `  W )   &    |-  J  =  ( TopOpen `  W )   =>    |-  (
 ( W  e.  CPreHil  /\  S  C_  V )  ->  ( O `  ( ( cls `  J ) `  S ) )  =  ( O `  S ) )
 
11.3.14  Convergence and completeness
 
Syntaxccfil 18641 Extend class notation with the set of Cauchy filters.
 class CauFil
 
Syntaxcca 18642 Extend class notation with a function on metric spaces whose value is the set of all Cauchy sequences of the space.
 class  Cau
 
Syntaxcms 18643 Extend class notation with class of complete metric spaces.
 class  CMet
 
Definitiondf-cfil 18644* Define the set of Cauchy filters on a metric space. A Cauchy filter is a filter on the set such that for every  0  <  x there is an element of the filter whose metric diameter is less than  x. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |- CauFil  =  ( d  e.  U. ran  * Met  |->  { f  e.  ( Fil `  dom  dom  d )  |  A. x  e.  RR+  E. y  e.  f  ( d " ( y  X.  y
 ) )  C_  (
 0 [,) x ) }
 )
 
Definitiondf-cau 18645* Define a function on metric spaces whose value is the set of Cauchy sequences of the space. (Contributed by NM, 8-Sep-2006.)
 |- 
 Cau  =  ( d  e.  U. ran  * Met  |->  { f  e.  ( dom 
 dom  d  ^pm  CC )  |  A. x  e.  RR+  E. j  e.  ZZ  ( f  |`  ( ZZ>= `  j ) ) : ( ZZ>= `  j ) --> ( ( f `  j ) ( ball `  d ) x ) } )
 
Definitiondf-cmet 18646* Define the class of complete metrics. (Contributed by Mario Carneiro, 1-May-2014.)
 |- 
 CMet  =  ( x  e.  _V  |->  { d  e.  ( Met `  x )  | 
 A. f  e.  (CauFil `  d ) ( (
 MetOpen `  d )  fLim  f )  =/=  (/) } )
 
Theoremlmmbr 18647* Express the binary relation "sequence  F converges to point  P " in a metric space. Definition 1.4-1 of [Kreyszig] p. 25. The condition  F  C_  ( CC  X.  X ) allows us to use objects more general than sequences when convenient; see the comment in df-lm 16922. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 1-May-2014.)
 |-  J  =  ( MetOpen `  D )   &    |-  ( ph  ->  D  e.  ( * Met `  X ) )   =>    |-  ( ph  ->  ( F ( ~~> t `  J ) P  <->  ( F  e.  ( X  ^pm  CC )  /\  P  e.  X  /\  A. x  e.  RR+  E. y  e.  ran  ZZ>= ( F  |`  y ) : y --> ( P ( ball `  D ) x ) ) ) )
 
Theoremlmmbr2 18648* Express the binary relation "sequence  F converges to point  P " in a metric space. Definition 1.4-1 of [Kreyszig] p. 25. The condition  F  C_  ( CC  X.  X ) allows us to use objects more general than sequences when convenient; see the comment in df-lm 16922. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 1-May-2014.)
 |-  J  =  ( MetOpen `  D )   &    |-  ( ph  ->  D  e.  ( * Met `  X ) )   =>    |-  ( ph  ->  ( F ( ~~> t `  J ) P  <->  ( F  e.  ( X  ^pm  CC )  /\  P  e.  X  /\  A. x  e.  RR+  E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j )
 ( k  e.  dom  F 
 /\  ( F `  k )  e.  X  /\  ( ( F `  k ) D P )  <  x ) ) ) )
 
Theoremlmmbr3 18649* Express the binary relation "sequence  F converges to point  P " in a metric space using an abitrary set of upper integers. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 1-May-2014.)
 |-  J  =  ( MetOpen `  D )   &    |-  ( ph  ->  D  e.  ( * Met `  X ) )   &    |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   =>    |-  ( ph  ->  ( F ( ~~> t `  J ) P  <->  ( F  e.  ( X  ^pm  CC )  /\  P  e.  X  /\  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
 ( k  e.  dom  F 
 /\  ( F `  k )  e.  X  /\  ( ( F `  k ) D P )  <  x ) ) ) )
 
Theoremlmmcvg 18650* Convergence property of a converging sequence. (Contributed by NM, 1-Jun-2007.) (Revised by Mario Carneiro, 1-May-2014.)
 |-  J  =  ( MetOpen `  D )   &    |-  ( ph  ->  D  e.  ( * Met `  X ) )   &    |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )   &    |-  ( ph  ->  F ( ~~> t `  J ) P )   &    |-  ( ph  ->  R  e.  RR+ )   =>    |-  ( ph  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
 ( A  e.  X  /\  ( A D P )  <  R ) )
 
Theoremlmmbrf 18651* Express the binary relation "sequence  F converges to point  P " in a metric space using an abitrary set of upper integers. This version of lmmbr2 18648 presupposes that  F is a function. (Contributed by NM, 20-Jul-2007.) (Revised by Mario Carneiro, 1-May-2014.)
 |-  J  =  ( MetOpen `  D )   &    |-  ( ph  ->  D  e.  ( * Met `  X ) )   &    |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )   &    |-  ( ph  ->  F : Z --> X )   =>    |-  ( ph  ->  ( F (
 ~~> t `  J ) P  <->  ( P  e.  X  /\  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( A D P )  < 
 x ) ) )
 
Theoremlmnn 18652* A condition that implies convergence. (Contributed by NM, 8-Jun-2007.) (Revised by Mario Carneiro, 1-May-2014.)
 |-  J  =  ( MetOpen `  D )   &    |-  ( ph  ->  D  e.  ( * Met `  X ) )   &    |-  ( ph  ->  P  e.  X )   &    |-  ( ph  ->  F : NN --> X )   &    |-  (
 ( ph  /\  k  e. 
 NN )  ->  (
 ( F `  k
 ) D P )  <  ( 1  /  k ) )   =>    |-  ( ph  ->  F ( ~~> t `  J ) P )
 
Theoremcfilfval 18653* The set of Cauchy filters on a metric space. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  ( D  e.  ( * Met `  X )  ->  (CauFil `  D )  =  { f  e.  ( Fil `  X )  | 
 A. x  e.  RR+  E. y  e.  f  ( D " ( y  X.  y ) ) 
 C_  ( 0 [,) x ) } )
 
Theoremiscfil 18654* The property of being a Cauchy filter. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  ( D  e.  ( * Met `  X )  ->  ( F  e.  (CauFil `  D )  <->  ( F  e.  ( Fil `  X )  /\  A. x  e.  RR+  E. y  e.  F  ( D " ( y  X.  y ) ) 
 C_  ( 0 [,) x ) ) ) )
 
Theoremiscfil2 18655* The property of being a Cauchy filter. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  ( D  e.  ( * Met `  X )  ->  ( F  e.  (CauFil `  D )  <->  ( F  e.  ( Fil `  X )  /\  A. x  e.  RR+  E. y  e.  F  A. z  e.  y  A. w  e.  y  (
 z D w )  <  x ) ) )
 
Theoremcfilfil 18656 A Cauchy filter is a filter. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  ( ( D  e.  ( * Met `  X )  /\  F  e.  (CauFil `  D ) )  ->  F  e.  ( Fil `  X ) )
 
Theoremcfili 18657* Property of a Cauchy filter. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  ( ( F  e.  (CauFil `  D )  /\  R  e.  RR+ )  ->  E. x  e.  F  A. y  e.  x  A. z  e.  x  (
 y D z )  <  R )
 
Theoremcfil3i 18658* A Cauchy filter contains balls of any pre-chosen size. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  ( ( D  e.  ( * Met `  X )  /\  F  e.  (CauFil `  D )  /\  R  e.  RR+ )  ->  E. x  e.  X  ( x (
 ball `  D ) R )  e.  F )
 
Theoremcfilss 18659 A filter finer than a Cauchy filter is Cauchy. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  ( ( ( D  e.  ( * Met `  X )  /\  F  e.  (CauFil `  D )
 )  /\  ( G  e.  ( Fil `  X )  /\  F  C_  G ) )  ->  G  e.  (CauFil `  D ) )
 
Theoremfgcfil 18660* The Cauchy filter condition for a filter base. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  ( ( D  e.  ( * Met `  X )  /\  B  e.  ( fBas `  X ) ) 
 ->  ( ( X filGen B )  e.  (CauFil `  D ) 
 <-> 
 A. x  e.  RR+  E. y  e.  B  A. z  e.  y  A. w  e.  y  (
 z D w )  <  x ) )
 
Theoremfmcfil 18661* The Cauchy filter condition for a filter map. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  ( ( D  e.  ( * Met `  X )  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X ) 
 ->  ( ( ( X 
 FilMap  F ) `  B )  e.  (CauFil `  D ) 
 <-> 
 A. x  e.  RR+  E. y  e.  B  A. z  e.  y  A. w  e.  y  (
 ( F `  z
 ) D ( F `
  w ) )  <  x ) )
 
Theoremiscfil3 18662* A filter is Cauchy iff it contains a ball of any chosen size. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  ( D  e.  ( * Met `  X )  ->  ( F  e.  (CauFil `  D )  <->  ( F  e.  ( Fil `  X )  /\  A. r  e.  RR+  E. x  e.  X  ( x ( ball `  D ) r )  e.  F ) ) )
 
Theoremcfilfcls 18663 Similar to ultrafilters (uffclsflim 17689), the cluster points and limit points of a Cauchy filter coincide. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  J  =  ( MetOpen `  D )   &    |-  X  =  dom  dom 
 D   =>    |-  ( F  e.  (CauFil `  D )  ->  ( J  fClus  F )  =  ( J  fLim  F ) )
 
Theoremcaufval 18664* The set of Cauchy sequences on a metric space. (Contributed by NM, 8-Sep-2006.) (Revised by Mario Carneiro, 5-Sep-2015.)
 |-  ( D  e.  ( * Met `  X )  ->  ( Cau `  D )  =  { f  e.  ( X  ^pm  CC )  |  A. x  e.  RR+  E. k  e.  ZZ  ( f  |`  ( ZZ>= `  k ) ) : ( ZZ>= `  k ) --> ( ( f `  k ) ( ball `  D ) x ) } )
 
Theoremiscau 18665* Express the property " F is a Cauchy sequence of metric  D." Part of Definition 1.4-3 of [Kreyszig] p. 28. The condition  F  C_  ( CC  X.  X ) allows us to use objects more general than sequences when convenient; see the comment in df-lm 16922. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)
 |-  ( D  e.  ( * Met `  X )  ->  ( F  e.  ( Cau `  D )  <->  ( F  e.  ( X  ^pm  CC )  /\  A. x  e.  RR+  E. k  e.  ZZ  ( F  |`  ( ZZ>= `  k
 ) ) : (
 ZZ>= `  k ) --> ( ( F `  k ) ( ball `  D ) x ) ) ) )
 
Theoremiscau2 18666* Express the property " F is a Cauchy sequence of metric  D," using an abitrary set of upper integers. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)
 |-  ( D  e.  ( * Met `  X )  ->  ( F  e.  ( Cau `  D )  <->  ( F  e.  ( X  ^pm  CC )  /\  A. x  e.  RR+  E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j ) ( k  e.  dom  F  /\  ( F `  k )  e.  X  /\  (
 ( F `  k
 ) D ( F `
  j ) )  <  x ) ) ) )
 
Theoremiscau3 18667* Express the Cauchy sequence property in the more conventional three-quantifier form. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  D  e.  ( * Met `  X ) )   &    |-  ( ph  ->  M  e.  ZZ )   =>    |-  ( ph  ->  ( F  e.  ( Cau `  D )  <->  ( F  e.  ( X  ^pm  CC )  /\  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( k  e.  dom  F  /\  ( F `  k )  e.  X  /\  A. m  e.  ( ZZ>= `  k ) ( ( F `  k ) D ( F `  m ) )  < 
 x ) ) ) )
 
Theoremiscau4 18668* Express the property " F is a Cauchy sequence of metric  D," using an arbitrary set of upper integers. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  D  e.  ( * Met `  X ) )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )   &    |-  ( ( ph  /\  j  e.  Z ) 
 ->  ( F `  j
 )  =  B )   =>    |-  ( ph  ->  ( F  e.  ( Cau `  D ) 
 <->  ( F  e.  ( X  ^pm  CC )  /\  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
 ( k  e.  dom  F 
 /\  A  e.  X  /\  ( A D B )  <  x ) ) ) )
 
Theoremiscauf 18669* Express the property " F is a Cauchy sequence of metric  D " presupposing  F is a function. (Contributed by NM, 24-Jul-2007.) (Revised by Mario Carneiro, 23-Dec-2013.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  D  e.  ( * Met `  X ) )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )   &    |-  ( ( ph  /\  j  e.  Z ) 
 ->  ( F `  j
 )  =  B )   &    |-  ( ph  ->  F : Z
 --> X )   =>    |-  ( ph  ->  ( F  e.  ( Cau `  D )  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( B D A )  < 
 x ) )
 
Theoremcaun0 18670 A metric with a Cauchy sequence cannot be empty. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 24-Dec-2013.)
 |-  ( ( D  e.  ( * Met `  X )  /\  F  e.  ( Cau `  D ) ) 
 ->  X  =/=  (/) )
 
Theoremcaufpm 18671 Inclusion of a Cauchy sequence, under our definition. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 24-Dec-2013.)
 |-  ( ( D  e.  ( * Met `  X )  /\  F  e.  ( Cau `  D ) ) 
 ->  F  e.  ( X 
 ^pm  CC ) )
 
Theoremcaucfil 18672 A Cauchy sequence predicate can be expressed in terms of the Cauchy filter predicate for a suitably chosen filter. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  L  =  ( ( X  FilMap  F ) `
  ( ZZ>= " Z ) )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  M  e.  ZZ  /\  F : Z --> X ) 
 ->  ( F  e.  ( Cau `  D )  <->  L  e.  (CauFil `  D ) ) )
 
Theoremiscmet 18673* The property " D is a complete metric." meaning all Cauchy filters converge to a point in the space. (Contributed by Mario Carneiro, 1-May-2014.) (Revised by Mario Carneiro, 13-Oct-2015.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( CMet `  X )  <->  ( D  e.  ( Met `  X )  /\  A. f  e.  (CauFil `  D ) ( J 
 fLim  f )  =/=  (/) ) )
 
Theoremcmetcvg 18674 The convergence of a Cauchy filter in a complete metric space. (Contributed by Mario Carneiro, 14-Oct-2015.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( ( D  e.  ( CMet `  X )  /\  F  e.  (CauFil `  D ) )  ->  ( J 
 fLim  F )  =/=  (/) )
 
Theoremcmetmet 18675 A complete metric space is a metric space. (Contributed by NM, 18-Dec-2006.) (Revised by Mario Carneiro, 29-Jan-2014.)
 |-  ( D  e.  ( CMet `  X )  ->  D  e.  ( Met `  X ) )
 
Theoremcmetmeti 18676 A complete metric space is a metric space. (Contributed by NM, 26-Oct-2007.)
 |-  D  e.  ( CMet `  X )   =>    |-  D  e.  ( Met `  X )
 
Theoremcmetcaulem 18677* Lemma for cmetcau 18678. (Contributed by Mario Carneiro, 14-Oct-2015.)
 |-  J  =  ( MetOpen `  D )   &    |-  ( ph  ->  D  e.  ( CMet `  X ) )   &    |-  ( ph  ->  P  e.  X )   &    |-  ( ph  ->  F  e.  ( Cau `  D ) )   &    |-  G  =  ( x  e.  NN  |->  if ( x  e. 
 dom  F ,  ( F `
  x ) ,  P ) )   =>    |-  ( ph  ->  F  e.  dom  ( ~~> t `  J ) )
 
Theoremcmetcau 18678 The convergence of a Cauchy sequence in a complete metric space. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 14-Oct-2015.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( ( D  e.  ( CMet `  X )  /\  F  e.  ( Cau `  D ) )  ->  F  e.  dom  ( ~~> t `  J ) )
 
Theoremiscmet3lem3 18679* Lemma for iscmet3 18682. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( ( 1  /  2 ) ^ k )  <  R )
 
Theoremiscmet3lem1 18680* Lemma for iscmet3 18682. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  J  =  (
 MetOpen `  D )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  D  e.  ( Met `  X ) )   &    |-  ( ph  ->  F : Z --> X )   &    |-  ( ph  ->  A. k  e. 
 ZZ  A. u  e.  ( S `  k ) A. v  e.  ( S `  k ) ( u D v )  < 
 ( ( 1  / 
 2 ) ^ k
 ) )   &    |-  ( ph  ->  A. k  e.  Z  A. n  e.  ( M ... k ) ( F `
  k )  e.  ( S `  n ) )   =>    |-  ( ph  ->  F  e.  ( Cau `  D ) )
 
Theoremiscmet3lem2 18681* Lemma for iscmet3 18682. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  J  =  (
 MetOpen `  D )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  D  e.  ( Met `  X ) )   &    |-  ( ph  ->  F : Z --> X )   &    |-  ( ph  ->  A. k  e. 
 ZZ  A. u  e.  ( S `  k ) A. v  e.  ( S `  k ) ( u D v )  < 
 ( ( 1  / 
 2 ) ^ k
 ) )   &    |-  ( ph  ->  A. k  e.  Z  A. n  e.  ( M ... k ) ( F `
  k )  e.  ( S `  n ) )   &    |-  ( ph  ->  G  e.  ( Fil `  X ) )   &    |-  ( ph  ->  S : ZZ --> G )   &    |-  ( ph  ->  F  e.  dom  ( ~~> t `  J ) )   =>    |-  ( ph  ->  ( J  fLim  G )  =/=  (/) )
 
Theoremiscmet3 18682* The property " D is a complete metric" expressed in terms of functions on  NN (or any other upper integer set). Thus we only have to look at functions on  NN, and not all possible Cauchy filters, to determine completeness. (The proof uses countable choice.) (Contributed by NM, 18-Dec-2006.) (Revised by Mario Carneiro, 5-May-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  J  =  (
 MetOpen `  D )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  D  e.  ( Met `  X ) )   =>    |-  ( ph  ->  ( D  e.  ( CMet `  X )  <->  A. f  e.  ( Cau `  D ) ( f : Z --> X  ->  f  e.  dom  ( ~~> t `  J ) ) ) )
 
Theoremiscmet2 18683 A metric  D is complete iff all Cauchy sequences converge to a point in the space. The proof uses countable choice. Part of Definition 1.4-3 of [Kreyszig] p. 28. (Contributed by NM, 7-Sep-2006.) (Revised by Mario Carneiro, 15-Oct-2015.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( CMet `  X )  <->  ( D  e.  ( Met `  X )  /\  ( Cau `  D )  C_  dom  ( ~~> t `  J ) ) )
 
Theoremcfilresi 18684 A Cauchy filter on a metric subspace extends to a Cauchy filter in the larger space. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  ( ( D  e.  ( * Met `  X )  /\  F  e.  (CauFil `  ( D  |`  ( Y  X.  Y ) ) ) )  ->  ( X filGen F )  e.  (CauFil `  D )
 )
 
Theoremcfilres 18685 Cauchy filter on a metric subspace. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  ( ( D  e.  ( * Met `  X )  /\  F  e.  ( Fil `  X )  /\  Y  e.  F )  ->  ( F  e.  (CauFil `  D )  <->  ( Ft  Y )  e.  (CauFil `  ( D  |`  ( Y  X.  Y ) ) ) ) )
 
Theoremcaussi 18686 Cauchy sequence on a metric subspace. (Contributed by NM, 30-Jan-2008.) (Revised by Mario Carneiro, 30-Dec-2013.)
 |-  ( D  e.  ( * Met `  X )  ->  ( Cau `  ( D  |`  ( Y  X.  Y ) ) ) 
 C_  ( Cau `  D ) )
 
Theoremcauss 18687 Cauchy sequence on a metric subspace. (Contributed by NM, 29-Jan-2008.) (Revised by Mario Carneiro, 30-Dec-2013.)
 |-  ( ( D  e.  ( * Met `  X )  /\  F : NN --> Y )  ->  ( F  e.  ( Cau `  D ) 
 <->  F  e.  ( Cau `  ( D  |`  ( Y  X.  Y ) ) ) ) )
 
Theoremequivcfil 18688* If the metric  D is "strongly finer" than  C (meaning that there is a positive real constant 
R such that  C ( x ,  y )  <_  R  x.  D (
x ,  y )), all the  D-Cauchy filters are also  C-Cauchy. (Using this theorem twice in each direction states that if two metrics are strongly equivalent, then they have the same Cauchy sequences.) (Contributed by Mario Carneiro, 14-Sep-2015.)
 |-  ( ph  ->  C  e.  ( Met `  X ) )   &    |-  ( ph  ->  D  e.  ( Met `  X ) )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  ( x C y )  <_  ( R  x.  ( x D y ) ) )   =>    |-  ( ph  ->  (CauFil `  D )  C_  (CauFil `  C ) )
 
Theoremequivcau 18689* If the metric  D is "strongly finer" than  C (meaning that there is a positive real constant 
R such that  C ( x ,  y )  <_  R  x.  D (
x ,  y )), all the  D-Cauchy sequences are also  C-Cauchy. (Using this theorem twice in each direction states that if two metrics are strongly equivalent, then they have the same Cauchy sequences.) (Contributed by Mario Carneiro, 14-Sep-2015.)
 |-  ( ph  ->  C  e.  ( Met `  X ) )   &    |-  ( ph  ->  D  e.  ( Met `  X ) )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  ( x C y )  <_  ( R  x.  ( x D y ) ) )   =>    |-  ( ph  ->  ( Cau `  D )  C_  ( Cau `  C )
 )
 
Theoremlmle 18690* If the distance from each member of a converging sequence to a given point is less than or equal to a given amount, so is the convergence value. (Contributed by NM, 23-Dec-2007.) (Proof shortened by Mario Carneiro, 1-May-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  J  =  (
 MetOpen `  D )   &    |-  ( ph  ->  D  e.  ( * Met `  X )
 )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F ( ~~> t `  J ) P )   &    |-  ( ph  ->  Q  e.  X )   &    |-  ( ph  ->  R  e.  RR* )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( Q D ( F `  k ) )  <_  R )   =>    |-  ( ph  ->  ( Q D P )  <_  R )
 
Theoremlmclim 18691 Relate a limit on the metric space of complex numbers to our complex number limit notation. (Contributed by NM, 9-Dec-2006.) (Revised by Mario Carneiro, 1-May-2014.)
 |-  J  =  ( TopOpen ` fld )   &    |-  Z  =  ( ZZ>= `  M )   =>    |-  (
 ( M  e.  ZZ  /\  Z  C_  dom  F ) 
 ->  ( F ( ~~> t `  J ) P  <->  ( F  e.  ( CC  ^pm  CC )  /\  F  ~~>  P ) ) )
 
Theoremlmclimf 18692 Relate a limit on the metric space of complex numbers to our complex number limit notation. (Contributed by NM, 24-Jul-2007.) (Revised by Mario Carneiro, 1-May-2014.)
 |-  J  =  ( TopOpen ` fld )   &    |-  Z  =  ( ZZ>= `  M )   =>    |-  (
 ( M  e.  ZZ  /\  F : Z --> CC )  ->  ( F ( ~~> t `  J ) P  <->  F  ~~>  P ) )
 
Theoremmetelcls 18693* A point belongs to the closure of a subset iff there is a sequence in the subset converging to it. Theorem 1.4-6(a) of [Kreyszig] p. 30. This proof uses countable choice ax-cc 8029. The statement can be generalized to first-countable spaces, not just metrizable spaces. (Contributed by NM, 8-Nov-2007.) (Proof shortened by Mario Carneiro, 1-May-2015.)
 |-  J  =  ( MetOpen `  D )   &    |-  ( ph  ->  D  e.  ( * Met `  X ) )   &    |-  ( ph  ->  S  C_  X )   =>    |-  ( ph  ->  ( P  e.  ( ( cls `  J ) `  S )  <->  E. f ( f : NN --> S  /\  f ( ~~> t `  J ) P ) ) )
 
Theoremmetcld 18694* A subset of a metric space is closed iff every convergent sequence on it converges to a point in the subset. Theorem 1.4-6(b) of [Kreyszig] p. 30. (Contributed by NM, 11-Nov-2007.) (Revised by Mario Carneiro, 1-May-2014.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X )  ->  ( S  e.  ( Clsd `  J )  <->  A. x A. f ( ( f : NN --> S  /\  f ( ~~> t `  J ) x ) 
 ->  x  e.  S ) ) )
 
Theoremmetcld2 18695 A subset of a metric space is closed iff every convergent sequence on it converges to a point in the subset. Theorem 1.4-6(b) of [Kreyszig] p. 30. (Contributed by Mario Carneiro, 1-May-2014.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X )  ->  ( S  e.  ( Clsd `  J )  <->  ( ( ~~> t `  J ) " ( S  ^m  NN ) )  C_  S ) )
 
Theoremcaubl 18696* Sufficient condition to ensure a sequence of nested balls is Cauchy. (Contributed by Mario Carneiro, 18-Jan-2014.) (Revised by Mario Carneiro, 1-May-2014.)
 |-  ( ph  ->  D  e.  ( * Met `  X ) )   &    |-  ( ph  ->  F : NN --> ( X  X.  RR+ ) )   &    |-  ( ph  ->  A. n  e.  NN  ( ( ball `  D ) `  ( F `  ( n  +  1
 ) ) )  C_  ( ( ball `  D ) `  ( F `  n ) ) )   &    |-  ( ph  ->  A. r  e.  RR+  E. n  e.  NN  ( 2nd `  ( F `  n ) )  < 
 r )   =>    |-  ( ph  ->  ( 1st  o.  F )  e.  ( Cau `  D ) )
 
Theoremcaublcls 18697* The convergent point of a sequence of nested balls is in the closures of any of the balls (i.e. it is in the intersection of the closures). Indeed, it is the only point in the intersection because a metric space is Hausdorff, but we don't prove this here. (Contributed by Mario Carneiro, 21-Jan-2014.) (Revised by Mario Carneiro, 1-May-2014.)
 |-  ( ph  ->  D  e.  ( * Met `  X ) )   &    |-  ( ph  ->  F : NN --> ( X  X.  RR+ ) )   &    |-  ( ph  ->  A. n  e.  NN  ( ( ball `  D ) `  ( F `  ( n  +  1
 ) ) )  C_  ( ( ball `  D ) `  ( F `  n ) ) )   &    |-  J  =  ( MetOpen `  D )   =>    |-  ( ( ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  P  e.  ( ( cls `  J ) `  ( ( ball `  D ) `  ( F `  A ) ) ) )
 
Theoremmetcnp4 18698* Two ways to say a mapping from metric  C to metric  D is continuous at point  P. Theorem 14-4.3 of [Gleason] p. 240. (Contributed by NM, 17-May-2007.) (Revised by Mario Carneiro, 4-May-2014.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   &    |-  ( ph  ->  C  e.  ( * Met `  X )
 )   &    |-  ( ph  ->  D  e.  ( * Met `  Y ) )   &    |-  ( ph  ->  P  e.  X )   =>    |-  ( ph  ->  ( F  e.  ( ( J  CnP  K ) `
  P )  <->  ( F : X
 --> Y  /\  A. f
 ( ( f : NN --> X  /\  f
 ( ~~> t `  J ) P )  ->  ( F  o.  f ) ( ~~> t `  K ) ( F `  P ) ) ) ) )
 
Theoremmetcn4 18699* Two ways to say a mapping from metric  C to metric  D is continuous. Theorem 10.3 of [Munkres] p. 128. (Contributed by NM, 13-Jun-2007.) (Revised by Mario Carneiro, 4-May-2014.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   &    |-  ( ph  ->  C  e.  ( * Met `  X )
 )   &    |-  ( ph  ->  D  e.  ( * Met `  Y ) )   &    |-  ( ph  ->  F : X --> Y )   =>    |-  ( ph  ->  ( F  e.  ( J  Cn  K ) 
 <-> 
 A. f ( f : NN --> X  ->  A. x ( f ( ~~> t `  J ) x  ->  ( F  o.  f ) ( ~~> t `  K ) ( F `
  x ) ) ) ) )
 
Theoremiscmet3i 18700* Properties that determine a complete metric space. (Contributed by NM, 15-Apr-2007.) (Revised by Mario Carneiro, 5-May-2014.)
 |-  J  =  ( MetOpen `  D )   &    |-  D  e.  ( Met `  X )   &    |-  (
 ( f  e.  ( Cau `  D )  /\  f : NN --> X ) 
 ->  f  e.  dom  (
 ~~> t `  J ) )   =>    |-  D  e.  ( CMet `  X )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31404
  Copyright terms: Public domain < Previous  Next >