HomeHome Metamath Proof Explorer
Theorem List (p. 187 of 322)
< 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
(1-21498)
  Hilbert Space Explorer  Hilbert Space Explorer
(21499-23021)
  Users' Mathboxes  Users' Mathboxes
(23022-32154)
 

Theorem List for Metamath Proof Explorer - 18601-18700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnmhmcn 18601 A linear operator over a normed complex module is bounded iff it is continuous. (Contributed by Mario Carneiro, 22-Oct-2015.)
 |-  J  =  ( TopOpen `  S )   &    |-  K  =  (
 TopOpen `  T )   &    |-  G  =  (Scalar `  S )   &    |-  B  =  ( Base `  G )   =>    |-  (
 ( S  e.  (NrmMod  i^i CMod )  /\  T  e.  (NrmMod  i^i CMod )  /\  QQ  C_  B )  ->  ( F  e.  ( S NMHom  T )  <->  ( F  e.  ( S LMHom  T )  /\  F  e.  ( J  Cn  K ) ) ) )
 
11.4.2  Complex pre-Hilbert space
 
Syntaxccph 18602 Extend class notation with a complex pre-Hilbert space.
 class  CPreHil
 
Syntaxctch 18603 Function to put a norm on a Hilbert space.
 class toCHil
 
Definitiondf-cph 18604* Define a complex pre-Hilbert space. By restricting the scalar field to a quadratically closed subfield of  CC, we have enough structure to define a norm, with the associated connection to a metric and topology. (Contributed by Mario Carneiro, 8-Oct-2015.)
 |-  CPreHil  =  { w  e.  ( PreHil  i^i NrmMod )  |  [. (Scalar `  w )  /  f ]. [. ( Base `  f )  /  k ]. ( f  =  (flds  k ) 
 /\  ( sqr " (
 k  i^i  ( 0 [,)  +oo ) ) ) 
 C_  k  /\  ( norm `  w )  =  ( x  e.  ( Base `  w )  |->  ( sqr `  ( x ( .i `  w ) x ) ) ) ) }
 
Definitiondf-tch 18605* Define a function to augment a (pre-)Hilbert space with a norm. No extra parameters are needed, but some conditions must be satisfied to ensure that this in fact creates a normed pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |- toCHil  =  ( w  e.  _V  |->  ( w toNrmGrp  ( x  e.  ( Base `  w )  |->  ( sqr `  ( x ( .i `  w ) x ) ) ) ) )
 
Theoremiscph 18606* A complex pre-Hilbert space is a pre-Hilbert space over a quadratically closed subfield of the complexes, with a norm defined (Contributed by Mario Carneiro, 8-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  .,  =  ( .i `  W )   &    |-  N  =  ( norm `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( W  e.  CPreHil  <->  ( ( W  e.  PreHil  /\  W  e. NrmMod  /\  F  =  (flds  K ) )  /\  ( sqr " ( K  i^i  ( 0 [,)  +oo ) ) )  C_  K  /\  N  =  ( x  e.  V  |->  ( sqr `  ( x  .,  x ) ) ) ) )
 
Theoremcphphl 18607 A complex pre-Hilbert space is a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  ( W  e.  CPreHil  ->  W  e.  PreHil )
 
Theoremcphnlm 18608 A complex pre-Hilbert space is a normed module. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  ( W  e.  CPreHil  ->  W  e. NrmMod )
 
Theoremcphngp 18609 A complex pre-Hilbert space is a normed group. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  ( W  e.  CPreHil  ->  W  e. NrmGrp )
 
Theoremcphlmod 18610 A complex pre-Hilbert space is a left module. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  ( W  e.  CPreHil  ->  W  e.  LMod )
 
Theoremcphlvec 18611 A complex pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  ( W  e.  CPreHil  ->  W  e.  LVec )
 
Theoremcphnvc 18612 A complex pre-Hilbert space is a normed vector space. (Contributed by Mario Carneiro, 8-Oct-2015.)
 |-  ( W  e.  CPreHil  ->  W  e. NrmVec )
 
Theoremcphsubrglem 18613 Lemma for cphsubrg 18616. (Contributed by Mario Carneiro, 9-Oct-2015.)
 |-  K  =  ( Base `  F )   &    |-  ( ph  ->  F  =  (flds  A ) )   &    |-  ( ph  ->  F  e.  DivRing )   =>    |-  ( ph  ->  ( F  =  (flds  K )  /\  K  =  ( A  i^i  CC )  /\  K  e.  (SubRing ` fld ) ) )
 
Theoremcphreccllem 18614 Lemma for cphreccl 18617. (Contributed by Mario Carneiro, 8-Oct-2015.)
 |-  K  =  ( Base `  F )   &    |-  ( ph  ->  F  =  (flds  A ) )   &    |-  ( ph  ->  F  e.  DivRing )   =>    |-  ( ( ph  /\  X  e.  K  /\  X  =/=  0 )  ->  ( 1 
 /  X )  e.  K )
 
Theoremcphsca 18615 A complex pre-Hilbert space is a vector space over a subfield of  CC. (Contributed by Mario Carneiro, 8-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( W  e.  CPreHil  ->  F  =  (flds  K ) )
 
Theoremcphsubrg 18616 The scalar field of a complex pre-Hilbert space is a subring of  CC. (Contributed by Mario Carneiro, 8-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( W  e.  CPreHil  ->  K  e.  (SubRing ` fld ) )
 
Theoremcphreccl 18617 The scalar field of a complex pre-Hilbert space is closed under reciprocal. (Contributed by Mario Carneiro, 8-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( ( W  e.  CPreHil  /\  A  e.  K  /\  A  =/=  0 )  ->  ( 1  /  A )  e.  K )
 
Theoremcphdivcl 18618 The scalar field of a complex pre-Hilbert space is closed under reciprocal. (Contributed by Mario Carneiro, 11-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( ( W  e.  CPreHil  /\  ( A  e.  K  /\  B  e.  K  /\  B  =/=  0 ) ) 
 ->  ( A  /  B )  e.  K )
 
Theoremcphcjcl 18619 The scalar field of a complex pre-Hilbert space is closed under conjugation. (Contributed by Mario Carneiro, 11-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( ( W  e.  CPreHil  /\  A  e.  K ) 
 ->  ( * `  A )  e.  K )
 
Theoremcphsqrcl 18620 The scalar field of a complex pre-Hilbert space is closed under square roots of positive reals (i.e. it is quadratically closed relative to  RR). (Contributed by Mario Carneiro, 8-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( ( W  e.  CPreHil  /\  ( A  e.  K  /\  A  e.  RR  /\  0  <_  A ) ) 
 ->  ( sqr `  A )  e.  K )
 
Theoremcphabscl 18621 The scalar field of a complex pre-Hilbert space is closed under the absolute value operation. (Contributed by Mario Carneiro, 11-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( ( W  e.  CPreHil  /\  A  e.  K ) 
 ->  ( abs `  A )  e.  K )
 
Theoremcphsqrcl2 18622 The scalar field of a complex pre-Hilbert space is closed under square roots of all numbers except possibly the negative reals. (Contributed by Mario Carneiro, 8-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( ( W  e.  CPreHil  /\  A  e.  K  /\  -.  -u A  e.  RR+ )  ->  ( sqr `  A )  e.  K )
 
Theoremcphsqrcl3 18623 If the scalar field contains  _i, it is completely closed under square roots (i.e. it is quadratically closed). (Contributed by Mario Carneiro, 11-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( ( W  e.  CPreHil  /\  _i  e.  K  /\  A  e.  K )  ->  ( sqr `  A )  e.  K )
 
Theoremcphqss 18624 The scalar field of a complex pre-Hilbert space contains all rational numbers. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( W  e.  CPreHil  ->  QQ  C_  K )
 
Theoremcphclm 18625 A complex pre-Hilbert space is a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  ( W  e.  CPreHil  ->  W  e. CMod )
 
Theoremcphnmvs 18626 Norm of a scalar product. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 norm `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( ( W  e.  CPreHil  /\  X  e.  K  /\  Y  e.  V )  ->  ( N `  ( X  .x.  Y ) )  =  ( ( abs `  X )  x.  ( N `  Y ) ) )
 
Theoremcphipcl 18627 An inner product is a member of the complex numbers. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  .,  =  ( .i `  W )   =>    |-  ( ( W  e.  CPreHil  /\  A  e.  V  /\  B  e.  V )  ->  ( A  .,  B )  e.  CC )
 
Theoremcphnmfval 18628* The value of the norm in a complex pre-Hilbert space is the square root of the inner product of a vector with itself. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  .,  =  ( .i `  W )   &    |-  N  =  ( norm `  W )   =>    |-  ( W  e.  CPreHil  ->  N  =  ( x  e.  V  |->  ( sqr `  ( x  .,  x ) ) ) )
 
Theoremcphnm 18629 The square of the norm is the norm of an inner product in a normed pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  .,  =  ( .i `  W )   &    |-  N  =  ( norm `  W )   =>    |-  (
 ( W  e.  CPreHil  /\  A  e.  V )  ->  ( N `  A )  =  ( sqr `  ( A  .,  A ) ) )
 
Theoremnmsq 18630 The square of the norm is the norm of an inner product in a normed pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  .,  =  ( .i `  W )   &    |-  N  =  ( norm `  W )   =>    |-  (
 ( W  e.  CPreHil  /\  A  e.  V )  ->  ( ( N `  A ) ^ 2
 )  =  ( A 
 .,  A ) )
 
Theoremcphnmf 18631 The norm of a vector is a member of the scalar field in a complex pre-Hilbert space. (Contributed by Mario Carneiro, 9-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  .,  =  ( .i `  W )   &    |-  N  =  ( norm `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( W  e.  CPreHil  ->  N : V
 --> K )
 
Theoremcphnmcl 18632 The norm of a vector is a member of the scalar field in a complex pre-Hilbert space. (Contributed by Mario Carneiro, 9-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  .,  =  ( .i `  W )   &    |-  N  =  ( norm `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  (
 ( W  e.  CPreHil  /\  A  e.  V )  ->  ( N `  A )  e.  K )
 
Theoremreipcl 18633 An inner product of an element with itself is real. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  .,  =  ( .i `  W )   =>    |-  ( ( W  e.  CPreHil  /\  A  e.  V )  ->  ( A 
 .,  A )  e. 
 RR )
 
Theoremipge0 18634 The inner product in a complex pre-Hilbert space is positive definite. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  .,  =  ( .i `  W )   =>    |-  ( ( W  e.  CPreHil  /\  A  e.  V )  ->  0  <_  ( A  .,  A ) )
 
Theoremcphipcj 18635 Conjugate of an inner product in a complex pre-Hilbert space. Complex version of ipcj 16538. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   =>    |-  (
 ( W  e.  CPreHil  /\  A  e.  V  /\  B  e.  V )  ->  ( * `  ( A  .,  B ) )  =  ( B  .,  A ) )
 
Theoremcphorthcom 18636 Orthogonality (meaning inner product is 0) is commutative. Complex version of iporthcom 16539. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   =>    |-  (
 ( W  e.  CPreHil  /\  A  e.  V  /\  B  e.  V )  ->  ( ( A  .,  B )  =  0  <->  ( B  .,  A )  =  0 ) )
 
Theoremcphip0l 18637 Inner product with a zero first argument. Part of proof of Theorem 6.44 of [Ponnusamy] p. 361. Complex version of ip0l 16540. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   =>    |-  ( ( W  e.  CPreHil  /\  A  e.  V ) 
 ->  (  .0.  .,  A )  =  0 )
 
Theoremcphip0r 18638 Inner product with a zero second argument. Complex version of ip0r 16541. (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 18639 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 16542. (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 18640 Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. Complex version of ipdir 16543. (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 18641 Distributive law for inner product. Complex version of ipdi 16544. (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 18642 Distributive law for inner product. Complex version of ip2di 16545. (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 18643 Distributive law for inner product subtraction. Complex version of ipsubdir 16546. (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 18644 Distributive law for inner product subtraction. Complex version of ipsubdi 16547. (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 18645 Distributive law for inner product subtraction. Complex version of ip2subdi 16548. (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 18646 Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. See ipass 16549, his5 21665. (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 18647 "Associative" law for second argument of inner product (compare cphass 18646). See ipassr 16550, 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 18648 Move scalar multiplication to outside of inner product. See his35 21667. (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 18649* Lemma for tchbas 18651 and similar theorems. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  V  =  ( Base `  W )   =>    |-  ( x  e.  V  |->  ( sqr `  ( x  .,  x ) ) )  e.  _V
 
Theoremtchval 18650* 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 18651 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 18652 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 18653 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 18654 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 18655 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 18656 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 18657 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 18658 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 18659* 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 18660 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 18661 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 18662 Lemma for tchcph 18667. (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 18663 Lemma for tchcph 18667: 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 18664* 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 18665* Lemma for tchcph 18667: 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 18666* Lemma for tchcph 18667: 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 18667* 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 18668 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 18669 Lemma for nmpar 18670. (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 18670 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 18671 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 18672* 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 18673 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 18674* Continuity of inner product; analogue of cnmpt12f 17360 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 18675* Continuity of inner product; analogue of cnmpt22f 17369 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 18676 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 18677 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 ) )
 
11.4.3  Convergence and completeness
 
Syntaxccfil 18678 Extend class notation with the set of Cauchy filters.
 class CauFil
 
Syntaxcca 18679 Extend class notation with a function on metric spaces whose value is the set of all Cauchy sequences of the space.
 class  Cau
 
Syntaxcms 18680 Extend class notation with class of complete metric spaces.
 class  CMet
 
Definitiondf-cfil 18681* 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 18682* 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 18683* 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 18684* 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 16959. (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 18685* 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 16959. (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 18686* 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 18687* 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 18688* 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 18685 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 18689* 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 18690* 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 18691* 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 18692* 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 18693 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 18694* 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 18695* 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 18696 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 18697* 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 18698* 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 18699* 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 18700 Similar to ultrafilters (uffclsflim 17726), 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 ) )
    < 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 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32154
  Copyright terms: Public domain < Previous  Next >