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Theorem List for Metamath Proof Explorer - 19101-19200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcphabscl 19101 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 19102 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 19103 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 19104 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 19105 A complex pre-Hilbert space is a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  ( W  e.  CPreHil  ->  W  e. CMod )
 
Theoremcphnmvs 19106 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 19107 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 19108* 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 19109 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 19110 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 19111 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 19112 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 19113 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 19114 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 19115 Conjugate of an inner product in a complex pre-Hilbert space. Complex version of ipcj 16820. (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 19116 Orthogonality (meaning inner product is 0) is commutative. Complex version of iporthcom 16821. (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 19117 Inner product with a zero first argument. Part of proof of Theorem 6.44 of [Ponnusamy] p. 361. Complex version of ip0l 16822. (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 19118 Inner product with a zero second argument. Complex version of ip0r 16823. (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 19119 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 16824. (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 19120 Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. Complex version of ipdir 16825. (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 19121 Distributive law for inner product. Complex version of ipdi 16826. (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 19122 Distributive law for inner product. Complex version of ip2di 16827. (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 19123 Distributive law for inner product subtraction. Complex version of ipsubdir 16828. (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 19124 Distributive law for inner product subtraction. Complex version of ipsubdi 16829. (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 19125 Distributive law for inner product subtraction. Complex version of ip2subdi 16830. (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 19126 Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. See ipass 16831, his5 22541. (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 19127 "Associative" law for second argument of inner product (compare cphass 19126). See ipassr 16832, 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 19128 Move scalar multiplication to outside of inner product. See his35 22543. (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 19129* Lemma for tchbas 19131 and similar theorems. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  V  =  ( Base `  W )   =>    |-  ( x  e.  V  |->  ( sqr `  ( x  .,  x ) ) )  e.  _V
 
Theoremtchval 19130* 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 19131 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 19132 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 19133 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 19134 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 19135 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 19136 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 19137 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 19138 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 19139* 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 19140 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 19141 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 19142 Lemma for tchcph 19147. (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 19143 Lemma for tchcph 19147: 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 19144* 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 19145* Lemma for tchcph 19147: 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 19146* Lemma for tchcph 19147: 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 19147* 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 19148 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 19149 Lemma for nmpar 19150. (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 19150 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 19151 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 19152* 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 19153 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 19154* Continuity of inner product; analogue of cnmpt12f 17651 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 19155* Continuity of inner product; analogue of cnmpt22f 17660 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 19156 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 19157 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 ) )
 
11.5.3  Convergence and completeness
 
Syntaxccfil 19158 Extend class notation with the set of Cauchy filters.
 class CauFil
 
Syntaxcca 19159 Extend class notation with a function on metric spaces whose value is the set of all Cauchy sequences of the space.
 class  Cau
 
Syntaxcms 19160 Extend class notation with class of complete metric spaces.
 class  CMet
 
Definitiondf-cfil 19161* 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 19162* 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 19163* 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 19164* 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 17247. (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 19165* 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 17247. (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 19166* 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 19167* 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 19168* 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 19165 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 19169* 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 19170* 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 19171* 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 19172* 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 19173 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 19174* 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 19175* 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 19176 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 19177* 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 19178* 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 19179* 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 19180 Similar to ultrafilters (uffclsflim 18016), 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 19181* 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 19182* 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 17247. (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 19183* 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 19184* 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 19185* 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 19186* 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 19187 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 19188 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 19189 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 19190* 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 19191 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 19192 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 19193 A complete metric space is a metric space. (Contributed by NM, 26-Oct-2007.)
 |-  D  e.  ( CMet `  X )   =>    |-  D  e.  ( Met `  X )
 
Theoremcmetcaulem 19194* Lemma for cmetcau 19195. (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 19195 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 19196* Lemma for iscmet3 19199. (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 19197* Lemma for iscmet3 19199. (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 19198* Lemma for iscmet3 19199. (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 19199* 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 19200 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 ) ) )
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