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Theorem List for Metamath Proof Explorer - 18401-18500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremclmfgrp 18401 The scalar ring of a complex module is a group. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e. CMod  ->  F  e.  Grp )
 
Theoremclm0 18402 The zero of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e. CMod  ->  0  =  ( 0g `  F ) )
 
Theoremclm1 18403 The identity of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e. CMod  ->  1  =  ( 1r `  F ) )
 
Theoremclmadd 18404 The addition of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e. CMod  ->  +  =  ( +g  `  F ) )
 
Theoremclmmul 18405 The multiplication of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e. CMod  ->  x.  =  ( .r `  F ) )
 
Theoremclmcj 18406 The conjugation of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e. CMod  ->  *  =  ( * r `
  F ) )
 
Theoremisclmi 18407 Reverse direction of isclm 18394. (Contributed by Mario Carneiro, 30-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( ( W  e.  LMod  /\  F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) )  ->  W  e. CMod )
 
Theoremclmzss 18408 The scalar ring of a complex module contains the integers. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( W  e. CMod  ->  ZZ  C_  K )
 
Theoremclmsscn 18409 The scalar ring of a complex module is a subset of the complexes. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( W  e. CMod  ->  K 
 C_  CC )
 
Theoremclmsub 18410 Subtraction in the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( ( W  e. CMod  /\  A  e.  K  /\  B  e.  K )  ->  ( A  -  B )  =  ( A ( -g `  F ) B ) )
 
Theoremclmneg 18411 Negation in the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( ( W  e. CMod  /\  A  e.  K ) 
 ->  -u A  =  ( ( inv g `  F ) `  A ) )
 
Theoremclmabs 18412 Norm in the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( ( W  e. CMod  /\  A  e.  K ) 
 ->  ( abs `  A )  =  ( ( norm `  F ) `  A ) )
 
Theoremclmacl 18413 Closure of ring addition for a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( ( W  e. CMod  /\  X  e.  K  /\  Y  e.  K )  ->  ( X  +  Y )  e.  K )
 
Theoremclmmcl 18414 Closure of ring multiplication for a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( ( W  e. CMod  /\  X  e.  K  /\  Y  e.  K )  ->  ( X  x.  Y )  e.  K )
 
Theoremclmsubcl 18415 Closure of ring subtraction for a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( ( W  e. CMod  /\  X  e.  K  /\  Y  e.  K )  ->  ( X  -  Y )  e.  K )
 
Theoremlmhmclm 18416 The domain of a linear operator is a complex module iff the range is. (Contributed by Mario Carneiro, 21-Oct-2015.)
 |-  ( F  e.  ( S LMHom  T )  ->  ( S  e. CMod  <->  T  e. CMod ) )
 
Theoremclmvsass 18417 Associative law for scalar product. (lmodvsass 15489 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( Base `  F )   =>    |-  (
 ( W  e. CMod  /\  ( Q  e.  K  /\  R  e.  K  /\  X  e.  V )
 )  ->  ( ( Q  x.  R )  .x.  X )  =  ( Q 
 .x.  ( R  .x.  X ) ) )
 
Theoremclmvsdir 18418 Distributive law for scalar product. (lmodvsdir 15487 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( Base `  F )   &    |-  .+  =  ( +g  `  W )   =>    |-  (
 ( W  e. CMod  /\  ( Q  e.  K  /\  R  e.  K  /\  X  e.  V )
 )  ->  ( ( Q  +  R )  .x.  X )  =  ( ( Q  .x.  X )  .+  ( R  .x.  X ) ) )
 
Theoremclmvs1 18419 Scalar product with ring unit. (lmodvs1 15493 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  .x.  =  ( .s `  W )   =>    |-  ( ( W  e. CMod  /\  X  e.  V )  ->  ( 1 
 .x.  X )  =  X )
 
Theoremclm0vs 18420 Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (lmod0vs 15498 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  .0.  =  ( 0g `  W )   =>    |-  ( ( W  e. CMod  /\  X  e.  V ) 
 ->  ( 0  .x.  X )  =  .0.  )
 
Theoremclmvneg1 18421 Minus 1 times a vector is the negative of the vector. Equation 2 of [Kreyszig] p. 51. (lmodvneg1 15502 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( inv g `  W )   &    |-  F  =  (Scalar `  W )   &    |- 
 .x.  =  ( .s `  W )   =>    |-  ( ( W  e. CMod  /\  X  e.  V ) 
 ->  ( -u 1  .x.  X )  =  ( N `  X ) )
 
Theoremclmvsneg 18422 Multiplication of a vector by a negated scalar. (lmodvsneg 15504 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  B  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  N  =  ( inv g `  W )   &    |-  K  =  (
 Base `  F )   &    |-  ( ph  ->  W  e. CMod )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  R  e.  K )   =>    |-  ( ph  ->  ( N `  ( R  .x.  X ) )  =  (
 -u R  .x.  X ) )
 
Theoremclmmulg 18423 The group multiple function matches the scalar multiplication function. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  .xb  =  (.g `  W )   &    |-  .x.  =  ( .s `  W )   =>    |-  ( ( W  e. CMod  /\  A  e.  ZZ  /\  B  e.  V )  ->  ( A  .xb  B )  =  ( A 
 .x.  B ) )
 
Theoremclmsubdir 18424 Scalar multiplication distributive law for subtraction. (lmodsubdir 15518 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  .-  =  ( -g `  W )   &    |-  ( ph  ->  W  e. CMod )   &    |-  ( ph  ->  A  e.  K )   &    |-  ( ph  ->  B  e.  K )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  (
 ( A  -  B )  .x.  X )  =  ( ( A  .x.  X )  .-  ( B  .x.  X ) ) )
 
Theoremzlmclm 18425 The  ZZ-module operation turns an arbitrary abelian group into a complex module. (Contributed by Mario Carneiro, 30-Oct-2015.)
 |-  W  =  ( ZMod `  G )   =>    |-  ( G  e.  Abel  <->  W  e. CMod )
 
Theoremclmzlmvsca 18426 The scalar product of a complex module matches the scalar product of the derived  ZZ-module, which implies, together with zlmbas 16304 and zlmplusg 16305, that any module over  ZZ is structure-equivalent to the canonical  ZZ-module  ZMod `  G. (Contributed by Mario Carneiro, 30-Oct-2015.)
 |-  W  =  ( ZMod `  G )   &    |-  X  =  (
 Base `  G )   =>    |-  ( ( G  e. CMod  /\  ( A  e.  ZZ  /\  B  e.  X ) )  ->  ( A ( .s `  G ) B )  =  ( A ( .s `  W ) B ) )
 
Theoremnmoleub2lem 18427* Lemma for nmoleub2a 18430 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.)
 |-  N  =  ( S
 normOp T )   &    |-  V  =  (
 Base `  S )   &    |-  L  =  ( norm `  S )   &    |-  M  =  ( norm `  T )   &    |-  G  =  (Scalar `  S )   &    |-  K  =  ( Base `  G )   &    |-  ( ph  ->  S  e.  (NrmMod  i^i CMod ) )   &    |-  ( ph  ->  T  e.  (NrmMod  i^i CMod )
 )   &    |-  ( ph  ->  F  e.  ( S LMHom  T ) )   &    |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ( ph  /\ 
 A. x  e.  V  ( ps  ->  ( ( M `  ( F `  x ) )  /  R )  <_  A ) )  ->  0  <_  A )   &    |-  ( ( ( ( ph  /\  A. x  e.  V  ( ps  ->  ( ( M `
  ( F `  x ) )  /  R )  <_  A ) )  /\  A  e.  RR )  /\  ( y  e.  V  /\  y  =/=  ( 0g `  S ) ) )  ->  ( M `  ( F `
  y ) ) 
 <_  ( A  x.  ( L `  y ) ) )   &    |-  ( ( ph  /\  x  e.  V ) 
 ->  ( ps  ->  ( L `  x )  <_  R ) )   =>    |-  ( ph  ->  ( ( N `  F )  <_  A  <->  A. x  e.  V  ( ps  ->  ( ( M `  ( F `  x ) )  /  R )  <_  A ) ) )
 
Theoremnmoleub2lem3 18428* Lemma for nmoleub2a 18430 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.)
 |-  N  =  ( S
 normOp T )   &    |-  V  =  (
 Base `  S )   &    |-  L  =  ( norm `  S )   &    |-  M  =  ( norm `  T )   &    |-  G  =  (Scalar `  S )   &    |-  K  =  ( Base `  G )   &    |-  ( ph  ->  S  e.  (NrmMod  i^i CMod ) )   &    |-  ( ph  ->  T  e.  (NrmMod  i^i CMod )
 )   &    |-  ( ph  ->  F  e.  ( S LMHom  T ) )   &    |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ph  ->  QQ  C_  K )   &    |-  .x.  =  ( .s `  S )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0  <_  A )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  B  =/=  ( 0g `  S ) )   &    |-  ( ph  ->  ( (
 r  .x.  B )  e.  V  ->  ( ( L `  ( r  .x.  B ) )  <  R  ->  ( ( M `  ( F `  ( r 
 .x.  B ) ) ) 
 /  R )  <_  A ) ) )   &    |-  ( ph  ->  -.  ( M `  ( F `  B ) )  <_  ( A  x.  ( L `  B ) ) )   =>    |- 
 -.  ph
 
Theoremnmoleub2lem2 18429* Lemma for nmoleub2a 18430 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.)
 |-  N  =  ( S
 normOp T )   &    |-  V  =  (
 Base `  S )   &    |-  L  =  ( norm `  S )   &    |-  M  =  ( norm `  T )   &    |-  G  =  (Scalar `  S )   &    |-  K  =  ( Base `  G )   &    |-  ( ph  ->  S  e.  (NrmMod  i^i CMod ) )   &    |-  ( ph  ->  T  e.  (NrmMod  i^i CMod )
 )   &    |-  ( ph  ->  F  e.  ( S LMHom  T ) )   &    |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ph  ->  QQ  C_  K )   &    |-  ( ( ( L `  x )  e.  RR  /\  R  e.  RR )  ->  (
 ( L `  x ) O R  ->  ( L `  x )  <_  R ) )   &    |-  (
 ( ( L `  x )  e.  RR  /\  R  e.  RR )  ->  ( ( L `  x )  <  R  ->  ( L `  x ) O R ) )   =>    |-  ( ph  ->  ( ( N `  F )  <_  A 
 <-> 
 A. x  e.  V  ( ( L `  x ) O R  ->  ( ( M `  ( F `  x ) )  /  R ) 
 <_  A ) ) )
 
Theoremnmoleub2a 18430* The operator norm is the supremum of the value of a linear operator in the closed unit ball. (Contributed by Mario Carneiro, 19-Oct-2015.)
 |-  N  =  ( S
 normOp T )   &    |-  V  =  (
 Base `  S )   &    |-  L  =  ( norm `  S )   &    |-  M  =  ( norm `  T )   &    |-  G  =  (Scalar `  S )   &    |-  K  =  ( Base `  G )   &    |-  ( ph  ->  S  e.  (NrmMod  i^i CMod ) )   &    |-  ( ph  ->  T  e.  (NrmMod  i^i CMod )
 )   &    |-  ( ph  ->  F  e.  ( S LMHom  T ) )   &    |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ph  ->  QQ  C_  K )   =>    |-  ( ph  ->  (
 ( N `  F )  <_  A  <->  A. x  e.  V  ( ( L `  x )  <_  R  ->  ( ( M `  ( F `  x ) ) 
 /  R )  <_  A ) ) )
 
Theoremnmoleub2b 18431* The operator norm is the supremum of the value of a linear operator in the open unit ball. (Contributed by Mario Carneiro, 19-Oct-2015.)
 |-  N  =  ( S
 normOp T )   &    |-  V  =  (
 Base `  S )   &    |-  L  =  ( norm `  S )   &    |-  M  =  ( norm `  T )   &    |-  G  =  (Scalar `  S )   &    |-  K  =  ( Base `  G )   &    |-  ( ph  ->  S  e.  (NrmMod  i^i CMod ) )   &    |-  ( ph  ->  T  e.  (NrmMod  i^i CMod )
 )   &    |-  ( ph  ->  F  e.  ( S LMHom  T ) )   &    |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ph  ->  QQ  C_  K )   =>    |-  ( ph  ->  (
 ( N `  F )  <_  A  <->  A. x  e.  V  ( ( L `  x )  <  R  ->  ( ( M `  ( F `  x ) ) 
 /  R )  <_  A ) ) )
 
Theoremnmoleub3 18432* The operator norm is the supremum of the value of a linear operator on the closed unit sphere. (Contributed by Mario Carneiro, 19-Oct-2015.)
 |-  N  =  ( S
 normOp T )   &    |-  V  =  (
 Base `  S )   &    |-  L  =  ( norm `  S )   &    |-  M  =  ( norm `  T )   &    |-  G  =  (Scalar `  S )   &    |-  K  =  ( Base `  G )   &    |-  ( ph  ->  S  e.  (NrmMod  i^i CMod ) )   &    |-  ( ph  ->  T  e.  (NrmMod  i^i CMod )
 )   &    |-  ( ph  ->  F  e.  ( S LMHom  T ) )   &    |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  RR  C_  K )   =>    |-  ( ph  ->  (
 ( N `  F )  <_  A  <->  A. x  e.  V  ( ( L `  x )  =  R  ->  ( ( M `  ( F `  x ) )  /  R ) 
 <_  A ) ) )
 
Theoremnmhmcn 18433 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 ) ) ) )
 
11.3.13  Complex pre-Hilbert space
 
Syntaxccph 18434 Extend class notation with a complex pre-Hilbert space.
 class  CPreHil
 
Syntaxctch 18435 Function to put a norm on a Hilbert space.
 class toCHil
 
Definitiondf-cph 18436* 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 18437* 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 18438* 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 18439 A complex pre-Hilbert space is a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  ( W  e.  CPreHil  ->  W  e.  PreHil )
 
Theoremcphnlm 18440 A complex pre-Hilbert space is a normed module. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  ( W  e.  CPreHil  ->  W  e. NrmMod )
 
Theoremcphngp 18441 A complex pre-Hilbert space is a normed group. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  ( W  e.  CPreHil  ->  W  e. NrmGrp )
 
Theoremcphlmod 18442 A complex pre-Hilbert space is a left module. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  ( W  e.  CPreHil  ->  W  e.  LMod )
 
Theoremcphlvec 18443 A complex pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  ( W  e.  CPreHil  ->  W  e.  LVec )
 
Theoremcphnvc 18444 A complex pre-Hilbert space is a normed vector space. (Contributed by Mario Carneiro, 8-Oct-2015.)
 |-  ( W  e.  CPreHil  ->  W  e. NrmVec )
 
Theoremcphsubrglem 18445 Lemma for cphsubrg 18448. (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 18446 Lemma for cphreccl 18449. (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 18447 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 18448 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 18449 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 18450 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 18451 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 18452 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 18453 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 18454 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 18455 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 18456 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 18457 A complex pre-Hilbert space is a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  ( W  e.  CPreHil  ->  W  e. CMod )
 
Theoremcphnmvs 18458 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 18459 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 18460* 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 18461 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 18462 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 18463 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 18464 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 18465 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 18466 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 18467 Conjugate of an inner product in a complex pre-Hilbert space. Complex version of ipcj 16370. (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 18468 Orthogonality (meaning inner product is 0) is commutative. Complex version of iporthcom 16371. (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 18469 Inner product with a zero first argument. Part of proof of Theorem 6.44 of [Ponnusamy] p. 361. Complex version of ip0l 16372. (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 18470 Inner product with a zero second argument. Complex version of ip0r 16373. (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 18471 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 16374. (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 18472 Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. Complex version of ipdir 16375. (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 18473 Distributive law for inner product. Complex version of ipdi 16376. (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 18474 Distributive law for inner product. Complex version of ip2di 16377. (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 18475 Distributive law for inner product subtraction. Complex version of ipsubdir 16378. (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 18476 Distributive law for inner product subtraction. Complex version of ipsubdi 16379. (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 18477 Distributive law for inner product subtraction. Complex version of ip2subdi 16380. (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 18478 Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. See ipass 16381, his5 21495. (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 18479 "Associative" law for second argument of inner product (compare cphass 18478). See ipassr 16382, 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 18480 Move scalar multiplication to outside of inner product. See his35 21497. (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 18481* Lemma for tchbas 18483 and similar theorems. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  V  =  ( Base `  W )   =>    |-  ( x  e.  V  |->  ( sqr `  ( x  .,  x ) ) )  e.  _V
 
Theoremtchval 18482* 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 18483 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 18484 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 18485 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 18486 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 18487 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 18488 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 18489 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 18490 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 18491* 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 18492 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 18493 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 18494 Lemma for tchcph 18499. (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 18495 Lemma for tchcph 18499: 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 18496* 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 18497* Lemma for tchcph 18499: 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 18498* Lemma for tchcph 18499: 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 18499* 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 18500 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 ) ) )
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