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Theorem List for Metamath Proof Explorer - 21001-21100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnvvc 21001 The vector space component of a normed complex vector space. (Contributed by NM, 28-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
 |-  W  =  ( 1st `  U )   =>    |-  ( U  e.  NrmCVec  ->  W  e.  CVec OLD )
 
Theoremnvablo 21002 The vector addition operation of a normed complex vector space is an Abelian group. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
 |-  G  =  ( +v
 `  U )   =>    |-  ( U  e.  NrmCVec  ->  G  e.  AbelOp )
 
Theoremnvgrp 21003 The vector addition operation of a normed complex vector space is a group. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
 |-  G  =  ( +v
 `  U )   =>    |-  ( U  e.  NrmCVec  ->  G  e.  GrpOp )
 
Theoremnvgf 21004 Mapping for the vector addition operation. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   =>    |-  ( U  e.  NrmCVec  ->  G : ( X  X.  X ) --> X )
 
Theoremnvsf 21005 Mapping for the scalar multiplication operation. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( U  e.  NrmCVec  ->  S : ( CC  X.  X ) --> X )
 
Theoremnvgcl 21006 Closure law for the vector addition (group) operation of a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
 
Theoremnvcom 21007 The vector addition (group) operation is commutative. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  =  ( B G A ) )
 
Theoremnvass 21008 The vector addition (group) operation is associative. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A G B ) G C )  =  ( A G ( B G C ) ) )
 
Theoremnvadd12 21009 Commutative/associative law for vector addition. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( A G ( B G C ) )  =  ( B G ( A G C ) ) )
 
Theoremnvadd32 21010 Commutative/associative law for vector addition. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A G B ) G C )  =  ( ( A G C ) G B ) )
 
Theoremnvrcan 21011 Right cancellation law for vector addition. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A G C )  =  ( B G C ) 
 <->  A  =  B ) )
 
Theoremnvlcan 21012 Left cancellation law for vector addition. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( C G A )  =  ( C G B ) 
 <->  A  =  B ) )
 
Theoremnvadd4 21013 Rearrangement of 4 terms in a vector sum. (Contributed by NM, 8-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X ) 
 /\  ( C  e.  X  /\  D  e.  X ) )  ->  ( ( A G B ) G ( C G D ) )  =  ( ( A G C ) G ( B G D ) ) )
 
Theoremnvscl 21014 Closure law for the scalar product operation of a normed complex vector space. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  ( A S B )  e.  X )
 
Theoremnvsid 21015 Identity element for the scalar product of a normed complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X ) 
 ->  ( 1 S A )  =  A )
 
Theoremnvsass 21016 Associative law for the scalar product of a normed complex vector space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X )
 )  ->  ( ( A  x.  B ) S C )  =  ( A S ( B S C ) ) )
 
Theoremnvscom 21017 Commutative law for the scalar product of a normed complex vector space. (Contributed by NM, 14-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X )
 )  ->  ( A S ( B S C ) )  =  ( B S ( A S C ) ) )
 
Theoremnvdi 21018 Distributive law for the scalar product of a complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  ->  ( A S ( B G C ) )  =  ( ( A S B ) G ( A S C ) ) )
 
Theoremnvdir 21019 Distributive law for the scalar product of a complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X ) )  ->  ( ( A  +  B ) S C )  =  ( ( A S C ) G ( B S C ) ) )
 
Theoremnv2 21020 A vector plus itself is two times the vector. (Contributed by NM, 9-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G A )  =  ( 2 S A ) )
 
Theoremvsfval 21021 Value of the function for the vector subtraction operation on a normed complex vector space. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 27-Dec-2014.) (New usage is discouraged.)
 |-  G  =  ( +v
 `  U )   &    |-  M  =  ( -v `  U )   =>    |-  M  =  (  /g  `  G )
 
Theoremnvzcl 21022 Closure law for the zero vector of a normed complex vector space. (Contributed by NM, 27-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Z  =  (
 0vec `  U )   =>    |-  ( U  e.  NrmCVec  ->  Z  e.  X )
 
Theoremnv0rid 21023 The zero vector is a right identity element. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  Z  =  ( 0vec `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X ) 
 ->  ( A G Z )  =  A )
 
Theoremnv0lid 21024 The zero vector is a left identity element. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  Z  =  ( 0vec `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X ) 
 ->  ( Z G A )  =  A )
 
Theoremnv0 21025 Zero times a vector is the zero vector. (Contributed by NM, 27-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  Z  =  ( 0vec `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X ) 
 ->  ( 0 S A )  =  Z )
 
Theoremnvsz 21026 Anything times the zero vector is the zero vector. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
 |-  S  =  ( .s
 OLD `  U )   &    |-  Z  =  ( 0vec `  U )   =>    |-  (
 ( U  e.  NrmCVec  /\  A  e.  CC )  ->  ( A S Z )  =  Z )
 
Theoremnvinv 21027 Minus 1 times a vector is the underlying group's inverse element. Equation 2 of [Kreyszig] p. 51. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  M  =  ( inv `  G )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X ) 
 ->  ( -u 1 S A )  =  ( M `  A ) )
 
Theoremnvinvfval 21028 Function for the negative of a vector on a normed complex vector space, in terms of the underlying addition group inverse. (We currently do not have a separate notation for the negative of a vector.) (Contributed by NM, 27-Mar-2008.) (New usage is discouraged.)
 |-  G  =  ( +v
 `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( S  o.  `' ( 2nd  |`  ( { -u 1 }  X.  _V ) ) )   =>    |-  ( U  e.  NrmCVec  ->  N  =  ( inv `  G ) )
 
Theoremnvm 21029 Vector subtraction in terms of group division operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  M  =  ( -v `  U )   &    |-  N  =  ( 
 /g  `  G )   =>    |-  (
 ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( A N B ) )
 
Theoremnvmval 21030 Value of vector subtraction on a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  M  =  ( -v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( A G ( -u 1 S B ) ) )
 
Theoremnvmval2 21031 Value of vector subtraction on a normed complex vector space. (Contributed by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  M  =  ( -v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( ( -u 1 S B ) G A ) )
 
Theoremnvmfval 21032* Value of the function for the vector subtraction operation on a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  M  =  ( -v `  U )   =>    |-  ( U  e.  NrmCVec  ->  M  =  ( x  e.  X ,  y  e.  X  |->  ( x G ( -u 1 S y ) ) ) )
 
Theoremnvzs 21033 Two ways to express the negative of a vector. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  Z  =  ( 0vec `  U )   =>    |-  (
 ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( Z M A )  =  ( -u 1 S A ) )
 
Theoremnvmf 21034 Mapping for the vector subtraction operation. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   =>    |-  ( U  e.  NrmCVec  ->  M : ( X  X.  X ) --> X )
 
Theoremnvmcl 21035 Closure law for the vector subtraction operation of a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  e.  X )
 
Theoremnvnnncan1 21036 Cancellation law for vector subtraction. (nnncan1 8963 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A M B ) M ( A M C ) )  =  ( C M B ) )
 
Theoremnvnnncan2 21037 Cancellation law for vector subtraction. (nnncan2 8964 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A M C ) M ( B M C ) )  =  ( A M B ) )
 
Theoremnvmdi 21038 Distributive law for scalar product over subtraction. (Contributed by NM, 14-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  ->  ( A S ( B M C ) )  =  ( ( A S B ) M ( A S C ) ) )
 
Theoremnvnegneg 21039 Double negative of a vector. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X ) 
 ->  ( -u 1 S (
 -u 1 S A ) )  =  A )
 
Theoremnvmul0or 21040 If a scalar product is zero, one of its factors must be zero. (Contributed by NM, 6-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  Z  =  ( 0vec `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  ( ( A S B )  =  Z  <->  ( A  =  0  \/  B  =  Z ) ) )
 
Theoremnvrinv 21041 A vector minus itself. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  Z  =  ( 0vec `  U )   =>    |-  (
 ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G (
 -u 1 S A ) )  =  Z )
 
Theoremnvlinv 21042 Minus a vector plus itself. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  Z  =  ( 0vec `  U )   =>    |-  (
 ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( ( -u 1 S A ) G A )  =  Z )
 
Theoremnvsubadd 21043 Relationship between vector subtraction and addition. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  M  =  ( -v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A M B )  =  C  <->  ( B G C )  =  A ) )
 
Theoremnvpncan2 21044 Cancellation law for vector subtraction. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  M  =  ( -v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( ( A G B ) M A )  =  B )
 
Theoremnvpncan 21045 Cancellation law for vector subtraction. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  M  =  ( -v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( ( A G B ) M B )  =  A )
 
Theoremnvaddsubass 21046 Associative-type law for vector addition and subtraction. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  M  =  ( -v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A G B ) M C )  =  ( A G ( B M C ) ) )
 
Theoremnvaddsub 21047 Commutative/associative law for vector addition and subtraction. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  M  =  ( -v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A G B ) M C )  =  ( ( A M C ) G B ) )
 
Theoremnvnpcan 21048 Cancellation law for a normed complex vector space. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  M  =  ( -v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( ( A M B ) G B )  =  A )
 
Theoremnvaddsub4 21049 Rearrangement of 4 terms in a mixed vector addition and subtraction. (Contributed by NM, 8-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  M  =  ( -v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X ) 
 /\  ( C  e.  X  /\  D  e.  X ) )  ->  ( ( A G B ) M ( C G D ) )  =  ( ( A M C ) G ( B M D ) ) )
 
Theoremnvsubsub23 21050 Swap subtrahend and result of vector subtraction. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A M B )  =  C  <->  ( A M C )  =  B ) )
 
Theoremnvnncan 21051 Cancellation law for a normed complex vector space. (Contributed by NM, 17-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M ( A M B ) )  =  B )
 
Theoremnvmeq0 21052 The difference between two vectors is zero iff they are equal. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   &    |-  Z  =  ( 0vec `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( ( A M B )  =  Z  <->  A  =  B ) )
 
Theoremnvmid 21053 A vector minus itself is the zero vector. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   &    |-  Z  =  ( 0vec `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X ) 
 ->  ( A M A )  =  Z )
 
Theoremnvf 21054 Mapping for the norm function. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  N  =  (
 normCV `  U )   =>    |-  ( U  e.  NrmCVec  ->  N : X --> RR )
 
Theoremnvcl 21055 The norm of a normed complex vector space is a real number. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  N  =  (
 normCV `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X ) 
 ->  ( N `  A )  e.  RR )
 
Theoremnvcli 21056 The norm of a normed complex vector space is a real number. (Contributed by NM, 20-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  N  =  (
 normCV `  U )   &    |-  U  e.  NrmCVec   &    |-  A  e.  X   =>    |-  ( N `  A )  e.  RR
 
Theoremnvdm 21057 Two ways to express the set of vectors in a normed complex vector space. (Contributed by NM, 31-Jan-2007.) (New usage is discouraged.)
 |-  G  =  ( +v
 `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( U  e.  NrmCVec  ->  ( X  =  dom  N  <->  X  =  ran  G ) )
 
Theoremnvs 21058 Proportionality property of the norm of a scalar product in a normed complex vector space. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  ( N `  ( A S B ) )  =  ( ( abs `  A )  x.  ( N `  B ) ) )
 
Theoremnvsge0 21059 The norm of a scalar product with a nonnegative real. (Contributed by NM, 1-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  RR  /\  0  <_  A )  /\  B  e.  X ) 
 ->  ( N `  ( A S B ) )  =  ( A  x.  ( N `  B ) ) )
 
Theoremnvm1 21060 The norm of the negative of a vector. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X ) 
 ->  ( N `  ( -u 1 S A ) )  =  ( N `
  A ) )
 
Theoremnvdif 21061 The norm of the difference between two vectors. (Contributed by NM, 1-Dec-2006.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A G ( -u 1 S B ) ) )  =  ( N `
  ( B G ( -u 1 S A ) ) ) )
 
Theoremnvpi 21062 The norm of a vector plus the imaginary scalar product of another. (Contributed by NM, 2-Feb-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A G ( _i S B ) ) )  =  ( N `
  ( B G ( -u _i S A ) ) ) )
 
Theoremnvsub 21063 The norm of the difference between two vectors. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A M B ) )  =  ( N `  ( B M A ) ) )
 
Theoremnvz0 21064 The norm of a zero vector is zero. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.)
 |-  Z  =  ( 0vec `  U )   &    |-  N  =  (
 normCV `  U )   =>    |-  ( U  e.  NrmCVec  ->  ( N `  Z )  =  0 )
 
Theoremnvz 21065 The norm of a vector is zero iff the vector is zero. First part of Problem 2 of [Kreyszig] p. 64. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Z  =  (
 0vec `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( ( N `  A )  =  0  <->  A  =  Z ) )
 
Theoremnvtri 21066 Triangle inequality for the norm of a normed complex vector space. (Contributed by NM, 11-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A G B ) ) 
 <_  ( ( N `  A )  +  ( N `  B ) ) )
 
Theoremnvmtri 21067 Triangle inequality for the norm of a vector difference. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A M B ) ) 
 <_  ( ( N `  A )  +  ( N `  B ) ) )
 
Theoremnvmtri2 21068 Triangle inequality for the norm of a vector difference. (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( N `  ( A M C ) )  <_  ( ( N `  ( A M B ) )  +  ( N `  ( B M C ) ) ) )
 
Theoremnvabs 21069 Norm difference property of a normed complex vector space. Problem 3 of [Kreyszig] p. 64. (Contributed by NM, 4-Dec-2006.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( abs `  (
 ( N `  A )  -  ( N `  B ) ) ) 
 <_  ( N `  ( A G ( -u 1 S B ) ) ) )
 
Theoremnvge0 21070 The norm of a normed complex vector space is nonnegative. Second part of Problem 2 of [Kreyszig] p. 64. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  N  =  (
 normCV `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X ) 
 ->  0  <_  ( N `
  A ) )
 
Theoremnvgt0 21071 A nonzero norm is positive. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Z  =  (
 0vec `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A  =/=  Z  <->  0  <  ( N `  A ) ) )
 
Theoremnv1 21072 From any nonzero vector, construct a vector whose norm is one. (Contributed by NM, 6-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  Z  =  ( 0vec `  U )   &    |-  N  =  (
 normCV `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  A  =/=  Z )  ->  ( N `  ( ( 1  /  ( N `
  A ) ) S A ) )  =  1 )
 
Theoremnvop 21073 A complex inner product space in terms of ordered pair components. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
 |-  G  =  ( +v
 `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( U  e.  NrmCVec  ->  U  =  <. <. G ,  S >. ,  N >. )
 
Theoremnvoprne 21074 The vector addition and scalar product operations are not identical. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  ( <. <. G ,  S >. ,  N >.  e.  NrmCVec  ->  G  =/=  S )
 
15.4.2  Examples of normed complex vector spaces
 
Theoremcnnv 21075 The set of complex numbers is a normed complex vector space. The vector operation is  +, the scalar product is  x., and the norm function is  abs. (Contributed by Steve Rodriguez, 3-Dec-2006.) (New usage is discouraged.)
 |-  U  =  <. <.  +  ,  x.  >. ,  abs >.   =>    |-  U  e.  NrmCVec
 
Theoremcnnvg 21076 The vector addition (group) operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +  ,  x.  >. ,  abs >.   =>    |-  +  =  ( +v `  U )
 
Theoremcnnvba 21077 The base set of the normed complex vector space of complex numbers. (Contributed by NM, 7-Nov-2007.) (New usage is discouraged.)
 |-  U  =  <. <.  +  ,  x.  >. ,  abs >.   =>    |-  CC  =  (
 BaseSet `  U )
 
Theoremcnnvdemo 21078 Derive the associative law for complex number addition addass 8704 to demonstrate the use of cnnv 21075, cnnvg 21076, and cnnvba 21077. (Contributed by NM, 12-Jan-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  +  C )  =  ( A  +  ( B  +  C ) ) )
 
Theoremcnnvs 21079 The scalar product operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +  ,  x.  >. ,  abs >.   =>    |-  x.  =  ( .s OLD `  U )
 
Theoremcnnvnm 21080 The norm operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +  ,  x.  >. ,  abs >.   =>    |-  abs  =  ( normCV `  U )
 
Theoremcnnvm 21081 The vector subtraction operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  U  =  <. <.  +  ,  x.  >. ,  abs >.   =>    |-  -  =  ( -v `  U )
 
Theoremelimnv 21082 Hypothesis elimination lemma for normed complex vector spaces to assist weak deduction theorem. (Contributed by NM, 16-May-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Z  =  (
 0vec `  U )   &    |-  U  e. 
 NrmCVec   =>    |-  if ( A  e.  X ,  A ,  Z )  e.  X
 
Theoremelimnvu 21083 Hypothesis elimination lemma for normed complex vector spaces to assist weak deduction theorem. (Contributed by NM, 16-May-2007.) (New usage is discouraged.)
 |- 
 if ( U  e.  NrmCVec ,  U ,  <. <.  +  ,  x.  >. ,  abs >. )  e.  NrmCVec
 
15.4.3  Induced metric of a normed complex vector space
 
Theoremimsval 21084 Value of the induced metric of a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  M  =  ( -v
 `  U )   &    |-  N  =  ( normCV `  U )   &    |-  D  =  ( IndMet `  U )   =>    |-  ( U  e.  NrmCVec  ->  D  =  ( N  o.  M ) )
 
Theoremimsdval 21085 Value of the induced metric (distance function) of a normed complex vector space. Equation 1 of [Kreyszig] p. 59. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 27-Dec-2014.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   &    |-  N  =  ( normCV `  U )   &    |-  D  =  ( IndMet `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( N `  ( A M B ) ) )
 
Theoremimsdval2 21086 Value of the distance function of the induced metric of a normed complex vector space. Equation 1 of [Kreyszig] p. 59. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   &    |-  D  =  ( IndMet `  U )   =>    |-  (
 ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( N `  ( A G (
 -u 1 S B ) ) ) )
 
Theoremnvnd 21087 The norm of a normed complex vector space expressed in terms of the distance function of its induced metric. Problem 1 of [Kreyszig] p. 63. (Contributed by NM, 4-Dec-2006.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Z  =  (
 0vec `  U )   &    |-  N  =  ( normCV `  U )   &    |-  D  =  ( IndMet `  U )   =>    |-  (
 ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  A )  =  ( A D Z ) )
 
Theoremimsdf 21088 Mapping for the induced metric distance function of a normed complex vector space. (Contributed by NM, 29-Nov-2006.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  D  =  (
 IndMet `  U )   =>    |-  ( U  e.  NrmCVec  ->  D : ( X  X.  X ) --> RR )
 
Theoremimsmetlem 21089 Lemma for imsmet 21090. (Contributed by NM, 29-Nov-2006.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  M  =  ( inv `  G )   &    |-  S  =  ( .s OLD `  U )   &    |-  Z  =  ( 0vec `  U )   &    |-  N  =  (
 normCV `  U )   &    |-  D  =  (
 IndMet `  U )   &    |-  U  e. 
 NrmCVec   =>    |-  D  e.  ( Met `  X )
 
Theoremimsmet 21090 The induced metric of a normed complex vector space is a metric space. Part of Definition 2.2-1 of [Kreyszig] p. 58. (Contributed by NM, 4-Dec-2006.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  D  =  (
 IndMet `  U )   =>    |-  ( U  e.  NrmCVec  ->  D  e.  ( Met `  X ) )
 
Theoremimsxmet 21091 The induced metric of a normed complex vector space is an extended metric space. (Contributed by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  D  =  (
 IndMet `  U )   =>    |-  ( U  e.  NrmCVec  ->  D  e.  ( * Met `  X ) )
 
Theoremnvelbl 21092 Membership of a vector in a ball. (Contributed by NM, 27-Dec-2007.) (Revised by Mario Carneiro, 10-Jan-2014.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   &    |-  N  =  ( normCV `  U )   &    |-  D  =  ( IndMet `  U )   =>    |-  ( ( ( U  e.  NrmCVec  /\  R  e.  RR+ )  /\  ( P  e.  X  /\  A  e.  X ) )  ->  ( A  e.  ( P ( ball `  D ) R )  <->  ( N `  ( A M P ) )  <  R ) )
 
Theoremnvelbl2 21093 Membership of an off-center vector in a ball. (Contributed by NM, 27-Dec-2007.) (Revised by Mario Carneiro, 10-Jan-2014.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  N  =  ( normCV `  U )   &    |-  D  =  ( IndMet `  U )   =>    |-  ( ( ( U  e.  NrmCVec  /\  R  e.  RR+ )  /\  ( P  e.  X  /\  A  e.  X ) )  ->  ( ( P G A )  e.  ( P ( ball `  D ) R )  <->  ( N `  A )  <  R ) )
 
Theoremnvlmcl 21094 Closure of the limit of a converging vector sequence. (Contributed by NM, 26-Dec-2007.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  D  =  (
 IndMet `  U )   &    |-  J  =  ( MetOpen `  D )   =>    |-  (
 ( U  e.  NrmCVec  /\  F ( ~~> t `  J ) P ) 
 ->  P  e.  X )
 
Theoremnvlmle 21095* If the norm of each member of a converging sequence is less than or equal to a given amount, so is the norm of the convergence value. (Contributed by NM, 25-Dec-2007.) (Revised by Mario Carneiro, 5-May-2014.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  D  =  (
 IndMet `  U )   &    |-  J  =  ( MetOpen `  D )   &    |-  N  =  ( normCV `  U )   &    |-  ( ph  ->  U  e.  NrmCVec )   &    |-  ( ph  ->  F : NN
 --> X )   &    |-  ( ph  ->  F ( ~~> t `  J ) P )   &    |-  ( ph  ->  R  e.  RR )   &    |-  (
 ( ph  /\  k  e. 
 NN )  ->  ( N `  ( F `  k ) )  <_  R )   =>    |-  ( ph  ->  ( N `  P )  <_  R )
 
Theoremcnims 21096 The metric induced on the complex numbers. cnmet 18113 proves that it is a metric. (Contributed by Steve Rodriguez, 5-Dec-2006.) (Revised by NM, 15-Jan-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +  ,  x.  >. ,  abs >.   &    |-  D  =  ( abs  o.  -  )   =>    |-  D  =  ( IndMet `  U )
 
Theoremvacn 21097 Vector addition is jointly continuous in both arguments. (Contributed by Jeffrey Hankins, 16-Jun-2009.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
 |-  C  =  ( IndMet `  U )   &    |-  J  =  (
 MetOpen `  C )   &    |-  G  =  ( +v `  U )   =>    |-  ( U  e.  NrmCVec  ->  G  e.  ( ( J  tX  J )  Cn  J ) )
 
Theoremnmcvcn 21098 The norm of a normed complex vector space is a continuous function. (Contributed by NM, 16-May-2007.) (Proof shortened by Mario Carneiro, 10-Jan-2014.) (New usage is discouraged.)
 |-  N  =  ( normCV `  U )   &    |-  C  =  ( IndMet `  U )   &    |-  J  =  (
 MetOpen `  C )   &    |-  K  =  ( topGen `  ran  (,) )   =>    |-  ( U  e.  NrmCVec  ->  N  e.  ( J  Cn  K ) )
 
Theoremnmcnc 21099 The norm of a normed complex vector space is a continuous function to  CC. (For  RR, see nmcvcn 21098.) (Contributed by NM, 12-Aug-2007.) (New usage is discouraged.)
 |-  N  =  ( normCV `  U )   &    |-  C  =  ( IndMet `  U )   &    |-  J  =  (
 MetOpen `  C )   &    |-  K  =  ( TopOpen ` fld )   =>    |-  ( U  e.  NrmCVec  ->  N  e.  ( J  Cn  K ) )
 
Theoremsmcnlem 21100* Lemma for smcn 21101. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
 |-  C  =  ( IndMet `  U )   &    |-  J  =  (
 MetOpen `  C )   &    |-  S  =  ( .s OLD `  U )   &    |-  K  =  ( TopOpen ` fld )   &    |-  X  =  ( BaseSet `  U )   &    |-  N  =  ( normCV `  U )   &    |-  U  e. 
 NrmCVec   &    |-  T  =  ( 1  /  ( 1  +  (
 ( ( ( N `
  y )  +  ( abs `  x )
 )  +  1 ) 
 /  r ) ) )   =>    |-  S  e.  ( ( K  tX  J )  Cn  J )
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