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Theorem List for Intuitionistic Logic Explorer - 9601-9700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfac4 9601 The factorial of 4. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( ! `  4
 )  = ; 2 4
 
Theoremfacnn2 9602 Value of the factorial function expressed recursively. (Contributed by NM, 2-Dec-2004.)
 |-  ( N  e.  NN  ->  ( ! `  N )  =  ( ( ! `  ( N  -  1 ) )  x.  N ) )
 
Theoremfaccl 9603 Closure of the factorial function. (Contributed by NM, 2-Dec-2004.)
 |-  ( N  e.  NN0  ->  ( ! `  N )  e.  NN )
 
Theoremfaccld 9604 Closure of the factorial function, deduction version of faccl 9603. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( ! `  N )  e. 
 NN )
 
Theoremfacne0 9605 The factorial function is nonzero. (Contributed by NM, 26-Apr-2005.)
 |-  ( N  e.  NN0  ->  ( ! `  N )  =/=  0 )
 
Theoremfacdiv 9606 A positive integer divides the factorial of an equal or larger number. (Contributed by NM, 2-May-2005.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN  /\  N  <_  M )  ->  ( ( ! `  M )  /  N )  e.  NN )
 
Theoremfacndiv 9607 No positive integer (greater than one) divides the factorial plus one of an equal or larger number. (Contributed by NM, 3-May-2005.)
 |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( 1  <  N  /\  N  <_  M ) )  ->  -.  ( ( ( ! `
  M )  +  1 )  /  N )  e.  ZZ )
 
Theoremfacwordi 9608 Ordering property of factorial. (Contributed by NM, 9-Dec-2005.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0  /\  M  <_  N )  ->  ( ! `  M )  <_  ( ! `  N ) )
 
Theoremfaclbnd 9609 A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M ^ ( N  +  1 )
 )  <_  ( ( M ^ M )  x.  ( ! `  N ) ) )
 
Theoremfaclbnd2 9610 A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.)
 |-  ( N  e.  NN0  ->  ( ( 2 ^ N )  /  2
 )  <_  ( ! `  N ) )
 
Theoremfaclbnd3 9611 A lower bound for the factorial function. (Contributed by NM, 19-Dec-2005.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M ^ N )  <_  ( ( M ^ M )  x.  ( ! `  N ) ) )
 
Theoremfaclbnd6 9612 Geometric lower bound for the factorial function, where N is usually held constant. (Contributed by Paul Chapman, 28-Dec-2007.)
 |-  ( ( N  e.  NN0  /\  M  e.  NN0 )  ->  ( ( ! `  N )  x.  (
 ( N  +  1 ) ^ M ) )  <_  ( ! `  ( N  +  M ) ) )
 
Theoremfacubnd 9613 An upper bound for the factorial function. (Contributed by Mario Carneiro, 15-Apr-2016.)
 |-  ( N  e.  NN0  ->  ( ! `  N ) 
 <_  ( N ^ N ) )
 
Theoremfacavg 9614 The product of two factorials is greater than or equal to the factorial of (the floor of) their average. (Contributed by NM, 9-Dec-2005.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( ! `  ( |_ `  ( ( M  +  N )  / 
 2 ) ) ) 
 <_  ( ( ! `  M )  x.  ( ! `  N ) ) )
 
3.6.8  The binomial coefficient operation
 
Syntaxcbc 9615 Extend class notation to include the binomial coefficient operation (combinatorial choose operation).
 class  _C
 
Definitiondf-bc 9616* Define the binomial coefficient operation. For example,  ( 5  _C  3 )  =  1 0 (ex-bc 10282).

In the literature, this function is often written as a column vector of the two arguments, or with the arguments as subscripts before and after the letter "C".  ( N  _C  K
) is read " N choose  K." Definition of binomial coefficient in [Gleason] p. 295. As suggested by Gleason, we define it to be 0 when  0  <_  k  <_  n does not hold. (Contributed by NM, 10-Jul-2005.)

 |- 
 _C  =  ( n  e.  NN0 ,  k  e. 
 ZZ  |->  if ( k  e.  ( 0 ... n ) ,  ( ( ! `  n )  /  ( ( ! `  ( n  -  k
 ) )  x.  ( ! `  k ) ) ) ,  0 ) )
 
Theorembcval 9617 Value of the binomial coefficient, 
N choose  K. Definition of binomial coefficient in [Gleason] p. 295. As suggested by Gleason, we define it to be 0 when  0  <_  K  <_  N does not hold. See bcval2 9618 for the value in the standard domain. (Contributed by NM, 10-Jul-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
 |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( N  _C  K )  =  if ( K  e.  ( 0 ... N ) ,  (
 ( ! `  N )  /  ( ( ! `
  ( N  -  K ) )  x.  ( ! `  K ) ) ) ,  0 ) )
 
Theorembcval2 9618 Value of the binomial coefficient, 
N choose  K, in its standard domain. (Contributed by NM, 9-Jun-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
 |-  ( K  e.  (
 0 ... N )  ->  ( N  _C  K )  =  ( ( ! `
  N )  /  ( ( ! `  ( N  -  K ) )  x.  ( ! `  K ) ) ) )
 
Theorembcval3 9619 Value of the binomial coefficient, 
N choose  K, outside of its standard domain. Remark in [Gleason] p. 295. (Contributed by NM, 14-Jul-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)
 |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0
 ... N ) ) 
 ->  ( N  _C  K )  =  0 )
 
Theorembcval4 9620 Value of the binomial coefficient, 
N choose  K, outside of its standard domain. Remark in [Gleason] p. 295. (Contributed by NM, 14-Jul-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
 |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  ( K  <  0  \/  N  <  K ) )  ->  ( N  _C  K )  =  0 )
 
Theorembcrpcl 9621 Closure of the binomial coefficient in the positive reals. (This is mostly a lemma before we have bccl2 9636.) (Contributed by Mario Carneiro, 10-Mar-2014.)
 |-  ( K  e.  (
 0 ... N )  ->  ( N  _C  K )  e.  RR+ )
 
Theorembccmpl 9622 "Complementing" its second argument doesn't change a binary coefficient. (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 5-Mar-2014.)
 |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( N  _C  K )  =  ( N  _C  ( N  -  K ) ) )
 
Theorembcn0 9623  N choose 0 is 1. Remark in [Gleason] p. 296. (Contributed by NM, 17-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)
 |-  ( N  e.  NN0  ->  ( N  _C  0
 )  =  1 )
 
Theorembc0k 9624 The binomial coefficient " 0 choose  K " is 0 for a positive integer K. Note that  ( 0  _C  0 )  =  1 (see bcn0 9623). (Contributed by Alexander van der Vekens, 1-Jan-2018.)
 |-  ( K  e.  NN  ->  ( 0  _C  K )  =  0 )
 
Theorembcnn 9625  N choose  N is 1. Remark in [Gleason] p. 296. (Contributed by NM, 17-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)
 |-  ( N  e.  NN0  ->  ( N  _C  N )  =  1 )
 
Theorembcn1 9626 Binomial coefficient:  N choose  1. (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)
 |-  ( N  e.  NN0  ->  ( N  _C  1
 )  =  N )
 
Theorembcnp1n 9627 Binomial coefficient:  N  +  1 choose  N. (Contributed by NM, 20-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)
 |-  ( N  e.  NN0  ->  ( ( N  +  1 )  _C  N )  =  ( N  +  1 ) )
 
Theorembcm1k 9628 The proportion of one binomial coefficient to another with  K decreased by 1. (Contributed by Mario Carneiro, 10-Mar-2014.)
 |-  ( K  e.  (
 1 ... N )  ->  ( N  _C  K )  =  ( ( N  _C  ( K  -  1 ) )  x.  ( ( N  -  ( K  -  1
 ) )  /  K ) ) )
 
Theorembcp1n 9629 The proportion of one binomial coefficient to another with  N increased by 1. (Contributed by Mario Carneiro, 10-Mar-2014.)
 |-  ( K  e.  (
 0 ... N )  ->  ( ( N  +  1 )  _C  K )  =  ( ( N  _C  K )  x.  ( ( N  +  1 )  /  (
 ( N  +  1 )  -  K ) ) ) )
 
Theorembcp1nk 9630 The proportion of one binomial coefficient to another with  N and  K increased by 1. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  ( K  e.  (
 0 ... N )  ->  ( ( N  +  1 )  _C  ( K  +  1 )
 )  =  ( ( N  _C  K )  x.  ( ( N  +  1 )  /  ( K  +  1
 ) ) ) )
 
Theoremibcval5 9631 Write out the top and bottom parts of the binomial coefficient  ( N  _C  K )  =  ( N  x.  ( N  -  1 )  x. 
...  x.  ( ( N  -  K )  +  1 ) )  /  K ! explicitly. In this form, it is valid even for  N  <  K, although it is no longer valid for nonpositive  K. (Contributed by Jim Kingdon, 6-Nov-2021.)
 |-  ( ( N  e.  NN0  /\  K  e.  NN )  ->  ( N  _C  K )  =  ( (  seq ( ( N  -  K )  +  1
 ) (  x.  ,  _I  ,  CC ) `  N )  /  ( ! `  K ) ) )
 
Theorembcn2 9632 Binomial coefficient:  N choose  2. (Contributed by Mario Carneiro, 22-May-2014.)
 |-  ( N  e.  NN0  ->  ( N  _C  2
 )  =  ( ( N  x.  ( N  -  1 ) ) 
 /  2 ) )
 
Theorembcp1m1 9633 Compute the binomial coefficient of 
( N  +  1 ) over  ( N  - 
1 ) (Contributed by Scott Fenton, 11-May-2014.) (Revised by Mario Carneiro, 22-May-2014.)
 |-  ( N  e.  NN0  ->  ( ( N  +  1 )  _C  ( N  -  1 ) )  =  ( ( ( N  +  1 )  x.  N )  / 
 2 ) )
 
Theorembcpasc 9634 Pascal's rule for the binomial coefficient, generalized to all integers  K. Equation 2 of [Gleason] p. 295. (Contributed by NM, 13-Jul-2005.) (Revised by Mario Carneiro, 10-Mar-2014.)
 |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( ( N  _C  K )  +  ( N  _C  ( K  -  1 ) ) )  =  ( ( N  +  1 )  _C  K ) )
 
Theorembccl 9635 A binomial coefficient, in its extended domain, is a nonnegative integer. (Contributed by NM, 10-Jul-2005.) (Revised by Mario Carneiro, 9-Nov-2013.)
 |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( N  _C  K )  e.  NN0 )
 
Theorembccl2 9636 A binomial coefficient, in its standard domain, is a positive integer. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 10-Mar-2014.)
 |-  ( K  e.  (
 0 ... N )  ->  ( N  _C  K )  e.  NN )
 
Theorembcn2m1 9637 Compute the binomial coefficient " N choose 2 " from " ( N  -  1 ) choose 2 ": (N-1) + ( (N-1) 2 ) = ( N 2 ). (Contributed by Alexander van der Vekens, 7-Jan-2018.)
 |-  ( N  e.  NN  ->  ( ( N  -  1 )  +  (
 ( N  -  1
 )  _C  2 )
 )  =  ( N  _C  2 ) )
 
Theorembcn2p1 9638 Compute the binomial coefficient " ( N  +  1
) choose 2 " from " N choose 2 ": N + ( N 2 ) = ( (N+1) 2 ). (Contributed by Alexander van der Vekens, 8-Jan-2018.)
 |-  ( N  e.  NN0  ->  ( N  +  ( N  _C  2 ) )  =  ( ( N  +  1 )  _C  2 ) )
 
Theorempermnn 9639 The number of permutations of  N  -  R objects from a collection of  N objects is a positive integer. (Contributed by Jason Orendorff, 24-Jan-2007.)
 |-  ( R  e.  (
 0 ... N )  ->  ( ( ! `  N )  /  ( ! `  R ) )  e.  NN )
 
Theorembcnm1 9640 The binomial coefficent of  ( N  -  1 ) is  N. (Contributed by Scott Fenton, 16-May-2014.)
 |-  ( N  e.  NN0  ->  ( N  _C  ( N  -  1 ) )  =  N )
 
Theorem4bc3eq4 9641 The value of four choose three. (Contributed by Scott Fenton, 11-Jun-2016.)
 |-  ( 4  _C  3
 )  =  4
 
Theorem4bc2eq6 9642 The value of four choose two. (Contributed by Scott Fenton, 9-Jan-2017.)
 |-  ( 4  _C  2
 )  =  6
 
3.7  Elementary real and complex functions
 
3.7.1  The "shift" operation
 
Syntaxcshi 9643 Extend class notation with function shifter.
 class  shift
 
Definitiondf-shft 9644* Define a function shifter. This operation offsets the value argument of a function (ordinarily on a subset of  CC) and produces a new function on  CC. See shftval 9654 for its value. (Contributed by NM, 20-Jul-2005.)
 |- 
 shift  =  ( f  e.  _V ,  x  e. 
 CC  |->  { <. y ,  z >.  |  ( y  e. 
 CC  /\  ( y  -  x ) f z ) } )
 
Theoremshftlem 9645* Two ways to write a shifted set  ( B  +  A
). (Contributed by Mario Carneiro, 3-Nov-2013.)
 |-  ( ( A  e.  CC  /\  B  C_  CC )  ->  { x  e. 
 CC  |  ( x  -  A )  e.  B }  =  { x  |  E. y  e.  B  x  =  ( y  +  A ) } )
 
Theoremshftuz 9646* A shift of the upper integers. (Contributed by Mario Carneiro, 5-Nov-2013.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  { x  e. 
 CC  |  ( x  -  A )  e.  ( ZZ>= `  B ) }  =  ( ZZ>= `  ( B  +  A ) ) )
 
Theoremshftfvalg 9647* The value of the sequence shifter operation is a function on  CC.  A is ordinarily an integer. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  ( ( A  e.  CC  /\  F  e.  V )  ->  ( F  shift  A )  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) }
 )
 
Theoremovshftex 9648 Existence of the result of applying shift. (Contributed by Jim Kingdon, 15-Aug-2021.)
 |-  ( ( F  e.  V  /\  A  e.  CC )  ->  ( F  shift  A )  e.  _V )
 
Theoremshftfibg 9649 Value of a fiber of the relation  F. (Contributed by Jim Kingdon, 15-Aug-2021.)
 |-  ( ( F  e.  V  /\  A  e.  CC  /\  B  e.  CC )  ->  ( ( F  shift  A ) " { B } )  =  ( F " { ( B  -  A ) }
 ) )
 
Theoremshftfval 9650* The value of the sequence shifter operation is a function on  CC.  A is ordinarily an integer. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( A  e.  CC  ->  ( F  shift  A )  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) }
 )
 
Theoremshftdm 9651* Domain of a relation shifted by  A. The set on the right is more commonly notated as  ( dom  F  +  A ) (meaning add  A to every element of  dom  F). (Contributed by Mario Carneiro, 3-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( A  e.  CC  ->  dom  ( F  shift  A )  =  { x  e.  CC  |  ( x  -  A )  e.  dom  F }
 )
 
Theoremshftfib 9652 Value of a fiber of the relation  F. (Contributed by Mario Carneiro, 4-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  (
 ( F  shift  A )
 " { B }
 )  =  ( F
 " { ( B  -  A ) }
 ) )
 
Theoremshftfn 9653* Functionality and domain of a sequence shifted by  A. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  ( F  shift  A )  Fn 
 { x  e.  CC  |  ( x  -  A )  e.  B }
 )
 
Theoremshftval 9654 Value of a sequence shifted by  A. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 4-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  (
 ( F  shift  A ) `
  B )  =  ( F `  ( B  -  A ) ) )
 
Theoremshftval2 9655 Value of a sequence shifted by  A  -  B. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( F  shift  ( A  -  B ) ) `  ( A  +  C ) )  =  ( F `  ( B  +  C ) ) )
 
Theoremshftval3 9656 Value of a sequence shifted by  A  -  B. (Contributed by NM, 20-Jul-2005.)
 |-  F  e.  _V   =>    |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  (
 ( F  shift  ( A  -  B ) ) `
  A )  =  ( F `  B ) )
 
Theoremshftval4 9657 Value of a sequence shifted by  -u A. (Contributed by NM, 18-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  (
 ( F  shift  -u A ) `  B )  =  ( F `  ( A  +  B )
 ) )
 
Theoremshftval5 9658 Value of a shifted sequence. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  (
 ( F  shift  A ) `
  ( B  +  A ) )  =  ( F `  B ) )
 
Theoremshftf 9659* Functionality of a shifted sequence. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( ( F : B --> C  /\  A  e.  CC )  ->  ( F  shift  A ) : { x  e. 
 CC  |  ( x  -  A )  e.  B } --> C )
 
Theorem2shfti 9660 Composite shift operations. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  (
 ( F  shift  A ) 
 shift  B )  =  ( F  shift  ( A  +  B ) ) )
 
Theoremshftidt2 9661 Identity law for the shift operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( F  shift  0 )  =  ( F  |`  CC )
 
Theoremshftidt 9662 Identity law for the shift operation. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( A  e.  CC  ->  ( ( F 
 shift  0 ) `  A )  =  ( F `  A ) )
 
Theoremshftcan1 9663 Cancellation law for the shift operation. (Contributed by NM, 4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  (
 ( ( F  shift  A )  shift  -u A ) `  B )  =  ( F `  B ) )
 
Theoremshftcan2 9664 Cancellation law for the shift operation. (Contributed by NM, 4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  (
 ( ( F  shift  -u A )  shift  A ) `
  B )  =  ( F `  B ) )
 
Theoremshftvalg 9665 Value of a sequence shifted by  A. (Contributed by Scott Fenton, 16-Dec-2017.)
 |-  ( ( F  e.  V  /\  A  e.  CC  /\  B  e.  CC )  ->  ( ( F  shift  A ) `  B )  =  ( F `  ( B  -  A ) ) )
 
Theoremshftval4g 9666 Value of a sequence shifted by  -u A. (Contributed by Jim Kingdon, 19-Aug-2021.)
 |-  ( ( F  e.  V  /\  A  e.  CC  /\  B  e.  CC )  ->  ( ( F  shift  -u A ) `  B )  =  ( F `  ( A  +  B ) ) )
 
3.7.2  Real and imaginary parts; conjugate
 
Syntaxccj 9667 Extend class notation to include complex conjugate function.
 class  *
 
Syntaxcre 9668 Extend class notation to include real part of a complex number.
 class  Re
 
Syntaxcim 9669 Extend class notation to include imaginary part of a complex number.
 class  Im
 
Definitiondf-cj 9670* Define the complex conjugate function. See cjcli 9741 for its closure and cjval 9673 for its value. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
 |-  *  =  ( x  e.  CC  |->  ( iota_ y  e.  CC  ( ( x  +  y )  e.  RR  /\  ( _i  x.  ( x  -  y ) )  e. 
 RR ) ) )
 
Definitiondf-re 9671 Define a function whose value is the real part of a complex number. See reval 9677 for its value, recli 9739 for its closure, and replim 9687 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
 |-  Re  =  ( x  e.  CC  |->  ( ( x  +  ( * `
  x ) ) 
 /  2 ) )
 
Definitiondf-im 9672 Define a function whose value is the imaginary part of a complex number. See imval 9678 for its value, imcli 9740 for its closure, and replim 9687 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
 |-  Im  =  ( x  e.  CC  |->  ( Re
 `  ( x  /  _i ) ) )
 
Theoremcjval 9673* The value of the conjugate of a complex number. (Contributed by Mario Carneiro, 6-Nov-2013.)
 |-  ( A  e.  CC  ->  ( * `  A )  =  ( iota_ x  e. 
 CC  ( ( A  +  x )  e. 
 RR  /\  ( _i  x.  ( A  -  x ) )  e.  RR ) ) )
 
Theoremcjth 9674 The defining property of the complex conjugate. (Contributed by Mario Carneiro, 6-Nov-2013.)
 |-  ( A  e.  CC  ->  ( ( A  +  ( * `  A ) )  e.  RR  /\  ( _i  x.  ( A  -  ( * `  A ) ) )  e.  RR ) )
 
Theoremcjf 9675 Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)
 |-  * : CC --> CC
 
Theoremcjcl 9676 The conjugate of a complex number is a complex number (closure law). (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
 |-  ( A  e.  CC  ->  ( * `  A )  e.  CC )
 
Theoremreval 9677 The value of the real part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
 |-  ( A  e.  CC  ->  ( Re `  A )  =  ( ( A  +  ( * `  A ) )  / 
 2 ) )
 
Theoremimval 9678 The value of the imaginary part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
 |-  ( A  e.  CC  ->  ( Im `  A )  =  ( Re `  ( A  /  _i ) ) )
 
Theoremimre 9679 The imaginary part of a complex number in terms of the real part function. (Contributed by NM, 12-May-2005.) (Revised by Mario Carneiro, 6-Nov-2013.)
 |-  ( A  e.  CC  ->  ( Im `  A )  =  ( Re `  ( -u _i  x.  A ) ) )
 
Theoremreim 9680 The real part of a complex number in terms of the imaginary part function. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  CC  ->  ( Re `  A )  =  ( Im `  ( _i  x.  A ) ) )
 
Theoremrecl 9681 The real part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
 |-  ( A  e.  CC  ->  ( Re `  A )  e.  RR )
 
Theoremimcl 9682 The imaginary part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
 |-  ( A  e.  CC  ->  ( Im `  A )  e.  RR )
 
Theoremref 9683 Domain and codomain of the real part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)
 |-  Re : CC --> RR
 
Theoremimf 9684 Domain and codomain of the imaginary part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)
 |-  Im : CC --> RR
 
Theoremcrre 9685 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 12-May-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Re `  ( A  +  ( _i  x.  B ) ) )  =  A )
 
Theoremcrim 9686 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 12-May-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Im `  ( A  +  ( _i  x.  B ) ) )  =  B )
 
Theoremreplim 9687 Reconstruct a complex number from its real and imaginary parts. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 7-Nov-2013.)
 |-  ( A  e.  CC  ->  A  =  ( ( Re `  A )  +  ( _i  x.  ( Im `  A ) ) ) )
 
Theoremremim 9688 Value of the conjugate of a complex number. The value is the real part minus  _i times the imaginary part. Definition 10-3.2 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 7-Nov-2013.)
 |-  ( A  e.  CC  ->  ( * `  A )  =  ( ( Re `  A )  -  ( _i  x.  ( Im `  A ) ) ) )
 
Theoremreim0 9689 The imaginary part of a real number is 0. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
 |-  ( A  e.  RR  ->  ( Im `  A )  =  0 )
 
Theoremreim0b 9690 A number is real iff its imaginary part is 0. (Contributed by NM, 26-Sep-2005.)
 |-  ( A  e.  CC  ->  ( A  e.  RR  <->  ( Im `  A )  =  0 ) )
 
Theoremrereb 9691 A number is real iff it equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 20-Aug-2008.)
 |-  ( A  e.  CC  ->  ( A  e.  RR  <->  ( Re `  A )  =  A ) )
 
Theoremmulreap 9692 A product with a real multiplier apart from zero is real iff the multiplicand is real. (Contributed by Jim Kingdon, 14-Jun-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B #  0 )  ->  ( A  e.  RR  <->  ( B  x.  A )  e. 
 RR ) )
 
Theoremrere 9693 A real number equals its real part. One direction of Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Paul Chapman, 7-Sep-2007.)
 |-  ( A  e.  RR  ->  ( Re `  A )  =  A )
 
Theoremcjreb 9694 A number is real iff it equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 2-Jul-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
 |-  ( A  e.  CC  ->  ( A  e.  RR  <->  ( * `  A )  =  A ) )
 
Theoremrecj 9695 Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014.)
 |-  ( A  e.  CC  ->  ( Re `  ( * `  A ) )  =  ( Re `  A ) )
 
Theoremreneg 9696 Real part of negative. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
 |-  ( A  e.  CC  ->  ( Re `  -u A )  =  -u ( Re
 `  A ) )
 
Theoremreadd 9697 Real part distributes over addition. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( Re `  ( A  +  B ) )  =  (
 ( Re `  A )  +  ( Re `  B ) ) )
 
Theoremresub 9698 Real part distributes over subtraction. (Contributed by NM, 17-Mar-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( Re `  ( A  -  B ) )  =  (
 ( Re `  A )  -  ( Re `  B ) ) )
 
Theoremremullem 9699 Lemma for remul 9700, immul 9707, and cjmul 9713. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( Re
 `  ( A  x.  B ) )  =  ( ( ( Re
 `  A )  x.  ( Re `  B ) )  -  (
 ( Im `  A )  x.  ( Im `  B ) ) ) 
 /\  ( Im `  ( A  x.  B ) )  =  (
 ( ( Re `  A )  x.  ( Im `  B ) )  +  ( ( Im
 `  A )  x.  ( Re `  B ) ) )  /\  ( * `  ( A  x.  B ) )  =  ( ( * `
  A )  x.  ( * `  B ) ) ) )
 
Theoremremul 9700 Real part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( Re `  ( A  x.  B ) )  =  (
 ( ( Re `  A )  x.  ( Re `  B ) )  -  ( ( Im
 `  A )  x.  ( Im `  B ) ) ) )
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