P. 104 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10301-10400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fnfzo0hash 10301	The size of a function on a half-open range of nonnegative integers equals the upper bound of this range. (Contributed by Alexander van der Vekens, 26-Jan-2018.) (Proof shortened by AV, 11-Apr-2021.)
|- ( ( N e. NN0 /\ F : ( 0..^ N ) --> B ) -> (♯ ` F ) = N )
Theorem	hashfacen 10302*	The number of bijections between two sets is a cardinal invariant. (Contributed by Mario Carneiro, 21-Jan-2015.)
|- ( ( A ~~ B /\ C ~~ D ) -> { f | f : A -1-1-onto-> C } ~~ { f | f : B -1-1-onto-> D } )
Theorem	leisorel 10303	Version of isorel 5601 for strictly increasing functions on the reals. (Contributed by Mario Carneiro, 6-Apr-2015.) (Revised by Mario Carneiro, 9-Sep-2015.)
|- ( ( F Isom < , < ( A , B ) /\ ( A C_ RR* /\ B C_ RR* ) /\ ( C e. A /\ D e. A ) ) -> ( C <_ D <-> ( F ` C ) <_ ( F ` D ) ) )
Theorem	zfz1isolemsplit 10304	Lemma for zfz1iso 10307. Removing one element from an integer range. (Contributed by Jim Kingdon, 8-Sep-2022.)
|- ( ph -> X e. Fin )   &    |- ( ph -> M e. X )   =>    |- ( ph -> ( 1 ... (♯ ` X ) ) = ( ( 1 ... (♯ ` ( X \ { M } ) ) ) u. { (♯ ` X ) } ) )
Theorem	zfz1isolemiso 10305*	Lemma for zfz1iso 10307. Adding one element to the order isomorphism. (Contributed by Jim Kingdon, 8-Sep-2022.)
|- ( ph -> X e. Fin )   &    |- ( ph -> X C_ ZZ )   &    |- ( ph -> M e. X )   &    |- ( ph -> A. z e. X z <_ M )   &    |- ( ph -> G Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) )   &    |- ( ph -> A e. ( 1 ... (♯ ` X ) ) )   &    |- ( ph -> B e. ( 1 ... (♯ ` X ) ) )   =>    |- ( ph -> ( A < B <-> ( ( G u. { <. (♯ ` X ) , M >. } ) ` A ) < ( ( G u. { <. (♯ ` X ) , M >. } ) ` B ) ) )
Theorem	zfz1isolem1 10306*	Lemma for zfz1iso 10307. Existence of an order isomorphism given the existence of shorter isomorphisms. (Contributed by Jim Kingdon, 7-Sep-2022.)
|- ( ph -> K e. om )   &    |- ( ph -> A. y ( ( ( y C_ ZZ /\ y e. Fin ) /\ y ~~ K ) -> E. f f Isom < , < ( ( 1 ... (♯ ` y ) ) , y ) ) )   &    |- ( ph -> X C_ ZZ )   &    |- ( ph -> X e. Fin )   &    |- ( ph -> X ~~ suc K )   &    |- ( ph -> M e. X )   &    |- ( ph -> A. z e. X z <_ M )   =>    |- ( ph -> E. f f Isom < , < ( ( 1 ... (♯ ` X ) ) , X ) )
Theorem	zfz1iso 10307*	A finite set of integers has an order isomorphism to a one-based finite sequence. (Contributed by Jim Kingdon, 3-Sep-2022.)
|- ( ( A C_ ZZ /\ A e. Fin ) -> E. f f Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) )
Theorem	iseqcoll 10308*	The function F contains a sparse set of nonzero values to be summed. The function G is an order isomorphism from the set of nonzero values of F to a 1-based finite sequence, and H collects these nonzero values together. Under these conditions, the sum over the values in H yields the same result as the sum over the original set F. (Contributed by Mario Carneiro, 2-Apr-2014.)
|- ( ( ph /\ k e. S ) -> ( Z .+ k ) = k )   &    |- ( ( ph /\ k e. S ) -> ( k .+ Z ) = k )   &    |- ( ( ph /\ ( k e. S /\ n e. S ) ) -> ( k .+ n ) e. S )   &    |- ( ph -> Z e. S )   &    |- ( ph -> G Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) )   &    |- ( ph -> N e. ( 1 ... (♯ ` A ) ) )   &    |- ( ph -> A C_ ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` 1 ) ) -> ( H ` k ) e. S )   &    |- ( ( ph /\ k e. ( ( M ... ( G ` (♯ ` A ) ) ) \ A ) ) -> ( F ` k ) = Z )   &    |- ( ( ph /\ n e. ( 1 ... (♯ ` A ) ) ) -> ( H ` n ) = ( F ` ( G ` n ) ) )   =>    |- ( ph -> ( seq M ( .+ , F , S ) ` ( G ` N ) ) = ( seq 1 ( .+ , H , S ) ` N ) )
3.7  Elementary real and complex functions
3.7.1  The "shift" operation
Syntax	cshi 10309	Extend class notation with function shifter.
class shift
Definition	df-shft 10310*	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 10320 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 ) } )
Theorem	shftlem 10311*	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 ) } )
Theorem	shftuz 10312*	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 ) ) )
Theorem	shftfvalg 10313*	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 ) } )
Theorem	ovshftex 10314	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 )
Theorem	shftfibg 10315	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 ) } ) )
Theorem	shftfval 10316*	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 ) } )
Theorem	shftdm 10317*	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 } )
Theorem	shftfib 10318	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 ) } ) )
Theorem	shftfn 10319*	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 } )
Theorem	shftval 10320	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 ) ) )
Theorem	shftval2 10321	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 ) ) )
Theorem	shftval3 10322	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 ) )
Theorem	shftval4 10323	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 ) ) )
Theorem	shftval5 10324	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 ) )
Theorem	shftf 10325*	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 )
Theorem	2shfti 10326	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 ) ) )
Theorem	shftidt2 10327	Identity law for the shift operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
|- F e. _V   =>    |- ( F shift 0 ) = ( F |` CC )
Theorem	shftidt 10328	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 ) )
Theorem	shftcan1 10329	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 ) )
Theorem	shftcan2 10330	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 ) )
Theorem	shftvalg 10331	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 ) ) )
Theorem	shftval4g 10332	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 ) ) )
Theorem	seq3shft 10333*	Shifting the index set of a sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 17-Oct-2022.)
|- ( ph -> F e. V )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ( ph /\ x e. ( ZZ>= ` ( M - N ) ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ph -> seq M ( .+ , ( F shift N ) ) = ( seq ( M - N ) ( .+ , F ) shift N ) )
3.7.2  Real and imaginary parts; conjugate
Syntax	ccj 10334	Extend class notation to include complex conjugate function.
class *
Syntax	cre 10335	Extend class notation to include real part of a complex number.
class Re
Syntax	cim 10336	Extend class notation to include imaginary part of a complex number.
class Im
Definition	df-cj 10337*	Define the complex conjugate function. See cjcli 10408 for its closure and cjval 10340 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 ) ) )
Definition	df-re 10338	Define a function whose value is the real part of a complex number. See reval 10344 for its value, recli 10406 for its closure, and replim 10354 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
|- Re = ( x e. CC |-> ( ( x + ( * ` x ) ) / 2 ) )
Definition	df-im 10339	Define a function whose value is the imaginary part of a complex number. See imval 10345 for its value, imcli 10407 for its closure, and replim 10354 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
|- Im = ( x e. CC |-> ( Re ` ( x / _i ) ) )
Theorem	cjval 10340*	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 ) ) )
Theorem	cjth 10341	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 ) )
Theorem	cjf 10342	Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)
|- * : CC --> CC
Theorem	cjcl 10343	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 )
Theorem	reval 10344	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 ) )
Theorem	imval 10345	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 ) ) )
Theorem	imre 10346	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 ) ) )
Theorem	reim 10347	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 ) ) )
Theorem	recl 10348	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 )
Theorem	imcl 10349	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 )
Theorem	ref 10350	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
Theorem	imf 10351	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
Theorem	crre 10352	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 )
Theorem	crim 10353	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 )
Theorem	replim 10354	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 ) ) ) )
Theorem	remim 10355	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 ) ) ) )
Theorem	reim0 10356	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 )
Theorem	reim0b 10357	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 ) )
Theorem	rereb 10358	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 ) )
Theorem	mulreap 10359	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 ) )
Theorem	rere 10360	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 )
Theorem	cjreb 10361	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 ) )
Theorem	recj 10362	Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014.)
|- ( A e. CC -> ( Re ` ( * ` A ) ) = ( Re ` A ) )
Theorem	reneg 10363	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 ) )
Theorem	readd 10364	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 ) ) )
Theorem	resub 10365	Real part distributes over subtraction. (Contributed by NM, 17-Mar-2005.)
|- ( ( A e. CC /\ B e. CC ) -> ( Re ` ( A - B ) ) = ( ( Re ` A ) - ( Re ` B ) ) )
Theorem	remullem 10366	Lemma for remul 10367, immul 10374, and cjmul 10380. (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 ) ) ) )
Theorem	remul 10367	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 ) ) ) )
Theorem	remul2 10368	Real part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
|- ( ( A e. RR /\ B e. CC ) -> ( Re ` ( A x. B ) ) = ( A x. ( Re ` B ) ) )
Theorem	redivap 10369	Real part of a division. Related to remul2 10368. (Contributed by Jim Kingdon, 14-Jun-2020.)
|- ( ( A e. CC /\ B e. RR /\ B # 0 ) -> ( Re ` ( A / B ) ) = ( ( Re ` A ) / B ) )
Theorem	imcj 10370	Imaginary part of a complex conjugate. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
|- ( A e. CC -> ( Im ` ( * ` A ) ) = -u ( Im ` A ) )
Theorem	imneg 10371	The imaginary part of a negative number. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
|- ( A e. CC -> ( Im ` -u A ) = -u ( Im ` A ) )
Theorem	imadd 10372	Imaginary part distributes over addition. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
|- ( ( A e. CC /\ B e. CC ) -> ( Im ` ( A + B ) ) = ( ( Im ` A ) + ( Im ` B ) ) )
Theorem	imsub 10373	Imaginary part distributes over subtraction. (Contributed by NM, 18-Mar-2005.)
|- ( ( A e. CC /\ B e. CC ) -> ( Im ` ( A - B ) ) = ( ( Im ` A ) - ( Im ` B ) ) )
Theorem	immul 10374	Imaginary part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
|- ( ( A e. CC /\ B e. CC ) -> ( Im ` ( A x. B ) ) = ( ( ( Re ` A ) x. ( Im ` B ) ) + ( ( Im ` A ) x. ( Re ` B ) ) ) )
Theorem	immul2 10375	Imaginary part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
|- ( ( A e. RR /\ B e. CC ) -> ( Im ` ( A x. B ) ) = ( A x. ( Im ` B ) ) )
Theorem	imdivap 10376	Imaginary part of a division. Related to immul2 10375. (Contributed by Jim Kingdon, 14-Jun-2020.)
|- ( ( A e. CC /\ B e. RR /\ B # 0 ) -> ( Im ` ( A / B ) ) = ( ( Im ` A ) / B ) )
Theorem	cjre 10377	A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 8-Oct-1999.)
|- ( A e. RR -> ( * ` A ) = A )
Theorem	cjcj 10378	The conjugate of the conjugate is the original complex number. Proposition 10-3.4(e) of [Gleason] p. 133. (Contributed by NM, 29-Jul-1999.) (Proof shortened by Mario Carneiro, 14-Jul-2014.)
|- ( A e. CC -> ( * ` ( * ` A ) ) = A )
Theorem	cjadd 10379	Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by NM, 31-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
|- ( ( A e. CC /\ B e. CC ) -> ( * ` ( A + B ) ) = ( ( * ` A ) + ( * ` B ) ) )
Theorem	cjmul 10380	Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by NM, 29-Jul-1999.) (Proof shortened by Mario Carneiro, 14-Jul-2014.)
|- ( ( A e. CC /\ B e. CC ) -> ( * ` ( A x. B ) ) = ( ( * ` A ) x. ( * ` B ) ) )
Theorem	ipcnval 10381	Standard inner product on complex numbers. (Contributed by NM, 29-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 ) ) ) )
Theorem	cjmulrcl 10382	A complex number times its conjugate is real. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
|- ( A e. CC -> ( A x. ( * ` A ) ) e. RR )
Theorem	cjmulval 10383	A complex number times its conjugate. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 14-Jul-2014.)
|- ( A e. CC -> ( A x. ( * ` A ) ) = ( ( ( Re ` A ) ^ 2 ) + ( ( Im ` A ) ^ 2 ) ) )
Theorem	cjmulge0 10384	A complex number times its conjugate is nonnegative. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
|- ( A e. CC -> 0 <_ ( A x. ( * ` A ) ) )
Theorem	cjneg 10385	Complex conjugate of negative. (Contributed by NM, 27-Feb-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
|- ( A e. CC -> ( * ` -u A ) = -u ( * ` A ) )
Theorem	addcj 10386	A number plus its conjugate is twice its real part. Compare Proposition 10-3.4(h) of [Gleason] p. 133. (Contributed by NM, 21-Jan-2007.) (Revised by Mario Carneiro, 14-Jul-2014.)
|- ( A e. CC -> ( A + ( * ` A ) ) = ( 2 x. ( Re ` A ) ) )
Theorem	cjsub 10387	Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.)
|- ( ( A e. CC /\ B e. CC ) -> ( * ` ( A - B ) ) = ( ( * ` A ) - ( * ` B ) ) )
Theorem	cjexp 10388	Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.)
|- ( ( A e. CC /\ N e. NN0 ) -> ( * ` ( A ^ N ) ) = ( ( * ` A ) ^ N ) )
Theorem	imval2 10389	The imaginary part of a number in terms of complex conjugate. (Contributed by NM, 30-Apr-2005.)
|- ( A e. CC -> ( Im ` A ) = ( ( A - ( * ` A ) ) / ( 2 x. _i ) ) )
Theorem	re0 10390	The real part of zero. (Contributed by NM, 27-Jul-1999.)
|- ( Re ` 0 ) = 0
Theorem	im0 10391	The imaginary part of zero. (Contributed by NM, 27-Jul-1999.)
|- ( Im ` 0 ) = 0
Theorem	re1 10392	The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
|- ( Re ` 1 ) = 1
Theorem	im1 10393	The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
|- ( Im ` 1 ) = 0
Theorem	rei 10394	The real part of _i. (Contributed by Scott Fenton, 9-Jun-2006.)
|- ( Re ` _i ) = 0
Theorem	imi 10395	The imaginary part of _i. (Contributed by Scott Fenton, 9-Jun-2006.)
|- ( Im ` _i ) = 1
Theorem	cj0 10396	The conjugate of zero. (Contributed by NM, 27-Jul-1999.)
|- ( * ` 0 ) = 0
Theorem	cji 10397	The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.)
|- ( * ` _i ) = -u _i
Theorem	cjreim 10398	The conjugate of a representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.)
|- ( ( A e. RR /\ B e. RR ) -> ( * ` ( A + ( _i x. B ) ) ) = ( A - ( _i x. B ) ) )
Theorem	cjreim2 10399	The conjugate of the representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)
|- ( ( A e. RR /\ B e. RR ) -> ( * ` ( A - ( _i x. B ) ) ) = ( A + ( _i x. B ) ) )
Theorem	cj11 10400	Complex conjugate is a one-to-one function. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Eric Schmidt, 2-Jul-2009.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( * ` A ) = ( * ` B ) <-> A = B ) )