P. 114 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 11301-11400   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-s7 11301	Define the length 7 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
|- <" A B C D E F G "> = ( <" A B C D E F "> ++ <" G "> )
Definition	df-s8 11302	Define the length 8 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
|- <" A B C D E F G H "> = ( <" A B C D E F G "> ++ <" H "> )
Theorem	cats1cld 11303	Closure of concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
|- T = ( S ++ <" X "> )   &    |- ( ph -> S e. Word A )   &    |- ( ph -> X e. A )   =>    |- ( ph -> T e. Word A )
Theorem	cats1fvn 11304	The last symbol of a concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
|- T = ( S ++ <" X "> )   &    |- S e. Word _V   &    |- (♯ ` S ) = M   =>    |- ( X e. V -> ( T ` M ) = X )
Theorem	cats1fvnd 11305	The last symbol of a concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.) (Revised by Jim Kingdon, 20-Jan-2026.)
|- T = ( S ++ <" X "> )   &    |- ( ph -> S e. Word _V )   &    |- ( ph -> X e. V )   &    |- ( ph -> (♯ ` S ) = M )   =>    |- ( ph -> ( T ` M ) = X )
Theorem	cats1fvd 11306	A symbol other than the last in a concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.) (Revised by Jim Kingdon, 20-Jan-2026.)
|- T = ( S ++ <" X "> )   &    |- ( ph -> S e. Word _V )   &    |- ( ph -> (♯ ` S ) = M )   &    |- ( ph -> Y e. V )   &    |- ( ph -> X e. W )   &    |- ( ph -> ( S ` N ) = Y )   &    |- ( ph -> N e. NN0 )   &    |- ( ph -> N < M )   =>    |- ( ph -> ( T ` N ) = Y )
Theorem	cats1lend 11307	The length of concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.) (Revised by Jim Kingdon, 19-Jan-2026.)
|- T = ( S ++ <" X "> )   &    |- ( ph -> S e. Word _V )   &    |- ( ph -> X e. W )   &    |- (♯ ` S ) = M   &    |- ( M + 1 ) = N   =>    |- ( ph -> (♯ ` T ) = N )
Theorem	cats1catd 11308	Closure of concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.) (Revised by Jim Kingdon, 19-Jan-2026.)
|- T = ( S ++ <" X "> )   &    |- ( ph -> A e. Word _V )   &    |- ( ph -> S e. Word _V )   &    |- ( ph -> X e. W )   &    |- ( ph -> C = ( B ++ <" X "> ) )   &    |- ( ph -> B = ( A ++ S ) )   =>    |- ( ph -> C = ( A ++ T ) )
Theorem	cats2catd 11309	Closure of concatenation of concatenations with singleton words. (Contributed by AV, 1-Mar-2021.) (Revised by Jim Kingdon, 19-Jan-2026.)
|- ( ph -> B e. Word _V )   &    |- ( ph -> D e. Word _V )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. W )   &    |- ( ph -> A = ( B ++ <" X "> ) )   &    |- ( ph -> C = ( <" Y "> ++ D ) )   =>    |- ( ph -> ( A ++ C ) = ( ( B ++ <" X Y "> ) ++ D ) )
Theorem	s2eqd 11310	Equality theorem for a doubleton word. (Contributed by Mario Carneiro, 27-Feb-2016.)
|- ( ph -> A = N )   &    |- ( ph -> B = O )   =>    |- ( ph -> <" A B "> = <" N O "> )
Theorem	s3eqd 11311	Equality theorem for a length 3 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
|- ( ph -> A = N )   &    |- ( ph -> B = O )   &    |- ( ph -> C = P )   =>    |- ( ph -> <" A B C "> = <" N O P "> )
Theorem	s4eqd 11312	Equality theorem for a length 4 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
|- ( ph -> A = N )   &    |- ( ph -> B = O )   &    |- ( ph -> C = P )   &    |- ( ph -> D = Q )   =>    |- ( ph -> <" A B C D "> = <" N O P Q "> )
Theorem	s5eqd 11313	Equality theorem for a length 5 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
|- ( ph -> A = N )   &    |- ( ph -> B = O )   &    |- ( ph -> C = P )   &    |- ( ph -> D = Q )   &    |- ( ph -> E = R )   =>    |- ( ph -> <" A B C D E "> = <" N O P Q R "> )
Theorem	s6eqd 11314	Equality theorem for a length 6 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
|- ( ph -> A = N )   &    |- ( ph -> B = O )   &    |- ( ph -> C = P )   &    |- ( ph -> D = Q )   &    |- ( ph -> E = R )   &    |- ( ph -> F = S )   =>    |- ( ph -> <" A B C D E F "> = <" N O P Q R S "> )
Theorem	s7eqd 11315	Equality theorem for a length 7 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
|- ( ph -> A = N )   &    |- ( ph -> B = O )   &    |- ( ph -> C = P )   &    |- ( ph -> D = Q )   &    |- ( ph -> E = R )   &    |- ( ph -> F = S )   &    |- ( ph -> G = T )   =>    |- ( ph -> <" A B C D E F G "> = <" N O P Q R S T "> )
Theorem	s8eqd 11316	Equality theorem for a length 8 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
|- ( ph -> A = N )   &    |- ( ph -> B = O )   &    |- ( ph -> C = P )   &    |- ( ph -> D = Q )   &    |- ( ph -> E = R )   &    |- ( ph -> F = S )   &    |- ( ph -> G = T )   &    |- ( ph -> H = U )   =>    |- ( ph -> <" A B C D E F G H "> = <" N O P Q R S T U "> )
Theorem	s3eq2 11317	Equality theorem for a length 3 word for the second symbol. (Contributed by AV, 4-Jan-2022.)
|- ( B = D -> <" A B C "> = <" A D C "> )
Theorem	s2cld 11318	A doubleton word is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
|- ( ph -> A e. X )   &    |- ( ph -> B e. X )   =>    |- ( ph -> <" A B "> e. Word X )
Theorem	s3cld 11319	A length 3 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
|- ( ph -> A e. X )   &    |- ( ph -> B e. X )   &    |- ( ph -> C e. X )   =>    |- ( ph -> <" A B C "> e. Word X )
Theorem	s4cld 11320	A length 4 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
|- ( ph -> A e. X )   &    |- ( ph -> B e. X )   &    |- ( ph -> C e. X )   &    |- ( ph -> D e. X )   =>    |- ( ph -> <" A B C D "> e. Word X )
Theorem	s5cld 11321	A length 5 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
|- ( ph -> A e. X )   &    |- ( ph -> B e. X )   &    |- ( ph -> C e. X )   &    |- ( ph -> D e. X )   &    |- ( ph -> E e. X )   =>    |- ( ph -> <" A B C D E "> e. Word X )
Theorem	s6cld 11322	A length 6 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
|- ( ph -> A e. X )   &    |- ( ph -> B e. X )   &    |- ( ph -> C e. X )   &    |- ( ph -> D e. X )   &    |- ( ph -> E e. X )   &    |- ( ph -> F e. X )   =>    |- ( ph -> <" A B C D E F "> e. Word X )
Theorem	s7cld 11323	A length 7 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
|- ( ph -> A e. X )   &    |- ( ph -> B e. X )   &    |- ( ph -> C e. X )   &    |- ( ph -> D e. X )   &    |- ( ph -> E e. X )   &    |- ( ph -> F e. X )   &    |- ( ph -> G e. X )   =>    |- ( ph -> <" A B C D E F G "> e. Word X )
Theorem	s8cld 11324	A length 8 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
|- ( ph -> A e. X )   &    |- ( ph -> B e. X )   &    |- ( ph -> C e. X )   &    |- ( ph -> D e. X )   &    |- ( ph -> E e. X )   &    |- ( ph -> F e. X )   &    |- ( ph -> G e. X )   &    |- ( ph -> H e. X )   =>    |- ( ph -> <" A B C D E F G H "> e. Word X )
Theorem	s2cl 11325	A doubleton word is a word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
|- ( ( A e. X /\ B e. X ) -> <" A B "> e. Word X )
Theorem	s3cl 11326	A length 3 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
|- ( ( A e. X /\ B e. X /\ C e. X ) -> <" A B C "> e. Word X )
Theorem	s2fv0g 11327	Extract the first symbol from a doubleton word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
|- ( ( A e. V /\ B e. W ) -> ( <" A B "> ` 0 ) = A )
Theorem	s2fv1g 11328	Extract the second symbol from a doubleton word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
|- ( ( A e. V /\ B e. W ) -> ( <" A B "> ` 1 ) = B )
Theorem	s2leng 11329	The length of a doubleton word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
|- ( ( A e. V /\ B e. W ) -> (♯ ` <" A B "> ) = 2 )
Theorem	s2dmg 11330	The domain of a doubleton word is an unordered pair. (Contributed by AV, 9-Jan-2020.)
|- ( ( A e. V /\ B e. W ) -> dom <" A B "> = { 0 , 1 } )
Theorem	s3fv0g 11331	Extract the first symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)
|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( <" A B C "> ` 0 ) = A )
Theorem	s3fv1g 11332	Extract the second symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)
|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( <" A B C "> ` 1 ) = B )
4.8  Elementary real and complex functions
4.8.1  The "shift" operation
Syntax	cshi 11333	Extend class notation with function shifter.
class shift
Definition	df-shft 11334*	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 11344 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 11335*	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 11336*	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 11337*	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 11338	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 11339	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 11340*	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 11341*	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 11342	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 11343*	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 11344	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 11345	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 11346	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 11347	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 11348	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 11349*	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 11350	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 11351	Identity law for the shift operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
|- F e. _V   =>    |- ( F shift 0 ) = ( F |` CC )
Theorem	shftidt 11352	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 11353	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 11354	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 11355	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 11356	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 11357*	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 ) )
4.8.2  Real and imaginary parts; conjugate
Syntax	ccj 11358	Extend class notation to include complex conjugate function.
class *
Syntax	cre 11359	Extend class notation to include real part of a complex number.
class Re
Syntax	cim 11360	Extend class notation to include imaginary part of a complex number.
class Im
Definition	df-cj 11361*	Define the complex conjugate function. See cjcli 11432 for its closure and cjval 11364 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 11362	Define a function whose value is the real part of a complex number. See reval 11368 for its value, recli 11430 for its closure, and replim 11378 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
|- Re = ( x e. CC |-> ( ( x + ( * ` x ) ) / 2 ) )
Definition	df-im 11363	Define a function whose value is the imaginary part of a complex number. See imval 11369 for its value, imcli 11431 for its closure, and replim 11378 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
|- Im = ( x e. CC |-> ( Re ` ( x / _i ) ) )
Theorem	cjval 11364*	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 11365	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 11366	Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)
|- * : CC --> CC
Theorem	cjcl 11367	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 11368	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 11369	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 11370	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 11371	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 11372	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 11373	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 11374	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 11375	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 11376	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 11377	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 11378	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 11379	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 11380	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 11381	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 11382	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 11383	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 11384	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 11385	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 11386	Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014.)
|- ( A e. CC -> ( Re ` ( * ` A ) ) = ( Re ` A ) )
Theorem	reneg 11387	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 11388	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 11389	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 11390	Lemma for remul 11391, immul 11398, and cjmul 11404. (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 11391	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 11392	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 11393	Real part of a division. Related to remul2 11392. (Contributed by Jim Kingdon, 14-Jun-2020.)
|- ( ( A e. CC /\ B e. RR /\ B # 0 ) -> ( Re ` ( A / B ) ) = ( ( Re ` A ) / B ) )
Theorem	imcj 11394	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 11395	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 11396	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 11397	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 11398	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 11399	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 11400	Imaginary part of a division. Related to immul2 11399. (Contributed by Jim Kingdon, 14-Jun-2020.)
|- ( ( A e. CC /\ B e. RR /\ B # 0 ) -> ( Im ` ( A / B ) ) = ( ( Im ` A ) / B ) )