P. 113 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 11201-11300   *Has distinct variable group(s)

Type	Label	Description
Statement
4.7.9  Subwords and concatenations
Theorem	pfxcctswrd 11201	The concatenation of the prefix of a word and the rest of the word yields the word itself. (Contributed by AV, 21-Oct-2018.) (Revised by AV, 9-May-2020.)
|- ( ( W e. Word V /\ M e. ( 0 ... (♯ ` W ) ) ) -> ( ( W prefix M ) ++ ( W substr <. M , (♯ ` W ) >. ) ) = W )
Theorem	lenpfxcctswrd 11202	The length of the concatenation of the prefix of a word and the rest of the word is the length of the word. (Contributed by AV, 21-Oct-2018.) (Revised by AV, 9-May-2020.)
|- ( ( W e. Word V /\ M e. ( 0 ... (♯ ` W ) ) ) -> (♯ ` ( ( W prefix M ) ++ ( W substr <. M , (♯ ` W ) >. ) ) ) = (♯ ` W ) )
Theorem	lenrevpfxcctswrd 11203	The length of the concatenation of the rest of a word and the prefix of the word is the length of the word. (Contributed by Alexander van der Vekens, 1-Apr-2018.) (Revised by AV, 9-May-2020.)
|- ( ( W e. Word V /\ M e. ( 0 ... (♯ ` W ) ) ) -> (♯ ` ( ( W substr <. M , (♯ ` W ) >. ) ++ ( W prefix M ) ) ) = (♯ ` W ) )
Theorem	pfxlswccat 11204	Reconstruct a nonempty word from its prefix and last symbol. (Contributed by Alexander van der Vekens, 5-Aug-2018.) (Revised by AV, 9-May-2020.)
|- ( ( W e. Word V /\ W =/= (/) ) -> ( ( W prefix ( (♯ ` W ) - 1 ) ) ++ <" (lastS ` W ) "> ) = W )
Theorem	ccats1pfxeq 11205	The last symbol of a word concatenated with the word with the last symbol removed results in the word itself. (Contributed by Alexander van der Vekens, 24-Oct-2018.) (Revised by AV, 9-May-2020.)
|- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( W = ( U prefix (♯ ` W ) ) -> U = ( W ++ <" (lastS ` U ) "> ) ) )
Theorem	ccats1pfxeqrex 11206*	There exists a symbol such that its concatenation after the prefix obtained by deleting the last symbol of a nonempty word results in the word itself. (Contributed by AV, 5-Oct-2018.) (Revised by AV, 9-May-2020.)
|- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( W = ( U prefix (♯ ` W ) ) -> E. s e. V U = ( W ++ <" s "> ) ) )
Theorem	ccatopth 11207	An opth 4299-like theorem for recovering the two halves of a concatenated word. (Contributed by Mario Carneiro, 1-Oct-2015.) (Proof shortened by AV, 12-Oct-2022.)
|- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` A ) = (♯ ` C ) ) -> ( ( A ++ B ) = ( C ++ D ) <-> ( A = C /\ B = D ) ) )
Theorem	ccatopth2 11208	An opth 4299-like theorem for recovering the two halves of a concatenated word. (Contributed by Mario Carneiro, 1-Oct-2015.)
|- ( ( ( A e. Word X /\ B e. Word X ) /\ ( C e. Word X /\ D e. Word X ) /\ (♯ ` B ) = (♯ ` D ) ) -> ( ( A ++ B ) = ( C ++ D ) <-> ( A = C /\ B = D ) ) )
Theorem	ccatlcan 11209	Concatenation of words is left-cancellative. (Contributed by Mario Carneiro, 2-Oct-2015.)
|- ( ( A e. Word X /\ B e. Word X /\ C e. Word X ) -> ( ( C ++ A ) = ( C ++ B ) <-> A = B ) )
Theorem	ccatrcan 11210	Concatenation of words is right-cancellative. (Contributed by Mario Carneiro, 2-Oct-2015.)
|- ( ( A e. Word X /\ B e. Word X /\ C e. Word X ) -> ( ( A ++ C ) = ( B ++ C ) <-> A = B ) )
Theorem	wrdeqs1cat 11211	Decompose a nonempty word by separating off the first symbol. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.) (Proof shortened by AV, 12-Oct-2022.)
|- ( ( W e. Word A /\ W =/= (/) ) -> W = ( <" ( W ` 0 ) "> ++ ( W substr <. 1 , (♯ ` W ) >. ) ) )
Theorem	cats1un 11212	Express a word with an extra symbol as the union of the word and the new value. (Contributed by Mario Carneiro, 28-Feb-2016.)
|- ( ( A e. Word X /\ B e. X ) -> ( A ++ <" B "> ) = ( A u. { <. (♯ ` A ) , B >. } ) )
Theorem	wrdind 11213*	Perform induction over the structure of a word. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Proof shortened by AV, 12-Oct-2022.)
|- ( x = (/) -> ( ph <-> ps ) )   &    |- ( x = y -> ( ph <-> ch ) )   &    |- ( x = ( y ++ <" z "> ) -> ( ph <-> th ) )   &    |- ( x = A -> ( ph <-> ta ) )   &    |- ps   &    |- ( ( y e. Word B /\ z e. B ) -> ( ch -> th ) )   =>    |- ( A e. Word B -> ta )
Theorem	wrd2ind 11214*	Perform induction over the structure of two words of the same length. (Contributed by AV, 23-Jan-2019.) (Proof shortened by AV, 12-Oct-2022.)
|- ( ( x = (/) /\ w = (/) ) -> ( ph <-> ps ) )   &    |- ( ( x = y /\ w = u ) -> ( ph <-> ch ) )   &    |- ( ( x = ( y ++ <" z "> ) /\ w = ( u ++ <" s "> ) ) -> ( ph <-> th ) )   &    |- ( x = A -> ( rh <-> ta ) )   &    |- ( w = B -> ( ph <-> rh ) )   &    |- ps   &    |- ( ( ( y e. Word X /\ z e. X ) /\ ( u e. Word Y /\ s e. Y ) /\ (♯ ` y ) = (♯ ` u ) ) -> ( ch -> th ) )   =>    |- ( ( A e. Word X /\ B e. Word Y /\ (♯ ` A ) = (♯ ` B ) ) -> ta )
4.7.10  Subwords of concatenations
Theorem	swrdccatfn 11215	The subword of a concatenation as function. (Contributed by Alexander van der Vekens, 27-May-2018.)
|- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( (♯ ` A ) + (♯ ` B ) ) ) ) ) -> ( ( A ++ B ) substr <. M , N >. ) Fn ( 0..^ ( N - M ) ) )
Theorem	swrdccatin1 11216	The subword of a concatenation of two words within the first of the concatenated words. (Contributed by Alexander van der Vekens, 28-Mar-2018.)
|- ( ( A e. Word V /\ B e. Word V ) -> ( ( M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` A ) ) ) -> ( ( A ++ B ) substr <. M , N >. ) = ( A substr <. M , N >. ) ) )
Theorem	pfxccatin12lem4 11217	Lemma 4 for pfxccatin12 11224. (Contributed by Alexander van der Vekens, 30-Mar-2018.) (Revised by Alexander van der Vekens, 23-May-2018.)
|- ( ( L e. NN0 /\ M e. NN0 /\ N e. ZZ ) -> ( ( K e. ( 0..^ ( N - M ) ) /\ -. K e. ( 0..^ ( L - M ) ) ) -> K e. ( ( L - M )..^ ( ( L - M ) + ( N - L ) ) ) ) )
Theorem	pfxccatin12lem2a 11218	Lemma for pfxccatin12lem2 11222. (Contributed by AV, 30-Mar-2018.) (Revised by AV, 27-May-2018.)
|- ( ( M e. ( 0 ... L ) /\ N e. ( L ... X ) ) -> ( ( K e. ( 0..^ ( N - M ) ) /\ -. K e. ( 0..^ ( L - M ) ) ) -> ( K + M ) e. ( L..^ X ) ) )
Theorem	pfxccatin12lem1 11219	Lemma 1 for pfxccatin12 11224. (Contributed by AV, 30-Mar-2018.) (Revised by AV, 9-May-2020.)
|- ( ( M e. ( 0 ... L ) /\ N e. ( L ... X ) ) -> ( ( K e. ( 0..^ ( N - M ) ) /\ -. K e. ( 0..^ ( L - M ) ) ) -> ( K - ( L - M ) ) e. ( 0..^ ( N - L ) ) ) )
Theorem	swrdccatin2 11220	The subword of a concatenation of two words within the second of the concatenated words. (Contributed by Alexander van der Vekens, 28-Mar-2018.) (Revised by Alexander van der Vekens, 27-May-2018.)
|- L = (♯ ` A )   =>    |- ( ( A e. Word V /\ B e. Word V ) -> ( ( M e. ( L ... N ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) -> ( ( A ++ B ) substr <. M , N >. ) = ( B substr <. ( M - L ) , ( N - L ) >. ) ) )
Theorem	pfxccatin12lem2c 11221	Lemma for pfxccatin12lem2 11222 and pfxccatin12lem3 11223. (Contributed by AV, 30-Mar-2018.) (Revised by AV, 27-May-2018.)
|- L = (♯ ` A )   =>    |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( ( A ++ B ) e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... (♯ ` ( A ++ B ) ) ) ) )
Theorem	pfxccatin12lem2 11222	Lemma 2 for pfxccatin12 11224. (Contributed by AV, 30-Mar-2018.) (Revised by AV, 9-May-2020.)
|- L = (♯ ` A )   =>    |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( ( K e. ( 0..^ ( N - M ) ) /\ -. K e. ( 0..^ ( L - M ) ) ) -> ( ( ( A ++ B ) substr <. M , N >. ) ` K ) = ( ( B prefix ( N - L ) ) ` ( K - (♯ ` ( A substr <. M , L >. ) ) ) ) ) )
Theorem	pfxccatin12lem3 11223	Lemma 3 for pfxccatin12 11224. (Contributed by AV, 30-Mar-2018.) (Revised by AV, 27-May-2018.)
|- L = (♯ ` A )   =>    |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) ) -> ( ( K e. ( 0..^ ( N - M ) ) /\ K e. ( 0..^ ( L - M ) ) ) -> ( ( ( A ++ B ) substr <. M , N >. ) ` K ) = ( ( A substr <. M , L >. ) ` K ) ) )
Theorem	pfxccatin12 11224	The subword of a concatenation of two words within both of the concatenated words. (Contributed by Alexander van der Vekens, 5-Apr-2018.) (Revised by AV, 9-May-2020.)
|- L = (♯ ` A )   =>    |- ( ( A e. Word V /\ B e. Word V ) -> ( ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) -> ( ( A ++ B ) substr <. M , N >. ) = ( ( A substr <. M , L >. ) ++ ( B prefix ( N - L ) ) ) ) )
Theorem	pfxccat3 11225	The subword of a concatenation is either a subword of the first concatenated word or a subword of the second concatenated word or a concatenation of a suffix of the first word with a prefix of the second word. (Contributed by Alexander van der Vekens, 30-Mar-2018.) (Revised by AV, 10-May-2020.)
|- L = (♯ ` A )   =>    |- ( ( A e. Word V /\ B e. Word V ) -> ( ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( L + (♯ ` B ) ) ) ) -> ( ( A ++ B ) substr <. M , N >. ) = if ( N <_ L , ( A substr <. M , N >. ) , if ( L <_ M , ( B substr <. ( M - L ) , ( N - L ) >. ) , ( ( A substr <. M , L >. ) ++ ( B prefix ( N - L ) ) ) ) ) ) )
Theorem	swrdccat 11226	The subword of a concatenation of two words as concatenation of subwords of the two concatenated words. (Contributed by Alexander van der Vekens, 29-May-2018.)
|- L = (♯ ` A )   =>    |- ( ( A e. Word V /\ B e. Word V ) -> ( ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( L + (♯ ` B ) ) ) ) -> ( ( A ++ B ) substr <. M , N >. ) = ( ( A substr <. M , if ( N <_ L , N , L ) >. ) ++ ( B substr <. if ( 0 <_ ( M - L ) , ( M - L ) , 0 ) , ( N - L ) >. ) ) ) )
Theorem	pfxccatpfx1 11227	A prefix of a concatenation being a prefix of the first concatenated word. (Contributed by AV, 10-May-2020.)
|- L = (♯ ` A )   =>    |- ( ( A e. Word V /\ B e. Word V /\ N e. ( 0 ... L ) ) -> ( ( A ++ B ) prefix N ) = ( A prefix N ) )
Theorem	pfxccatpfx2 11228	A prefix of a concatenation of two words being the first word concatenated with a prefix of the second word. (Contributed by AV, 10-May-2020.)
|- L = (♯ ` A )   &    |- M = (♯ ` B )   =>    |- ( ( A e. Word V /\ B e. Word V /\ N e. ( ( L + 1 ) ... ( L + M ) ) ) -> ( ( A ++ B ) prefix N ) = ( A ++ ( B prefix ( N - L ) ) ) )
Theorem	pfxccat3a 11229	A prefix of a concatenation is either a prefix of the first concatenated word or a concatenation of the first word with a prefix of the second word. (Contributed by Alexander van der Vekens, 31-Mar-2018.) (Revised by AV, 10-May-2020.)
|- L = (♯ ` A )   &    |- M = (♯ ` B )   =>    |- ( ( A e. Word V /\ B e. Word V ) -> ( N e. ( 0 ... ( L + M ) ) -> ( ( A ++ B ) prefix N ) = if ( N <_ L , ( A prefix N ) , ( A ++ ( B prefix ( N - L ) ) ) ) ) )
Theorem	swrdccat3blem 11230	Lemma for swrdccat3b 11231. (Contributed by AV, 30-May-2018.)
|- L = (♯ ` A )   =>    |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) /\ ( L + (♯ ` B ) ) <_ L ) -> if ( L <_ M , ( B substr <. ( M - L ) , (♯ ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( A substr <. M , ( L + (♯ ` B ) ) >. ) )
Theorem	swrdccat3b 11231	A suffix of a concatenation is either a suffix of the second concatenated word or a concatenation of a suffix of the first word with the second word. (Contributed by Alexander van der Vekens, 31-Mar-2018.) (Revised by Alexander van der Vekens, 30-May-2018.) (Proof shortened by AV, 14-Oct-2022.)
|- L = (♯ ` A )   =>    |- ( ( A e. Word V /\ B e. Word V ) -> ( M e. ( 0 ... ( L + (♯ ` B ) ) ) -> ( ( A ++ B ) substr <. M , ( L + (♯ ` B ) ) >. ) = if ( L <_ M , ( B substr <. ( M - L ) , (♯ ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) ) )
Theorem	pfxccatid 11232	A prefix of a concatenation of length of the first concatenated word is the first word itself. (Contributed by Alexander van der Vekens, 20-Sep-2018.) (Revised by AV, 10-May-2020.)
|- ( ( A e. Word V /\ B e. Word V /\ N = (♯ ` A ) ) -> ( ( A ++ B ) prefix N ) = A )
Theorem	ccats1pfxeqbi 11233	A word is a prefix of a word with length greater by 1 than the first word iff the second word is the first word concatenated with the last symbol of the second word. (Contributed by AV, 24-Oct-2018.) (Revised by AV, 10-May-2020.)
|- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( W = ( U prefix (♯ ` W ) ) <-> U = ( W ++ <" (lastS ` U ) "> ) ) )
Theorem	swrdccatin1d 11234	The subword of a concatenation of two words within the first of the concatenated words. (Contributed by AV, 31-May-2018.) (Revised by Mario Carneiro/AV, 21-Oct-2018.)
|- ( ph -> (♯ ` A ) = L )   &    |- ( ph -> ( A e. Word V /\ B e. Word V ) )   &    |- ( ph -> M e. ( 0 ... N ) )   &    |- ( ph -> N e. ( 0 ... L ) )   =>    |- ( ph -> ( ( A ++ B ) substr <. M , N >. ) = ( A substr <. M , N >. ) )
Theorem	swrdccatin2d 11235	The subword of a concatenation of two words within the second of the concatenated words. (Contributed by AV, 31-May-2018.) (Revised by Mario Carneiro/AV, 21-Oct-2018.)
|- ( ph -> (♯ ` A ) = L )   &    |- ( ph -> ( A e. Word V /\ B e. Word V ) )   &    |- ( ph -> M e. ( L ... N ) )   &    |- ( ph -> N e. ( L ... ( L + (♯ ` B ) ) ) )   =>    |- ( ph -> ( ( A ++ B ) substr <. M , N >. ) = ( B substr <. ( M - L ) , ( N - L ) >. ) )
Theorem	pfxccatin12d 11236	The subword of a concatenation of two words within both of the concatenated words. (Contributed by AV, 31-May-2018.) (Revised by AV, 10-May-2020.)
|- ( ph -> (♯ ` A ) = L )   &    |- ( ph -> ( A e. Word V /\ B e. Word V ) )   &    |- ( ph -> M e. ( 0 ... L ) )   &    |- ( ph -> N e. ( L ... ( L + (♯ ` B ) ) ) )   =>    |- ( ph -> ( ( A ++ B ) substr <. M , N >. ) = ( ( A substr <. M , L >. ) ++ ( B prefix ( N - L ) ) ) )
Theorem	reuccatpfxs1lem 11237*	Lemma for reuccatpfxs1 11238. (Contributed by Alexander van der Vekens, 5-Oct-2018.) (Revised by AV, 9-May-2020.)
|- ( ( ( W e. Word V /\ U e. X ) /\ A. s e. V ( ( W ++ <" s "> ) e. X -> S = s ) /\ A. x e. X ( x e. Word V /\ (♯ ` x ) = ( (♯ ` W ) + 1 ) ) ) -> ( W = ( U prefix (♯ ` W ) ) -> U = ( W ++ <" S "> ) ) )
Theorem	reuccatpfxs1 11238*	There is a unique word having the length of a given word increased by 1 with the given word as prefix if there is a unique symbol which extends the given word. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 21-Jan-2022.) (Revised by AV, 13-Oct-2022.)
|- F/_ v X   =>    |- ( ( W e. Word V /\ A. x e. X ( x e. Word V /\ (♯ ` x ) = ( (♯ ` W ) + 1 ) ) ) -> ( E! v e. V ( W ++ <" v "> ) e. X -> E! x e. X W = ( x prefix (♯ ` W ) ) ) )
Theorem	reuccatpfxs1v 11239*	There is a unique word having the length of a given word increased by 1 with the given word as prefix if there is a unique symbol which extends the given word. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 21-Jan-2022.) (Revised by AV, 10-May-2022.) (Proof shortened by AV, 13-Oct-2022.)
|- ( ( W e. Word V /\ A. x e. X ( x e. Word V /\ (♯ ` x ) = ( (♯ ` W ) + 1 ) ) ) -> ( E! v e. V ( W ++ <" v "> ) e. X -> E! x e. X W = ( x prefix (♯ ` W ) ) ) )
4.8  Elementary real and complex functions
4.8.1  The "shift" operation
Syntax	cshi 11240	Extend class notation with function shifter.
class shift
Definition	df-shft 11241*	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 11251 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 11242*	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 11243*	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 11244*	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 11245	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 11246	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 11247*	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 11248*	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 11249	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 11250*	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 11251	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 11252	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 11253	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 11254	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 11255	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 11256*	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 11257	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 11258	Identity law for the shift operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
|- F e. _V   =>    |- ( F shift 0 ) = ( F |` CC )
Theorem	shftidt 11259	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 11260	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 11261	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 11262	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 11263	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 11264*	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 11265	Extend class notation to include complex conjugate function.
class *
Syntax	cre 11266	Extend class notation to include real part of a complex number.
class Re
Syntax	cim 11267	Extend class notation to include imaginary part of a complex number.
class Im
Definition	df-cj 11268*	Define the complex conjugate function. See cjcli 11339 for its closure and cjval 11271 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 11269	Define a function whose value is the real part of a complex number. See reval 11275 for its value, recli 11337 for its closure, and replim 11285 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
|- Re = ( x e. CC |-> ( ( x + ( * ` x ) ) / 2 ) )
Definition	df-im 11270	Define a function whose value is the imaginary part of a complex number. See imval 11276 for its value, imcli 11338 for its closure, and replim 11285 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
|- Im = ( x e. CC |-> ( Re ` ( x / _i ) ) )
Theorem	cjval 11271*	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 11272	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 11273	Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)
|- * : CC --> CC
Theorem	cjcl 11274	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 11275	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 11276	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 11277	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 11278	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 11279	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 11280	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 11281	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 11282	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 11283	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 11284	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 11285	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 11286	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 11287	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 11288	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 11289	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 11290	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 11291	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 11292	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 11293	Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014.)
|- ( A e. CC -> ( Re ` ( * ` A ) ) = ( Re ` A ) )
Theorem	reneg 11294	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 11295	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 11296	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 11297	Lemma for remul 11298, immul 11305, and cjmul 11311. (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 11298	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 11299	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 11300	Real part of a division. Related to remul2 11299. (Contributed by Jim Kingdon, 14-Jun-2020.)
|- ( ( A e. CC /\ B e. RR /\ B # 0 ) -> ( Re ` ( A / B ) ) = ( ( Re ` A ) / B ) )