P. 108 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10701-10800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fihasheqf1oi 10701	The size of two finite sets is equal if there is a bijection mapping one of the sets onto the other. (Contributed by Jim Kingdon, 21-Feb-2022.)
|- ( ( A e. Fin /\ F : A -1-1-onto-> B ) -> (♯ ` A ) = (♯ ` B ) )
Theorem	fihashf1rn 10702	The size of a finite set which is a one-to-one function is equal to the size of the function's range. (Contributed by Jim Kingdon, 21-Feb-2022.)
|- ( ( A e. Fin /\ F : A -1-1-> B ) -> (♯ ` F ) = (♯ ` ran F ) )
Theorem	fihasheqf1od 10703	The size of two finite sets is equal if there is a bijection mapping one of the sets onto the other. (Contributed by Jim Kingdon, 21-Feb-2022.)
|- ( ph -> A e. Fin )   &    |- ( ph -> F : A -1-1-onto-> B )   =>    |- ( ph -> (♯ ` A ) = (♯ ` B ) )
Theorem	fz1eqb 10704	Two possibly-empty 1-based finite sets of sequential integers are equal iff their endpoints are equal. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 29-Mar-2014.)
|- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( 1 ... M ) = ( 1 ... N ) <-> M = N ) )
Theorem	filtinf 10705	The size of an infinite set is greater than the size of a finite set. (Contributed by Jim Kingdon, 21-Feb-2022.)
|- ( ( A e. Fin /\ om ~<_ B ) -> (♯ ` A ) < (♯ ` B ) )
Theorem	isfinite4im 10706	A finite set is equinumerous to the range of integers from one up to the hash value of the set. (Contributed by Jim Kingdon, 22-Feb-2022.)
|- ( A e. Fin -> ( 1 ... (♯ ` A ) ) ~~ A )
Theorem	fihasheq0 10707	Two ways of saying a finite set is empty. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 27-Jul-2014.) (Intuitionized by Jim Kingdon, 23-Feb-2022.)
|- ( A e. Fin -> ( (♯ ` A ) = 0 <-> A = (/) ) )
Theorem	fihashneq0 10708	Two ways of saying a finite set is not empty. Also, "A is inhabited" would be equivalent by fin0 6851. (Contributed by Alexander van der Vekens, 23-Sep-2018.) (Intuitionized by Jim Kingdon, 23-Feb-2022.)
|- ( A e. Fin -> ( 0 < (♯ ` A ) <-> A =/= (/) ) )
Theorem	hashnncl 10709	Positive natural closure of the hash function. (Contributed by Mario Carneiro, 16-Jan-2015.)
|- ( A e. Fin -> ( (♯ ` A ) e. NN <-> A =/= (/) ) )
Theorem	hash0 10710	The empty set has size zero. (Contributed by Mario Carneiro, 8-Jul-2014.)
|- (♯ ` (/) ) = 0
Theorem	hashsng 10711	The size of a singleton. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 13-Feb-2013.)
|- ( A e. V -> (♯ ` { A } ) = 1 )
Theorem	fihashen1 10712	A finite set has size 1 if and only if it is equinumerous to the ordinal 1. (Contributed by AV, 14-Apr-2019.) (Intuitionized by Jim Kingdon, 23-Feb-2022.)
|- ( A e. Fin -> ( (♯ ` A ) = 1 <-> A ~~ 1o ) )
Theorem	fihashfn 10713	A function on a finite set is equinumerous to its domain. (Contributed by Mario Carneiro, 12-Mar-2015.) (Intuitionized by Jim Kingdon, 24-Feb-2022.)
|- ( ( F Fn A /\ A e. Fin ) -> (♯ ` F ) = (♯ ` A ) )
Theorem	fseq1hash 10714	The value of the size function on a finite 1-based sequence. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 12-Mar-2015.)
|- ( ( N e. NN0 /\ F Fn ( 1 ... N ) ) -> (♯ ` F ) = N )
Theorem	omgadd 10715	Mapping ordinal addition to integer addition. (Contributed by Jim Kingdon, 24-Feb-2022.)
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   =>    |- ( ( A e. om /\ B e. om ) -> ( G ` ( A +o B ) ) = ( ( G ` A ) + ( G ` B ) ) )
Theorem	fihashdom 10716	Dominance relation for the size function. (Contributed by Jim Kingdon, 24-Feb-2022.)
|- ( ( A e. Fin /\ B e. Fin ) -> ( (♯ ` A ) <_ (♯ ` B ) <-> A ~<_ B ) )
Theorem	hashunlem 10717	Lemma for hashun 10718. Ordinal size of the union. (Contributed by Jim Kingdon, 25-Feb-2022.)
|- ( ph -> A e. Fin )   &    |- ( ph -> B e. Fin )   &    |- ( ph -> ( A i^i B ) = (/) )   &    |- ( ph -> N e. om )   &    |- ( ph -> M e. om )   &    |- ( ph -> A ~~ N )   &    |- ( ph -> B ~~ M )   =>    |- ( ph -> ( A u. B ) ~~ ( N +o M ) )
Theorem	hashun 10718	The size of the union of disjoint finite sets is the sum of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.) (Revised by Mario Carneiro, 15-Sep-2013.)
|- ( ( A e. Fin /\ B e. Fin /\ ( A i^i B ) = (/) ) -> (♯ ` ( A u. B ) ) = ( (♯ ` A ) + (♯ ` B ) ) )
Theorem	1elfz0hash 10719	1 is an element of the finite set of sequential nonnegative integers bounded by the size of a nonempty finite set. (Contributed by AV, 9-May-2020.)
|- ( ( A e. Fin /\ A =/= (/) ) -> 1 e. ( 0 ... (♯ ` A ) ) )
Theorem	hashunsng 10720	The size of the union of a finite set with a disjoint singleton is one more than the size of the set. (Contributed by Paul Chapman, 30-Nov-2012.)
|- ( B e. V -> ( ( A e. Fin /\ -. B e. A ) -> (♯ ` ( A u. { B } ) ) = ( (♯ ` A ) + 1 ) ) )
Theorem	hashprg 10721	The size of an unordered pair. (Contributed by Mario Carneiro, 27-Sep-2013.) (Revised by Mario Carneiro, 5-May-2016.) (Revised by AV, 18-Sep-2021.)
|- ( ( A e. V /\ B e. W ) -> ( A =/= B <-> (♯ ` { A , B } ) = 2 ) )
Theorem	prhash2ex 10722	There is (at least) one set with two different elements: the unordered pair containing 0 and 1. In contrast to pr0hash2ex 10728, numbers are used instead of sets because their representation is shorter (and more comprehensive). (Contributed by AV, 29-Jan-2020.)
|- (♯ ` { 0 , 1 } ) = 2
Theorem	hashp1i 10723	Size of a natural number ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
|- A e. om   &    |- B = suc A   &    |- (♯ ` A ) = M   &    |- ( M + 1 ) = N   =>    |- (♯ ` B ) = N
Theorem	hash1 10724	Size of a natural number ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
|- (♯ ` 1o ) = 1
Theorem	hash2 10725	Size of a natural number ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
|- (♯ ` 2o ) = 2
Theorem	hash3 10726	Size of a natural number ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
|- (♯ ` 3o ) = 3
Theorem	hash4 10727	Size of a natural number ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
|- (♯ ` 4o ) = 4
Theorem	pr0hash2ex 10728	There is (at least) one set with two different elements: the unordered pair containing the empty set and the singleton containing the empty set. (Contributed by AV, 29-Jan-2020.)
|- (♯ ` { (/) , { (/) } } ) = 2
Theorem	fihashss 10729	The size of a subset is less than or equal to the size of its superset. (Contributed by Alexander van der Vekens, 14-Jul-2018.)
|- ( ( A e. Fin /\ B e. Fin /\ B C_ A ) -> (♯ ` B ) <_ (♯ ` A ) )
Theorem	fiprsshashgt1 10730	The size of a superset of a proper unordered pair is greater than 1. (Contributed by AV, 6-Feb-2021.)
|- ( ( ( A e. V /\ B e. W /\ A =/= B ) /\ C e. Fin ) -> ( { A , B } C_ C -> 2 <_ (♯ ` C ) ) )
Theorem	fihashssdif 10731	The size of the difference of a finite set and a finite subset is the set's size minus the subset's. (Contributed by Jim Kingdon, 31-May-2022.)
|- ( ( A e. Fin /\ B e. Fin /\ B C_ A ) -> (♯ ` ( A \ B ) ) = ( (♯ ` A ) - (♯ ` B ) ) )
Theorem	hashdifsn 10732	The size of the difference of a finite set and a singleton subset is the set's size minus 1. (Contributed by Alexander van der Vekens, 6-Jan-2018.)
|- ( ( A e. Fin /\ B e. A ) -> (♯ ` ( A \ { B } ) ) = ( (♯ ` A ) - 1 ) )
Theorem	hashdifpr 10733	The size of the difference of a finite set and a proper ordered pair subset is the set's size minus 2. (Contributed by AV, 16-Dec-2020.)
|- ( ( A e. Fin /\ ( B e. A /\ C e. A /\ B =/= C ) ) -> (♯ ` ( A \ { B , C } ) ) = ( (♯ ` A ) - 2 ) )
Theorem	hashfz 10734	Value of the numeric cardinality of a nonempty integer range. (Contributed by Stefan O'Rear, 12-Sep-2014.) (Proof shortened by Mario Carneiro, 15-Apr-2015.)
|- ( B e. ( ZZ>= ` A ) -> (♯ ` ( A ... B ) ) = ( ( B - A ) + 1 ) )
Theorem	hashfzo 10735	Cardinality of a half-open set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
|- ( B e. ( ZZ>= ` A ) -> (♯ ` ( A..^ B ) ) = ( B - A ) )
Theorem	hashfzo0 10736	Cardinality of a half-open set of integers based at zero. (Contributed by Stefan O'Rear, 15-Aug-2015.)
|- ( B e. NN0 -> (♯ ` ( 0..^ B ) ) = B )
Theorem	hashfzp1 10737	Value of the numeric cardinality of a (possibly empty) integer range. (Contributed by AV, 19-Jun-2021.)
|- ( B e. ( ZZ>= ` A ) -> (♯ ` ( ( A + 1 ) ... B ) ) = ( B - A ) )
Theorem	hashfz0 10738	Value of the numeric cardinality of a nonempty range of nonnegative integers. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
|- ( B e. NN0 -> (♯ ` ( 0 ... B ) ) = ( B + 1 ) )
Theorem	hashxp 10739	The size of the Cartesian product of two finite sets is the product of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.)
|- ( ( A e. Fin /\ B e. Fin ) -> (♯ ` ( A X. B ) ) = ( (♯ ` A ) x. (♯ ` B ) ) )
Theorem	fimaxq 10740*	A finite set of rational numbers has a maximum. (Contributed by Jim Kingdon, 6-Sep-2022.)
|- ( ( A C_ QQ /\ A e. Fin /\ A =/= (/) ) -> E. x e. A A. y e. A y <_ x )
Theorem	fiubm 10741*	Lemma for fiubz 10742 and fiubnn 10743. A general form of those theorems. (Contributed by Jim Kingdon, 29-Oct-2024.)
|- ( ph -> A C_ B )   &    |- ( ph -> B C_ QQ )   &    |- ( ph -> C e. B )   &    |- ( ph -> A e. Fin )   =>    |- ( ph -> E. x e. B A. y e. A y <_ x )
Theorem	fiubz 10742*	A finite set of integers has an upper bound which is an integer. (Contributed by Jim Kingdon, 29-Oct-2024.)
|- ( ( A C_ ZZ /\ A e. Fin ) -> E. x e. ZZ A. y e. A y <_ x )
Theorem	fiubnn 10743*	A finite set of natural numbers has an upper bound which is a a natural number. (Contributed by Jim Kingdon, 29-Oct-2024.)
|- ( ( A C_ NN /\ A e. Fin ) -> E. x e. NN A. y e. A y <_ x )
Theorem	resunimafz0 10744	The union of a restriction by an image over an open range of nonnegative integers and a singleton of an ordered pair is a restriction by an image over an interval of nonnegative integers. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 20-Feb-2021.)
|- ( ph -> Fun I )   &    |- ( ph -> F : ( 0..^ (♯ ` F ) ) --> dom I )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   =>    |- ( ph -> ( I |` ( F " ( 0 ... N ) ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) u. { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } ) )
Theorem	fnfz0hash 10745	The size of a function on a finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 25-Jun-2018.)
|- ( ( N e. NN0 /\ F Fn ( 0 ... N ) ) -> (♯ ` F ) = ( N + 1 ) )
Theorem	ffz0hash 10746	The size of a function on a finite set of sequential nonnegative integers equals the upper bound of the sequence increased by 1. (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Proof shortened by AV, 11-Apr-2021.)
|- ( ( N e. NN0 /\ F : ( 0 ... N ) --> B ) -> (♯ ` F ) = ( N + 1 ) )
Theorem	ffzo0hash 10747	The size of a function on a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 25-Mar-2018.)
|- ( ( N e. NN0 /\ F Fn ( 0..^ N ) ) -> (♯ ` F ) = N )
Theorem	fnfzo0hash 10748	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 10749*	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 10750	Version of isorel 5776 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 10751	Lemma for zfz1iso 10754. 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 10752*	Lemma for zfz1iso 10754. 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 10753*	Lemma for zfz1iso 10754. 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 10754*	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	seq3coll 10755*	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.) (Revised by Jim Kingdon, 9-Apr-2023.)
|- ( ( 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 ) ` ( G ` N ) ) = ( seq 1 ( .+ , H ) ` N ) )
4.7  Elementary real and complex functions
4.7.1  The "shift" operation
Syntax	cshi 10756	Extend class notation with function shifter.
class shift
Definition	df-shft 10757*	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 10767 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 10758*	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 10759*	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 10760*	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 10761	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 10762	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 10763*	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 10764*	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 10765	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 10766*	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 10767	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 10768	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 10769	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 10770	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 10771	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 10772*	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 10773	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 10774	Identity law for the shift operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
|- F e. _V   =>    |- ( F shift 0 ) = ( F |` CC )
Theorem	shftidt 10775	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 10776	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 10777	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 10778	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 10779	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 10780*	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 ) )
4.7.2  Real and imaginary parts; conjugate
Syntax	ccj 10781	Extend class notation to include complex conjugate function.
class *
Syntax	cre 10782	Extend class notation to include real part of a complex number.
class Re
Syntax	cim 10783	Extend class notation to include imaginary part of a complex number.
class Im
Definition	df-cj 10784*	Define the complex conjugate function. See cjcli 10855 for its closure and cjval 10787 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 10785	Define a function whose value is the real part of a complex number. See reval 10791 for its value, recli 10853 for its closure, and replim 10801 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
|- Re = ( x e. CC |-> ( ( x + ( * ` x ) ) / 2 ) )
Definition	df-im 10786	Define a function whose value is the imaginary part of a complex number. See imval 10792 for its value, imcli 10854 for its closure, and replim 10801 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
|- Im = ( x e. CC |-> ( Re ` ( x / _i ) ) )
Theorem	cjval 10787*	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 10788	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 10789	Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)
|- * : CC --> CC
Theorem	cjcl 10790	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 10791	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 10792	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 10793	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 10794	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 10795	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 10796	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 10797	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 10798	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 10799	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 10800	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 )