P. 375 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37401-37500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mptelpm 37401*	A function in maps-to notation is a partial map . (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- ( ( ph /\ x e. A ) -> B e. C )   &    |- ( ph -> A C_ D )   &    |- ( ph -> C e. V )   &    |- ( ph -> D e. W )   =>    |- ( ph -> ( x e. A |-> B ) e. ( C ^pm D ) )
Theorem	f1oeq3d 37402	Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> A = B )   =>    |- ( ph -> ( F : C -1-1-onto-> A <-> F : C -1-1-onto-> B ) )
Theorem	fimass 37403	The image of a class is a subset of its codomain. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( F : A --> B -> ( F " X ) C_ B )
Theorem	rnmptpr 37404*	Range of a function defined on an unordered pair. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- F = ( x e. { A , B } |-> C )   &    |- ( x = A -> C = D )   &    |- ( x = B -> C = E )   =>    |- ( ph -> ran F = { D , E } )
Theorem	ffnd 37405	A mapping is a function. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> F : A --> B )   =>    |- ( ph -> F Fn A )
Theorem	resmpti 37406*	Restriction of the mapping operation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- B C_ A   =>    |- ( ( x e. A |-> C ) |` B ) = ( x e. B |-> C )
Theorem	founiiun 37407*	Union expressed as an indexed union, when a map onto is given. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( F : A -onto-> B -> U. B = U_ x e. A ( F ` x ) )
Theorem	f1oeq2d 37408	Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> A = B )   =>    |- ( ph -> ( F : A -1-1-onto-> C <-> F : B -1-1-onto-> C ) )
Theorem	rnresun 37409	Distribution law for range of a restriction over a union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ran ( F |` ( A u. B ) ) = ( ran ( F |` A ) u. ran ( F |` B ) )
Theorem	f1oeq1d 37410	Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> F = G )   =>    |- ( ph -> ( F : A -1-1-onto-> B <-> G : A -1-1-onto-> B ) )
Theorem	dffo3f 37411*	An onto mapping expressed in terms of function values. As dffo3 6052 but with less disjoint vars constraints. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- F/_ x F   =>    |- ( F : A -onto-> B <-> ( F : A --> B /\ A. y e. B E. x e. A y = ( F ` x ) ) )
Theorem	rnresss 37412	The range of a restriction is a subset of the whole range. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ran ( A |` B ) C_ ran A
Theorem	mpteq1i 37413*	An equality theorem for the maps to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- A = B   =>    |- ( x e. A |-> C ) = ( x e. B |-> C )
Theorem	elrnmptd 37414*	The range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- F = ( x e. A |-> B )   &    |- ( ph -> E. x e. A C = B )   &    |- ( ph -> C e. V )   =>    |- ( ph -> C e. ran F )
Theorem	elrnmptf 37415	The range of a function in maps-to notation. Same as elrnmpt 5100, but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- F/_ x C   &    |- F = ( x e. A |-> B )   =>    |- ( C e. V -> ( C e. ran F <-> E. x e. A C = B ) )
Theorem	mptss 37416*	Sufficient condition for inclusion in map-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( A C_ B -> ( x e. A |-> C ) C_ ( x e. B |-> C ) )
Theorem	rnmptssrn 37417*	Inclusion relation for two ranges expressed in map-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ( ph /\ x e. A ) -> B e. V )   &    |- ( ( ph /\ x e. A ) -> E. y e. C B = D )   =>    |- ( ph -> ran ( x e. A |-> B ) C_ ran ( y e. C |-> D ) )
Theorem	disjf1 37418*	A 1 to 1 mapping built from disjoint, nonempty sets . (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- F/ x ph   &    |- F = ( x e. A |-> B )   &    |- ( ( ph /\ x e. A ) -> B e. V )   &    |- ( ( ph /\ x e. A ) -> B =/= (/) )   &    |- ( ph -> Disj x e. A B )   =>    |- ( ph -> F : A -1-1-> V )
Theorem	rnsnf 37419	The range of a function whose domain is a singleton. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> A e. V )   &    |- ( ph -> F : { A } --> B )   =>    |- ( ph -> ran F = { ( F ` A ) } )
Theorem	wessf1ornlem 37420*	Given a function F on a well ordered domain A there exists a subset of A such that F restricted to such subset is injective and onto the range of F (without using the axiom of choice) (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> F Fn A )   &    |- ( ph -> A e. V )   &    |- ( ph -> R We A )   &    |- G = ( y e. ran F |-> ( iota_ x e. ( `' F " { y } ) A. z e. ( `' F " { y } ) -. z R x ) )   =>    |- ( ph -> E. x e. ~P A ( F |` x ) : x -1-1-onto-> ran F )
Theorem	wessf1orn 37421*	Given a function F on a well ordered domain A there exists a subset of A such that F restricted to such subset is injective and onto the range of F (without using the axiom of choice) (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> F Fn A )   &    |- ( ph -> A e. V )   &    |- ( ph -> R We A )   =>    |- ( ph -> E. x e. ~P A ( F |` x ) : x -1-1-onto-> ran F )
Theorem	foelrnf 37422*	Property of a surjective function. As foelrn 6056 but with less disjoint vars constraints. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- F/_ x F   =>    |- ( ( F : A -onto-> B /\ C e. B ) -> E. x e. A C = ( F ` x ) )
Theorem	nelrnres 37423	If A is not in the range, it is not in the range of any restriction. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( -. A e. ran B -> -. A e. ran ( B |` C ) )
Theorem	disjrnmpt2 37424*	Disjointness of the range of a function in map-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- F = ( x e. A |-> B )   =>    |- (Disj x e. A B -> Disj y e. ran F y )
Theorem	elrnmpt1sf 37425*	Elementhood in an image set. Same as elrnmpt1s 5101, but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- F/_ x C   &    |- F = ( x e. A |-> B )   &    |- ( x = D -> B = C )   =>    |- ( ( D e. A /\ C e. V ) -> C e. ran F )
Theorem	founiiun0 37426*	Union expressed as an indexed union, when a map onto is given. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( F : A -onto-> ( B u. { (/) } ) -> U. B = U_ x e. A ( F ` x ) )
Theorem	disjf1o 37427*	A bijection built from disjoint sets. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- F/ x ph   &    |- F = ( x e. A |-> B )   &    |- ( ( ph /\ x e. A ) -> B e. V )   &    |- ( ph -> Disj x e. A B )   &    |- C = { x e. A | B =/= (/) }   &    |- D = ( ran F \ { (/) } )   =>    |- ( ph -> ( F |` C ) : C -1-1-onto-> D )
Theorem	fompt 37428*	Express being onto for a mapping operation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- F = ( x e. A |-> C )   =>    |- ( F : A -onto-> B <-> ( A. x e. A C e. B /\ A. y e. B E. x e. A y = C ) )
Theorem	disjinfi 37429*	Only a finite number of disjoint sets can have a non empty intersection with a finite set C (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ( ph /\ x e. A ) -> B e. V )   &    |- ( ph -> Disj x e. A B )   &    |- ( ph -> C e. Fin )   =>    |- ( ph -> { x e. A | ( B i^i C ) =/= (/) } e. Fin )
Theorem	fvovco 37430	Value of the composition of an operator, with a given function. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> F : X --> ( V X. W ) )   &    |- ( ph -> Y e. X )   =>    |- ( ph -> ( ( O o. F ) ` Y ) = ( ( 1st ` ( F ` Y ) ) O ( 2nd ` ( F ` Y ) ) ) )
Theorem	ssnnf1octb 37431*	There exists a bijection between a subset of NN and a given nonempty countable set. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ( A ~<_ om /\ A =/= (/) ) -> E. f ( dom f C_ NN /\ f : dom f -1-1-onto-> A ) )
Theorem	mapdm0 37432	The empty set is the only map with empty domain. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( A e. V -> ( A ^m (/) ) = { (/) } )
Theorem	nnf1oxpnn 37433	There is a bijection between the set of positive integers and the Cartesian product of the set of positive integers with itself. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- E. f f : NN -1-1-onto-> ( NN X. NN )
Theorem	rnmptssd 37434*	The range of an operation given by the "maps to" notation as a subset. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- F/ x ph   &    |- F = ( x e. A |-> B )   &    |- ( ( ph /\ x e. A ) -> B e. C )   =>    |- ( ph -> ran F C_ C )
Theorem	projf1o 37435*	A biijection from a set to a projection in a two dimensional space. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> A e. V )   &    |- F = ( x e. B |-> <. A , x >. )   =>    |- ( ph -> F : B -1-1-onto-> ( { A } X. B ) )
Theorem	fvmap 37436	Function value for a member of a set exponentiation. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( ph -> F e. ( A ^m B ) )   &    |- ( ph -> C e. B )   =>    |- ( ph -> ( F ` C ) e. A )
21.30.3  Ordering on real numbers - Real and complex numbers basic operations
Theorem	sub2times 37437	Subtracting from a number, twice the number itself, gives negative the number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( A e. CC -> ( A - ( 2 x. A ) ) = -u A )
Theorem	xrltled 37438	'Less than' implies 'less than or equal to', for extended reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. RR* )   &    |- ( ph -> B e. RR* )   &    |- ( ph -> A < B )   =>    |- ( ph -> A <_ B )
Theorem	abssubrp 37439	The distance of two distinct complex number is a strictly positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( A e. CC /\ B e. CC /\ A =/= B ) -> ( abs ` ( A - B ) ) e. RR+ )
Theorem	elfzfzo 37440	Relationship between membership in a half open finite set of sequential integers and membership in a finite set of sequential intergers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( A e. ( M..^ N ) <-> ( A e. ( M ... N ) /\ A < N ) )
Theorem	oddfl 37441	Odd number representation by using the floor function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( K e. ZZ /\ ( K mod 2 ) =/= 0 ) -> K = ( ( 2 x. ( |_ ` ( K / 2 ) ) ) + 1 ) )
Theorem	abscosbd 37442	Bound for the absolute value of the cosine of a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( A e. RR -> ( abs ` ( cos ` A ) ) <_ 1 )
Theorem	mul13d 37443	Commutative/associative law that swaps the first and the third factor in a triple product. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( A x. ( B x. C ) ) = ( C x. ( B x. A ) ) )
Theorem	negpilt0 37444	Negative pi is negative. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- -u pi < 0
Theorem	dstregt0 37445*	A complex number A that is not real, has a distance from the reals that is strictly larger than 0. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. ( CC \ RR ) )   =>    |- ( ph -> E. x e. RR+ A. y e. RR x < ( abs ` ( A - y ) ) )
Theorem	subadd4b 37446	Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> D e. CC )   =>    |- ( ph -> ( ( A - B ) + ( C - D ) ) = ( ( A - D ) + ( C - B ) ) )
Theorem	xrlttri5d 37447	Not equal and not larger implies smaller. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. RR* )   &    |- ( ph -> B e. RR* )   &    |- ( ph -> A =/= B )   &    |- ( ph -> -. B < A )   =>    |- ( ph -> A < B )
Theorem	neglt 37448	The negative of a positive number is less than the number itself. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( A e. RR+ -> -u A < A )
Theorem	zltlesub 37449	If an integer N is smaller or equal to a real, and we subtract a quantity smaller than 1, then N is smaller or equal to the result. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> N e. ZZ )   &    |- ( ph -> A e. RR )   &    |- ( ph -> N <_ A )   &    |- ( ph -> B e. RR )   &    |- ( ph -> B < 1 )   &    |- ( ph -> ( A - B ) e. ZZ )   =>    |- ( ph -> N <_ ( A - B ) )
Theorem	divlt0gt0d 37450	The ratio of a negative numerator and a positive denominator is negative. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR+ )   &    |- ( ph -> A < 0 )   =>    |- ( ph -> ( A / B ) < 0 )
Theorem	3rp 37451	3 is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- 3 e. RR+
Theorem	leimnltdOLD 37452	'Less than or equal to' implies 'not less than'. (Contributed by Glauco Siliprandi, 11-Dec-2019.) Obsolete as of 18-Jul-2020. Use lensymd 9793 instead, which is the same. (New usage is discouraged.) (Proof modification is discouraged.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A <_ B )   =>    |- ( ph -> -. B < A )
Theorem	lt2addmuld 37453	If two real numbers are less than a third real number, the sum of the two real numbers is less than twice the third real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> A < C )   &    |- ( ph -> B < C )   =>    |- ( ph -> ( A + B ) < ( 2 x. C ) )
Theorem	subsub23d 37454	Swap subtrahend and result of subtraction. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( ( A - B ) = C <-> ( A - C ) = B ) )
Theorem	2timesgt 37455	Double of a positive real is larger than the real itself. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( A e. RR+ -> A < ( 2 x. A ) )
Theorem	reopn 37456	The reals are open with respect to the standard topology. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- RR e. ( topGen ` ran (,) )
Theorem	elfzop1le2 37457	A member in a half-open integer interval plus 1 is less or equal than the upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( K e. ( M..^ N ) -> ( K + 1 ) <_ N )
Theorem	sub31 37458	Swap the first and third terms in a double subtraction. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A - ( B - C ) ) = ( C - ( B - A ) ) )
Theorem	nnne1ge2 37459	A positive integer which is not 1 is greater than or equal to 2. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( N e. NN /\ N =/= 1 ) -> 2 <_ N )
Theorem	lefldiveq 37460	A closed enough, smaller real C has the same floor of A when both are divided by B. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR+ )   &    |- ( ph -> C e. ( ( A - ( A mod B ) ) [,] A ) )   =>    |- ( ph -> ( |_ ` ( A / B ) ) = ( |_ ` ( C / B ) ) )
Theorem	negsubdi3d 37461	Distribution of negative over subtraction. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> -u ( A - B ) = ( -u A - -u B ) )
Theorem	ltaddneg 37462	Adding a negative number to another number decreases it. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( A e. RR /\ B e. RR ) -> ( A < 0 <-> ( B + A ) < B ) )
Theorem	ltdiv2dd 37463	Division of a positive number by both sides of 'less than'. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. RR+ )   &    |- ( ph -> C e. RR+ )   &    |- ( ph -> A < B )   =>    |- ( ph -> ( C / B ) < ( C / A ) )
Theorem	adddirp1d 37464	Distributive law, plus 1 version. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( ( A + 1 ) x. B ) = ( ( A x. B ) + B ) )
Theorem	absnpncand 37465	Triangular inequality, combined with cancellation law for subtraction. (Contributed by Glauco Siliprandi, 11-Dec-2019.) TODO (NM): usage (2 times) should be replaced by abs3difd 13521, and absnpncand 37465 should be deleted afterwards.
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( abs ` ( A - C ) ) <_ ( ( abs ` ( A - B ) ) + ( abs ` ( B - C ) ) ) )
Theorem	abssinbd 37466	Bound for the absolute value of the sine of a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( A e. RR -> ( abs ` ( sin ` A ) ) <_ 1 )
Theorem	halffl 37467	Floor of ( 1 / 2 ). (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( |_ ` ( 1 / 2 ) ) = 0
Theorem	monoords 37468*	Ordering relation for a strictly monotonic sequence, increasing case. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. RR )   &    |- ( ( ph /\ k e. ( M..^ N ) ) -> ( F ` k ) < ( F ` ( k + 1 ) ) )   &    |- ( ph -> I e. ( M ... N ) )   &    |- ( ph -> J e. ( M ... N ) )   &    |- ( ph -> I < J )   =>    |- ( ph -> ( F ` I ) < ( F ` J ) )
Theorem	hashssle 37469	The size of a subset of a finite set is less than the size of the containing set. (Contributed by Glauco Siliprandi, 11-Dec-2019.) TODO (NM): usage (2 times) should be replaced by hashss 12592, and hashssle 37469 should be deleted afterwards.
|- ( ( A e. Fin /\ B C_ A ) -> ( # ` B ) <_ ( # ` A ) )
Theorem	flltnz 37470	If A is not an integer, the floor of A is less than A (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( A e. RR /\ -. A e. ZZ ) -> ( |_ ` A ) < A )
Theorem	lttri5d 37471	Not equal and not larger implies smaller. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A =/= B )   &    |- ( ph -> -. B < A )   =>    |- ( ph -> A < B )
Theorem	fzisoeu 37472*	A finite ordered set has a unique order isomorphism to a generic finite sequence of integers. This theorem generalizes fz1iso 12629 for the base index and also states the uniqueness condition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> H e. Fin )   &    |- ( ph -> < Or H )   &    |- ( ph -> M e. ZZ )   &    |- N = ( ( # ` H ) + ( M - 1 ) )   =>    |- ( ph -> E! f f Isom < , < ( ( M ... N ) , H ) )
Theorem	lt3addmuld 37473	If three real numbers are less than a fourth real number, the sum of the three real numbers is less than three times the third real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> D e. RR )   &    |- ( ph -> A < D )   &    |- ( ph -> B < D )   &    |- ( ph -> C < D )   =>    |- ( ph -> ( ( A + B ) + C ) < ( 3 x. D ) )
Theorem	absnpncan2d 37474	Triangular inequality, combined with cancellation law for subtraction (applied twice). (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> D e. CC )   =>    |- ( ph -> ( abs ` ( A - D ) ) <_ ( ( ( abs ` ( A - B ) ) + ( abs ` ( B - C ) ) ) + ( abs ` ( C - D ) ) ) )
Theorem	fperiodmullem 37475*	A function with period T is also periodic with period nonnegative multiple of T. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> F : RR --> CC )   &    |- ( ph -> T e. RR )   &    |- ( ph -> N e. NN0 )   &    |- ( ph -> X e. RR )   &    |- ( ( ph /\ x e. RR ) -> ( F ` ( x + T ) ) = ( F ` x ) )   =>    |- ( ph -> ( F ` ( X + ( N x. T ) ) ) = ( F ` X ) )
Theorem	fperiodmul 37476*	A function with period T is also periodic with period multiple of T. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> F : RR --> CC )   &    |- ( ph -> T e. RR )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> X e. RR )   &    |- ( ( ph /\ x e. RR ) -> ( F ` ( x + T ) ) = ( F ` x ) )   =>    |- ( ph -> ( F ` ( X + ( N x. T ) ) ) = ( F ` X ) )
Theorem	upbdrech 37477*	Choice of an upper bound for a non empty bunded set (image set version). (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A =/= (/) )   &    |- ( ( ph /\ x e. A ) -> B e. RR )   &    |- ( ph -> E. y e. RR A. x e. A B <_ y )   &    |- C = sup ( { z | E. x e. A z = B } , RR , < )   =>    |- ( ph -> ( C e. RR /\ A. x e. A B <_ C ) )
Theorem	lt4addmuld 37478	If four real numbers are less than a fifth real number, the sum of the four real numbers is less than four times the fifth real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> D e. RR )   &    |- ( ph -> E e. RR )   &    |- ( ph -> A < E )   &    |- ( ph -> B < E )   &    |- ( ph -> C < E )   &    |- ( ph -> D < E )   =>    |- ( ph -> ( ( ( A + B ) + C ) + D ) < ( 4 x. E ) )
Theorem	absnpncan3d 37479	Triangular inequality, combined with cancellation law for subtraction (applied three times). (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> D e. CC )   &    |- ( ph -> E e. CC )   =>    |- ( ph -> ( abs ` ( A - E ) ) <_ ( ( ( ( abs ` ( A - B ) ) + ( abs ` ( B - C ) ) ) + ( abs ` ( C - D ) ) ) + ( abs ` ( D - E ) ) ) )
Theorem	upbdrech2 37480*	Choice of an upper bound for a possibly empty bunded set (image set version) (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( ph /\ x e. A ) -> B e. RR )   &    |- ( ph -> E. y e. RR A. x e. A B <_ y )   &    |- C = if ( A = (/) , 0 , sup ( { z | E. x e. A z = B } , RR , < ) )   =>    |- ( ph -> ( C e. RR /\ A. x e. A B <_ C ) )
Theorem	ssfiunibd 37481*	A finite union of bounded sets is bounded. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ z e. U. A ) -> B e. RR )   &    |- ( ( ph /\ x e. A ) -> E. y e. RR A. z e. x B <_ y )   &    |- ( ph -> C C_ U. A )   =>    |- ( ph -> E. w e. RR A. z e. C B <_ w )
Theorem	fz1ssfz0 37482	Subset relationship for finite sets of sequential integers. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- ( 1 ... N ) C_ ( 0 ... N )
Theorem	subdir2d 37483	Distribution of multiplication over subtraction. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( C x. ( A - B ) ) = ( ( C x. A ) - ( C x. B ) ) )
Theorem	fzdifsuc2 37484	Remove a successor from the end of a finite set of sequential integers. Similar to fzdifsuc 11862, but with a weaker condition. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- ( N e. ( ZZ>= ` ( M - 1 ) ) -> ( M ... N ) = ( ( M ... ( N + 1 ) ) \ { ( N + 1 ) } ) )
Theorem	fzsscn 37485	A finite sequence of integers is a set of complex numbers. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- ( M ... N ) C_ CC
Theorem	divcan8d 37486	A cancellation law for division. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> A =/= 0 )   &    |- ( ph -> B =/= 0 )   =>    |- ( ph -> ( B / ( A x. B ) ) = ( 1 / A ) )
Theorem	faccld 37487	Closure of the factorial function. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- ( ph -> N e. NN0 )   =>    |- ( ph -> ( ! ` N ) e. NN )
Theorem	dmmcand 37488	Cancellation law for division and multiplication. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> B =/= 0 )   =>    |- ( ph -> ( ( A / B ) x. ( B x. C ) ) = ( A x. C ) )
Theorem	fzssre 37489	A finite sequence of integers is a set of real numbers. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- ( M ... N ) C_ RR
Theorem	elfzelzd 37490	A member of a finite set of sequential integer is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- ( ph -> K e. ( M ... N ) )   =>    |- ( ph -> K e. ZZ )
Theorem	bccld 37491	A binomial coefficient, in its extended domain, is a nonnegative integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- ( ph -> N e. NN0 )   &    |- ( ph -> K e. ZZ )   =>    |- ( ph -> ( N _C K ) e. NN0 )
Theorem	leadd12dd 37492	Addition to both sides of 'less than or equal to'. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> D e. RR )   &    |- ( ph -> A <_ C )   &    |- ( ph -> B <_ D )   =>    |- ( ph -> ( A + B ) <_ ( C + D ) )
Theorem	fzssnn0 37493	A finite set of sequential integers that is a subset of NN0. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- ( 0 ... N ) C_ NN0
Theorem	xreqle 37494	Equality implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ( A e. RR* /\ A = B ) -> A <_ B )
Theorem	xaddid2d 37495	0 is a left identity for extended real addition. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> A e. RR* )   =>    |- ( ph -> ( 0 +e A ) = A )
Theorem	xadd0ge 37496	A number is less than or equal to itself plus a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> A e. RR* )   &    |- ( ph -> B e. ( 0 [,] +oo ) )   =>    |- ( ph -> A <_ ( A +e B ) )
Theorem	xaddid1d 37497	0 is a right identity for extended real addition. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> A e. RR* )   =>    |- ( ph -> ( A +e 0 ) = A )
Theorem	elfzolem1 37498	A member in a half-open integer interval is less than or equal to the upper bound minus 1 . (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( K e. ( M..^ N ) -> K <_ ( N - 1 ) )
Theorem	xrgtned 37499	'Greater than' implies not equal. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> A e. RR* )   &    |- ( ph -> B e. RR* )   &    |- ( ph -> A < B )   =>    |- ( ph -> B =/= A )
Theorem	xrleneltd 37500	'Less than or equal to' and 'not equals' implies 'less than', for extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> A e. RR* )   &    |- ( ph -> B e. RR* )   &    |- ( ph -> A <_ B )   &    |- ( ph -> A =/= B )   =>    |- ( ph -> A < B )