P. 374 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37301-37400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rexsngf 37301*	Restricted existential quantification over a singleton. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- F/ x ps   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. V -> ( E. x e. { A } ph <-> ps ) )
Theorem	uzwo4 37302*	Well-ordering principle: any nonempty subset of an upper set of integers has the least element. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- F/ j ps   &    |- ( j = k -> ( ph <-> ps ) )   =>    |- ( ( S C_ ( ZZ>= ` M ) /\ E. j e. S ph ) -> E. j e. S ( ph /\ A. k e. S ( k < j -> -. ps ) ) )
Theorem	0in 37303	The intersection of the empty set with a class is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( (/) i^i A ) = (/)
Theorem	unisn0 37304	The union of the singleton of the empty set is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- U. { (/) } = (/)
Theorem	ssin0 37305	If two classes are disjoint, two respective subclasses are disjoint. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ( ( A i^i B ) = (/) /\ C C_ A /\ D C_ B ) -> ( C i^i D ) = (/) )
Theorem	inabs3 37306	Absorption law for intersection. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( C C_ B -> ( ( A i^i B ) i^i C ) = ( A i^i C ) )
Theorem	pwpwuni 37307	Relationship between power class and union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( A e. V -> ( A e. ~P ~P B <-> U. A e. ~P B ) )
Theorem	disjiun2 37308*	In a disjoint collection, an indexed union is disjoint from an additional term. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> Disj x e. A B )   &    |- ( ph -> C C_ A )   &    |- ( ph -> D e. ( A \ C ) )   &    |- ( x = D -> B = E )   =>    |- ( ph -> ( U_ x e. C B i^i E ) = (/) )
Theorem	0pwfi 37309	The empty set is in any power set, and it's finite. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- (/) e. ( ~P A i^i Fin )
Theorem	ssinss2d 37310	Intersection preserves subclass relationship. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> B C_ C )   =>    |- ( ph -> ( A i^i B ) C_ C )
Theorem	zct 37311	The set of integer numbers is countable. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ZZ ~<_ om
Theorem	iunxsngf2 37312*	A singleton index picks out an instance of an indexed union's argument. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- F/_ x C   &    |- ( x = A -> B = C )   =>    |- ( A e. V -> U_ x e. { A } B = C )
Theorem	pwfin0 37313	A finite set always belongs to a power class. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ~P A i^i Fin ) =/= (/)
Theorem	uzct 37314	An upper integer set is countable. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- Z = ( ZZ>= ` N )   =>    |- Z ~<_ om
Theorem	iunxsnf 37315*	A singleton index picks out an instance of an indexed union's argument. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- F/_ x C   &    |- A e. _V   &    |- ( x = A -> B = C )   =>    |- U_ x e. { A } B = C
Theorem	fiiuncl 37316*	If a set is closed under the union of two sets, then it is closed under finite indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- F/ x ph   &    |- ( ( ph /\ x e. A ) -> B e. D )   &    |- ( ( ph /\ y e. D /\ z e. D ) -> ( y u. z ) e. D )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> A =/= (/) )   =>    |- ( ph -> U_ x e. A B e. D )
Theorem	iunp1 37317*	The addition of the next set to a union indexed by a finite set of sequential integers. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- F/_ k B   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( k = ( N + 1 ) -> A = B )   =>    |- ( ph -> U_ k e. ( M ... ( N + 1 ) ) A = ( U_ k e. ( M ... N ) A u. B ) )
Theorem	fiunicl 37318*	If a set is closed under the union of two sets, then it is closed under finite union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ( ph /\ x e. A /\ y e. A ) -> ( x u. y ) e. A )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> A =/= (/) )   =>    |- ( ph -> U. A e. A )
Theorem	nnnfi 37319	The set of positive integers is infinite. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- -. NN e. Fin
Theorem	ixpeq2d 37320	Equality theorem for infinite Cartesian product. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- F/ x ph   &    |- ( ( ph /\ x e. A ) -> B = C )   =>    |- ( ph -> X_ x e. A B = X_ x e. A C )
Theorem	disjxp1 37321*	The sets of a cartesian product are disjoint if the sets in the first argument are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> Disj x e. A B )   =>    |- ( ph -> Disj x e. A ( B X. C ) )
Theorem	elexd 37322	If a class is a member of another class, it is a set. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> A e. V )   =>    |- ( ph -> A e. _V )
Theorem	elpwd 37323	Membership in a power class. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> A e. V )   &    |- ( ph -> A C_ B )   =>    |- ( ph -> A e. ~P B )
Theorem	disjsnxp 37324*	The sets in the cartesian product of singletons with other sets, are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- Disj j e. A ( { j } X. B )
Theorem	ssfid 37325	A subset of a finite set is finite. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
|- ( ph -> A e. Fin )   &    |- ( ph -> B C_ A )   =>    |- ( ph -> B e. Fin )
21.30.2  Functions
Theorem	unima 37326	Image of a union. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( F Fn A /\ B C_ A /\ C C_ A ) -> ( F " ( B u. C ) ) = ( ( F " B ) u. ( F " C ) ) )
Theorem	feq1dd 37327	Equality deduction for functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> F = G )   &    |- ( ph -> F : A --> B )   =>    |- ( ph -> G : A --> B )
Theorem	fnresdmss 37328	A function does not change when restricted to a set that contains its domain. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( F Fn A /\ A C_ B ) -> ( F |` B ) = F )
Theorem	fmptsnxp 37329*	Maps-to notation and cross product for a singleton function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( A e. V /\ B e. W ) -> ( x e. { A } |-> B ) = ( { A } X. { B } ) )
Theorem	mptex2 37330*	If a class given as a map-to notation is a set, it's image values are set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> ( t e. A |-> B ) : A --> C )   =>    |- ( ( ph /\ t e. A ) -> B e. C )
Theorem	fvmpt2bd 37331*	Value of a function given by the "maps to" notation. Deduction version. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> F = ( x e. A |-> B ) )   =>    |- ( ( ph /\ x e. A /\ B e. C ) -> ( F ` x ) = B )
Theorem	rnmptfi 37332*	The range of a function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- A = ( x e. B |-> C )   =>    |- ( B e. Fin -> ran A e. Fin )
Theorem	fresin2 37333	Restriction of a function with respect to the intersection with its domain. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( F : A --> B -> ( F |` ( C i^i A ) ) = ( F |` C ) )
Theorem	rnmptc 37334*	Range of a constant function in map to notation. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- F = ( x e. A |-> B )   &    |- ( ( ph /\ x e. A ) -> B e. C )   &    |- ( ph -> A =/= (/) )   =>    |- ( ph -> ran F = { B } )
Theorem	ffi 37335	A function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( F : A --> B /\ A e. Fin ) -> F e. Fin )
Theorem	suprnmpt 37336*	An explicit bound for the range of a bounded function. (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 )   &    |- F = ( x e. A |-> B )   &    |- C = sup ( ran F , RR , < )   =>    |- ( ph -> ( C e. RR /\ A. x e. A B <_ C ) )
Theorem	rnffi 37337	The range of a function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( F : A --> B /\ A e. Fin ) -> ran F e. Fin )
Theorem	mptelpm 37338*	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 37339	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 37340	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 37341*	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 37342	A mapping is a function. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> F : A --> B )   =>    |- ( ph -> F Fn A )
Theorem	resmpti 37343*	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 37344*	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 37345	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 37346	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 37347	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 37348*	An onto mapping expressed in terms of function values. As dffo3 5989 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 37349	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 37350*	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 37351*	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 37352	The range of a function in maps-to notation. Same as elrnmpt 5036, 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 37353*	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 37354*	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 37355*	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 37356	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 37357*	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 37358*	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 37359*	Property of a surjective function. As foelrn 5993 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 37360	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 37361*	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 37362*	Elementhood in an image set. Same as elrnmpt1s 5037, 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 37363*	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 37364*	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 37365*	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 37366*	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 37367	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 37368*	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 37369	The empty set is the only map with empty domain. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( A e. V -> ( A ^m (/) ) = { (/) } )
Theorem	nnf1oxpnn 37370	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 37371*	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 37372*	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 37373	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 37374	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 37375	'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 37376	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 37377	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 37378	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 37379	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 37380	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 37381	Negative pi is negative. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- -u pi < 0
Theorem	dstregt0 37382*	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 37383	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 37384	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 37385	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 37386	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 37387	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 37388	3 is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- 3 e. RR+
Theorem	leimnltdOLD 37389	'Less than or equal to' implies 'not less than'. (Contributed by Glauco Siliprandi, 11-Dec-2019.) Obsolete as of 18-Jul-2020. Use lensymd 9730 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 37390	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 37391	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 37392	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 37393	The reals are open with respect to the standard topology. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- RR e. ( topGen ` ran (,) )
Theorem	elfzop1le2 37394	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 37395	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 37396	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 37397	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 37398	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 37399	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 37400	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 ) )