P. 150 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14901-15000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rescncf 14901	A continuous complex function restricted to a subset is continuous. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 25-Aug-2014.)
|- ( C C_ A -> ( F e. ( A -cn-> B ) -> ( F |` C ) e. ( C -cn-> B ) ) )
Theorem	cncfcdm 14902	Change the codomain of a continuous complex function. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 1-May-2015.)
|- ( ( C C_ CC /\ F e. ( A -cn-> B ) ) -> ( F e. ( A -cn-> C ) <-> F : A --> C ) )
Theorem	cncfss 14903	The set of continuous functions is expanded when the codomain is expanded. (Contributed by Mario Carneiro, 30-Aug-2014.)
|- ( ( B C_ C /\ C C_ CC ) -> ( A -cn-> B ) C_ ( A -cn-> C ) )
Theorem	climcncf 14904	Image of a limit under a continuous map. (Contributed by Mario Carneiro, 7-Apr-2015.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F e. ( A -cn-> B ) )   &    |- ( ph -> G : Z --> A )   &    |- ( ph -> G ~~> D )   &    |- ( ph -> D e. A )   =>    |- ( ph -> ( F o. G ) ~~> ( F ` D ) )
Theorem	abscncf 14905	Absolute value is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
|- abs e. ( CC -cn-> RR )
Theorem	recncf 14906	Real part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
|- Re e. ( CC -cn-> RR )
Theorem	imcncf 14907	Imaginary part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
|- Im e. ( CC -cn-> RR )
Theorem	cjcncf 14908	Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
|- * e. ( CC -cn-> CC )
Theorem	mulc1cncf 14909*	Multiplication by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
|- F = ( x e. CC |-> ( A x. x ) )   =>    |- ( A e. CC -> F e. ( CC -cn-> CC ) )
Theorem	divccncfap 14910*	Division by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Jim Kingdon, 9-Jan-2023.)
|- F = ( x e. CC |-> ( x / A ) )   =>    |- ( ( A e. CC /\ A # 0 ) -> F e. ( CC -cn-> CC ) )
Theorem	cncfco 14911	The composition of two continuous maps on complex numbers is also continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 25-Aug-2014.)
|- ( ph -> F e. ( A -cn-> B ) )   &    |- ( ph -> G e. ( B -cn-> C ) )   =>    |- ( ph -> ( G o. F ) e. ( A -cn-> C ) )
Theorem	cncfmet 14912	Relate complex function continuity to metric space continuity. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.)
|- C = ( ( abs o. - ) |` ( A X. A ) )   &    |- D = ( ( abs o. - ) |` ( B X. B ) )   &    |- J = ( MetOpen ` C )   &    |- K = ( MetOpen ` D )   =>    |- ( ( A C_ CC /\ B C_ CC ) -> ( A -cn-> B ) = ( J Cn K ) )
Theorem	cncfcncntop 14913	Relate complex function continuity to topological continuity. (Contributed by Mario Carneiro, 17-Feb-2015.)
|- J = ( MetOpen ` ( abs o. - ) )   &    |- K = ( J ↾t A )   &    |- L = ( J ↾t B )   =>    |- ( ( A C_ CC /\ B C_ CC ) -> ( A -cn-> B ) = ( K Cn L ) )
Theorem	cncfcn1cntop 14914	Relate complex function continuity to topological continuity. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.) (Revised by Jim Kingdon, 16-Jun-2023.)
|- J = ( MetOpen ` ( abs o. - ) )   =>    |- ( CC -cn-> CC ) = ( J Cn J )
Theorem	cncfcn1 14915	Relate complex function continuity to topological continuity. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.)
|- J = ( TopOpen ` ℂfld )   =>    |- ( CC -cn-> CC ) = ( J Cn J )
Theorem	cncfmptc 14916*	A constant function is a continuous function on CC. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Sep-2015.)
|- ( ( A e. T /\ S C_ CC /\ T C_ CC ) -> ( x e. S |-> A ) e. ( S -cn-> T ) )
Theorem	cncfmptid 14917*	The identity function is a continuous function on CC. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 17-May-2016.)
|- ( ( S C_ T /\ T C_ CC ) -> ( x e. S |-> x ) e. ( S -cn-> T ) )
Theorem	cncfmpt1f 14918*	Composition of continuous functions. -cn-> analogue of cnmpt11f 14604. (Contributed by Mario Carneiro, 3-Sep-2014.)
|- ( ph -> F e. ( CC -cn-> CC ) )   &    |- ( ph -> ( x e. X |-> A ) e. ( X -cn-> CC ) )   =>    |- ( ph -> ( x e. X |-> ( F ` A ) ) e. ( X -cn-> CC ) )
Theorem	cncfmpt2fcntop 14919*	Composition of continuous functions. -cn-> analogue of cnmpt12f 14606. (Contributed by Mario Carneiro, 3-Sep-2014.)
|- J = ( MetOpen ` ( abs o. - ) )   &    |- ( ph -> F e. ( ( J tX J ) Cn J ) )   &    |- ( ph -> ( x e. X |-> A ) e. ( X -cn-> CC ) )   &    |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> CC ) )   =>    |- ( ph -> ( x e. X |-> ( A F B ) ) e. ( X -cn-> CC ) )
Theorem	addccncf 14920*	Adding a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- F = ( x e. CC |-> ( x + A ) )   =>    |- ( A e. CC -> F e. ( CC -cn-> CC ) )
Theorem	idcncf 14921	The identity function is a continuous function on CC. (Contributed by Jeff Madsen, 11-Jun-2010.) (Moved into main set.mm as cncfmptid 14917 and may be deleted by mathbox owner, JM. --MC 12-Sep-2015.) (Revised by Mario Carneiro, 12-Sep-2015.)
|- F = ( x e. CC |-> x )   =>    |- F e. ( CC -cn-> CC )
Theorem	sub1cncf 14922*	Subtracting a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
|- F = ( x e. CC |-> ( x - A ) )   =>    |- ( A e. CC -> F e. ( CC -cn-> CC ) )
Theorem	sub2cncf 14923*	Subtraction from a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
|- F = ( x e. CC |-> ( A - x ) )   =>    |- ( A e. CC -> F e. ( CC -cn-> CC ) )
Theorem	cdivcncfap 14924*	Division with a constant numerator is continuous. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 26-May-2023.)
|- F = ( x e. { y e. CC | y # 0 } |-> ( A / x ) )   =>    |- ( A e. CC -> F e. ( { y e. CC | y # 0 } -cn-> CC ) )
Theorem	negcncf 14925*	The negative function is continuous. (Contributed by Mario Carneiro, 30-Dec-2016.)
|- F = ( x e. A |-> -u x )   =>    |- ( A C_ CC -> F e. ( A -cn-> CC ) )
Theorem	negfcncf 14926*	The negative of a continuous complex function is continuous. (Contributed by Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.)
|- G = ( x e. A |-> -u ( F ` x ) )   =>    |- ( F e. ( A -cn-> CC ) -> G e. ( A -cn-> CC ) )
Theorem	mulcncflem 14927*	Lemma for mulcncf 14928. (Contributed by Jim Kingdon, 29-May-2023.)
|- ( ph -> ( x e. X |-> A ) e. ( X -cn-> CC ) )   &    |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> CC ) )   &    |- ( ph -> V e. X )   &    |- ( ph -> E e. RR+ )   &    |- ( ph -> F e. RR+ )   &    |- ( ph -> G e. RR+ )   &    |- ( ph -> S e. RR+ )   &    |- ( ph -> T e. RR+ )   &    |- ( ph -> A. u e. X ( ( abs ` ( u - V ) ) < S -> ( abs ` ( ( ( x e. X |-> A ) ` u ) - ( ( x e. X |-> A ) ` V ) ) ) < F ) )   &    |- ( ph -> A. u e. X ( ( abs ` ( u - V ) ) < T -> ( abs ` ( ( ( x e. X |-> B ) ` u ) - ( ( x e. X |-> B ) ` V ) ) ) < G ) )   &    |- ( ph -> A. u e. X ( ( ( abs ` ( [_ u / x ]_ A - [_ V / x ]_ A ) ) < F /\ ( abs ` ( [_ u / x ]_ B - [_ V / x ]_ B ) ) < G ) -> ( abs ` ( ( [_ u / x ]_ A x. [_ u / x ]_ B ) - ( [_ V / x ]_ A x. [_ V / x ]_ B ) ) ) < E ) )   =>    |- ( ph -> E. d e. RR+ A. u e. X ( ( abs ` ( u - V ) ) < d -> ( abs ` ( ( ( x e. X |-> ( A x. B ) ) ` u ) - ( ( x e. X |-> ( A x. B ) ) ` V ) ) ) < E ) )
Theorem	mulcncf 14928*	The multiplication of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
|- ( ph -> ( x e. X |-> A ) e. ( X -cn-> CC ) )   &    |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> CC ) )   =>    |- ( ph -> ( x e. X |-> ( A x. B ) ) e. ( X -cn-> CC ) )
Theorem	expcncf 14929*	The power function on complex numbers, for fixed exponent N, is continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
|- ( N e. NN0 -> ( x e. CC |-> ( x ^ N ) ) e. ( CC -cn-> CC ) )
Theorem	cnrehmeocntop 14930*	The canonical bijection from ( RR X. RR ) to CC described in cnref1o 9742 is in fact a homeomorphism of the usual topologies on these sets. (It is also an isometry, if ( RR X. RR ) is metrized with the l<SUP>2</SUP> norm.) (Contributed by Mario Carneiro, 25-Aug-2014.)
|- F = ( x e. RR , y e. RR |-> ( x + ( _i x. y ) ) )   &    |- J = ( topGen ` ran (,) )   &    |- K = ( MetOpen ` ( abs o. - ) )   =>    |- F e. ( ( J tX J ) Homeo K )
Theorem	cnopnap 14931*	The complex numbers apart from a given complex number form an open set. (Contributed by Jim Kingdon, 14-Dec-2023.)
|- ( A e. CC -> { w e. CC | w # A } e. ( MetOpen ` ( abs o. - ) ) )
PART 10  BASIC REAL AND COMPLEX ANALYSIS
10.1  Continuity
Theorem	addcncf 14932*	The addition of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> ( x e. X |-> A ) e. ( X -cn-> CC ) )   &    |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> CC ) )   =>    |- ( ph -> ( x e. X |-> ( A + B ) ) e. ( X -cn-> CC ) )
Theorem	subcncf 14933*	The subtraction of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> ( x e. X |-> A ) e. ( X -cn-> CC ) )   &    |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> CC ) )   =>    |- ( ph -> ( x e. X |-> ( A - B ) ) e. ( X -cn-> CC ) )
Theorem	divcncfap 14934*	The quotient of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> ( x e. X |-> A ) e. ( X -cn-> CC ) )   &    |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> { y e. CC | y # 0 } ) )   =>    |- ( ph -> ( x e. X |-> ( A / B ) ) e. ( X -cn-> CC ) )
Theorem	maxcncf 14935*	The maximum of two continuous real functions is continuous. (Contributed by Jim Kingdon, 18-Jul-2025.)
|- ( ph -> ( x e. X |-> A ) e. ( X -cn-> RR ) )   &    |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> RR ) )   =>    |- ( ph -> ( x e. X |-> sup ( { A , B } , RR , < ) ) e. ( X -cn-> RR ) )
Theorem	mincncf 14936*	The minimum of two continuous real functions is continuous. (Contributed by Jim Kingdon, 19-Jul-2025.)
|- ( ph -> ( x e. X |-> A ) e. ( X -cn-> RR ) )   &    |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> RR ) )   =>    |- ( ph -> ( x e. X |-> inf ( { A , B } , RR , < ) ) e. ( X -cn-> RR ) )
10.1.1  Dedekind cuts
Theorem	dedekindeulemuub 14937*	Lemma for dedekindeu 14943. Any element of the upper cut is an upper bound for the lower cut. (Contributed by Jim Kingdon, 2-Feb-2024.)
|- ( ph -> L C_ RR )   &    |- ( ph -> U C_ RR )   &    |- ( ph -> E. q e. RR q e. L )   &    |- ( ph -> E. r e. RR r e. U )   &    |- ( ph -> A. q e. RR ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. RR ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. RR A. r e. RR ( q < r -> ( q e. L \/ r e. U ) ) )   &    |- ( ph -> A e. U )   =>    |- ( ph -> A. z e. L z < A )
Theorem	dedekindeulemub 14938*	Lemma for dedekindeu 14943. The lower cut has an upper bound. (Contributed by Jim Kingdon, 31-Jan-2024.)
|- ( ph -> L C_ RR )   &    |- ( ph -> U C_ RR )   &    |- ( ph -> E. q e. RR q e. L )   &    |- ( ph -> E. r e. RR r e. U )   &    |- ( ph -> A. q e. RR ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. RR ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. RR A. r e. RR ( q < r -> ( q e. L \/ r e. U ) ) )   =>    |- ( ph -> E. x e. RR A. y e. L y < x )
Theorem	dedekindeulemloc 14939*	Lemma for dedekindeu 14943. The set L is located. (Contributed by Jim Kingdon, 31-Jan-2024.)
|- ( ph -> L C_ RR )   &    |- ( ph -> U C_ RR )   &    |- ( ph -> E. q e. RR q e. L )   &    |- ( ph -> E. r e. RR r e. U )   &    |- ( ph -> A. q e. RR ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. RR ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. RR A. r e. RR ( q < r -> ( q e. L \/ r e. U ) ) )   =>    |- ( ph -> A. x e. RR A. y e. RR ( x < y -> ( E. z e. L x < z \/ A. z e. L z < y ) ) )
Theorem	dedekindeulemlub 14940*	Lemma for dedekindeu 14943. The set L has a least upper bound. (Contributed by Jim Kingdon, 31-Jan-2024.)
|- ( ph -> L C_ RR )   &    |- ( ph -> U C_ RR )   &    |- ( ph -> E. q e. RR q e. L )   &    |- ( ph -> E. r e. RR r e. U )   &    |- ( ph -> A. q e. RR ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. RR ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. RR A. r e. RR ( q < r -> ( q e. L \/ r e. U ) ) )   =>    |- ( ph -> E. x e. RR ( A. y e. L -. x < y /\ A. y e. RR ( y < x -> E. z e. L y < z ) ) )
Theorem	dedekindeulemlu 14941*	Lemma for dedekindeu 14943. There is a number which separates the lower and upper cuts. (Contributed by Jim Kingdon, 31-Jan-2024.)
|- ( ph -> L C_ RR )   &    |- ( ph -> U C_ RR )   &    |- ( ph -> E. q e. RR q e. L )   &    |- ( ph -> E. r e. RR r e. U )   &    |- ( ph -> A. q e. RR ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. RR ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. RR A. r e. RR ( q < r -> ( q e. L \/ r e. U ) ) )   =>    |- ( ph -> E. x e. RR ( A. q e. L q < x /\ A. r e. U x < r ) )
Theorem	dedekindeulemeu 14942*	Lemma for dedekindeu 14943. Part of proving uniqueness. (Contributed by Jim Kingdon, 31-Jan-2024.)
|- ( ph -> L C_ RR )   &    |- ( ph -> U C_ RR )   &    |- ( ph -> E. q e. RR q e. L )   &    |- ( ph -> E. r e. RR r e. U )   &    |- ( ph -> A. q e. RR ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. RR ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. RR A. r e. RR ( q < r -> ( q e. L \/ r e. U ) ) )   &    |- ( ph -> A e. RR )   &    |- ( ph -> ( A. q e. L q < A /\ A. r e. U A < r ) )   &    |- ( ph -> B e. RR )   &    |- ( ph -> ( A. q e. L q < B /\ A. r e. U B < r ) )   &    |- ( ph -> A < B )   =>    |- ( ph -> F. )
Theorem	dedekindeu 14943*	A Dedekind cut identifies a unique real number. Similar to df-inp 7550 except that the the Dedekind cut is formed by sets of reals (rather than positive rationals). But in both cases the defining property of a Dedekind cut is that it is inhabited (bounded), rounded, disjoint, and located. (Contributed by Jim Kingdon, 5-Jan-2024.)
|- ( ph -> L C_ RR )   &    |- ( ph -> U C_ RR )   &    |- ( ph -> E. q e. RR q e. L )   &    |- ( ph -> E. r e. RR r e. U )   &    |- ( ph -> A. q e. RR ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. RR ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. RR A. r e. RR ( q < r -> ( q e. L \/ r e. U ) ) )   =>    |- ( ph -> E! x e. RR ( A. q e. L q < x /\ A. r e. U x < r ) )
Theorem	suplociccreex 14944*	An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 8116 but that one is for the entire real line rather than a closed interval. (Contributed by Jim Kingdon, 14-Feb-2024.)
|- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> B < C )   &    |- ( ph -> A C_ ( B [,] C ) )   &    |- ( ph -> E. x x e. A )   &    |- ( ph -> A. x e. ( B [,] C ) A. y e. ( B [,] C ) ( x < y -> ( E. z e. A x < z \/ A. z e. A z < y ) ) )   =>    |- ( ph -> E. x e. RR ( A. y e. A -. x < y /\ A. y e. RR ( y < x -> E. z e. A y < z ) ) )
Theorem	suplociccex 14945*	An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 8116 but that one is for the entire real line rather than a closed interval. (Contributed by Jim Kingdon, 14-Feb-2024.)
|- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> B < C )   &    |- ( ph -> A C_ ( B [,] C ) )   &    |- ( ph -> E. x x e. A )   &    |- ( ph -> A. x e. ( B [,] C ) A. y e. ( B [,] C ) ( x < y -> ( E. z e. A x < z \/ A. z e. A z < y ) ) )   =>    |- ( ph -> E. x e. ( B [,] C ) ( A. y e. A -. x < y /\ A. y e. ( B [,] C ) ( y < x -> E. z e. A y < z ) ) )
Theorem	dedekindicclemuub 14946*	Lemma for dedekindicc 14953. Any element of the upper cut is an upper bound for the lower cut. (Contributed by Jim Kingdon, 15-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> L C_ ( A [,] B ) )   &    |- ( ph -> U C_ ( A [,] B ) )   &    |- ( ph -> E. q e. ( A [,] B ) q e. L )   &    |- ( ph -> E. r e. ( A [,] B ) r e. U )   &    |- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. U ) ) )   &    |- ( ph -> C e. U )   =>    |- ( ph -> A. z e. L z < C )
Theorem	dedekindicclemub 14947*	Lemma for dedekindicc 14953. The lower cut has an upper bound. (Contributed by Jim Kingdon, 15-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> L C_ ( A [,] B ) )   &    |- ( ph -> U C_ ( A [,] B ) )   &    |- ( ph -> E. q e. ( A [,] B ) q e. L )   &    |- ( ph -> E. r e. ( A [,] B ) r e. U )   &    |- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. U ) ) )   =>    |- ( ph -> E. x e. ( A [,] B ) A. y e. L y < x )
Theorem	dedekindicclemloc 14948*	Lemma for dedekindicc 14953. The set L is located. (Contributed by Jim Kingdon, 15-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> L C_ ( A [,] B ) )   &    |- ( ph -> U C_ ( A [,] B ) )   &    |- ( ph -> E. q e. ( A [,] B ) q e. L )   &    |- ( ph -> E. r e. ( A [,] B ) r e. U )   &    |- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. U ) ) )   =>    |- ( ph -> A. x e. ( A [,] B ) A. y e. ( A [,] B ) ( x < y -> ( E. z e. L x < z \/ A. z e. L z < y ) ) )
Theorem	dedekindicclemlub 14949*	Lemma for dedekindicc 14953. The set L has a least upper bound. (Contributed by Jim Kingdon, 15-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> L C_ ( A [,] B ) )   &    |- ( ph -> U C_ ( A [,] B ) )   &    |- ( ph -> E. q e. ( A [,] B ) q e. L )   &    |- ( ph -> E. r e. ( A [,] B ) r e. U )   &    |- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. U ) ) )   &    |- ( ph -> A < B )   =>    |- ( ph -> E. x e. ( A [,] B ) ( A. y e. L -. x < y /\ A. y e. ( A [,] B ) ( y < x -> E. z e. L y < z ) ) )
Theorem	dedekindicclemlu 14950*	Lemma for dedekindicc 14953. There is a number which separates the lower and upper cuts. (Contributed by Jim Kingdon, 15-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> L C_ ( A [,] B ) )   &    |- ( ph -> U C_ ( A [,] B ) )   &    |- ( ph -> E. q e. ( A [,] B ) q e. L )   &    |- ( ph -> E. r e. ( A [,] B ) r e. U )   &    |- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. U ) ) )   &    |- ( ph -> A < B )   =>    |- ( ph -> E. x e. ( A [,] B ) ( A. q e. L q < x /\ A. r e. U x < r ) )
Theorem	dedekindicclemeu 14951*	Lemma for dedekindicc 14953. Part of proving uniqueness. (Contributed by Jim Kingdon, 15-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> L C_ ( A [,] B ) )   &    |- ( ph -> U C_ ( A [,] B ) )   &    |- ( ph -> E. q e. ( A [,] B ) q e. L )   &    |- ( ph -> E. r e. ( A [,] B ) r e. U )   &    |- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. U ) ) )   &    |- ( ph -> A < B )   &    |- ( ph -> C e. ( A [,] B ) )   &    |- ( ph -> ( A. q e. L q < C /\ A. r e. U C < r ) )   &    |- ( ph -> D e. ( A [,] B ) )   &    |- ( ph -> ( A. q e. L q < D /\ A. r e. U D < r ) )   &    |- ( ph -> C < D )   =>    |- ( ph -> F. )
Theorem	dedekindicclemicc 14952*	Lemma for dedekindicc 14953. Same as dedekindicc 14953, except that we merely show x to be an element of ( A [,] B ). Later we will strengthen that to ( A (,) B ). (Contributed by Jim Kingdon, 5-Jan-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> L C_ ( A [,] B ) )   &    |- ( ph -> U C_ ( A [,] B ) )   &    |- ( ph -> E. q e. ( A [,] B ) q e. L )   &    |- ( ph -> E. r e. ( A [,] B ) r e. U )   &    |- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. U ) ) )   &    |- ( ph -> A < B )   =>    |- ( ph -> E! x e. ( A [,] B ) ( A. q e. L q < x /\ A. r e. U x < r ) )
Theorem	dedekindicc 14953*	A Dedekind cut identifies a unique real number. Similar to df-inp 7550 except that the Dedekind cut is formed by sets of reals (rather than positive rationals). But in both cases the defining property of a Dedekind cut is that it is inhabited (bounded), rounded, disjoint, and located. (Contributed by Jim Kingdon, 19-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> L C_ ( A [,] B ) )   &    |- ( ph -> U C_ ( A [,] B ) )   &    |- ( ph -> E. q e. ( A [,] B ) q e. L )   &    |- ( ph -> E. r e. ( A [,] B ) r e. U )   &    |- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. U ) ) )   &    |- ( ph -> A < B )   =>    |- ( ph -> E! x e. ( A (,) B ) ( A. q e. L q < x /\ A. r e. U x < r ) )
10.1.2  Intermediate value theorem
Theorem	ivthinclemlm 14954*	Lemma for ivthinc 14963. The lower cut is bounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )   &    |- L = { w e. ( A [,] B ) | ( F ` w ) < U }   &    |- R = { w e. ( A [,] B ) | U < ( F ` w ) }   =>    |- ( ph -> E. q e. ( A [,] B ) q e. L )
Theorem	ivthinclemum 14955*	Lemma for ivthinc 14963. The upper cut is bounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )   &    |- L = { w e. ( A [,] B ) | ( F ` w ) < U }   &    |- R = { w e. ( A [,] B ) | U < ( F ` w ) }   =>    |- ( ph -> E. r e. ( A [,] B ) r e. R )
Theorem	ivthinclemlopn 14956*	Lemma for ivthinc 14963. The lower cut is open. (Contributed by Jim Kingdon, 6-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )   &    |- L = { w e. ( A [,] B ) | ( F ` w ) < U }   &    |- R = { w e. ( A [,] B ) | U < ( F ` w ) }   &    |- ( ph -> Q e. L )   =>    |- ( ph -> E. r e. L Q < r )
Theorem	ivthinclemlr 14957*	Lemma for ivthinc 14963. The lower cut is rounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )   &    |- L = { w e. ( A [,] B ) | ( F ` w ) < U }   &    |- R = { w e. ( A [,] B ) | U < ( F ` w ) }   =>    |- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )
Theorem	ivthinclemuopn 14958*	Lemma for ivthinc 14963. The upper cut is open. (Contributed by Jim Kingdon, 19-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )   &    |- L = { w e. ( A [,] B ) | ( F ` w ) < U }   &    |- R = { w e. ( A [,] B ) | U < ( F ` w ) }   &    |- ( ph -> S e. R )   =>    |- ( ph -> E. q e. R q < S )
Theorem	ivthinclemur 14959*	Lemma for ivthinc 14963. The upper cut is rounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )   &    |- L = { w e. ( A [,] B ) | ( F ` w ) < U }   &    |- R = { w e. ( A [,] B ) | U < ( F ` w ) }   =>    |- ( ph -> A. r e. ( A [,] B ) ( r e. R <-> E. q e. R q < r ) )
Theorem	ivthinclemdisj 14960*	Lemma for ivthinc 14963. The lower and upper cuts are disjoint. (Contributed by Jim Kingdon, 18-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )   &    |- L = { w e. ( A [,] B ) | ( F ` w ) < U }   &    |- R = { w e. ( A [,] B ) | U < ( F ` w ) }   =>    |- ( ph -> ( L i^i R ) = (/) )
Theorem	ivthinclemloc 14961*	Lemma for ivthinc 14963. Locatedness. (Contributed by Jim Kingdon, 18-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )   &    |- L = { w e. ( A [,] B ) | ( F ` w ) < U }   &    |- R = { w e. ( A [,] B ) | U < ( F ` w ) }   =>    |- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. R ) ) )
Theorem	ivthinclemex 14962*	Lemma for ivthinc 14963. Existence of a number between the lower cut and the upper cut. (Contributed by Jim Kingdon, 20-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )   &    |- L = { w e. ( A [,] B ) | ( F ` w ) < U }   &    |- R = { w e. ( A [,] B ) | U < ( F ` w ) }   =>    |- ( ph -> E! z e. ( A (,) B ) ( A. q e. L q < z /\ A. r e. R z < r ) )
Theorem	ivthinc 14963*	The intermediate value theorem, increasing case, for a strictly monotonic function. Theorem 5.5 of [Bauer], p. 494. This is Metamath 100 proof #79. (Contributed by Jim Kingdon, 5-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )   =>    |- ( ph -> E. c e. ( A (,) B ) ( F ` c ) = U )
Theorem	ivthdec 14964*	The intermediate value theorem, decreasing case, for a strictly monotonic function. (Contributed by Jim Kingdon, 20-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` B ) < U /\ U < ( F ` A ) ) )   &    |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` y ) < ( F ` x ) )   =>    |- ( ph -> E. c e. ( A (,) B ) ( F ` c ) = U )
Theorem	ivthreinc 14965*	Restating the intermediate value theorem. Given a hypothesis stating the intermediate value theorem (in a strong form which is not provable given our axioms alone), provide a conclusion similar to the theorem as stated in the Metamath Proof Explorer (which is also similar to how we state the theorem for a strictly monotonic function at ivthinc 14963). Being able to have a hypothesis stating the intermediate value theorem will be helpful when it comes time to show that it implies a constructive taboo. This version of the theorem requires that the function F is continuous on the entire real line, not just ( A [,] B ) which may be an unnecessary condition but which is sufficient for the way we want to use it. (Contributed by Jim Kingdon, 7-Jul-2025.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> F e. ( RR -cn-> RR ) )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- ( ph -> A. f ( f e. ( RR -cn-> RR ) -> A. a e. RR A. b e. RR ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( f ` x ) = 0 ) ) ) )   =>    |- ( ph -> E. c e. ( A (,) B ) ( F ` c ) = U )
Theorem	hovercncf 14966	The hover function is continuous. By hover function, we mean a a function which starts out as a line of slope one, is constant at zero from zero to one, and then resumes as a slope of one. (Contributed by Jim Kingdon, 20-Jul-2025.)
|- F = ( x e. RR |-> sup ( {inf ( { x , 0 } , RR , < ) , ( x - 1 ) } , RR , < ) )   =>    |- F e. ( RR -cn-> RR )
Theorem	hovera 14967*	A point at which the hover function is less than a given value. (Contributed by Jim Kingdon, 21-Jul-2025.)
|- F = ( x e. RR |-> sup ( {inf ( { x , 0 } , RR , < ) , ( x - 1 ) } , RR , < ) )   =>    |- ( Z e. RR -> ( F ` ( Z - 1 ) ) < Z )
Theorem	hoverb 14968*	A point at which the hover function is greater than a given value. (Contributed by Jim Kingdon, 21-Jul-2025.)
|- F = ( x e. RR |-> sup ( {inf ( { x , 0 } , RR , < ) , ( x - 1 ) } , RR , < ) )   =>    |- ( Z e. RR -> Z < ( F ` ( Z + 2 ) ) )
Theorem	hoverlt1 14969*	The hover function evaluated at a point less than one. (Contributed by Jim Kingdon, 22-Jul-2025.)
|- F = ( x e. RR |-> sup ( {inf ( { x , 0 } , RR , < ) , ( x - 1 ) } , RR , < ) )   =>    |- ( ( C e. RR /\ C < 1 ) -> ( F ` C ) <_ 0 )
Theorem	hovergt0 14970*	The hover function evaluated at a point greater than zero. (Contributed by Jim Kingdon, 22-Jul-2025.)
|- F = ( x e. RR |-> sup ( {inf ( { x , 0 } , RR , < ) , ( x - 1 ) } , RR , < ) )   =>    |- ( ( C e. RR /\ 0 < C ) -> 0 <_ ( F ` C ) )
Theorem	ivthdichlem 14971*	Lemma for ivthdich 14973. The result, with a few notational conveniences. (Contributed by Jim Kingdon, 22-Jul-2025.)
|- F = ( x e. RR |-> sup ( {inf ( { x , 0 } , RR , < ) , ( x - 1 ) } , RR , < ) )   &    |- ( ph -> Z e. RR )   &    |- ( ph -> A. f ( f e. ( RR -cn-> RR ) -> A. a e. RR A. b e. RR ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( f ` x ) = 0 ) ) ) )   =>    |- ( ph -> ( Z <_ 0 \/ 0 <_ Z ) )
Theorem	dich0 14972*	Real number dichotomy stated in terms of two real numbers or a real number and zero. (Contributed by Jim Kingdon, 22-Jul-2025.)
|- ( A. z e. RR ( z <_ 0 \/ 0 <_ z ) <-> A. x e. RR A. y e. RR ( x <_ y \/ y <_ x ) )
Theorem	ivthdich 14973*	The intermediate value theorem implies real number dichotomy. Because real number dichotomy (also known as analytic LLPO) is a constructive taboo, this means we will be unable to prove the intermediate value theorem as stated here (although versions with additional conditions, such as ivthinc 14963 for strictly monotonic functions, can be proved). The proof is via a function which we call the hover function and which is also described in Section 5.1 of [Bauer], p. 493. Consider any real number z. We want to show that z <_ 0 \/ 0 <_ z. Because of hovercncf 14966, hovera 14967, and hoverb 14968, we are able to apply the intermediate value theorem to get a value c such that the hover function at c equals z. By axltwlin 8111, c < 1 or 0 < c, and that leads to z <_ 0 by hoverlt1 14969 or 0 <_ z by hovergt0 14970. (Contributed by Jim Kingdon and Mario Carneiro, 22-Jul-2025.)
|- ( A. f ( f e. ( RR -cn-> RR ) -> A. a e. RR A. b e. RR ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( f ` x ) = 0 ) ) ) -> A. r e. RR A. s e. RR ( r <_ s \/ s <_ r ) )
10.2  Derivatives
10.2.1  Real and complex differentiation
10.2.1.1  Derivatives of functions of one complex or real variable
Syntax	climc 14974	The limit operator.
class lim CC
Syntax	cdv 14975	The derivative operator.
class _D
Definition	df-limced 14976*	Define the set of limits of a complex function at a point. Under normal circumstances, this will be a singleton or empty, depending on whether the limit exists. (Contributed by Mario Carneiro, 24-Dec-2016.) (Revised by Jim Kingdon, 3-Jun-2023.)
|- lim CC = ( f e. ( CC ^pm CC ) , x e. CC |-> { y e. CC | ( ( f : dom f --> CC /\ dom f C_ CC ) /\ ( x e. CC /\ A. e e. RR+ E. d e. RR+ A. z e. dom f ( ( z # x /\ ( abs ` ( z - x ) ) < d ) -> ( abs ` ( ( f ` z ) - y ) ) < e ) ) ) } )
Definition	df-dvap 14977*	Define the derivative operator. This acts on functions to produce a function that is defined where the original function is differentiable, with value the derivative of the function at these points. The set s here is the ambient topological space under which we are evaluating the continuity of the difference quotient. Although the definition is valid for any subset of CC and is well-behaved when s contains no isolated points, we will restrict our attention to the cases s = RR or s = CC for the majority of the development, these corresponding respectively to real and complex differentiation. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 25-Jun-2023.)
|- _D = ( s e. ~P CC , f e. ( CC ^pm s ) |-> U_ x e. ( ( int ` ( ( MetOpen ` ( abs o. - ) ) ↾t s ) ) ` dom f ) ( { x } X. ( ( z e. { w e. dom f | w # x } |-> ( ( ( f ` z ) - ( f ` x ) ) / ( z - x ) ) ) lim CC x ) ) )
Theorem	limcrcl 14978	Reverse closure for the limit operator. (Contributed by Mario Carneiro, 28-Dec-2016.)
|- ( C e. ( F lim CC B ) -> ( F : dom F --> CC /\ dom F C_ CC /\ B e. CC ) )
Theorem	limccl 14979	Closure of the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.)
|- ( F lim CC B ) C_ CC
Theorem	ellimc3apf 14980*	Write the epsilon-delta definition of a limit. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 4-Nov-2023.)
|- ( ph -> F : A --> CC )   &    |- ( ph -> A C_ CC )   &    |- ( ph -> B e. CC )   &    |- F/_ z F   =>    |- ( ph -> ( C e. ( F lim CC B ) <-> ( C e. CC /\ A. x e. RR+ E. y e. RR+ A. z e. A ( ( z # B /\ ( abs ` ( z - B ) ) < y ) -> ( abs ` ( ( F ` z ) - C ) ) < x ) ) ) )
Theorem	ellimc3ap 14981*	Write the epsilon-delta definition of a limit. (Contributed by Mario Carneiro, 28-Dec-2016.) Use apartness. (Revised by Jim Kingdon, 3-Jun-2023.)
|- ( ph -> F : A --> CC )   &    |- ( ph -> A C_ CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( C e. ( F lim CC B ) <-> ( C e. CC /\ A. x e. RR+ E. y e. RR+ A. z e. A ( ( z # B /\ ( abs ` ( z - B ) ) < y ) -> ( abs ` ( ( F ` z ) - C ) ) < x ) ) ) )
Theorem	limcdifap 14982*	It suffices to consider functions which are not defined at B to define the limit of a function. In particular, the value of the original function F at B does not affect the limit of F. (Contributed by Mario Carneiro, 25-Dec-2016.) (Revised by Jim Kingdon, 3-Jun-2023.)
|- ( ph -> F : A --> CC )   &    |- ( ph -> A C_ CC )   =>    |- ( ph -> ( F lim CC B ) = ( ( F |` { x e. A | x # B } ) lim CC B ) )
Theorem	limcmpted 14983*	Express the limit operator for a function defined by a mapping, via epsilon-delta. (Contributed by Jim Kingdon, 3-Nov-2023.)
|- ( ph -> A C_ CC )   &    |- ( ph -> B e. CC )   &    |- ( ( ph /\ z e. A ) -> D e. CC )   =>    |- ( ph -> ( C e. ( ( z e. A |-> D ) lim CC B ) <-> ( C e. CC /\ A. x e. RR+ E. y e. RR+ A. z e. A ( ( z # B /\ ( abs ` ( z - B ) ) < y ) -> ( abs ` ( D - C ) ) < x ) ) ) )
Theorem	limcimolemlt 14984*	Lemma for limcimo 14985. (Contributed by Jim Kingdon, 3-Jul-2023.)
|- ( ph -> F : A --> CC )   &    |- ( ph -> A C_ CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> B e. C )   &    |- ( ph -> B e. S )   &    |- ( ph -> C e. ( K ↾t S ) )   &    |- ( ph -> S e. { RR , CC } )   &    |- ( ph -> { q e. C | q # B } C_ A )   &    |- K = ( MetOpen ` ( abs o. - ) )   &    |- ( ph -> D e. RR+ )   &    |- ( ph -> X e. ( F lim CC B ) )   &    |- ( ph -> Y e. ( F lim CC B ) )   &    |- ( ph -> A. z e. A ( ( z # B /\ ( abs ` ( z - B ) ) < D ) -> ( abs ` ( ( F ` z ) - X ) ) < ( ( abs ` ( X - Y ) ) / 2 ) ) )   &    |- ( ph -> G e. RR+ )   &    |- ( ph -> A. w e. A ( ( w # B /\ ( abs ` ( w - B ) ) < G ) -> ( abs ` ( ( F ` w ) - Y ) ) < ( ( abs ` ( X - Y ) ) / 2 ) ) )   =>    |- ( ph -> ( abs ` ( X - Y ) ) < ( abs ` ( X - Y ) ) )
Theorem	limcimo 14985*	Conditions which ensure there is at most one limit value of F at B. (Contributed by Mario Carneiro, 25-Dec-2016.) (Revised by Jim Kingdon, 8-Jul-2023.)
|- ( ph -> F : A --> CC )   &    |- ( ph -> A C_ CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> B e. C )   &    |- ( ph -> B e. S )   &    |- ( ph -> C e. ( K ↾t S ) )   &    |- ( ph -> S e. { RR , CC } )   &    |- ( ph -> { q e. C | q # B } C_ A )   &    |- K = ( MetOpen ` ( abs o. - ) )   =>    |- ( ph -> E* x x e. ( F lim CC B ) )
Theorem	limcresi 14986	Any limit of F is also a limit of the restriction of F. (Contributed by Mario Carneiro, 28-Dec-2016.)
|- ( F lim CC B ) C_ ( ( F |` C ) lim CC B )
Theorem	cnplimcim 14987	If a function is continuous at B, its limit at B equals the value of the function there. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 14-Jun-2023.)
|- K = ( MetOpen ` ( abs o. - ) )   &    |- J = ( K ↾t A )   =>    |- ( ( A C_ CC /\ B e. A ) -> ( F e. ( ( J CnP K ) ` B ) -> ( F : A --> CC /\ ( F ` B ) e. ( F lim CC B ) ) ) )
Theorem	cnplimclemle 14988	Lemma for cnplimccntop 14990. Satisfying the epsilon condition for continuity. (Contributed by Mario Carneiro and Jim Kingdon, 17-Nov-2023.)
|- K = ( MetOpen ` ( abs o. - ) )   &    |- J = ( K ↾t A )   &    |- ( ph -> A C_ CC )   &    |- ( ph -> F : A --> CC )   &    |- ( ph -> B e. A )   &    |- ( ph -> ( F ` B ) e. ( F lim CC B ) )   &    |- ( ph -> E e. RR+ )   &    |- ( ph -> D e. RR+ )   &    |- ( ph -> Z e. A )   &    |- ( ( ph /\ Z # B /\ ( abs ` ( Z - B ) ) < D ) -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) < ( E / 2 ) )   &    |- ( ph -> ( abs ` ( Z - B ) ) < D )   =>    |- ( ph -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) < E )
Theorem	cnplimclemr 14989	Lemma for cnplimccntop 14990. The reverse direction. (Contributed by Mario Carneiro and Jim Kingdon, 17-Nov-2023.)
|- K = ( MetOpen ` ( abs o. - ) )   &    |- J = ( K ↾t A )   &    |- ( ph -> A C_ CC )   &    |- ( ph -> F : A --> CC )   &    |- ( ph -> B e. A )   &    |- ( ph -> ( F ` B ) e. ( F lim CC B ) )   =>    |- ( ph -> F e. ( ( J CnP K ) ` B ) )
Theorem	cnplimccntop 14990	A function is continuous at B iff its limit at B equals the value of the function there. (Contributed by Mario Carneiro, 28-Dec-2016.)
|- K = ( MetOpen ` ( abs o. - ) )   &    |- J = ( K ↾t A )   =>    |- ( ( A C_ CC /\ B e. A ) -> ( F e. ( ( J CnP K ) ` B ) <-> ( F : A --> CC /\ ( F ` B ) e. ( F lim CC B ) ) ) )
Theorem	cnlimcim 14991*	If F is a continuous function, the limit of the function at each point equals the value of the function. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 16-Jun-2023.)
|- ( A C_ CC -> ( F e. ( A -cn-> CC ) -> ( F : A --> CC /\ A. x e. A ( F ` x ) e. ( F lim CC x ) ) ) )
Theorem	cnlimc 14992*	F is a continuous function iff the limit of the function at each point equals the value of the function. (Contributed by Mario Carneiro, 28-Dec-2016.)
|- ( A C_ CC -> ( F e. ( A -cn-> CC ) <-> ( F : A --> CC /\ A. x e. A ( F ` x ) e. ( F lim CC x ) ) ) )
Theorem	cnlimci 14993	If F is a continuous function, then the limit of the function at any point equals its value. (Contributed by Mario Carneiro, 28-Dec-2016.)
|- ( ph -> F e. ( A -cn-> D ) )   &    |- ( ph -> B e. A )   =>    |- ( ph -> ( F ` B ) e. ( F lim CC B ) )
Theorem	cnmptlimc 14994*	If F is a continuous function, then the limit of the function at any point equals its value. (Contributed by Mario Carneiro, 28-Dec-2016.)
|- ( ph -> ( x e. A |-> X ) e. ( A -cn-> D ) )   &    |- ( ph -> B e. A )   &    |- ( x = B -> X = Y )   =>    |- ( ph -> Y e. ( ( x e. A |-> X ) lim CC B ) )
Theorem	limccnpcntop 14995	If the limit of F at B is C and G is continuous at C, then the limit of G o. F at B is G ( C ). (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 18-Jun-2023.)
|- ( ph -> F : A --> D )   &    |- ( ph -> D C_ CC )   &    |- K = ( MetOpen ` ( abs o. - ) )   &    |- J = ( K ↾t D )   &    |- ( ph -> C e. ( F lim CC B ) )   &    |- ( ph -> G e. ( ( J CnP K ) ` C ) )   =>    |- ( ph -> ( G ` C ) e. ( ( G o. F ) lim CC B ) )
Theorem	limccnp2lem 14996*	Lemma for limccnp2cntop 14997. This is most of the result, expressed in epsilon-delta form, with a large number of hypotheses so that lengthy expressions do not need to be repeated. (Contributed by Jim Kingdon, 9-Nov-2023.)
|- ( ( ph /\ x e. A ) -> R e. X )   &    |- ( ( ph /\ x e. A ) -> S e. Y )   &    |- ( ph -> X C_ CC )   &    |- ( ph -> Y C_ CC )   &    |- K = ( MetOpen ` ( abs o. - ) )   &    |- J = ( ( K tX K ) ↾t ( X X. Y ) )   &    |- ( ph -> C e. ( ( x e. A |-> R ) lim CC B ) )   &    |- ( ph -> D e. ( ( x e. A |-> S ) lim CC B ) )   &    |- ( ph -> H e. ( ( J CnP K ) ` <. C , D >. ) )   &    |- F/ x ph   &    |- ( ph -> E e. RR+ )   &    |- ( ph -> L e. RR+ )   &    |- ( ph -> A. r e. X A. s e. Y ( ( ( C ( ( abs o. - ) |` ( X X. X ) ) r ) < L /\ ( D ( ( abs o. - ) |` ( Y X. Y ) ) s ) < L ) -> ( ( C H D ) ( abs o. - ) ( r H s ) ) < E ) )   &    |- ( ph -> F e. RR+ )   &    |- ( ph -> A. x e. A ( ( x # B /\ ( abs ` ( x - B ) ) < F ) -> ( abs ` ( R - C ) ) < L ) )   &    |- ( ph -> G e. RR+ )   &    |- ( ph -> A. x e. A ( ( x # B /\ ( abs ` ( x - B ) ) < G ) -> ( abs ` ( S - D ) ) < L ) )   =>    |- ( ph -> E. d e. RR+ A. x e. A ( ( x # B /\ ( abs ` ( x - B ) ) < d ) -> ( abs ` ( ( R H S ) - ( C H D ) ) ) < E ) )
Theorem	limccnp2cntop 14997*	The image of a convergent sequence under a continuous map is convergent to the image of the original point. Binary operation version. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 14-Nov-2023.)
|- ( ( ph /\ x e. A ) -> R e. X )   &    |- ( ( ph /\ x e. A ) -> S e. Y )   &    |- ( ph -> X C_ CC )   &    |- ( ph -> Y C_ CC )   &    |- K = ( MetOpen ` ( abs o. - ) )   &    |- J = ( ( K tX K ) ↾t ( X X. Y ) )   &    |- ( ph -> C e. ( ( x e. A |-> R ) lim CC B ) )   &    |- ( ph -> D e. ( ( x e. A |-> S ) lim CC B ) )   &    |- ( ph -> H e. ( ( J CnP K ) ` <. C , D >. ) )   =>    |- ( ph -> ( C H D ) e. ( ( x e. A |-> ( R H S ) ) lim CC B ) )
Theorem	limccoap 14998*	Composition of two limits. This theorem is only usable in the case where x # X implies R(x) # C so it is less general than might appear at first. (Contributed by Mario Carneiro, 29-Dec-2016.) (Revised by Jim Kingdon, 18-Dec-2023.)
|- ( ( ph /\ x e. { w e. A | w # X } ) -> R e. { w e. B | w # C } )   &    |- ( ( ph /\ y e. { w e. B | w # C } ) -> S e. CC )   &    |- ( ph -> C e. ( ( x e. { w e. A | w # X } |-> R ) lim CC X ) )   &    |- ( ph -> D e. ( ( y e. { w e. B | w # C } |-> S ) lim CC C ) )   &    |- ( y = R -> S = T )   =>    |- ( ph -> D e. ( ( x e. { w e. A | w # X } |-> T ) lim CC X ) )
Theorem	reldvg 14999	The derivative function is a relation. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 25-Jun-2023.)
|- ( ( S C_ CC /\ F e. ( CC ^pm S ) ) -> Rel ( S _D F ) )
Theorem	dvlemap 15000*	Closure for a difference quotient. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)
|- ( ph -> F : D --> CC )   &    |- ( ph -> D C_ CC )   &    |- ( ph -> B e. D )   =>    |- ( ( ph /\ A e. { w e. D | w # B } ) -> ( ( ( F ` A ) - ( F ` B ) ) / ( A - B ) ) e. CC )