P. 151 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 15001-15100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	imcncf 15001	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 15002	Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
|- * e. ( CC -cn-> CC )
Theorem	mulc1cncf 15003*	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 15004*	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 15005	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 15006	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 15007	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 15008	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 15009	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 15010*	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 15011*	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 15012*	Composition of continuous functions. -cn-> analogue of cnmpt11f 14698. (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 15013*	Composition of continuous functions. -cn-> analogue of cnmpt12f 14700. (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 15014*	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 15015	The identity function is a continuous function on CC. (Contributed by Jeff Madsen, 11-Jun-2010.) (Moved into main set.mm as cncfmptid 15011 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 15016*	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 15017*	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 15018*	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 15019*	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 15020*	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 15021*	Lemma for mulcncf 15022. (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 15022*	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 15023*	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 15024*	The canonical bijection from ( RR X. RR ) to CC described in cnref1o 9771 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 15025*	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 15026*	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 15027*	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 15028*	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 15029*	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 15030*	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 15031*	Lemma for dedekindeu 15037. 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 15032*	Lemma for dedekindeu 15037. 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 15033*	Lemma for dedekindeu 15037. 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 15034*	Lemma for dedekindeu 15037. 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 15035*	Lemma for dedekindeu 15037. 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 15036*	Lemma for dedekindeu 15037. 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 15037*	A Dedekind cut identifies a unique real number. Similar to df-inp 7578 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 15038*	An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 8144 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 15039*	An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 8144 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 15040*	Lemma for dedekindicc 15047. 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 15041*	Lemma for dedekindicc 15047. 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 15042*	Lemma for dedekindicc 15047. 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 15043*	Lemma for dedekindicc 15047. 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 15044*	Lemma for dedekindicc 15047. 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 15045*	Lemma for dedekindicc 15047. 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 15046*	Lemma for dedekindicc 15047. Same as dedekindicc 15047, 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 15047*	A Dedekind cut identifies a unique real number. Similar to df-inp 7578 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 15048*	Lemma for ivthinc 15057. 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 15049*	Lemma for ivthinc 15057. 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 15050*	Lemma for ivthinc 15057. 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 15051*	Lemma for ivthinc 15057. 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 15052*	Lemma for ivthinc 15057. 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 15053*	Lemma for ivthinc 15057. 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 15054*	Lemma for ivthinc 15057. 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 15055*	Lemma for ivthinc 15057. 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 15056*	Lemma for ivthinc 15057. 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 15057*	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 15058*	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 15059*	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 15057). 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 15060	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 15061*	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 15062*	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 15063*	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 15064*	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 15065*	Lemma for ivthdich 15067. 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 15066*	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 15067*	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 15057 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 15060, hovera 15061, and hoverb 15062, 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 8139, c < 1 or 0 < c, and that leads to z <_ 0 by hoverlt1 15063 or 0 <_ z by hovergt0 15064. (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 15068	The limit operator.
class lim CC
Syntax	cdv 15069	The derivative operator.
class _D
Definition	df-limced 15070*	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 15071*	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 15072	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 15073	Closure of the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.)
|- ( F lim CC B ) C_ CC
Theorem	ellimc3apf 15074*	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 15075*	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 15076*	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 15077*	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 15078*	Lemma for limcimo 15079. (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 15079*	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 15080	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 15081	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 15082	Lemma for cnplimccntop 15084. 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 15083	Lemma for cnplimccntop 15084. 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 15084	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 15085*	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 15086*	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 15087	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 15088*	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 15089	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 15090*	Lemma for limccnp2cntop 15091. 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 15091*	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 15092*	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 15093	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 15094*	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 )
Theorem	dvfvalap 15095*	Value and set bounds on the derivative operator. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)
|- T = ( K ↾t S )   &    |- K = ( MetOpen ` ( abs o. - ) )   =>    |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> ( ( S _D F ) = U_ x e. ( ( int ` T ) ` A ) ( { x } X. ( ( z e. { w e. A | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) ) /\ ( S _D F ) C_ ( ( ( int ` T ) ` A ) X. CC ) ) )
Theorem	eldvap 15096*	The differentiable predicate. A function F is differentiable at B with derivative C iff F is defined in a neighborhood of B and the difference quotient has limit C at B. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)
|- T = ( K ↾t S )   &    |- K = ( MetOpen ` ( abs o. - ) )   &    |- G = ( z e. { w e. A | w # B } |-> ( ( ( F ` z ) - ( F ` B ) ) / ( z - B ) ) )   &    |- ( ph -> S C_ CC )   &    |- ( ph -> F : A --> CC )   &    |- ( ph -> A C_ S )   =>    |- ( ph -> ( B ( S _D F ) C <-> ( B e. ( ( int ` T ) ` A ) /\ C e. ( G lim CC B ) ) ) )
Theorem	dvcl 15097	The derivative function takes values in the complex numbers. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)
|- ( ph -> S C_ CC )   &    |- ( ph -> F : A --> CC )   &    |- ( ph -> A C_ S )   =>    |- ( ( ph /\ B ( S _D F ) C ) -> C e. CC )
Theorem	dvbssntrcntop 15098	The set of differentiable points is a subset of the interior of the domain of the function. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)
|- ( ph -> S C_ CC )   &    |- ( ph -> F : A --> CC )   &    |- ( ph -> A C_ S )   &    |- J = ( K ↾t S )   &    |- K = ( MetOpen ` ( abs o. - ) )   =>    |- ( ph -> dom ( S _D F ) C_ ( ( int ` J ) ` A ) )
Theorem	dvbss 15099	The set of differentiable points is a subset of the domain of the function. (Contributed by Mario Carneiro, 6-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)
|- ( ph -> S C_ CC )   &    |- ( ph -> F : A --> CC )   &    |- ( ph -> A C_ S )   =>    |- ( ph -> dom ( S _D F ) C_ A )
Theorem	dvbsssg 15100	The set of differentiable points is a subset of the ambient topology. (Contributed by Mario Carneiro, 18-Mar-2015.) (Revised by Jim Kingdon, 28-Jun-2023.)
|- ( ( S C_ CC /\ F e. ( CC ^pm S ) ) -> dom ( S _D F ) C_ S )