P. 152 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 15101-15200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	tgioo 15101	The topology generated by open intervals of reals is the same as the open sets of the standard metric space on the reals. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
|- D = ( ( abs o. - ) |` ( RR X. RR ) )   &    |- J = ( MetOpen ` D )   =>    |- ( topGen ` ran (,) ) = J
Theorem	tgqioo 15102	The topology generated by open intervals of reals with rational endpoints is the same as the open sets of the standard metric space on the reals. In particular, this proves that the standard topology on the reals is second-countable. (Contributed by Mario Carneiro, 17-Jun-2014.)
|- Q = ( topGen ` ( (,) " ( QQ X. QQ ) ) )   =>    |- ( topGen ` ran (,) ) = Q
Theorem	resubmet 15103	The subspace topology induced by a subset of the reals. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Aug-2014.)
|- R = ( topGen ` ran (,) )   &    |- J = ( MetOpen ` ( ( abs o. - ) |` ( A X. A ) ) )   =>    |- ( A C_ RR -> J = ( R ↾t A ) )
Theorem	tgioo2cntop 15104	The standard topology on the reals is a subspace of the complex metric topology. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Jim Kingdon, 6-Aug-2023.)
|- J = ( MetOpen ` ( abs o. - ) )   =>    |- ( topGen ` ran (,) ) = ( J ↾t RR )
Theorem	rerestcntop 15105	The subspace topology induced by a subset of the reals. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Jim Kingdon, 6-Aug-2023.)
|- J = ( MetOpen ` ( abs o. - ) )   &    |- R = ( topGen ` ran (,) )   =>    |- ( A C_ RR -> ( J ↾t A ) = ( R ↾t A ) )
Theorem	tgioo2 15106	The standard topology on the reals is a subspace of the complex metric topology. (Contributed by Mario Carneiro, 13-Aug-2014.)
|- J = ( TopOpen ` ℂfld )   =>    |- ( topGen ` ran (,) ) = ( J ↾t RR )
Theorem	rerest 15107	The subspace topology induced by a subset of the reals. (Contributed by Mario Carneiro, 13-Aug-2014.)
|- J = ( TopOpen ` ℂfld )   &    |- R = ( topGen ` ran (,) )   =>    |- ( A C_ RR -> ( J ↾t A ) = ( R ↾t A ) )
Theorem	addcncntoplem 15108*	Lemma for addcncntop 15109, subcncntop 15110, and mulcncntop 15111. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Jim Kingdon, 22-Oct-2023.)
|- J = ( MetOpen ` ( abs o. - ) )   &    |- .+ : ( CC X. CC ) --> CC   &    |- ( ( a e. RR+ /\ b e. CC /\ c e. CC ) -> E. y e. RR+ E. z e. RR+ A. u e. CC A. v e. CC ( ( ( abs ` ( u - b ) ) < y /\ ( abs ` ( v - c ) ) < z ) -> ( abs ` ( ( u .+ v ) - ( b .+ c ) ) ) < a ) )   =>    |- .+ e. ( ( J tX J ) Cn J )
Theorem	addcncntop 15109	Complex number addition is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by NM, 30-Jul-2007.) (Proof shortened by Mario Carneiro, 5-May-2014.)
|- J = ( MetOpen ` ( abs o. - ) )   =>    |- + e. ( ( J tX J ) Cn J )
Theorem	subcncntop 15110	Complex number subtraction is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by NM, 4-Aug-2007.) (Proof shortened by Mario Carneiro, 5-May-2014.)
|- J = ( MetOpen ` ( abs o. - ) )   =>    |- - e. ( ( J tX J ) Cn J )
Theorem	mulcncntop 15111	Complex number multiplication is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by NM, 30-Jul-2007.) (Proof shortened by Mario Carneiro, 5-May-2014.)
|- J = ( MetOpen ` ( abs o. - ) )   =>    |- x. e. ( ( J tX J ) Cn J )
Theorem	divcnap 15112*	Complex number division is a continuous function, when the second argument is apart from zero. (Contributed by Mario Carneiro, 12-Aug-2014.) (Revised by Jim Kingdon, 25-Oct-2023.)
|- J = ( MetOpen ` ( abs o. - ) )   &    |- K = ( J ↾t { x e. CC | x # 0 } )   =>    |- ( y e. CC , z e. { x e. CC | x # 0 } |-> ( y / z ) ) e. ( ( J tX K ) Cn J )
Theorem	mpomulcn 15113*	Complex number multiplication is a continuous function. (Contributed by GG, 16-Mar-2025.)
|- J = ( TopOpen ` ℂfld )   =>    |- ( x e. CC , y e. CC |-> ( x x. y ) ) e. ( ( J tX J ) Cn J )
Theorem	fsumcncntop 15114*	A finite sum of functions to complex numbers from a common topological space is continuous. The class expression for B normally contains free variables k and x to index it. (Contributed by NM, 8-Aug-2007.) (Revised by Mario Carneiro, 23-Aug-2014.)
|- K = ( MetOpen ` ( abs o. - ) )   &    |- ( ph -> J e. (TopOn ` X ) )   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> ( x e. X |-> B ) e. ( J Cn K ) )   =>    |- ( ph -> ( x e. X |-> sum_ k e. A B ) e. ( J Cn K ) )
Theorem	fsumcn 15115*	A finite sum of functions to complex numbers from a common topological space is continuous. The class expression for B normally contains free variables k and x to index it. (Contributed by NM, 8-Aug-2007.) (Revised by Mario Carneiro, 23-Aug-2014.)
|- K = ( TopOpen ` ℂfld )   &    |- ( ph -> J e. (TopOn ` X ) )   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> ( x e. X |-> B ) e. ( J Cn K ) )   =>    |- ( ph -> ( x e. X |-> sum_ k e. A B ) e. ( J Cn K ) )
Theorem	expcn 15116*	The power function on complex numbers, for fixed exponent N, is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) Avoid ax-mulf 8068. (Revised by GG, 16-Mar-2025.)
|- J = ( TopOpen ` ℂfld )   =>    |- ( N e. NN0 -> ( x e. CC |-> ( x ^ N ) ) e. ( J Cn J ) )
9.2.7  Topological definitions using the reals
Syntax	ccncf 15117	Extend class notation to include the operation which returns a class of continuous complex functions.
class -cn->
Definition	df-cncf 15118*	Define the operation whose value is a class of continuous complex functions. (Contributed by Paul Chapman, 11-Oct-2007.)
|- -cn-> = ( a e. ~P CC , b e. ~P CC |-> { f e. ( b ^m a ) | A. x e. a A. e e. RR+ E. d e. RR+ A. y e. a ( ( abs ` ( x - y ) ) < d -> ( abs ` ( ( f ` x ) - ( f ` y ) ) ) < e ) } )
Theorem	cncfval 15119*	The value of the continuous complex function operation is the set of continuous functions from A to B. (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.)
|- ( ( A C_ CC /\ B C_ CC ) -> ( A -cn-> B ) = { f e. ( B ^m A ) | A. x e. A A. y e. RR+ E. z e. RR+ A. w e. A ( ( abs ` ( x - w ) ) < z -> ( abs ` ( ( f ` x ) - ( f ` w ) ) ) < y ) } )
Theorem	elcncf 15120*	Membership in the set of continuous complex functions from A to B. (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.)
|- ( ( A C_ CC /\ B C_ CC ) -> ( F e. ( A -cn-> B ) <-> ( F : A --> B /\ A. x e. A A. y e. RR+ E. z e. RR+ A. w e. A ( ( abs ` ( x - w ) ) < z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) ) ) )
Theorem	elcncf2 15121*	Version of elcncf 15120 with arguments commuted. (Contributed by Mario Carneiro, 28-Apr-2014.)
|- ( ( A C_ CC /\ B C_ CC ) -> ( F e. ( A -cn-> B ) <-> ( F : A --> B /\ A. x e. A A. y e. RR+ E. z e. RR+ A. w e. A ( ( abs ` ( w - x ) ) < z -> ( abs ` ( ( F ` w ) - ( F ` x ) ) ) < y ) ) ) )
Theorem	cncfrss 15122	Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
|- ( F e. ( A -cn-> B ) -> A C_ CC )
Theorem	cncfrss2 15123	Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
|- ( F e. ( A -cn-> B ) -> B C_ CC )
Theorem	cncff 15124	A continuous complex function's domain and codomain. (Contributed by Paul Chapman, 17-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.)
|- ( F e. ( A -cn-> B ) -> F : A --> B )
Theorem	cncfi 15125*	Defining property of a continuous function. (Contributed by Mario Carneiro, 30-Apr-2014.) (Revised by Mario Carneiro, 25-Aug-2014.)
|- ( ( F e. ( A -cn-> B ) /\ C e. A /\ R e. RR+ ) -> E. z e. RR+ A. w e. A ( ( abs ` ( w - C ) ) < z -> ( abs ` ( ( F ` w ) - ( F ` C ) ) ) < R ) )
Theorem	elcncf1di 15126*	Membership in the set of continuous complex functions from A to B. (Contributed by Paul Chapman, 26-Nov-2007.)
|- ( ph -> F : A --> B )   &    |- ( ph -> ( ( x e. A /\ y e. RR+ ) -> Z e. RR+ ) )   &    |- ( ph -> ( ( ( x e. A /\ w e. A ) /\ y e. RR+ ) -> ( ( abs ` ( x - w ) ) < Z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) ) )   =>    |- ( ph -> ( ( A C_ CC /\ B C_ CC ) -> F e. ( A -cn-> B ) ) )
Theorem	elcncf1ii 15127*	Membership in the set of continuous complex functions from A to B. (Contributed by Paul Chapman, 26-Nov-2007.)
|- F : A --> B   &    |- ( ( x e. A /\ y e. RR+ ) -> Z e. RR+ )   &    |- ( ( ( x e. A /\ w e. A ) /\ y e. RR+ ) -> ( ( abs ` ( x - w ) ) < Z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) )   =>    |- ( ( A C_ CC /\ B C_ CC ) -> F e. ( A -cn-> B ) )
Theorem	rescncf 15128	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 15129	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 15130	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 15131	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 15132	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 15133	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 15134	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 15135	Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
|- * e. ( CC -cn-> CC )
Theorem	mulc1cncf 15136*	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 15137*	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 15138	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 15139	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 15140	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 15141	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 15142	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 15143*	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 15144*	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 15145*	Composition of continuous functions. -cn-> analogue of cnmpt11f 14831. (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 15146*	Composition of continuous functions. -cn-> analogue of cnmpt12f 14833. (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 15147*	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 15148	The identity function is a continuous function on CC. (Contributed by Jeff Madsen, 11-Jun-2010.) (Moved into main set.mm as cncfmptid 15144 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 15149*	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 15150*	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 15151*	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 15152*	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 15153*	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 15154*	Lemma for mulcncf 15155. (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 15155*	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 15156*	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 15157*	The canonical bijection from ( RR X. RR ) to CC described in cnref1o 9792 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 15158*	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 15159*	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 15160*	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 15161*	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 15162*	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 15163*	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 15164*	Lemma for dedekindeu 15170. 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 15165*	Lemma for dedekindeu 15170. 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 15166*	Lemma for dedekindeu 15170. 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 15167*	Lemma for dedekindeu 15170. 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 15168*	Lemma for dedekindeu 15170. 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 15169*	Lemma for dedekindeu 15170. 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 15170*	A Dedekind cut identifies a unique real number. Similar to df-inp 7599 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 15171*	An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 8165 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 15172*	An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 8165 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 15173*	Lemma for dedekindicc 15180. 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 15174*	Lemma for dedekindicc 15180. 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 15175*	Lemma for dedekindicc 15180. 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 15176*	Lemma for dedekindicc 15180. 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 15177*	Lemma for dedekindicc 15180. 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 15178*	Lemma for dedekindicc 15180. 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 15179*	Lemma for dedekindicc 15180. Same as dedekindicc 15180, 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 15180*	A Dedekind cut identifies a unique real number. Similar to df-inp 7599 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 15181*	Lemma for ivthinc 15190. 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 15182*	Lemma for ivthinc 15190. 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 15183*	Lemma for ivthinc 15190. 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 15184*	Lemma for ivthinc 15190. 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 15185*	Lemma for ivthinc 15190. 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 15186*	Lemma for ivthinc 15190. 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 15187*	Lemma for ivthinc 15190. 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 15188*	Lemma for ivthinc 15190. 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 15189*	Lemma for ivthinc 15190. 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 15190*	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 15191*	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 15192*	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 15190). 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 15193	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 15194*	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 15195*	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 15196*	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 15197*	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 15198*	Lemma for ivthdich 15200. 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 15199*	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 15200*	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 15190 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 15193, hovera 15194, and hoverb 15195, 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 8160, c < 1 or 0 < c, and that leads to z <_ 0 by hoverlt1 15196 or 0 <_ z by hovergt0 15197. (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 ) )