P. 157 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 15601-15700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	metcnpi3 15601*	Epsilon-delta property of a metric space function continuous at P. A variation of metcnpi2 15600 with non-strict ordering. (Contributed by NM, 16-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
|- J = ( MetOpen ` C )   &    |- K = ( MetOpen ` D )   =>    |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) ) /\ ( F e. ( ( J CnP K ) ` P ) /\ A e. RR+ ) ) -> E. x e. RR+ A. y e. X ( ( y C P ) <_ x -> ( ( F ` y ) D ( F ` P ) ) <_ A ) )
Theorem	txmetcnp 15602*	Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.) (Revised by Jim Kingdon, 22-Oct-2023.)
|- J = ( MetOpen ` C )   &    |- K = ( MetOpen ` D )   &    |- L = ( MetOpen ` E )   =>    |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) -> ( F e. ( ( ( J tX K ) CnP L ) ` <. A , B >. ) <-> ( F : ( X X. Y ) --> Z /\ A. z e. RR+ E. w e. RR+ A. u e. X A. v e. Y ( ( ( A C u ) < w /\ ( B D v ) < w ) -> ( ( A F B ) E ( u F v ) ) < z ) ) ) )
Theorem	txmetcn 15603*	Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- J = ( MetOpen ` C )   &    |- K = ( MetOpen ` D )   &    |- L = ( MetOpen ` E )   =>    |- ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) -> ( F e. ( ( J tX K ) Cn L ) <-> ( F : ( X X. Y ) --> Z /\ A. x e. X A. y e. Y A. z e. RR+ E. w e. RR+ A. u e. X A. v e. Y ( ( ( x C u ) < w /\ ( y D v ) < w ) -> ( ( x F y ) E ( u F v ) ) < z ) ) ) )
Theorem	metcnpd 15604*	Two ways to say a mapping from metric C to metric D is continuous at point P. (Contributed by Jim Kingdon, 14-Jun-2023.)
|- ( ph -> J = ( MetOpen ` C ) )   &    |- ( ph -> K = ( MetOpen ` D ) )   &    |- ( ph -> C e. ( *Met ` X ) )   &    |- ( ph -> D e. ( *Met ` Y ) )   &    |- ( ph -> P e. X )   =>    |- ( ph -> ( F e. ( ( J CnP K ) ` P ) <-> ( F : X --> Y /\ A. y e. RR+ E. z e. RR+ A. w e. X ( ( P C w ) < z -> ( ( F ` P ) D ( F ` w ) ) < y ) ) ) )
9.2.6  Topology on the reals
Theorem	qtopbasss 15605*	The set of open intervals with endpoints in a subset forms a basis for a topology. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Jim Kingdon, 22-May-2023.)
|- S C_ RR*   &    |- ( ( x e. S /\ y e. S ) -> sup ( { x , y } , RR* , < ) e. S )   &    |- ( ( x e. S /\ y e. S ) -> inf ( { x , y } , RR* , < ) e. S )   =>    |- ( (,) " ( S X. S ) ) e. TopBases
Theorem	qtopbas 15606	The set of open intervals with rational endpoints forms a basis for a topology. (Contributed by NM, 8-Mar-2007.)
|- ( (,) " ( QQ X. QQ ) ) e. TopBases
Theorem	retopbas 15607	A basis for the standard topology on the reals. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 17-Jun-2014.)
|- ran (,) e. TopBases
Theorem	retop 15608	The standard topology on the reals. (Contributed by FL, 4-Jun-2007.)
|- ( topGen ` ran (,) ) e. Top
Theorem	uniretop 15609	The underlying set of the standard topology on the reals is the reals. (Contributed by FL, 4-Jun-2007.)
|- RR = U. ( topGen ` ran (,) )
Theorem	retopon 15610	The standard topology on the reals is a topology on the reals. (Contributed by Mario Carneiro, 28-Aug-2015.)
|- ( topGen ` ran (,) ) e. (TopOn ` RR )
Theorem	retps 15611	The standard topological space on the reals. (Contributed by NM, 19-Oct-2012.)
|- K = { <. ( Base ` ndx ) , RR >. , <. (TopSet ` ndx ) , ( topGen ` ran (,) ) >. }   =>    |- K e. TopSp
Theorem	iooretopg 15612	Open intervals are open sets of the standard topology on the reals . (Contributed by FL, 18-Jun-2007.) (Revised by Jim Kingdon, 23-May-2023.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( A (,) B ) e. ( topGen ` ran (,) ) )
Theorem	cnmetdval 15613	Value of the distance function of the metric space of complex numbers. (Contributed by NM, 9-Dec-2006.) (Revised by Mario Carneiro, 27-Dec-2014.)
|- D = ( abs o. - )   =>    |- ( ( A e. CC /\ B e. CC ) -> ( A D B ) = ( abs ` ( A - B ) ) )
Theorem	cnmet 15614	The absolute value metric determines a metric space on the complex numbers. This theorem provides a link between complex numbers and metrics spaces, making metric space theorems available for use with complex numbers. (Contributed by FL, 9-Oct-2006.)
|- ( abs o. - ) e. ( Met ` CC )
Theorem	cnxmet 15615	The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
|- ( abs o. - ) e. ( *Met ` CC )
Theorem	cntoptopon 15616	The topology of the complex numbers is a topology. (Contributed by Jim Kingdon, 6-Jun-2023.)
|- J = ( MetOpen ` ( abs o. - ) )   =>    |- J e. (TopOn ` CC )
Theorem	cntoptop 15617	The topology of the complex numbers is a topology. (Contributed by Jim Kingdon, 6-Jun-2023.)
|- J = ( MetOpen ` ( abs o. - ) )   =>    |- J e. Top
Theorem	cnbl0 15618	Two ways to write the open ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.)
|- D = ( abs o. - )   =>    |- ( R e. RR* -> ( `' abs " ( 0 [,) R ) ) = ( 0 ( ball ` D ) R ) )
Theorem	cnblcld 15619*	Two ways to write the closed ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.)
|- D = ( abs o. - )   =>    |- ( R e. RR* -> ( `' abs " ( 0 [,] R ) ) = { x e. CC | ( 0 D x ) <_ R } )
Theorem	cnfldms 15620	The complex number field is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
|- ℂfld e. MetSp
Theorem	cnfldxms 15621	The complex number field is a topological space. (Contributed by Mario Carneiro, 28-Aug-2015.)
|- ℂfld e. *MetSp
Theorem	cnfldtps 15622	The complex number field is a topological space. (Contributed by Mario Carneiro, 28-Aug-2015.)
|- ℂfld e. TopSp
Theorem	cnfldtopn 15623	The topology of the complex numbers. (Contributed by Mario Carneiro, 28-Aug-2015.)
|- J = ( TopOpen ` ℂfld )   =>    |- J = ( MetOpen ` ( abs o. - ) )
Theorem	cnfldtopon 15624	The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- J = ( TopOpen ` ℂfld )   =>    |- J e. (TopOn ` CC )
Theorem	cnfldtop 15625	The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- J = ( TopOpen ` ℂfld )   =>    |- J e. Top
Theorem	unicntopcntop 15626	The underlying set of the standard topology on the complex numbers is the set of complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by Jim Kingdon, 12-Dec-2023.)
|- CC = U. ( MetOpen ` ( abs o. - ) )
Theorem	unicntop 15627	The underlying set of the standard topology on the complex numbers is the set of complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- CC = U. ( TopOpen ` ℂfld )
Theorem	cnopncntop 15628	The set of complex numbers is open with respect to the standard topology on complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by Jim Kingdon, 12-Dec-2023.)
|- CC e. ( MetOpen ` ( abs o. - ) )
Theorem	cnopn 15629	The set of complex numbers is open with respect to the standard topology on complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- CC e. ( TopOpen ` ℂfld )
Theorem	reopnap 15630*	The real numbers apart from a given real number form an open set. (Contributed by Jim Kingdon, 13-Dec-2023.)
|- ( A e. RR -> { w e. RR | w # A } e. ( topGen ` ran (,) ) )
Theorem	remetdval 15631	Value of the distance function of the metric space of real numbers. (Contributed by NM, 16-May-2007.)
|- D = ( ( abs o. - ) |` ( RR X. RR ) )   =>    |- ( ( A e. RR /\ B e. RR ) -> ( A D B ) = ( abs ` ( A - B ) ) )
Theorem	remet 15632	The absolute value metric determines a metric space on the reals. (Contributed by NM, 10-Feb-2007.)
|- D = ( ( abs o. - ) |` ( RR X. RR ) )   =>    |- D e. ( Met ` RR )
Theorem	rexmet 15633	The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
|- D = ( ( abs o. - ) |` ( RR X. RR ) )   =>    |- D e. ( *Met ` RR )
Theorem	bl2ioo 15634	A ball in terms of an open interval of reals. (Contributed by NM, 18-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
|- D = ( ( abs o. - ) |` ( RR X. RR ) )   =>    |- ( ( A e. RR /\ B e. RR ) -> ( A ( ball ` D ) B ) = ( ( A - B ) (,) ( A + B ) ) )
Theorem	ioo2bl 15635	An open interval of reals in terms of a ball. (Contributed by NM, 18-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
|- D = ( ( abs o. - ) |` ( RR X. RR ) )   =>    |- ( ( A e. RR /\ B e. RR ) -> ( A (,) B ) = ( ( ( A + B ) / 2 ) ( ball ` D ) ( ( B - A ) / 2 ) ) )
Theorem	ioo2blex 15636	An open interval of reals in terms of a ball. (Contributed by Mario Carneiro, 14-Nov-2013.)
|- D = ( ( abs o. - ) |` ( RR X. RR ) )   =>    |- ( ( A e. RR /\ B e. RR ) -> ( A (,) B ) e. ran ( ball ` D ) )
Theorem	blssioo 15637	The balls of the standard real metric space are included in the open real intervals. (Contributed by NM, 8-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
|- D = ( ( abs o. - ) |` ( RR X. RR ) )   =>    |- ran ( ball ` D ) C_ ran (,)
Theorem	tgioo 15638	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 15639	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 15640	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 15641	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 15642	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 15643	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 15644	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 15645*	Lemma for addcncntop 15646, subcncntop 15647, and mulcncntop 15648. (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 15646	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 15647	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 15648	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 15649*	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 15650*	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 15651*	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 15652*	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 15653*	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 8296. (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 15654	Extend class notation to include the operation which returns a class of continuous complex functions.
class -cn->
Definition	df-cncf 15655*	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 15656*	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 15657*	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 15658*	Version of elcncf 15657 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 15659	Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
|- ( F e. ( A -cn-> B ) -> A C_ CC )
Theorem	cncfrss2 15660	Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
|- ( F e. ( A -cn-> B ) -> B C_ CC )
Theorem	cncff 15661	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 15662*	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 15663*	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 15664*	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 15665	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 15666	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 15667	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 15668	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 15669	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 15670	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 15671	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 15672	Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
|- * e. ( CC -cn-> CC )
Theorem	mulc1cncf 15673*	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 15674*	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 15675	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 15676	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 15677	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 15678	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 15679	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 15680*	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 15681*	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 15682*	Composition of continuous functions. -cn-> analogue of cnmpt11f 15368. (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 15683*	Composition of continuous functions. -cn-> analogue of cnmpt12f 15370. (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 15684*	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 15685	The identity function is a continuous function on CC. (Contributed by Jeff Madsen, 11-Jun-2010.) (Moved into main set.mm as cncfmptid 15681 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 15686*	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 15687*	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 15688*	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 15689*	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 15690*	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 15691*	Lemma for mulcncf 15692. (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 15692*	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 15693*	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 15694*	The canonical bijection from ( RR X. RR ) to CC described in cnref1o 10034 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 15695*	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 15696*	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 15697*	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 15698*	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 15699*	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 15700*	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 ) )