P. 154 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 15301-15400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cnplimclemle 15301	Lemma for cnplimccntop 15303. 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 15302	Lemma for cnplimccntop 15303. 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 15303	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 15304*	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 15305*	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 15306	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 15307*	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 15308	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 15309*	Lemma for limccnp2cntop 15310. 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 15310*	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 15311*	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 15312	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 15313*	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 15314*	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 15315*	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 15316	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 15317	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 15318	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 15319	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 )
Theorem	recnprss 15320	Both RR and CC are subsets of CC. (Contributed by Mario Carneiro, 10-Feb-2015.)
|- ( S e. { RR , CC } -> S C_ CC )
Theorem	dvfgg 15321	Explicitly write out the functionality condition on derivative for S = RR and CC. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Jim Kingdon, 28-Jun-2023.)
|- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) -> ( S _D F ) : dom ( S _D F ) --> CC )
Theorem	dvfpm 15322	The derivative is a function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 28-Jul-2023.)
|- ( F e. ( CC ^pm RR ) -> ( RR _D F ) : dom ( RR _D F ) --> CC )
Theorem	dvfcnpm 15323	The derivative is a function. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Jim Kingdon, 28-Jul-2023.)
|- ( F e. ( CC ^pm CC ) -> ( CC _D F ) : dom ( CC _D F ) --> CC )
Theorem	dvidlemap 15324*	Lemma for dvid 15328 and dvconst 15327. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 2-Aug-2023.)
|- ( ph -> F : CC --> CC )   &    |- ( ( ph /\ ( x e. CC /\ z e. CC /\ z # x ) ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )   &    |- B e. CC   =>    |- ( ph -> ( CC _D F ) = ( CC X. { B } ) )
Theorem	dvidrelem 15325*	Lemma for dvidre 15330 and dvconstre 15329. Analogue of dvidlemap 15324 for real numbers rather than complex numbers. (Contributed by Jim Kingdon, 3-Oct-2025.)
|- ( ph -> F : RR --> CC )   &    |- ( ( ph /\ ( x e. RR /\ z e. RR /\ z # x ) ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )   &    |- B e. CC   =>    |- ( ph -> ( RR _D F ) = ( RR X. { B } ) )
Theorem	dvidsslem 15326*	Lemma for dvconstss 15331. Analogue of dvidlemap 15324 where F is defined on an open subset of the real or complex numbers. (Contributed by Jim Kingdon, 3-Oct-2025.)
|- ( ph -> S e. { RR , CC } )   &    |- J = ( K ↾t S )   &    |- K = ( MetOpen ` ( abs o. - ) )   &    |- ( ph -> F : X --> CC )   &    |- ( ph -> X e. J )   &    |- ( ( ph /\ ( x e. X /\ z e. X /\ z # x ) ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )   &    |- B e. CC   =>    |- ( ph -> ( S _D F ) = ( X X. { B } ) )
Theorem	dvconst 15327	Derivative of a constant function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 2-Aug-2023.)
|- ( A e. CC -> ( CC _D ( CC X. { A } ) ) = ( CC X. { 0 } ) )
Theorem	dvid 15328	Derivative of the identity function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 2-Aug-2023.)
|- ( CC _D ( _I |` CC ) ) = ( CC X. { 1 } )
Theorem	dvconstre 15329	Real derivative of a constant function. (Contributed by Jim Kingdon, 3-Oct-2025.)
|- ( A e. CC -> ( RR _D ( RR X. { A } ) ) = ( RR X. { 0 } ) )
Theorem	dvidre 15330	Real derivative of the identity function. (Contributed by Jim Kingdon, 3-Oct-2025.)
|- ( RR _D ( _I |` RR ) ) = ( RR X. { 1 } )
Theorem	dvconstss 15331	Derivative of a constant function defined on an open set. (Contributed by Jim Kingdon, 6-Oct-2025.)
|- ( ph -> S e. { RR , CC } )   &    |- J = ( K ↾t S )   &    |- K = ( MetOpen ` ( abs o. - ) )   &    |- ( ph -> X e. J )   &    |- ( ph -> A e. CC )   =>    |- ( ph -> ( S _D ( X X. { A } ) ) = ( X X. { 0 } ) )
Theorem	dvcnp2cntop 15332	A function is continuous at each point for which it is differentiable. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
|- J = ( K ↾t A )   &    |- K = ( MetOpen ` ( abs o. - ) )   =>    |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ B e. dom ( S _D F ) ) -> F e. ( ( J CnP K ) ` B ) )
Theorem	dvcn 15333	A differentiable function is continuous. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-Sep-2015.)
|- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ dom ( S _D F ) = A ) -> F e. ( A -cn-> CC ) )
Theorem	dvaddxxbr 15334	The sum rule for derivatives at a point. That is, if the derivative of F at C is K and the derivative of G at C is L, then the derivative of the pointwise sum of those two functions at C is K + L. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 25-Nov-2023.)
|- ( ph -> F : X --> CC )   &    |- ( ph -> X C_ S )   &    |- ( ph -> G : X --> CC )   &    |- ( ph -> S C_ CC )   &    |- ( ph -> C ( S _D F ) K )   &    |- ( ph -> C ( S _D G ) L )   &    |- J = ( MetOpen ` ( abs o. - ) )   =>    |- ( ph -> C ( S _D ( F oF + G ) ) ( K + L ) )
Theorem	dvmulxxbr 15335	The product rule for derivatives at a point. For the (simpler but more limited) function version, see dvmulxx 15337. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 1-Dec-2023.)
|- ( ph -> F : X --> CC )   &    |- ( ph -> X C_ S )   &    |- ( ph -> G : X --> CC )   &    |- ( ph -> S C_ CC )   &    |- ( ph -> C ( S _D F ) K )   &    |- ( ph -> C ( S _D G ) L )   &    |- J = ( MetOpen ` ( abs o. - ) )   =>    |- ( ph -> C ( S _D ( F oF x. G ) ) ( ( K x. ( G ` C ) ) + ( L x. ( F ` C ) ) ) )
Theorem	dvaddxx 15336	The sum rule for derivatives at a point. For the (more general) relation version, see dvaddxxbr 15334. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 25-Nov-2023.)
|- ( ph -> F : X --> CC )   &    |- ( ph -> X C_ S )   &    |- ( ph -> G : X --> CC )   &    |- ( ph -> S e. { RR , CC } )   &    |- ( ph -> C e. dom ( S _D F ) )   &    |- ( ph -> C e. dom ( S _D G ) )   =>    |- ( ph -> ( ( S _D ( F oF + G ) ) ` C ) = ( ( ( S _D F ) ` C ) + ( ( S _D G ) ` C ) ) )
Theorem	dvmulxx 15337	The product rule for derivatives at a point. For the (more general) relation version, see dvmulxxbr 15335. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 2-Dec-2023.)
|- ( ph -> F : X --> CC )   &    |- ( ph -> X C_ S )   &    |- ( ph -> G : X --> CC )   &    |- ( ph -> S e. { RR , CC } )   &    |- ( ph -> C e. dom ( S _D F ) )   &    |- ( ph -> C e. dom ( S _D G ) )   =>    |- ( ph -> ( ( S _D ( F oF x. G ) ) ` C ) = ( ( ( ( S _D F ) ` C ) x. ( G ` C ) ) + ( ( ( S _D G ) ` C ) x. ( F ` C ) ) ) )
Theorem	dviaddf 15338	The sum rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
|- ( ph -> S e. { RR , CC } )   &    |- ( ph -> X C_ S )   &    |- ( ph -> F : X --> CC )   &    |- ( ph -> G : X --> CC )   &    |- ( ph -> dom ( S _D F ) = X )   &    |- ( ph -> dom ( S _D G ) = X )   =>    |- ( ph -> ( S _D ( F oF + G ) ) = ( ( S _D F ) oF + ( S _D G ) ) )
Theorem	dvimulf 15339	The product rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
|- ( ph -> S e. { RR , CC } )   &    |- ( ph -> X C_ S )   &    |- ( ph -> F : X --> CC )   &    |- ( ph -> G : X --> CC )   &    |- ( ph -> dom ( S _D F ) = X )   &    |- ( ph -> dom ( S _D G ) = X )   =>    |- ( ph -> ( S _D ( F oF x. G ) ) = ( ( ( S _D F ) oF x. G ) oF + ( ( S _D G ) oF x. F ) ) )
Theorem	dvcoapbr 15340*	The chain rule for derivatives at a point. The u # C -> ( G ` u ) # ( G ` C ) hypothesis constrains what functions work for G. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 21-Dec-2023.)
|- ( ph -> F : X --> CC )   &    |- ( ph -> X C_ S )   &    |- ( ph -> G : Y --> X )   &    |- ( ph -> Y C_ T )   &    |- ( ph -> A. u e. Y ( u # C -> ( G ` u ) # ( G ` C ) ) )   &    |- ( ph -> S C_ CC )   &    |- ( ph -> T C_ CC )   &    |- ( ph -> ( G ` C ) ( S _D F ) K )   &    |- ( ph -> C ( T _D G ) L )   &    |- J = ( MetOpen ` ( abs o. - ) )   =>    |- ( ph -> C ( T _D ( F o. G ) ) ( K x. L ) )
Theorem	dvcjbr 15341	The derivative of the conjugate of a function. For the (simpler but more limited) function version, see dvcj 15342. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
|- ( ph -> F : X --> CC )   &    |- ( ph -> X C_ RR )   &    |- ( ph -> C e. dom ( RR _D F ) )   =>    |- ( ph -> C ( RR _D ( * o. F ) ) ( * ` ( ( RR _D F ) ` C ) ) )
Theorem	dvcj 15342	The derivative of the conjugate of a function. For the (more general) relation version, see dvcjbr 15341. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
|- ( ( F : X --> CC /\ X C_ RR ) -> ( RR _D ( * o. F ) ) = ( * o. ( RR _D F ) ) )
Theorem	dvfre 15343	The derivative of a real function is real. (Contributed by Mario Carneiro, 1-Sep-2014.)
|- ( ( F : A --> RR /\ A C_ RR ) -> ( RR _D F ) : dom ( RR _D F ) --> RR )
Theorem	dvexp 15344*	Derivative of a power function. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
|- ( N e. NN -> ( CC _D ( x e. CC |-> ( x ^ N ) ) ) = ( x e. CC |-> ( N x. ( x ^ ( N - 1 ) ) ) ) )
Theorem	dvexp2 15345*	Derivative of an exponential, possibly zero power. (Contributed by Stefan O'Rear, 13-Nov-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
|- ( N e. NN0 -> ( CC _D ( x e. CC |-> ( x ^ N ) ) ) = ( x e. CC |-> if ( N = 0 , 0 , ( N x. ( x ^ ( N - 1 ) ) ) ) ) )
Theorem	dvrecap 15346*	Derivative of the reciprocal function. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Mario Carneiro, 28-Dec-2016.)
|- ( A e. CC -> ( CC _D ( x e. { w e. CC | w # 0 } |-> ( A / x ) ) ) = ( x e. { w e. CC | w # 0 } |-> -u ( A / ( x ^ 2 ) ) ) )
Theorem	dvmptidcn 15347	Function-builder for derivative: derivative of the identity. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 30-Dec-2023.)
|- ( CC _D ( x e. CC |-> x ) ) = ( x e. CC |-> 1 )
Theorem	dvmptccn 15348*	Function-builder for derivative: derivative of a constant. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 30-Dec-2023.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( CC _D ( x e. CC |-> A ) ) = ( x e. CC |-> 0 ) )
Theorem	dvmptid 15349*	Function-builder for derivative: derivative of the identity. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
|- ( ph -> S e. { RR , CC } )   =>    |- ( ph -> ( S _D ( x e. S |-> x ) ) = ( x e. S |-> 1 ) )
Theorem	dvmptc 15350*	Function-builder for derivative: derivative of a constant. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
|- ( ph -> S e. { RR , CC } )   &    |- ( ph -> A e. CC )   =>    |- ( ph -> ( S _D ( x e. S |-> A ) ) = ( x e. S |-> 0 ) )
Theorem	dvmptclx 15351*	Closure lemma for dvmptmulx 15353 and other related theorems. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
|- ( ph -> S e. { RR , CC } )   &    |- ( ( ph /\ x e. X ) -> A e. CC )   &    |- ( ( ph /\ x e. X ) -> B e. V )   &    |- ( ph -> ( S _D ( x e. X |-> A ) ) = ( x e. X |-> B ) )   &    |- ( ph -> X C_ S )   =>    |- ( ( ph /\ x e. X ) -> B e. CC )
Theorem	dvmptaddx 15352*	Function-builder for derivative, addition rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
|- ( ph -> S e. { RR , CC } )   &    |- ( ( ph /\ x e. X ) -> A e. CC )   &    |- ( ( ph /\ x e. X ) -> B e. V )   &    |- ( ph -> ( S _D ( x e. X |-> A ) ) = ( x e. X |-> B ) )   &    |- ( ph -> X C_ S )   &    |- ( ( ph /\ x e. X ) -> C e. CC )   &    |- ( ( ph /\ x e. X ) -> D e. W )   &    |- ( ph -> ( S _D ( x e. X |-> C ) ) = ( x e. X |-> D ) )   =>    |- ( ph -> ( S _D ( x e. X |-> ( A + C ) ) ) = ( x e. X |-> ( B + D ) ) )
Theorem	dvmptmulx 15353*	Function-builder for derivative, product rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
|- ( ph -> S e. { RR , CC } )   &    |- ( ( ph /\ x e. X ) -> A e. CC )   &    |- ( ( ph /\ x e. X ) -> B e. V )   &    |- ( ph -> ( S _D ( x e. X |-> A ) ) = ( x e. X |-> B ) )   &    |- ( ph -> X C_ S )   &    |- ( ( ph /\ x e. X ) -> C e. CC )   &    |- ( ( ph /\ x e. X ) -> D e. W )   &    |- ( ph -> ( S _D ( x e. X |-> C ) ) = ( x e. X |-> D ) )   =>    |- ( ph -> ( S _D ( x e. X |-> ( A x. C ) ) ) = ( x e. X |-> ( ( B x. C ) + ( D x. A ) ) ) )
Theorem	dvmptcmulcn 15354*	Function-builder for derivative, product rule for constant multiplier. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 31-Dec-2023.)
|- ( ( ph /\ x e. CC ) -> A e. CC )   &    |- ( ( ph /\ x e. CC ) -> B e. V )   &    |- ( ph -> ( CC _D ( x e. CC |-> A ) ) = ( x e. CC |-> B ) )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( CC _D ( x e. CC |-> ( C x. A ) ) ) = ( x e. CC |-> ( C x. B ) ) )
Theorem	dvmptnegcn 15355*	Function-builder for derivative, product rule for negatives. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 31-Dec-2023.)
|- ( ( ph /\ x e. CC ) -> A e. CC )   &    |- ( ( ph /\ x e. CC ) -> B e. V )   &    |- ( ph -> ( CC _D ( x e. CC |-> A ) ) = ( x e. CC |-> B ) )   =>    |- ( ph -> ( CC _D ( x e. CC |-> -u A ) ) = ( x e. CC |-> -u B ) )
Theorem	dvmptsubcn 15356*	Function-builder for derivative, subtraction rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 31-Dec-2023.)
|- ( ( ph /\ x e. CC ) -> A e. CC )   &    |- ( ( ph /\ x e. CC ) -> B e. V )   &    |- ( ph -> ( CC _D ( x e. CC |-> A ) ) = ( x e. CC |-> B ) )   &    |- ( ( ph /\ x e. CC ) -> C e. CC )   &    |- ( ( ph /\ x e. CC ) -> D e. W )   &    |- ( ph -> ( CC _D ( x e. CC |-> C ) ) = ( x e. CC |-> D ) )   =>    |- ( ph -> ( CC _D ( x e. CC |-> ( A - C ) ) ) = ( x e. CC |-> ( B - D ) ) )
Theorem	dvmptcjx 15357*	Function-builder for derivative, conjugate rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 24-May-2024.)
|- ( ( ph /\ x e. X ) -> A e. CC )   &    |- ( ( ph /\ x e. X ) -> B e. V )   &    |- ( ph -> ( RR _D ( x e. X |-> A ) ) = ( x e. X |-> B ) )   &    |- ( ph -> X C_ RR )   =>    |- ( ph -> ( RR _D ( x e. X |-> ( * ` A ) ) ) = ( x e. X |-> ( * ` B ) ) )
Theorem	dvmptfsum 15358*	Function-builder for derivative, finite sums rule. (Contributed by Stefan O'Rear, 12-Nov-2014.)
|- J = ( K ↾t S )   &    |- K = ( TopOpen ` ℂfld )   &    |- ( ph -> S e. { RR , CC } )   &    |- ( ph -> X e. J )   &    |- ( ph -> I e. Fin )   &    |- ( ( ph /\ i e. I /\ x e. X ) -> A e. CC )   &    |- ( ( ph /\ i e. I /\ x e. X ) -> B e. CC )   &    |- ( ( ph /\ i e. I ) -> ( S _D ( x e. X |-> A ) ) = ( x e. X |-> B ) )   =>    |- ( ph -> ( S _D ( x e. X |-> sum_ i e. I A ) ) = ( x e. X |-> sum_ i e. I B ) )
Theorem	dveflem 15359	Derivative of the exponential function at 0. The key step in the proof is eftlub 12162, to show that abs ( exp ( x ) - 1 - x ) <_ abs ( x ) ^ 2 x. ( 3 / 4 ). (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
|- 0 ( CC _D exp ) 1
Theorem	dvef 15360	Derivative of the exponential function. (Contributed by Mario Carneiro, 9-Aug-2014.) (Proof shortened by Mario Carneiro, 10-Feb-2015.)
|- ( CC _D exp ) = exp
PART 11  BASIC REAL AND COMPLEX FUNCTIONS
11.1  Polynomials
11.1.1  Elementary properties of complex polynomials
Syntax	cply 15361	Extend class notation to include the set of complex polynomials.
class Poly
Syntax	cidp 15362	Extend class notation to include the identity polynomial.
class Xp
Definition	df-ply 15363*	Define the set of polynomials on the complex numbers with coefficients in the given subset. (Contributed by Mario Carneiro, 17-Jul-2014.)
|- Poly = ( x e. ~P CC |-> { f | E. n e. NN0 E. a e. ( ( x u. { 0 } ) ^m NN0 ) f = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) } )
Definition	df-idp 15364	Define the identity polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
|- Xp = ( _I |` CC )
Theorem	plyval 15365*	Value of the polynomial set function. (Contributed by Mario Carneiro, 17-Jul-2014.)
|- ( S C_ CC -> (Poly ` S ) = { f | E. n e. NN0 E. a e. ( ( S u. { 0 } ) ^m NN0 ) f = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) } )
Theorem	plybss 15366	Reverse closure of the parameter S of the polynomial set function. (Contributed by Mario Carneiro, 22-Jul-2014.)
|- ( F e. (Poly ` S ) -> S C_ CC )
Theorem	elply 15367*	Definition of a polynomial with coefficients in S. (Contributed by Mario Carneiro, 17-Jul-2014.)
|- ( F e. (Poly ` S ) <-> ( S C_ CC /\ E. n e. NN0 E. a e. ( ( S u. { 0 } ) ^m NN0 ) F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) )
Theorem	elply2 15368*	The coefficient function can be assumed to have zeroes outside 0 ... n. (Contributed by Mario Carneiro, 20-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
|- ( F e. (Poly ` S ) <-> ( S C_ CC /\ E. n e. NN0 E. a e. ( ( S u. { 0 } ) ^m NN0 ) ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) ) )
Theorem	plyun0 15369	The set of polynomials is unaffected by the addition of zero. (This is built into the definition because all higher powers of a polynomial are effectively zero, so we require that the coefficient field contain zero to simplify some of our closure theorems.) (Contributed by Mario Carneiro, 17-Jul-2014.)
|- (Poly ` ( S u. { 0 } ) ) = (Poly ` S )
Theorem	plyf 15370	A polynomial is a function on the complex numbers. (Contributed by Mario Carneiro, 22-Jul-2014.)
|- ( F e. (Poly ` S ) -> F : CC --> CC )
Theorem	plyss 15371	The polynomial set function preserves the subset relation. (Contributed by Mario Carneiro, 17-Jul-2014.)
|- ( ( S C_ T /\ T C_ CC ) -> (Poly ` S ) C_ (Poly ` T ) )
Theorem	plyssc 15372	Every polynomial ring is contained in the ring of polynomials over CC. (Contributed by Mario Carneiro, 22-Jul-2014.)
|- (Poly ` S ) C_ (Poly ` CC )
Theorem	elplyr 15373*	Sufficient condition for elementhood in the set of polynomials. (Contributed by Mario Carneiro, 17-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
|- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) e. (Poly ` S ) )
Theorem	elplyd 15374*	Sufficient condition for elementhood in the set of polynomials. (Contributed by Mario Carneiro, 17-Jul-2014.)
|- ( ph -> S C_ CC )   &    |- ( ph -> N e. NN0 )   &    |- ( ( ph /\ k e. ( 0 ... N ) ) -> A e. S )   =>    |- ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... N ) ( A x. ( z ^ k ) ) ) e. (Poly ` S ) )
Theorem	ply1termlem 15375*	Lemma for ply1term 15376. (Contributed by Mario Carneiro, 26-Jul-2014.)
|- F = ( z e. CC |-> ( A x. ( z ^ N ) ) )   =>    |- ( ( A e. CC /\ N e. NN0 ) -> F = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( if ( k = N , A , 0 ) x. ( z ^ k ) ) ) )
Theorem	ply1term 15376*	A one-term polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
|- F = ( z e. CC |-> ( A x. ( z ^ N ) ) )   =>    |- ( ( S C_ CC /\ A e. S /\ N e. NN0 ) -> F e. (Poly ` S ) )
Theorem	plypow 15377*	A power is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
|- ( ( S C_ CC /\ 1 e. S /\ N e. NN0 ) -> ( z e. CC |-> ( z ^ N ) ) e. (Poly ` S ) )
Theorem	plyconst 15378	A constant function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
|- ( ( S C_ CC /\ A e. S ) -> ( CC X. { A } ) e. (Poly ` S ) )
Theorem	plyid 15379	The identity function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
|- ( ( S C_ CC /\ 1 e. S ) -> Xp e. (Poly ` S ) )
Theorem	plyaddlem1 15380*	Derive the coefficient function for the sum of two polynomials. (Contributed by Mario Carneiro, 23-Jul-2014.)
|- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> G e. (Poly ` S ) )   &    |- ( ph -> M e. NN0 )   &    |- ( ph -> N e. NN0 )   &    |- ( ph -> A : NN0 --> CC )   &    |- ( ph -> B : NN0 --> CC )   &    |- ( ph -> ( A " ( ZZ>= ` ( M + 1 ) ) ) = { 0 } )   &    |- ( ph -> ( B " ( ZZ>= ` ( N + 1 ) ) ) = { 0 } )   &    |- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... M ) ( ( A ` k ) x. ( z ^ k ) ) ) )   &    |- ( ph -> G = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( B ` k ) x. ( z ^ k ) ) ) )   =>    |- ( ph -> ( F oF + G ) = ( z e. CC |-> sum_ k e. ( 0 ... if ( M <_ N , N , M ) ) ( ( ( A oF + B ) ` k ) x. ( z ^ k ) ) ) )
Theorem	plymullem1 15381*	Derive the coefficient function for the product of two polynomials. (Contributed by Mario Carneiro, 23-Jul-2014.)
|- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> G e. (Poly ` S ) )   &    |- ( ph -> M e. NN0 )   &    |- ( ph -> N e. NN0 )   &    |- ( ph -> A : NN0 --> CC )   &    |- ( ph -> B : NN0 --> CC )   &    |- ( ph -> ( A " ( ZZ>= ` ( M + 1 ) ) ) = { 0 } )   &    |- ( ph -> ( B " ( ZZ>= ` ( N + 1 ) ) ) = { 0 } )   &    |- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... M ) ( ( A ` k ) x. ( z ^ k ) ) ) )   &    |- ( ph -> G = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( B ` k ) x. ( z ^ k ) ) ) )   =>    |- ( ph -> ( F oF x. G ) = ( z e. CC |-> sum_ n e. ( 0 ... ( M + N ) ) ( sum_ k e. ( 0 ... n ) ( ( A ` k ) x. ( B ` ( n - k ) ) ) x. ( z ^ n ) ) ) )
Theorem	plyaddlem 15382*	Lemma for plyadd 15384. (Contributed by Mario Carneiro, 21-Jul-2014.)
|- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> G e. (Poly ` S ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )   &    |- ( ph -> M e. NN0 )   &    |- ( ph -> N e. NN0 )   &    |- ( ph -> A e. ( ( S u. { 0 } ) ^m NN0 ) )   &    |- ( ph -> B e. ( ( S u. { 0 } ) ^m NN0 ) )   &    |- ( ph -> ( A " ( ZZ>= ` ( M + 1 ) ) ) = { 0 } )   &    |- ( ph -> ( B " ( ZZ>= ` ( N + 1 ) ) ) = { 0 } )   &    |- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... M ) ( ( A ` k ) x. ( z ^ k ) ) ) )   &    |- ( ph -> G = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( B ` k ) x. ( z ^ k ) ) ) )   =>    |- ( ph -> ( F oF + G ) e. (Poly ` S ) )
Theorem	plymullem 15383*	Lemma for plymul 15385. (Contributed by Mario Carneiro, 21-Jul-2014.)
|- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> G e. (Poly ` S ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )   &    |- ( ph -> M e. NN0 )   &    |- ( ph -> N e. NN0 )   &    |- ( ph -> A e. ( ( S u. { 0 } ) ^m NN0 ) )   &    |- ( ph -> B e. ( ( S u. { 0 } ) ^m NN0 ) )   &    |- ( ph -> ( A " ( ZZ>= ` ( M + 1 ) ) ) = { 0 } )   &    |- ( ph -> ( B " ( ZZ>= ` ( N + 1 ) ) ) = { 0 } )   &    |- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... M ) ( ( A ` k ) x. ( z ^ k ) ) ) )   &    |- ( ph -> G = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( B ` k ) x. ( z ^ k ) ) ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )   =>    |- ( ph -> ( F oF x. G ) e. (Poly ` S ) )
Theorem	plyadd 15384*	The sum of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
|- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> G e. (Poly ` S ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )   =>    |- ( ph -> ( F oF + G ) e. (Poly ` S ) )
Theorem	plymul 15385*	The product of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
|- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> G e. (Poly ` S ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )   =>    |- ( ph -> ( F oF x. G ) e. (Poly ` S ) )
Theorem	plysub 15386*	The difference of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
|- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> G e. (Poly ` S ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )   &    |- ( ph -> -u 1 e. S )   =>    |- ( ph -> ( F oF - G ) e. (Poly ` S ) )
Theorem	plyaddcl 15387	The sum of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( F oF + G ) e. (Poly ` CC ) )
Theorem	plymulcl 15388	The product of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( F oF x. G ) e. (Poly ` CC ) )
Theorem	plysubcl 15389	The difference of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( F oF - G ) e. (Poly ` CC ) )
Theorem	plycoeid3 15390*	Reconstruct a polynomial as an explicit sum of the coefficient function up to an index no smaller than the degree of the polynomial. (Contributed by Jim Kingdon, 17-Oct-2025.)
|- ( ph -> D e. NN0 )   &    |- ( ph -> A : NN0 --> CC )   &    |- ( ph -> ( A " ( ZZ>= ` ( D + 1 ) ) ) = { 0 } )   &    |- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... D ) ( ( A ` k ) x. ( z ^ k ) ) ) )   &    |- ( ph -> M e. ( ZZ>= ` D ) )   &    |- ( ph -> X e. CC )   =>    |- ( ph -> ( F ` X ) = sum_ j e. ( 0 ... M ) ( ( A ` j ) x. ( X ^ j ) ) )
Theorem	plycolemc 15391*	Lemma for plyco 15392. The result expressed as a sum, with a degree and coefficients for F specified as hypotheses. (Contributed by Jim Kingdon, 20-Sep-2025.)
|- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> G e. (Poly ` S ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )   &    |- ( ph -> N e. NN0 )   &    |- ( ph -> A : NN0 --> ( S u. { 0 } ) )   &    |- ( ph -> ( A " ( ZZ>= ` ( N + 1 ) ) ) = { 0 } )   &    |- ( ph -> F = ( x e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( x ^ k ) ) ) )   =>    |- ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) )
Theorem	plyco 15392*	The composition of two polynomials is a polynomial. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
|- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> G e. (Poly ` S ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )   =>    |- ( ph -> ( F o. G ) e. (Poly ` S ) )
Theorem	plycjlemc 15393*	Lemma for plycj 15394. (Contributed by Mario Carneiro, 24-Jul-2014.) (Revised by Jim Kingdon, 22-Sep-2025.)
|- ( ph -> N e. NN0 )   &    |- G = ( ( * o. F ) o. * )   &    |- ( ph -> A : NN0 --> ( S u. { 0 } ) )   &    |- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) )   &    |- ( ph -> F e. (Poly ` S ) )   =>    |- ( ph -> G = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( * o. A ) ` k ) x. ( z ^ k ) ) ) )
Theorem	plycj 15394*	The double conjugation of a polynomial is a polynomial. (The single conjugation is not because our definition of polynomial includes only holomorphic functions, i.e. no dependence on ( * ` z ) independently of z.) (Contributed by Mario Carneiro, 24-Jul-2014.)
|- G = ( ( * o. F ) o. * )   &    |- ( ( ph /\ x e. S ) -> ( * ` x ) e. S )   &    |- ( ph -> F e. (Poly ` S ) )   =>    |- ( ph -> G e. (Poly ` S ) )
Theorem	plycn 15395	A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.) Avoid ax-mulf 8085. (Revised by GG, 16-Mar-2025.)
|- ( F e. (Poly ` S ) -> F e. ( CC -cn-> CC ) )
Theorem	plyrecj 15396	A polynomial with real coefficients distributes under conjugation. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- ( ( F e. (Poly ` RR ) /\ A e. CC ) -> ( * ` ( F ` A ) ) = ( F ` ( * ` A ) ) )
Theorem	plyreres 15397	Real-coefficient polynomials restrict to real functions. (Contributed by Stefan O'Rear, 16-Nov-2014.)
|- ( F e. (Poly ` RR ) -> ( F |` RR ) : RR --> RR )
Theorem	dvply1 15398*	Derivative of a polynomial, explicit sum version. (Contributed by Stefan O'Rear, 13-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
|- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) )   &    |- ( ph -> G = ( z e. CC |-> sum_ k e. ( 0 ... ( N - 1 ) ) ( ( B ` k ) x. ( z ^ k ) ) ) )   &    |- ( ph -> A : NN0 --> CC )   &    |- B = ( k e. NN0 |-> ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) )   &    |- ( ph -> N e. NN0 )   =>    |- ( ph -> ( CC _D F ) = G )
Theorem	dvply2g 15399	The derivative of a polynomial with coefficients in a subring is a polynomial with coefficients in the same ring. (Contributed by Mario Carneiro, 1-Jan-2017.) (Revised by GG, 30-Apr-2025.)
|- ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) -> ( CC _D F ) e. (Poly ` S ) )
Theorem	dvply2 15400	The derivative of a polynomial is a polynomial. (Contributed by Stefan O'Rear, 14-Nov-2014.) (Proof shortened by Mario Carneiro, 1-Jan-2017.)
|- ( F e. (Poly ` S ) -> ( CC _D F ) e. (Poly ` CC ) )