P. 158 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 15701-15800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dvcoapbr 15701*	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 15702	The derivative of the conjugate of a function. For the (simpler but more limited) function version, see dvcj 15703. (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 15703	The derivative of the conjugate of a function. For the (more general) relation version, see dvcjbr 15702. (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 15704	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 15705*	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 15706*	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 15707*	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 15708	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 15709*	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 15710*	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 15711*	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 15712*	Closure lemma for dvmptmulx 15714 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 15713*	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 15714*	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 15715*	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 15716*	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 15717*	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 15718*	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 15719*	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 15720	Derivative of the exponential function at 0. The key step in the proof is eftlub 12404, 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 15721	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 15722	Extend class notation to include the set of complex polynomials.
class Poly
Syntax	cidp 15723	Extend class notation to include the identity polynomial.
class Xp
Definition	df-ply 15724*	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 15725	Define the identity polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
|- Xp = ( _I |` CC )
Theorem	plyval 15726*	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 15727	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 15728*	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 15729*	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 15730	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 15731	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 15732	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 15733	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 15734*	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 15735*	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 15736*	Lemma for ply1term 15737. (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 15737*	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 15738*	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 15739	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 15740	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 15741*	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 15742*	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 15743*	Lemma for plyadd 15745. (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 15744*	Lemma for plymul 15746. (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 15745*	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 15746*	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 15747*	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 15748	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 15749	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 15750	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 15751*	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 15752*	Lemma for plyco 15753. 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 15753*	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 15754*	Lemma for plycj 15755. (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 15755*	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 15756	A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.) Avoid ax-mulf 8266. (Revised by GG, 16-Mar-2025.)
|- ( F e. (Poly ` S ) -> F e. ( CC -cn-> CC ) )
Theorem	plyrecj 15757	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 15758	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 15759*	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 15760	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 15761	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 ) )
11.2  Basic trigonometry
11.2.1  The exponential, sine, and cosine functions (cont.)
Theorem	efcn 15762	The exponential function is continuous. (Contributed by Paul Chapman, 15-Sep-2007.) (Revised by Mario Carneiro, 20-Jun-2015.)
|- exp e. ( CC -cn-> CC )
Theorem	sincn 15763	Sine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)
|- sin e. ( CC -cn-> CC )
Theorem	coscn 15764	Cosine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)
|- cos e. ( CC -cn-> CC )
Theorem	reeff1olem 15765*	Lemma for reeff1o 15767. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
|- ( ( U e. RR /\ 1 < U ) -> E. x e. RR ( exp ` x ) = U )
Theorem	reeff1oleme 15766*	Lemma for reeff1o 15767. (Contributed by Jim Kingdon, 15-May-2024.)
|- ( U e. ( 0 (,) _e ) -> E. x e. RR ( exp ` x ) = U )
Theorem	reeff1o 15767	The real exponential function is one-to-one onto. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 10-Nov-2013.)
|- ( exp |` RR ) : RR -1-1-onto-> RR+
Theorem	efltlemlt 15768	Lemma for eflt 15769. The converse of efltim 12412 plus the epsilon-delta setup. (Contributed by Jim Kingdon, 22-May-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> ( exp ` A ) < ( exp ` B ) )   &    |- ( ph -> D e. RR+ )   &    |- ( ph -> ( ( abs ` ( A - B ) ) < D -> ( abs ` ( ( exp ` A ) - ( exp ` B ) ) ) < ( ( exp ` B ) - ( exp ` A ) ) ) )   =>    |- ( ph -> A < B )
Theorem	eflt 15769	The exponential function on the reals is strictly increasing. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Jim Kingdon, 21-May-2024.)
|- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> ( exp ` A ) < ( exp ` B ) ) )
Theorem	efle 15770	The exponential function on the reals is nondecreasing. (Contributed by Mario Carneiro, 11-Mar-2014.)
|- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> ( exp ` A ) <_ ( exp ` B ) ) )
Theorem	reefiso 15771	The exponential function on the reals determines an isomorphism from reals onto positive reals. (Contributed by Steve Rodriguez, 25-Nov-2007.) (Revised by Mario Carneiro, 11-Mar-2014.)
|- ( exp |` RR ) Isom < , < ( RR , RR+ )
Theorem	reapef 15772	Apartness and the exponential function for reals. (Contributed by Jim Kingdon, 11-Jul-2024.)
|- ( ( A e. RR /\ B e. RR ) -> ( A # B <-> ( exp ` A ) # ( exp ` B ) ) )
11.2.2  Properties of pi = 3.14159...
Theorem	pilem1 15773	Lemma for pire , pigt2lt4 and sinpi . (Contributed by Mario Carneiro, 9-May-2014.)
|- ( A e. ( RR+ i^i ( `' sin " { 0 } ) ) <-> ( A e. RR+ /\ ( sin ` A ) = 0 ) )
Theorem	cosz12 15774	Cosine has a zero between 1 and 2. (Contributed by Mario Carneiro and Jim Kingdon, 7-Mar-2024.)
|- E. p e. ( 1 (,) 2 ) ( cos ` p ) = 0
Theorem	sin0pilem1 15775*	Lemma for pi related theorems. (Contributed by Mario Carneiro and Jim Kingdon, 8-Mar-2024.)
|- E. p e. ( 1 (,) 2 ) ( ( cos ` p ) = 0 /\ A. x e. ( p (,) ( 2 x. p ) ) 0 < ( sin ` x ) )
Theorem	sin0pilem2 15776*	Lemma for pi related theorems. (Contributed by Mario Carneiro and Jim Kingdon, 8-Mar-2024.)
|- E. q e. ( 2 (,) 4 ) ( ( sin ` q ) = 0 /\ A. x e. ( 0 (,) q ) 0 < ( sin ` x ) )
Theorem	pilem3 15777	Lemma for pi related theorems. (Contributed by Jim Kingdon, 9-Mar-2024.)
|- ( pi e. ( 2 (,) 4 ) /\ ( sin ` pi ) = 0 )
Theorem	pigt2lt4 15778	pi is between 2 and 4. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
|- ( 2 < pi /\ pi < 4 )
Theorem	sinpi 15779	The sine of pi is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( sin ` pi ) = 0
Theorem	pire 15780	pi is a real number. (Contributed by Paul Chapman, 23-Jan-2008.)
|- pi e. RR
Theorem	picn 15781	pi is a complex number. (Contributed by David A. Wheeler, 6-Dec-2018.)
|- pi e. CC
Theorem	pipos 15782	pi is positive. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
|- 0 < pi
Theorem	pirp 15783	pi is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- pi e. RR+
Theorem	negpicn 15784	-u pi is a real number. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u pi e. CC
Theorem	sinhalfpilem 15785	Lemma for sinhalfpi 15790 and coshalfpi 15791. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( ( sin ` ( pi / 2 ) ) = 1 /\ ( cos ` ( pi / 2 ) ) = 0 )
Theorem	halfpire 15786	pi / 2 is real. (Contributed by David Moews, 28-Feb-2017.)
|- ( pi / 2 ) e. RR
Theorem	neghalfpire 15787	-u pi / 2 is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u ( pi / 2 ) e. RR
Theorem	neghalfpirx 15788	-u pi / 2 is an extended real. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u ( pi / 2 ) e. RR*
Theorem	pidiv2halves 15789	Adding pi / 2 to itself gives pi. See 2halves 9487. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( ( pi / 2 ) + ( pi / 2 ) ) = pi
Theorem	sinhalfpi 15790	The sine of pi / 2 is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( sin ` ( pi / 2 ) ) = 1
Theorem	coshalfpi 15791	The cosine of pi / 2 is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( cos ` ( pi / 2 ) ) = 0
Theorem	cosneghalfpi 15792	The cosine of -u pi / 2 is zero. (Contributed by David Moews, 28-Feb-2017.)
|- ( cos ` -u ( pi / 2 ) ) = 0
Theorem	efhalfpi 15793	The exponential of _i pi / 2 is _i. (Contributed by Mario Carneiro, 9-May-2014.)
|- ( exp ` ( _i x. ( pi / 2 ) ) ) = _i
Theorem	cospi 15794	The cosine of pi is -u 1. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( cos ` pi ) = -u 1
Theorem	efipi 15795	The exponential of _i x. pi is -u 1. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
|- ( exp ` ( _i x. pi ) ) = -u 1
Theorem	eulerid 15796	Euler's identity. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
|- ( ( exp ` ( _i x. pi ) ) + 1 ) = 0
Theorem	sin2pi 15797	The sine of 2 pi is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( sin ` ( 2 x. pi ) ) = 0
Theorem	cos2pi 15798	The cosine of 2 pi is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( cos ` ( 2 x. pi ) ) = 1
Theorem	ef2pi 15799	The exponential of 2 pi _i is 1. (Contributed by Mario Carneiro, 9-May-2014.)
|- ( exp ` ( _i x. ( 2 x. pi ) ) ) = 1
Theorem	ef2kpi 15800	If K is an integer, then the exponential of 2 K pi _i is 1. (Contributed by Mario Carneiro, 9-May-2014.)
|- ( K e. ZZ -> ( exp ` ( ( _i x. ( 2 x. pi ) ) x. K ) ) = 1 )