P. 155 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 15401-15500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dvmptc 15401*	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 15402*	Closure lemma for dvmptmulx 15404 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 15403*	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 15404*	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 15405*	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 15406*	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 15407*	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 15408*	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 15409*	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 15410	Derivative of the exponential function at 0. The key step in the proof is eftlub 12211, 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 15411	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 15412	Extend class notation to include the set of complex polynomials.
class Poly
Syntax	cidp 15413	Extend class notation to include the identity polynomial.
class Xp
Definition	df-ply 15414*	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 15415	Define the identity polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
|- Xp = ( _I |` CC )
Theorem	plyval 15416*	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 15417	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 15418*	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 15419*	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 15420	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 15421	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 15422	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 15423	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 15424*	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 15425*	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 15426*	Lemma for ply1term 15427. (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 15427*	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 15428*	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 15429	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 15430	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 15431*	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 15432*	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 15433*	Lemma for plyadd 15435. (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 15434*	Lemma for plymul 15436. (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 15435*	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 15436*	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 15437*	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 15438	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 15439	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 15440	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 15441*	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 15442*	Lemma for plyco 15443. 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 15443*	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 15444*	Lemma for plycj 15445. (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 15445*	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 15446	A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.) Avoid ax-mulf 8130. (Revised by GG, 16-Mar-2025.)
|- ( F e. (Poly ` S ) -> F e. ( CC -cn-> CC ) )
Theorem	plyrecj 15447	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 15448	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 15449*	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 15450	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 15451	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 15452	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 15453	Sine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)
|- sin e. ( CC -cn-> CC )
Theorem	coscn 15454	Cosine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)
|- cos e. ( CC -cn-> CC )
Theorem	reeff1olem 15455*	Lemma for reeff1o 15457. (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 15456*	Lemma for reeff1o 15457. (Contributed by Jim Kingdon, 15-May-2024.)
|- ( U e. ( 0 (,) _e ) -> E. x e. RR ( exp ` x ) = U )
Theorem	reeff1o 15457	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 15458	Lemma for eflt 15459. The converse of efltim 12219 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 15459	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 15460	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 15461	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 15462	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 15463	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 15464	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 15465*	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 15466*	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 15467	Lemma for pi related theorems. (Contributed by Jim Kingdon, 9-Mar-2024.)
|- ( pi e. ( 2 (,) 4 ) /\ ( sin ` pi ) = 0 )
Theorem	pigt2lt4 15468	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 15469	The sine of pi is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( sin ` pi ) = 0
Theorem	pire 15470	pi is a real number. (Contributed by Paul Chapman, 23-Jan-2008.)
|- pi e. RR
Theorem	picn 15471	pi is a complex number. (Contributed by David A. Wheeler, 6-Dec-2018.)
|- pi e. CC
Theorem	pipos 15472	pi is positive. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
|- 0 < pi
Theorem	pirp 15473	pi is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- pi e. RR+
Theorem	negpicn 15474	-u pi is a real number. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u pi e. CC
Theorem	sinhalfpilem 15475	Lemma for sinhalfpi 15480 and coshalfpi 15481. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( ( sin ` ( pi / 2 ) ) = 1 /\ ( cos ` ( pi / 2 ) ) = 0 )
Theorem	halfpire 15476	pi / 2 is real. (Contributed by David Moews, 28-Feb-2017.)
|- ( pi / 2 ) e. RR
Theorem	neghalfpire 15477	-u pi / 2 is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u ( pi / 2 ) e. RR
Theorem	neghalfpirx 15478	-u pi / 2 is an extended real. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u ( pi / 2 ) e. RR*
Theorem	pidiv2halves 15479	Adding pi / 2 to itself gives pi. See 2halves 9348. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( ( pi / 2 ) + ( pi / 2 ) ) = pi
Theorem	sinhalfpi 15480	The sine of pi / 2 is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( sin ` ( pi / 2 ) ) = 1
Theorem	coshalfpi 15481	The cosine of pi / 2 is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( cos ` ( pi / 2 ) ) = 0
Theorem	cosneghalfpi 15482	The cosine of -u pi / 2 is zero. (Contributed by David Moews, 28-Feb-2017.)
|- ( cos ` -u ( pi / 2 ) ) = 0
Theorem	efhalfpi 15483	The exponential of _i pi / 2 is _i. (Contributed by Mario Carneiro, 9-May-2014.)
|- ( exp ` ( _i x. ( pi / 2 ) ) ) = _i
Theorem	cospi 15484	The cosine of pi is -u 1. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( cos ` pi ) = -u 1
Theorem	efipi 15485	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 15486	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 15487	The sine of 2 pi is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( sin ` ( 2 x. pi ) ) = 0
Theorem	cos2pi 15488	The cosine of 2 pi is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( cos ` ( 2 x. pi ) ) = 1
Theorem	ef2pi 15489	The exponential of 2 pi _i is 1. (Contributed by Mario Carneiro, 9-May-2014.)
|- ( exp ` ( _i x. ( 2 x. pi ) ) ) = 1
Theorem	ef2kpi 15490	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 )
Theorem	efper 15491	The exponential function is periodic. (Contributed by Paul Chapman, 21-Apr-2008.) (Proof shortened by Mario Carneiro, 10-May-2014.)
|- ( ( A e. CC /\ K e. ZZ ) -> ( exp ` ( A + ( ( _i x. ( 2 x. pi ) ) x. K ) ) ) = ( exp ` A ) )
Theorem	sinperlem 15492	Lemma for sinper 15493 and cosper 15494. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
|- ( A e. CC -> ( F ` A ) = ( ( ( exp ` ( _i x. A ) ) O ( exp ` ( -u _i x. A ) ) ) / D ) )   &    |- ( ( A + ( K x. ( 2 x. pi ) ) ) e. CC -> ( F ` ( A + ( K x. ( 2 x. pi ) ) ) ) = ( ( ( exp ` ( _i x. ( A + ( K x. ( 2 x. pi ) ) ) ) ) O ( exp ` ( -u _i x. ( A + ( K x. ( 2 x. pi ) ) ) ) ) ) / D ) )   =>    |- ( ( A e. CC /\ K e. ZZ ) -> ( F ` ( A + ( K x. ( 2 x. pi ) ) ) ) = ( F ` A ) )
Theorem	sinper 15493	The sine function is periodic. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
|- ( ( A e. CC /\ K e. ZZ ) -> ( sin ` ( A + ( K x. ( 2 x. pi ) ) ) ) = ( sin ` A ) )
Theorem	cosper 15494	The cosine function is periodic. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
|- ( ( A e. CC /\ K e. ZZ ) -> ( cos ` ( A + ( K x. ( 2 x. pi ) ) ) ) = ( cos ` A ) )
Theorem	sin2kpi 15495	If K is an integer, then the sine of 2 K pi is 0. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
|- ( K e. ZZ -> ( sin ` ( K x. ( 2 x. pi ) ) ) = 0 )
Theorem	cos2kpi 15496	If K is an integer, then the cosine of 2 K pi is 1. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
|- ( K e. ZZ -> ( cos ` ( K x. ( 2 x. pi ) ) ) = 1 )
Theorem	sin2pim 15497	Sine of a number subtracted from 2 x. pi. (Contributed by Paul Chapman, 15-Mar-2008.)
|- ( A e. CC -> ( sin ` ( ( 2 x. pi ) - A ) ) = -u ( sin ` A ) )
Theorem	cos2pim 15498	Cosine of a number subtracted from 2 x. pi. (Contributed by Paul Chapman, 15-Mar-2008.)
|- ( A e. CC -> ( cos ` ( ( 2 x. pi ) - A ) ) = ( cos ` A ) )
Theorem	sinmpi 15499	Sine of a number less pi. (Contributed by Paul Chapman, 15-Mar-2008.)
|- ( A e. CC -> ( sin ` ( A - pi ) ) = -u ( sin ` A ) )
Theorem	cosmpi 15500	Cosine of a number less pi. (Contributed by Paul Chapman, 15-Mar-2008.)
|- ( A e. CC -> ( cos ` ( A - pi ) ) = -u ( cos ` A ) )