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	plyaddcl 15301	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 15302	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 15303	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 15304*	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 15305*	Lemma for plyco 15306. 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 15306*	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 15307*	Lemma for plycj 15308. (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 15308*	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 15309	A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.) Avoid ax-mulf 8068. (Revised by GG, 16-Mar-2025.)
|- ( F e. (Poly ` S ) -> F e. ( CC -cn-> CC ) )
Theorem	plyrecj 15310	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 15311	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 15312*	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 15313	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 15314	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 15315	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 15316	Sine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)
|- sin e. ( CC -cn-> CC )
Theorem	coscn 15317	Cosine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)
|- cos e. ( CC -cn-> CC )
Theorem	reeff1olem 15318*	Lemma for reeff1o 15320. (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 15319*	Lemma for reeff1o 15320. (Contributed by Jim Kingdon, 15-May-2024.)
|- ( U e. ( 0 (,) _e ) -> E. x e. RR ( exp ` x ) = U )
Theorem	reeff1o 15320	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 15321	Lemma for eflt 15322. The converse of efltim 12084 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 15322	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 15323	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 15324	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 15325	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 15326	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 15327	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 15328*	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 15329*	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 15330	Lemma for pi related theorems. (Contributed by Jim Kingdon, 9-Mar-2024.)
|- ( pi e. ( 2 (,) 4 ) /\ ( sin ` pi ) = 0 )
Theorem	pigt2lt4 15331	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 15332	The sine of pi is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( sin ` pi ) = 0
Theorem	pire 15333	pi is a real number. (Contributed by Paul Chapman, 23-Jan-2008.)
|- pi e. RR
Theorem	picn 15334	pi is a complex number. (Contributed by David A. Wheeler, 6-Dec-2018.)
|- pi e. CC
Theorem	pipos 15335	pi is positive. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
|- 0 < pi
Theorem	pirp 15336	pi is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- pi e. RR+
Theorem	negpicn 15337	-u pi is a real number. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u pi e. CC
Theorem	sinhalfpilem 15338	Lemma for sinhalfpi 15343 and coshalfpi 15344. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( ( sin ` ( pi / 2 ) ) = 1 /\ ( cos ` ( pi / 2 ) ) = 0 )
Theorem	halfpire 15339	pi / 2 is real. (Contributed by David Moews, 28-Feb-2017.)
|- ( pi / 2 ) e. RR
Theorem	neghalfpire 15340	-u pi / 2 is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u ( pi / 2 ) e. RR
Theorem	neghalfpirx 15341	-u pi / 2 is an extended real. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u ( pi / 2 ) e. RR*
Theorem	pidiv2halves 15342	Adding pi / 2 to itself gives pi. See 2halves 9286. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( ( pi / 2 ) + ( pi / 2 ) ) = pi
Theorem	sinhalfpi 15343	The sine of pi / 2 is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( sin ` ( pi / 2 ) ) = 1
Theorem	coshalfpi 15344	The cosine of pi / 2 is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( cos ` ( pi / 2 ) ) = 0
Theorem	cosneghalfpi 15345	The cosine of -u pi / 2 is zero. (Contributed by David Moews, 28-Feb-2017.)
|- ( cos ` -u ( pi / 2 ) ) = 0
Theorem	efhalfpi 15346	The exponential of _i pi / 2 is _i. (Contributed by Mario Carneiro, 9-May-2014.)
|- ( exp ` ( _i x. ( pi / 2 ) ) ) = _i
Theorem	cospi 15347	The cosine of pi is -u 1. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( cos ` pi ) = -u 1
Theorem	efipi 15348	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 15349	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 15350	The sine of 2 pi is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( sin ` ( 2 x. pi ) ) = 0
Theorem	cos2pi 15351	The cosine of 2 pi is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( cos ` ( 2 x. pi ) ) = 1
Theorem	ef2pi 15352	The exponential of 2 pi _i is 1. (Contributed by Mario Carneiro, 9-May-2014.)
|- ( exp ` ( _i x. ( 2 x. pi ) ) ) = 1
Theorem	ef2kpi 15353	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 15354	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 15355	Lemma for sinper 15356 and cosper 15357. (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 15356	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 15357	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 15358	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 15359	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 15360	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 15361	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 15362	Sine of a number less pi. (Contributed by Paul Chapman, 15-Mar-2008.)
|- ( A e. CC -> ( sin ` ( A - pi ) ) = -u ( sin ` A ) )
Theorem	cosmpi 15363	Cosine of a number less pi. (Contributed by Paul Chapman, 15-Mar-2008.)
|- ( A e. CC -> ( cos ` ( A - pi ) ) = -u ( cos ` A ) )
Theorem	sinppi 15364	Sine of a number plus pi. (Contributed by NM, 10-Aug-2008.)
|- ( A e. CC -> ( sin ` ( A + pi ) ) = -u ( sin ` A ) )
Theorem	cosppi 15365	Cosine of a number plus pi. (Contributed by NM, 18-Aug-2008.)
|- ( A e. CC -> ( cos ` ( A + pi ) ) = -u ( cos ` A ) )
Theorem	efimpi 15366	The exponential function at _i times a real number less pi. (Contributed by Paul Chapman, 15-Mar-2008.)
|- ( A e. CC -> ( exp ` ( _i x. ( A - pi ) ) ) = -u ( exp ` ( _i x. A ) ) )
Theorem	sinhalfpip 15367	The sine of pi / 2 plus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
|- ( A e. CC -> ( sin ` ( ( pi / 2 ) + A ) ) = ( cos ` A ) )
Theorem	sinhalfpim 15368	The sine of pi / 2 minus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
|- ( A e. CC -> ( sin ` ( ( pi / 2 ) - A ) ) = ( cos ` A ) )
Theorem	coshalfpip 15369	The cosine of pi / 2 plus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
|- ( A e. CC -> ( cos ` ( ( pi / 2 ) + A ) ) = -u ( sin ` A ) )
Theorem	coshalfpim 15370	The cosine of pi / 2 minus a number. (Contributed by Paul Chapman, 24-Jan-2008.)
|- ( A e. CC -> ( cos ` ( ( pi / 2 ) - A ) ) = ( sin ` A ) )
Theorem	ptolemy 15371	Ptolemy's Theorem. This theorem is named after the Greek astronomer and mathematician Ptolemy (Claudius Ptolemaeus). This particular version is expressed using the sine function. It is proved by expanding all the multiplication of sines to a product of cosines of differences using sinmul 12130, then using algebraic simplification to show that both sides are equal. This formalization is based on the proof in "Trigonometry" by Gelfand and Saul. This is Metamath 100 proof #95. (Contributed by David A. Wheeler, 31-May-2015.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) /\ ( ( A + B ) + ( C + D ) ) = pi ) -> ( ( ( sin ` A ) x. ( sin ` B ) ) + ( ( sin ` C ) x. ( sin ` D ) ) ) = ( ( sin ` ( B + C ) ) x. ( sin ` ( A + C ) ) ) )
Theorem	sincosq1lem 15372	Lemma for sincosq1sgn 15373. (Contributed by Paul Chapman, 24-Jan-2008.)
|- ( ( A e. RR /\ 0 < A /\ A < ( pi / 2 ) ) -> 0 < ( sin ` A ) )
Theorem	sincosq1sgn 15373	The signs of the sine and cosine functions in the first quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)
|- ( A e. ( 0 (,) ( pi / 2 ) ) -> ( 0 < ( sin ` A ) /\ 0 < ( cos ` A ) ) )
Theorem	sincosq2sgn 15374	The signs of the sine and cosine functions in the second quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)
|- ( A e. ( ( pi / 2 ) (,) pi ) -> ( 0 < ( sin ` A ) /\ ( cos ` A ) < 0 ) )
Theorem	sincosq3sgn 15375	The signs of the sine and cosine functions in the third quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)
|- ( A e. ( pi (,) ( 3 x. ( pi / 2 ) ) ) -> ( ( sin ` A ) < 0 /\ ( cos ` A ) < 0 ) )
Theorem	sincosq4sgn 15376	The signs of the sine and cosine functions in the fourth quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)
|- ( A e. ( ( 3 x. ( pi / 2 ) ) (,) ( 2 x. pi ) ) -> ( ( sin ` A ) < 0 /\ 0 < ( cos ` A ) ) )
Theorem	sinq12gt0 15377	The sine of a number strictly between 0 and pi is positive. (Contributed by Paul Chapman, 15-Mar-2008.)
|- ( A e. ( 0 (,) pi ) -> 0 < ( sin ` A ) )
Theorem	sinq34lt0t 15378	The sine of a number strictly between pi and 2 x. pi is negative. (Contributed by NM, 17-Aug-2008.)
|- ( A e. ( pi (,) ( 2 x. pi ) ) -> ( sin ` A ) < 0 )
Theorem	cosq14gt0 15379	The cosine of a number strictly between -u pi / 2 and pi / 2 is positive. (Contributed by Mario Carneiro, 25-Feb-2015.)
|- ( A e. ( -u ( pi / 2 ) (,) ( pi / 2 ) ) -> 0 < ( cos ` A ) )
Theorem	cosq23lt0 15380	The cosine of a number in the second and third quadrants is negative. (Contributed by Jim Kingdon, 14-Mar-2024.)
|- ( A e. ( ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) -> ( cos ` A ) < 0 )
Theorem	coseq0q4123 15381	Location of the zeroes of cosine in ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ). (Contributed by Jim Kingdon, 14-Mar-2024.)
|- ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) -> ( ( cos ` A ) = 0 <-> A = ( pi / 2 ) ) )
Theorem	coseq00topi 15382	Location of the zeroes of cosine in ( 0 [,] pi ). (Contributed by David Moews, 28-Feb-2017.)
|- ( A e. ( 0 [,] pi ) -> ( ( cos ` A ) = 0 <-> A = ( pi / 2 ) ) )
Theorem	coseq0negpitopi 15383	Location of the zeroes of cosine in ( -u pi (,] pi ). (Contributed by David Moews, 28-Feb-2017.)
|- ( A e. ( -u pi (,] pi ) -> ( ( cos ` A ) = 0 <-> A e. { ( pi / 2 ) , -u ( pi / 2 ) } ) )
Theorem	tanrpcl 15384	Positive real closure of the tangent function. (Contributed by Mario Carneiro, 29-Jul-2014.)
|- ( A e. ( 0 (,) ( pi / 2 ) ) -> ( tan ` A ) e. RR+ )
Theorem	tangtx 15385	The tangent function is greater than its argument on positive reals in its principal domain. (Contributed by Mario Carneiro, 29-Jul-2014.)
|- ( A e. ( 0 (,) ( pi / 2 ) ) -> A < ( tan ` A ) )
Theorem	sincosq1eq 15386	Complementarity of the sine and cosine functions in the first quadrant. (Contributed by Paul Chapman, 25-Jan-2008.)
|- ( ( A e. CC /\ B e. CC /\ ( A + B ) = 1 ) -> ( sin ` ( A x. ( pi / 2 ) ) ) = ( cos ` ( B x. ( pi / 2 ) ) ) )
Theorem	sincos4thpi 15387	The sine and cosine of pi / 4. (Contributed by Paul Chapman, 25-Jan-2008.)
|- ( ( sin ` ( pi / 4 ) ) = ( 1 / ( sqr ` 2 ) ) /\ ( cos ` ( pi / 4 ) ) = ( 1 / ( sqr ` 2 ) ) )
Theorem	tan4thpi 15388	The tangent of pi / 4. (Contributed by Mario Carneiro, 5-Apr-2015.)
|- ( tan ` ( pi / 4 ) ) = 1
Theorem	sincos6thpi 15389	The sine and cosine of pi / 6. (Contributed by Paul Chapman, 25-Jan-2008.) (Revised by Wolf Lammen, 24-Sep-2020.)
|- ( ( sin ` ( pi / 6 ) ) = ( 1 / 2 ) /\ ( cos ` ( pi / 6 ) ) = ( ( sqr ` 3 ) / 2 ) )
Theorem	sincos3rdpi 15390	The sine and cosine of pi / 3. (Contributed by Mario Carneiro, 21-May-2016.)
|- ( ( sin ` ( pi / 3 ) ) = ( ( sqr ` 3 ) / 2 ) /\ ( cos ` ( pi / 3 ) ) = ( 1 / 2 ) )
Theorem	pigt3 15391	pi is greater than 3. (Contributed by Brendan Leahy, 21-Aug-2020.)
|- 3 < pi
Theorem	pige3 15392	pi is greater than or equal to 3. (Contributed by Mario Carneiro, 21-May-2016.)
|- 3 <_ pi
Theorem	abssinper 15393	The absolute value of sine has period pi. (Contributed by NM, 17-Aug-2008.)
|- ( ( A e. CC /\ K e. ZZ ) -> ( abs ` ( sin ` ( A + ( K x. pi ) ) ) ) = ( abs ` ( sin ` A ) ) )
Theorem	sinkpi 15394	The sine of an integer multiple of pi is 0. (Contributed by NM, 11-Aug-2008.)
|- ( K e. ZZ -> ( sin ` ( K x. pi ) ) = 0 )
Theorem	coskpi 15395	The absolute value of the cosine of an integer multiple of pi is 1. (Contributed by NM, 19-Aug-2008.)
|- ( K e. ZZ -> ( abs ` ( cos ` ( K x. pi ) ) ) = 1 )
Theorem	cosordlem 15396	Cosine is decreasing over the closed interval from 0 to pi. (Contributed by Mario Carneiro, 10-May-2014.)
|- ( ph -> A e. ( 0 [,] pi ) )   &    |- ( ph -> B e. ( 0 [,] pi ) )   &    |- ( ph -> A < B )   =>    |- ( ph -> ( cos ` B ) < ( cos ` A ) )
Theorem	cosq34lt1 15397	Cosine is less than one in the third and fourth quadrants. (Contributed by Jim Kingdon, 19-Mar-2024.)
|- ( A e. ( pi [,) ( 2 x. pi ) ) -> ( cos ` A ) < 1 )
Theorem	cos02pilt1 15398	Cosine is less than one between zero and 2 x. pi. (Contributed by Jim Kingdon, 19-Mar-2024.)
|- ( A e. ( 0 (,) ( 2 x. pi ) ) -> ( cos ` A ) < 1 )
Theorem	cos0pilt1 15399	Cosine is between minus one and one on the open interval between zero and pi. (Contributed by Jim Kingdon, 7-May-2024.)
|- ( A e. ( 0 (,) pi ) -> ( cos ` A ) e. ( -u 1 (,) 1 ) )
Theorem	cos11 15400	Cosine is one-to-one over the closed interval from 0 to pi. (Contributed by Paul Chapman, 16-Mar-2008.) (Revised by Jim Kingdon, 6-May-2024.)
|- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> ( A = B <-> ( cos ` A ) = ( cos ` B ) ) )