P. 152 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 15101-15200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	plycoeid3 15101*	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 15102*	Lemma for plyco 15103. 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 15103*	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 15104*	Lemma for plycj 15105. (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 15105*	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 15106	A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.) Avoid ax-mulf 8021. (Revised by GG, 16-Mar-2025.)
|- ( F e. (Poly ` S ) -> F e. ( CC -cn-> CC ) )
Theorem	plyrecj 15107	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 15108	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 15109*	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 15110	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 15111	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 15112	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 15113	Sine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)
|- sin e. ( CC -cn-> CC )
Theorem	coscn 15114	Cosine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)
|- cos e. ( CC -cn-> CC )
Theorem	reeff1olem 15115*	Lemma for reeff1o 15117. (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 15116*	Lemma for reeff1o 15117. (Contributed by Jim Kingdon, 15-May-2024.)
|- ( U e. ( 0 (,) _e ) -> E. x e. RR ( exp ` x ) = U )
Theorem	reeff1o 15117	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 15118	Lemma for eflt 15119. The converse of efltim 11882 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 15119	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 15120	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 15121	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 15122	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 15123	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 15124	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 15125*	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 15126*	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 15127	Lemma for pi related theorems. (Contributed by Jim Kingdon, 9-Mar-2024.)
|- ( pi e. ( 2 (,) 4 ) /\ ( sin ` pi ) = 0 )
Theorem	pigt2lt4 15128	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 15129	The sine of pi is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( sin ` pi ) = 0
Theorem	pire 15130	pi is a real number. (Contributed by Paul Chapman, 23-Jan-2008.)
|- pi e. RR
Theorem	picn 15131	pi is a complex number. (Contributed by David A. Wheeler, 6-Dec-2018.)
|- pi e. CC
Theorem	pipos 15132	pi is positive. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
|- 0 < pi
Theorem	pirp 15133	pi is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- pi e. RR+
Theorem	negpicn 15134	-u pi is a real number. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u pi e. CC
Theorem	sinhalfpilem 15135	Lemma for sinhalfpi 15140 and coshalfpi 15141. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( ( sin ` ( pi / 2 ) ) = 1 /\ ( cos ` ( pi / 2 ) ) = 0 )
Theorem	halfpire 15136	pi / 2 is real. (Contributed by David Moews, 28-Feb-2017.)
|- ( pi / 2 ) e. RR
Theorem	neghalfpire 15137	-u pi / 2 is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u ( pi / 2 ) e. RR
Theorem	neghalfpirx 15138	-u pi / 2 is an extended real. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u ( pi / 2 ) e. RR*
Theorem	pidiv2halves 15139	Adding pi / 2 to itself gives pi. See 2halves 9239. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( ( pi / 2 ) + ( pi / 2 ) ) = pi
Theorem	sinhalfpi 15140	The sine of pi / 2 is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( sin ` ( pi / 2 ) ) = 1
Theorem	coshalfpi 15141	The cosine of pi / 2 is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( cos ` ( pi / 2 ) ) = 0
Theorem	cosneghalfpi 15142	The cosine of -u pi / 2 is zero. (Contributed by David Moews, 28-Feb-2017.)
|- ( cos ` -u ( pi / 2 ) ) = 0
Theorem	efhalfpi 15143	The exponential of _i pi / 2 is _i. (Contributed by Mario Carneiro, 9-May-2014.)
|- ( exp ` ( _i x. ( pi / 2 ) ) ) = _i
Theorem	cospi 15144	The cosine of pi is -u 1. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( cos ` pi ) = -u 1
Theorem	efipi 15145	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 15146	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 15147	The sine of 2 pi is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( sin ` ( 2 x. pi ) ) = 0
Theorem	cos2pi 15148	The cosine of 2 pi is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ( cos ` ( 2 x. pi ) ) = 1
Theorem	ef2pi 15149	The exponential of 2 pi _i is 1. (Contributed by Mario Carneiro, 9-May-2014.)
|- ( exp ` ( _i x. ( 2 x. pi ) ) ) = 1
Theorem	ef2kpi 15150	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 15151	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 15152	Lemma for sinper 15153 and cosper 15154. (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 15153	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 15154	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 15155	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 15156	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 15157	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 15158	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 15159	Sine of a number less pi. (Contributed by Paul Chapman, 15-Mar-2008.)
|- ( A e. CC -> ( sin ` ( A - pi ) ) = -u ( sin ` A ) )
Theorem	cosmpi 15160	Cosine of a number less pi. (Contributed by Paul Chapman, 15-Mar-2008.)
|- ( A e. CC -> ( cos ` ( A - pi ) ) = -u ( cos ` A ) )
Theorem	sinppi 15161	Sine of a number plus pi. (Contributed by NM, 10-Aug-2008.)
|- ( A e. CC -> ( sin ` ( A + pi ) ) = -u ( sin ` A ) )
Theorem	cosppi 15162	Cosine of a number plus pi. (Contributed by NM, 18-Aug-2008.)
|- ( A e. CC -> ( cos ` ( A + pi ) ) = -u ( cos ` A ) )
Theorem	efimpi 15163	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 15164	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 15165	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 15166	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 15167	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 15168	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 11928, 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 15169	Lemma for sincosq1sgn 15170. (Contributed by Paul Chapman, 24-Jan-2008.)
|- ( ( A e. RR /\ 0 < A /\ A < ( pi / 2 ) ) -> 0 < ( sin ` A ) )
Theorem	sincosq1sgn 15170	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 15171	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 15172	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 15173	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 15174	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 15175	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 15176	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 15177	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 15178	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 15179	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 15180	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 15181	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 15182	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 15183	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 15184	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 15185	The tangent of pi / 4. (Contributed by Mario Carneiro, 5-Apr-2015.)
|- ( tan ` ( pi / 4 ) ) = 1
Theorem	sincos6thpi 15186	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 15187	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 15188	pi is greater than 3. (Contributed by Brendan Leahy, 21-Aug-2020.)
|- 3 < pi
Theorem	pige3 15189	pi is greater than or equal to 3. (Contributed by Mario Carneiro, 21-May-2016.)
|- 3 <_ pi
Theorem	abssinper 15190	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 15191	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 15192	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 15193	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 15194	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 15195	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 15196	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 15197	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 ) ) )
Theorem	ioocosf1o 15198	The cosine function is a bijection when restricted to its principal domain. (Contributed by Mario Carneiro, 12-May-2014.) (Revised by Jim Kingdon, 7-May-2024.)
|- ( cos |` ( 0 (,) pi ) ) : ( 0 (,) pi ) -1-1-onto-> ( -u 1 (,) 1 )
Theorem	negpitopissre 15199	The interval ( -u pi (,] pi ) is a subset of the reals. (Contributed by David Moews, 28-Feb-2017.)
|- ( -u pi (,] pi ) C_ RR
11.2.3  The natural logarithm on complex numbers
Syntax	clog 15200	Extend class notation with the natural logarithm function on complex numbers.
class log