mmtheorems88 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 8701-8800 - Page 88 of 108

Type	Label	Description
Statement
Theorem	cos2pim 8701	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 8702	Sine of a number less pi. (Contributed by Paul Chapman, 15-Mar-2008.)
|- (A e. CC -> (sin` (A - pi)) = -u(sin` A))
Theorem	cosmpi 8703	Cosine of a number less pi. (Contributed by Paul Chapman, 15-Mar-2008.)
|- (A e. CC -> (cos` (A - pi)) = -u(cos` A))
Theorem	sinppi 8704	Sine of a number plus pi.
|- (A e. CC -> (sin` (A + pi)) = -u(sin` A))
Theorem	sinkpi 8705	The sine of an integer multiple of pi is 0.
|- (K e. ZZ -> (sin` (K x. pi)) = 0)
Theorem	efimpi 8706	The exponential function of 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 8707	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 8708	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 8709	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 8710	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	sincosq1lem 8711	Lemma for sincosq1sgn 8712.
Theorem	sincosq1sgn 8712	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 8713	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 8714	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 8715	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	sinq12gt0t 8716	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	sincosq1eq 8717	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 8718	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	sincos6thpi 8719	The sine and cosine of pi / 6. (Contributed by Paul Chapman, 25-Jan-2008.)
|- ((sin` (pi / 6)) = (1 / 2) /\ (cos` (pi / 6)) = ((sqr` 3) / 2))
Theorem	abssinper 8720	The absolute value of sine has period pi.
|- ((A e. CC /\ K e. ZZ) -> (abs` (sin` (A + (K x. pi)))) = (abs` (sin` A)))
Theorem	sineq0 8721	A real number whose sine is zero is an integer multiple of pi.
|- ((A e. RR /\ (sin` A) = 0) -> A = ((|_` (A / pi)) x. pi))
Theorem	cosh111lem1 8722	Lemma for cosh111t 8725.
Theorem	cosh111lem2 8723	Lemma for cosh111t 8725.
Theorem	cosh111lem3 8724	Lemma for cosh111t 8725.
Theorem	cosh111t 8725	Cosine is one-to-one over the closed-below, open-above interval from 0 to pi. (Contributed by Paul Chapman, 16-Mar-2008.)
|- ((A e. (0[,)pi) /\ B e. (0[,)pi)) -> (A = B <-> (cos` A) = (cos` B)))
Mapping of the exponential function
Theorem	efgh 8726	The exponential function of a scaled complex number is a group homomorphism from the group of complex numbers under addition to the set of complex numbers under multiplication. (Contributed by Paul Chapman, 25-Apr-2008.)
|- F = {<.x, y>. | (x e. CC /\ y = (exp` (A x. x)))}   =>   |- ((A e. CC /\ B e. CC /\ C e. CC) -> (F` (B + C)) = ((F` B) x. (F` C)))
Theorem	efghgrpilem 8727	Lemma for efghgrpi 8728,
Theorem	efghgrpi 8728	The image of a subgroup of the group +, under the exponential function of a scaled complex number, is an Abelian group. (Contributed by Paul Chapman, 25-Apr-2008.)
|- S = {y | E.x e. X y = (exp` (A x. x))}   &   |- G = ( x. |` (S X. S))   &   |- A e. CC   &   |- X (_ CC   &   |- ( + |` (X X. X)) e. (SubGrp` + )   =>   |- G e. Abel
Theorem	efif 8729	The exponential function of an imaginary number maps the closed-below, open-above interval from 0 to 2 x. pi to the unit circle. (Contributed by Paul Chapman, 16-Mar-2008.)
|- F = {<.x, y>. | (x e. (0[,)(2 x. pi)) /\ y = (exp` (i x. x)))}   &   |- S = {z e. CC | (abs` z) = 1}   =>   |- F:(0[,)(2 x. pi))-->S
Theorem	efifolem1 8730	Lemma for efifo 8737.
Theorem	efifolem2 8731	Lemma for efifo 8737.
Theorem	efifolem3 8732	Lemma for efifo 8737.
Theorem	efifolem4 8733	Lemma for efifo 8737.
Theorem	efifolem5 8734	Lemma for efifo 8737.
Theorem	efifolem6 8735	Lemma for efifo 8737.
Theorem	efifolem7 8736	Lemma for efifo 8737.
Theorem	efifo 8737	The exponential function of an imaginary number maps the closed-below, open-above interval from 0 to 2 x. pi onto the unit circle. (Contributed by Paul Chapman, 16-Mar-2008.)
|- F = {<.x, y>. | (x e. (0[,)(2 x. pi)) /\ y = (exp` (i x. x)))}   &   |- S = {z e. CC | (abs` z) = 1}   =>   |- F:(0[,)(2 x. pi))-onto->S
Theorem	efif1lem1 8738	Lemma for efif1 8745.
Theorem	efif1lem2 8739	Lemma for efif1 8745.
Theorem	efif1lem3 8740	Lemma for efif1 8745.
Theorem	efif1lem4 8741	Lemma for efif1 8745.
Theorem	efif1lem5 8742	Lemma for efif1 8745.
Theorem	efif1lem6 8743	Lemma for efif1 8745.
Theorem	efif1lem7 8744	Lemma for efif1 8745.
Theorem	efif1 8745	The exponential function of an imaginary number maps the closed-below, open-above interval from 0 to 2 x. pi one-to-one to the unit circle. (Contributed by Paul Chapman, 16-Mar-2008.)
|- F = {<.x, y>. | (x e. (0[,)(2 x. pi)) /\ y = (exp` (i x. x)))}   &   |- S = {z e. CC | (abs` z) = 1}   =>   |- F:(0[,)(2 x. pi))-1-1->S
Theorem	efif1o 8746	The exponential function of an imaginary number maps the closed-below, open-above interval from 0 to 2 x. pi one-to-one onto the unit circle. (Contributed by Paul Chapman, 16-Mar-2008.)
|- F = {<.x, y>. | (x e. (0[,)(2 x. pi)) /\ y = (exp` (i x. x)))}   &   |- S = {z e. CC | (abs` z) = 1}   =>   |- F:(0[,)(2 x. pi))-1-1-onto->S
Theorem	efielcirc 8747	Membership of the exponential function of i times a real number in the unit circle. (Contributed by Paul Chapman, 25-Mar-2008.)
|- C = {w e. CC | (abs` w) = 1}   =>   |- (A e. RR -> (exp` (i x. A)) e. C)
Theorem	circgrp 8748	The circle group T is an Abelian group. (Contributed by Paul Chapman, 25-Mar-2008.)
|- C = {w e. CC | (abs` w) = 1}   &   |- T = ( x. |` (C X. C))   =>   |- T e. Abel
Theorem	shftefif1olem 8749	Lemma for shftefif1o 8750.
Theorem	shftefif1o 8750	The exponential function of i times a real number maps any closed-below, open-above real interval of length 2 x. pi one-to-one onto the unit circle. (Contributed by Paul Chapman, 25-Mar-2008.)
|- D = (A[,)(A + (2 x. pi)))   &   |- G = {<.x, y>. | (x e. D /\ y = (exp` (i x. x)))}   &   |- C = {w e. CC | (abs` w) = 1}   =>   |- (A e. RR -> G:D-1-1-onto->C)
Theorem	eff1lem 8751	Lemma for eff1i 8752.
Theorem	eff1i 8752	The exponential function maps the set S, of complex numbers with imaginary part in a closed-below, open-above real interval of length 2 x. pi starting at A, one-to-one to the nonzero complex numbers. (Contributed by Paul Chapman, 16-Apr-2008.)
|- A e. RR   &   |- D = (A[,)(A + (2 x. pi)))   &   |- S = {v e. CC | (Im` v) e. D}   &   |- F = {<.z, w>. | (z e. D /\ w = (exp` (i x. z)))}   &   |- C = {v e. CC | (abs` v) = 1}   =>   |- (exp |` S):S-1-1->(CC \ {0})
Theorem	effoi 8753	The exponential function maps the set S, of complex numbers with imaginary part in a closed-below, open-above real interval of length 2 x. pi starting at A, onto the nonzero complex numbers. (Contributed by Paul Chapman, 16-Apr-2008.)
|- A e. RR   &   |- D = (A[,)(A + (2 x. pi)))   &   |- S = {v e. CC | (Im` v) e. D}   &   |- F = {<.z, w>. | (z e. D /\ w = (exp` (i x. z)))}   &   |- C = {v e. CC | (abs` v) = 1}   =>   |- (exp |` S):S-onto->(CC \ {0})
Theorem	eff1oi 8754	The exponential function maps the set S, of complex numbers with imaginary part in a closed-below, open-above real interval of length 2 x. pi starting at A, one-to-one onto the nonzero complex numbers. A would normally be fixed at 0 or -upi, according to choice of principal domain for the exponential function. (Contributed by Paul Chapman, 16-Apr-2008.)
|- A e. RR   &   |- D = (A[,)(A + (2 x. pi)))   &   |- S = {v e. CC | (Im` v) e. D}   =>   |- (exp |` S):S-1-1-onto->(CC \ {0})
Theorem	efper 8755	The exponential function is periodic. (Contributed by Paul Chapman, 21-Apr-2008.)
|- ((A e. CC /\ K e. ZZ) -> (exp` (A + ((i x. (2 x. pi)) x. K))) = (exp` A))
Theorem	eff1o 8756	The exponential function restricted to its principal domain maps one-to-one onto the nonzero complex numbers. (Contributed by Paul Chapman, 21-Apr-2008.)
|- (exp |` {