mmtheorems90 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 8901-9000 - Page 90 of 123

Type	Label	Description
Statement
Definition	df-ps 8901	Define the class of all posets (partially ordered sets) with weak ordering (e.g. "less than or equal to" instead of "less than"). A poset is a relation which is transitive, reflexive, and antisymmetric.
|- Poset = {r | (Rel r /\ (r o. r) (_ r /\ (r i^i `'r) = (I |` U.U.r))}
Definition	df-spw 8902	Define suprema under weak orderings. Unlike df-sup 4717 for strong orderings, supw is evaluates to a member of the field of R iff the supremum exists. Read R supw A as the R -supremum of set A.
|- supw = {<.<.r, x>., y>. | E.z(z = {w e. U.U.r | (A.v e. x vrw /\ A.v e. U.U.r(A.u e. x urv -> wrv))} /\ y = if(z =/= (/), U.z, P~U.U.U.r))}
Definition	df-nfw 8903	Define the class of all infima of a weak ordering relation.
|- infw = {<.<.r, x>., y>. | y = (`'r supw x)}
Definition	df-jn 8904	Define the class of all join operations on weak orderings.
|- join = {<.r, w>. | w = {<.<.x, y>., z>. | ((x e. U.U.r /\ y e. U.U.r) /\ z = (r supw {x, y}))}}
Definition	df-mee 8905	Define the class of all meet operations on weak orderings.
|- meet = {<.r, w>. | w = {<.<.x, y>., z>. | ((x e. U.U.r /\ y e. U.U.r) /\ z = (r infw {x, y}))}}
Definition	df-la 8906	Define the class of all lattices, which are posets in which every two elements have upper and lower bounds.
|- Lat = {r e. Poset | A.x e. dom rA.y e. dom r((r supw {x, y}) e. dom r /\ (r infw {x, y}) e. dom r)}
Theorem	isps 8907	The predicate "is a poset" i.e. a transitive, reflexive, antisymmetric relation.
|- (R e. A -> (R e. Poset <-> (Rel R /\ (R o. R) (_ R /\ (R i^i `'R) = (I |` U.U.R))))
Theorem	psrel 8908	A poset is a relation.
|- (A e. Poset -> Rel A)
Theorem	pslem 8909	Lemma for psref 8911 and others.
Theorem	psdmrn 8910	The domain and range of a poset equal its field.
|- (R e. Poset -> (dom R = U.U.R /\ ran R = U.U.R))
Theorem	psref 8911	A poset is reflexive.
|- X = dom R   =>   |- ((R e. Poset /\ A e. X) -> ARA)
Theorem	psrn 8912	The range of a poset equals it domain.
|- X = dom R   =>   |- (R e. Poset -> X = ran R)
Theorem	psasym 8913	A poset is antisymmetric.
|- ((R e. Poset /\ ARB /\ BRA) -> A = B)
Theorem	pstr 8914	A poset is transitive.
|- ((R e. Poset /\ ARB /\ BRC) -> ARC)
Theorem	spwval2 8915	Value of supremum under a weak ordering. Read R supw A as "the R -supremum of A." U.U.R is the field of a relation R by relfld 3620. Unlike df-sup 4717 for strong orderings, the supremum exists iff R supw A belongs to the field.
|- X = U.U.R   &   |- Z = {x e. X | (A.y e. A yRx /\ A.y e. X (A.z e. A zRy -> xRy))}   =>   |- ((R e. U /\ A e. W) -> (R supw A) = if(Z =/= (/), U.Z, P~U.X))
Theorem	spwval3 8916	Value of a supremum.
|- X = U.U.R   &   |- (ph <-> (A.y e. A yRx /\ A.y e. X (A.z e. A zRy -> xRy)))   =>   |- ((R e. U /\ A e. W /\ E.x e. X ph) -> (R supw A) = U.{x e. X | ph})
Theorem	spwnex3 8917	When the supremum of set A doesn't exist, R supw A isn't in the the field of order relation R.
|- X = U.U.R   &   |- (ph <-> (A.y e. A yRx /\ A.y e. X (A.z e. A zRy -> xRy)))   =>   |- ((R e. U /\ A e. W /\ -. E.x e. X ph) -> -. (R supw A) e. X)
Theorem	spwmo 8918	A poset has at most one supremum.
|- (ph <-> (A.y e. A yRx /\ A.y e. X (A.z e. A zRy -> xRy)))   =>   |- (R e. Poset -> E*x(x e. X /\ ph))
Theorem	spweu 8919	A supremum is unique.
|- (ph <-> (A.y e. A yRx /\ A.y e. X (A.z e. A zRy -> xRy)))   =>   |- ((R e. Poset /\ E.x e. X ph) -> E!x e. X ph)
Theorem	spwpr2 8920	Property of supremum defining condition for an unordered pair.
|- (ph <-> (A.y e. A yRx /\ A.y e. X (A.z e. A zRy -> xRy)))   =>   |- (((R e. T /\ A = {B, C}) /\ (B e. U /\ C e. W)) -> (ph <-> ((BRx /\ CRx) /\ A.y e. X ((BRy /\ CRy) -> xRy))))
Theorem	spwval 8921	Value of a supremum.
|- X = dom R   &   |- (ph <-> (A.y e. A yRx /\ A.y e. X (A.z e. A zRy -> xRy)))   =>   |- ((R e. Poset /\ A e. W /\ E.x e. X ph) -> (R supw A) = U.{x e. X | ph})
Theorem	spwcl 8922	Closure of a supremum.
|- X = dom R   &   |- (ph <-> (A.y e. A yRx /\ A.y e. X (A.z e. A zRy -> xRy)))   =>   |- ((R e. Poset /\ A e. W /\ E.x e. X ph) -> (R supw A) e. X)
Theorem	spwnex 8923	Non-closure when the supremum doesn't exist.
|- X = dom R   &   |- (ph <-> (A.y e. A yRx /\ A.y e. X (A.z e. A zRy -> xRy)))   =>   |- ((R e. Poset /\ A e. W /\ -. E.x e. X ph) -> -. (R supw A) e. X)
Theorem	spwex 8924	A supremum exists iff R supw A belongs to the domain of R.
|- X = dom R   &   |- (ph <-> (A.y e. A yRx /\ A.y e. X (A.z e. A zRy -> xRy)))   =>   |- ((R e. Poset /\ A e. W) -> (E.x e. X ph <-> (R supw A) e. X))
Theorem	spwpr4 8925	Supremum of an unordered pair.
|- X = dom R   =>   |- (((R e. Poset /\ C e. D) /\ (ARC /\ BRC) /\ A.x e. X ((ARx /\ BRx) -> CRx)) -> (R supw {A, B}) = C)
Theorem	spwpr4OLD 8926	Supremum of an unordered pair.
|- X = dom R   =>   |- (((R e. Poset /\ C e. X) /\ (ARC /\ BRC) /\ A.x e. X ((ARx /\ BRx) -> CRx)) -> (R supw {A, B}) = C)
Theorem	spwpr4aOLD 8927	Supremum of an unordered pair.
|- X = dom R   &   |- (ph <-> (A.y e. {A, B}yRx /\ A.y e. X (A.z e. {A, B}zRy -> xRy)))   =>   |- (((R e. Poset /\ C e. X) /\ (ARC /\ BRC) /\ A.y e. X ((ARy /\ BRy) -> CRy)) -> (R supw {A, B}) = C)
Theorem	spwpr4c 8928	Supremum of an unordered pair of comparable elements.
|- ((R e. Poset /\ B e. U /\ ARB) -> (R supw {A, B}) = B)
Theorem	isla 8929	The predicate "is a lattice" i.e. a poset in which any two elements have upper and lower bounds.
|- X = dom R   =>   |- (R e. Lat <-> (R e. Poset /\ A.x e. X A.y e. X ((R supw {x, y}) e. X /\ (R infw {x, y}) e. X)))
Theorem	laspwcl 8930	Closure of the supremum (join) of two lattice elements.
|- X = dom R   =>   |- ((R e. Lat /\ A e. X /\ B e. X) -> (R supw {A, B}) e. X)
Theorem	lanfwcl 8931	Closure of the infimum (meet) of two lattice elements.
|- X = dom R   =>   |- ((R e. Lat /\ A e. X /\ B e. X) -> (R infw {A, B}) e. X)
Real and complex numbers (cont.)
The exponential, sine, and cosine functions (cont.)
Theorem	sincolem 8932	Lemma for sinco 8934 and cosco 8935.
Theorem	sincnlem 8933	Lemma for sincn 8936 and coscn 8937.
Theorem	sinco 8934	Sine expressed as a function composition. (Contributed by Paul Chapman, 28-Nov-2007.)
|- F = {<.x, y>. | (x e. CC /\ y = (i x. x))}   &   |- G = {<.x, y>. | (x e. CC /\ y = (-ui x. x))}   &   |- J = {<.x, y>. | (x e. CC /\ y = (x / (2 x. i)))}   &   |- H = {<.w, v>. | (w e. CC /\ v = (((exp o. F)` w) - ((exp o. G)` w)))}   =>   |- sin = (J o. H)
Theorem	cosco 8935	Cosine expressed as a function composition. (Contributed by Paul Chapman, 28-Nov-2007.)
|- F = {<.x, y>. | (x e. CC /\ y = (i x. x))}   &   |- G = {<.x, y>. | (x e. CC /\ y = (-ui x. x))}   &   |- J = {<.x, y>. | (x e. CC /\ y = (x / 2))}   &   |- H = {<.w, v>. | (w e. CC /\ v = (((exp o. F)` w) + ((exp o. G)` w)))}   =>   |- cos = (J o. H)
Theorem	sincn 8936	Sine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.)
|- sin e. (CC-cn->CC)
Theorem	coscn 8937	Cosine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.)
|- cos e. (CC-cn->CC)
Properties of pi = 3.14159...
Theorem	pilem1 8938	Lemma for pire 8944, pigt2lt4 8942 and sinpi 8943.
Theorem	pilem2 8939	Lemma for pire 8944, pigt2lt4 8942 and sinpi 8943.
Theorem	pilem3 8940	Lemma for pire 8944, pigt2lt4 8942 and sinpi 8943.
Theorem	pilem4 8941	Lemma for pire 8944, pigt2lt4 8942 and sinpi 8943.
Theorem	pigt2lt4 8942	pi is between 2 and 4. (Contributed by Paul Chapman, 23-Jan-2008.)
|- (2 < pi /\ pi < 4)
Theorem	sinpi 8943	The sine of pi is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
|- (sin` pi) = 0
Theorem	pire 8944	pi is a real number. (Contributed by Paul Chapman, 23-Jan-2008.)
|- pi e. RR
Theorem	pipos 8945	pi is positive. (Contributed by Paul Chapman, 23-Jan-2008.)
|- 0 < pi
Theorem	sinhalfpilem 8946	Lemma for sinhalfpi 8947 and coshalfpi 8948.
Theorem	sinhalfpi 8947	The sine of pi / 2 is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
|- (sin` (pi / 2)) = 1
Theorem	coshalfpi 8948	The cosine of pi / 2 is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
|- (cos` (pi / 2)) = 0
Theorem	cospi 8949	The cosine of pi is -u1. (Contributed by Paul Chapman, 23-Jan-2008.)
|- (cos` pi) = -u1
Theorem	eulerid 8950	Euler's identity. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ((exp` (i x. pi)) + 1) = 0
Theorem	sin2pi 8951	The sine of 2pi is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
|- (sin` (2 x. pi)) = 0
Theorem	cos2pi 8952	The cosine of 2pi is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
|- (cos` (2 x. pi)) = 1
Theorem	sinperlem1 8953	Lemma for sin2kpi 8955 and cos2kpi 8956.
Theorem	sinperlem2 8954	Lemma for sin2kpi 8955 and cos2kpi 8956.
Theorem	sin2kpi 8955	If K is an integer, the sine of 2Kpi is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
|- (K e. ZZ -> (sin` (K x. (2 x. pi))) = 0)
Theorem	cos2kpi 8956	If K is an integer, the cosine of 2Kpi is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
|- (K e. ZZ -> (cos` (K x. (2 x. pi))) = 1)
Theorem	sinper 8957	The sine function is periodic. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ((A e. CC /\ K e. ZZ) -> (sin` (A + (K x. (2 x. pi)))) = (sin` A))
Theorem	cosper 8958	The cosine function is periodic. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ((A e. CC /\ K e. ZZ) -> (cos` (A + (K x. (2 x. pi)))) = (cos` A))
Theorem	sin2pim 8959	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 8960	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 8961	Sine of a number less pi. (Contributed by Paul Chapman, 15-Mar-2008.)
|- (A e. CC -> (sin` (A - pi)) = -u(sin` A))
Theorem	cosmpi 8962	Cosine of a number less pi. (Contributed by Paul Chapman, 15-Mar-2008.)
|- (A e. CC -> (cos` (A - pi)) = -u(cos` A))
Theorem	sinppi 8963	Sine of a number plus pi.
|- (A e. CC -> (sin` (A + pi)) = -u(sin` A))
Theorem	cosppi 8964	Cosine of a complex number plus pi.
|- (A e. CC -> (cos` (A + pi)) = -u(cos` A))
Theorem	efimpi 8965	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 8966	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 8967	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 8968	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 8969	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 8970	Lemma for sincosq1sgn 8971.
Theorem	sincosq1sgn 8971	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 8972	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 8973	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 8974	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 8975	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 8976	The sine of a number strictly between pi and 2 x. pi is negative.
|- (A e. (pi(,)(2 x. pi)) -> (sin` A) < 0)
Theorem	sincosq1eq 8977	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 8978	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 8979	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 8980	The absolute value of sine has period pi.
|- ((A e. CC /\ K e. ZZ) -> (abs` (sin` (A + (K x. pi)))) = (abs` (sin` A)))
Theorem	sinkpi 8981	The sine of an integer multiple of pi is 0.
|- (K e. ZZ -> (sin` (K x. pi)) = 0)
Theorem	coskpi 8982	The absolute value of the cosine of an integer multiple of pi is 1.
|- (K e. ZZ -> (abs` (cos` (K x. pi))) = 1)
Theorem	sineq0 8983	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	sineq0OLD 8984	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	sineq0re 8985	A number whose sine is zero is real. This theorem can be used to extend sineq0 8983 to complex numbers.
|- ((A e. CC /\ (sin` A) = 0) -> A e. RR)
Theorem	cosh111lem1 8986	Lemma for cosh111 8989.
Theorem	cosh111lem2 8987	Lemma for cosh111 8989.
Theorem	cosh111lem3 8988	Lemma for cosh111 8989.
Theorem	cosh111 8989	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 8990	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 8991	Lemma for efghgrpi 8992,
Theorem	efghgrpi 8992	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 8993	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 8994	Lemma for efifo 9001.
Theorem	efifolem2 8995	Lemma for efifo 9001.
Theorem	efifolem3 8996	Lemma for efifo 9001.
Theorem	efifolem4 8997	Lemma for efifo 9001.
Theorem	efifolem5 8998	Lemma for efifo 9001.
Theorem	efifolem6 8999	Lemma for efifo 9001.
Theorem	efifolem7 9000	Lemma for efifo 9001.