mmtheorems87 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 8601-8700 - Page 87 of 107

Type	Label	Description
Statement
Theorem	isps 8601	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 8602	A poset is a relation.
|- (A e. Poset -> Rel A)
Theorem	pslem 8603	Lemma for psref 8605 and others.
Theorem	psdmrn 8604	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 8605	A poset is reflexive.
|- X = dom R   =>   |- ((R e. Poset /\ A e. X) -> ARA)
Theorem	psasym 8606	A poset is antisymmetric.
|- ((R e. Poset /\ ARB /\ BRA) -> A = B)
Theorem	pstr 8607	A poset is transitive.
|- ((R e. Poset /\ ARB /\ BRC) -> ARC)
Theorem	spwval2 8608	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 3515. Unlike df-sup 4565 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 8609	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 8610	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 8611	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 8612	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	spwval 8613	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 8614	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 8615	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)
Real and complex numbers (cont.)
The exponential, sine, and cosine functions (cont.)
Theorem	sincolem 8616	Lemma for sinco 8618 and cosco 8619.
Theorem	sincnlem 8617	Lemma for sincn 8620 and coscn 8621.
Theorem	sinco 8618	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 8619	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 8620	Sine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.)
|- sin e. (CC-cn->CC)
Theorem	coscn 8621	Cosine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.)
|- cos e. (CC-cn->CC)
Properties of pi = 3.14159...
Theorem	pilem1 8622	Lemma for pire 8628, pigt2lt4 8626 and sinpi 8627.
Theorem	pilem2 8623	Lemma for pire 8628, pigt2lt4 8626 and sinpi 8627.
Theorem	pilem3 8624	Lemma for pire 8628, pigt2lt4 8626 and sinpi 8627.
Theorem	pilem4 8625	Lemma for pire 8628, pigt2lt4 8626 and sinpi 8627.
Theorem	pigt2lt4 8626	pi is between 2 and 4. (Contributed by Paul Chapman, 23-Jan-2008.)
|- (2 < pi /\ pi < 4)
Theorem	sinpi 8627	The sine of pi is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
|- (sin` pi) = 0
Theorem	pire 8628	pi is a real number. (Contributed by Paul Chapman, 23-Jan-2008.)
|- pi e. RR
Theorem	pipos 8629	pi is positive. (Contributed by Paul Chapman, 23-Jan-2008.)
|- 0 < pi
Theorem	sinhalfpilem 8630	Lemma for sinhalfpi 8631 and coshalfpi 8632.
Theorem	sinhalfpi 8631	The sine of pi / 2 is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
|- (sin` (pi / 2)) = 1
Theorem	coshalfpi 8632	The cosine of pi / 2 is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
|- (cos` (pi / 2)) = 0
Theorem	cospi 8633	The cosine of pi is -u1. (Contributed by Paul Chapman, 23-Jan-2008.)
|- (cos` pi) = -u1
Theorem	eulerid 8634	Euler's identity. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ((exp` (i x. pi)) + 1) = 0
Theorem	sin2pi 8635	The sine of 2pi is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
|- (sin` (2 x. pi)) = 0
Theorem	cos2pi 8636	The cosine of 2pi is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
|- (cos` (2 x. pi)) = 1
Theorem	sinperlem1 8637	Lemma for sin2kpi 8639 and cos2kpi 8640.
Theorem	sinperlem2 8638	Lemma for sin2kpi 8639 and cos2kpi 8640.
Theorem	sin2kpi 8639	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 8640	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 8641	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 8642	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 8643	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 8644	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 8645	Sine of a number less pi. (Contributed by Paul Chapman, 15-Mar-2008.)
|- (A e. CC -> (sin` (A - pi)) = -u(sin` A))
Theorem	cosmpi 8646	Cosine of a number less pi. (Contributed by Paul Chapman, 15-Mar-2008.)
|- (A e. CC -> (cos` (A - pi)) = -u(cos` A))
Theorem	efimpi 8647	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 8648	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 8649	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 8650	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 8651	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 8652	Lemma for sincosq1sgn 8653.
Theorem	sincosq1sgn 8653	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 8654	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 8655	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 8656	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 8657	The sine of a number strictly between 0 and pi is positive. (Contributed by Paul Chapman, 15-Mar-2008.)