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	psrel 8601	A poset is a relation.
|- (A e. Poset -> Rel A)
Theorem	pslem 8602	Lemma for psref 8604 and others.
Theorem	psdmrn 8603	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 8604	A poset is reflexive.
|- X = dom R   =>   |- ((R e. Poset /\ A e. X) -> ARA)
Theorem	psasym 8605	A poset is antisymmetric.
|- ((R e. Poset /\ ARB /\ BRA) -> A = B)
Theorem	pstr 8606	A poset is transitive.
|- ((R e. Poset /\ ARB /\ BRC) -> ARC)
Theorem	spwval2 8607	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 3514. Unlike df-sup 4564 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 8608	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 8609	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 8610	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 8611	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 8612	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 8613	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 8614	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 8615	Lemma for sinco 8617 and cosco 8618.
Theorem	sincnlem 8616	Lemma for sincn 8619 and coscn 8620.
Theorem	sinco 8617	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 8618	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 8619	Sine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.)
|- sin e. (CC-cn->CC)
Theorem	coscn 8620	Cosine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.)
|- cos e. (CC-cn->CC)
Properties of pi = 3.14159...
Theorem	pilem1 8621	Lemma for pire 8627, pigt2lt4 8625 and sinpi 8626.
Theorem	pilem2 8622	Lemma for pire 8627, pigt2lt4 8625 and sinpi 8626.
Theorem	pilem3 8623	Lemma for pire 8627, pigt2lt4 8625 and sinpi 8626.
Theorem	pilem4 8624	Lemma for pire 8627, pigt2lt4 8625 and sinpi 8626.
Theorem	pigt2lt4 8625	pi is between 2 and 4. (Contributed by Paul Chapman, 23-Jan-2008.)
|- (2 < pi /\ pi < 4)
Theorem	sinpi 8626	The sine of pi is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
|- (sin` pi) = 0
Theorem	pire 8627	pi is a real number. (Contributed by Paul Chapman, 23-Jan-2008.)
|- pi e. RR
Theorem	pipos 8628	pi is positive. (Contributed by Paul Chapman, 23-Jan-2008.)
|- 0 < pi
Theorem	sinhalfpilem 8629	Lemma for sinhalfpi 8630 and coshalfpi 8631.
Theorem	sinhalfpi 8630	The sine of pi / 2 is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
|- (sin` (pi / 2)) = 1
Theorem	coshalfpi 8631	The cosine of pi / 2 is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
|- (cos` (pi / 2)) = 0
Theorem	cospi 8632	The cosine of pi is -u1. (Contributed by Paul Chapman, 23-Jan-2008.)
|- (cos` pi) = -u1
Theorem	eulerid 8633	Euler's identity. (Contributed by Paul Chapman, 23-Jan-2008.)
|- ((exp` (i x. pi)) + 1) = 0
Theorem	sin2pi 8634	The sine of 2pi is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
|- (sin` (2 x. pi)) = 0
Theorem	cos2pi 8635	The cosine of 2pi is 1. (Contributed by Paul Chapman, 23-Jan-2008.)
|- (cos` (2 x. pi)) = 1
Theorem	sinperlem1 8636	Lemma for sin2kpi 8638 and cos2kpi 8639.
Theorem	sinperlem2 8637	Lemma for sin2kpi 8638 and cos2kpi 8639.
Theorem	sin2kpi 8638	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 8639	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 8640	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 8641	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 8642	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 8643	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 8644	Sine of a number less pi. (Contributed by Paul Chapman, 15-Mar-2008.)
|- (A e. CC -> (sin` (A - pi)) = -u(sin` A))
Theorem	cosmpi 8645	Cosine of a number less pi. (Contributed by Paul Chapman, 15-Mar-2008.)
|- (A e. CC -> (cos` (A - pi)) = -u(cos` A))
Theorem	efimpi 8646	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 8647	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 8648	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 8649	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 8650	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 8651	Lemma for sincosq1sgn 8652.
Theorem	sincosq1sgn 8652	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 8653	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 8654	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 8655	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 8656	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 8657	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) -> (