Metamath Proof Explorer - mmtheorems41

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Statement List for Metamath Proof Explorer - 4001-4100 - Page 41 of 58

Type	Label	Description
Statement
Theorem	1idsr 4001	1 is an identity element for multiplication.
|- (A e. R. -> (A .R 1R) = A)
Theorem	00sr 4002	A signed real times 0 is 0.
|- (A e. R. -> (A .R 0R) = 0R)
Theorem	ltasr 4003	Ordering property of addition.
|- A e. V   &   |- B e. V   =>   |- (C e. R. -> (A <R B <-> (C +R A) <R (C +R B)))
Theorem	pn0sr 4004	A signed real plus its negative is zero.
|- (A e. R. -> (A +R (A .R -1R)) = 0R)
Theorem	negexsr 4005	Existence of negative signed real. Part of Proposition 9-4.3 of [Gleason] p. 126.
|- (A e. R. -> E.x(x e. R. /\ (A +R x) = 0R))
Theorem	recexsrlem 4006	The reciprocal of a positive signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126.
|- A e. V   =>   |- (0R <R A -> E.x(x e. R. /\ (A .R x) = 1R))
Theorem	addgt0sr 4007	The sum of two positive signed reals is positive.
|- A e. V   &   |- B e. V   =>   |- ((0R <R A /\ 0R <R B) -> 0R <R (A +R B))
Theorem	mulgt0sr 4008	The product of two positive signed reals is positive.
|- A e. V   &   |- B e. V   =>   |- ((0R <R A /\ 0R <R B) -> 0R <R (A .R B))
Theorem	sqgt0sr 4009	The square of a nonzero signed real is positive.
|- A e. V   =>   |- (A e. R. -> (-. A = 0R -> 0R <R (A .R A)))
Theorem	recexsr 4010	The reciprocal of a nonzero signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126.
|- A e. V   =>   |- (A e. R. -> (-. A = 0R -> E.x(x e. R. /\ (A .R x) = 1R)))
Theorem	ssgt0sr 4011	The sum of squares of signed reals is positive if one is nonzero.
|- A e. V   &   |- B e. V   =>   |- ((A e. R. /\ B e. R.) -> (-. (A = 0R /\ B = 0R) -> 0R <R ((A .R A) +R (B .R B))))
Theorem	mappsrpr 4012	Mapping from positive signed reals to positive reals.
|- A e. V   =>   |- (0R <R [<.(A +P. 1P), 1P>.] ~R <-> A e. P.)
Theorem	ltpsrpr 4013	Mapping of order from positive signed reals to positive reals.
|- A e. V   &   |- B e. V   =>   |- ([<.(A +P. 1P), 1P>.] ~R <R [<.(B +P. 1P), 1P>.] ~R <-> A <P B)
Theorem	map2psrpr 4014	Equivalence for positive signed real.
|- A e. V   =>   |- (0R <R A <-> E.x(x e. P. /\ [<.(x +P. 1P), 1P>.] ~R = A))
Theorem	suppsrlem 4015	Mapping of non-empty subset from positive reals to positive signed reals.
|- B = {w | [<.(w +P. 1P), 1P>.] ~R e. A}   =>   |- ((A.x(x e. A -> 0R <R x) /\ -. A = (/)) -> (B (_ P. /\ -. B = (/)))
Theorem	suppsr 4016	A non-empty, bounded set of positive signed reals has a supremum.
|- B = {w | [<.(w +P. 1P), 1P>.] ~R e. A}   =>   |- (((A.x(x e. A -> 0R <R x) /\ -. A = (/)) /\ E.x(0R <R x /\ A.y(0R <R y -> (y e. A -> y <R x)))) -> E.x(0R <R x /\ A.y(0R <R y -> ((y e. A -> -. x <R y) /\ (y <R x -> E.z(0R <R z /\ (z e. A /\ y <R z)))))))
Theorem	suppsr2 4017	A non-empty, bounded set of positive signed reals has a supremum. (Converts quantifier restrictions to all reals.)
|- (((A.x(x e. A -> 0R <R x) /\ -. A = (/)) /\ E.x(x e. R. /\ A.y(y e. R. -> (y e. A -> y <R x)))) -> E.x(x e. R. /\ A.y(y e. R. -> ((y e. A -> -. x <R y) /\ (y <R x -> E.z(z e. R. /\ (z e. A /\ y <R z)))))))
Theorem	suppsr3 4018	A non-empty, bounded set with at least one positive real has a supremum.
|- B = {y | (y e. A /\ 0R <R y)}   =>   |- ((E.y(y e. A /\ 0R <R y) /\ E.x(x e. R. /\ A.y(y e. R. -> (y e. A -> y <R x)))) -> E.x(x e. R. /\ A.y(y e. R. -> ((y e. A -> -. x <R y) /\ (y <R x -> E.z(z e. R. /\ (z e. A /\ y <R z)))))))
Theorem	supsrlem1 4019	Lemma for supremum theorem.
|- C e. R.   =>   |- ((C +R (A +R -1R)) e. R. <-> A e. R.)
Theorem	supsrlem2 4020	Lemma for supremum theorem.
|- C e. R.   =>   |- (A e. R. <-> E.x(x e. R. /\ (C +R (x +R -1R)) = A))
Theorem	supsrlem3 4021	Lemma for supremum theorem.
|- C e. R.   &   |- A e. V   &   |- B e. V   =>   |- ((C +R (A +R -1R)) <R (C +R (B +R -1R)) <-> A <R B)
Theorem	supsrlem4 4022	Lemma for supremum theorem.
|- C e. R.   &   |- B = {w | (C +R (w +R -1R)) e. A}   &   |- D e. V   =>   |- (D e. B <-> (C +R (D +R -1R)) e. A)
Theorem	supsrlem5 4023	Lemma for supremum theorem.
|- C e. R.   &   |- B = {w | (C +R (w +R -1R)) e. A}   =>   |- (C e. A -> E.y(y e. B /\ 0R <R y))
Theorem	supsrlem6 4024	Lemma for supremum theorem.
|- C e. R.   &   |- B = {w | (C +R (w +R -1R)) e. A}   =>   |- ((C e. A /\ E.x(x e. R. /\ A.y(y e. R. -> (y e. A -> y <R x)))) -> E.x(x e. R. /\ A.y(y e. R. -> ((y e. A -> -. x <R y) /\ (y <R x -> E.z(z e. R. /\ (z e. A /\ y <R z)))))))
Theorem	supsr 4025	A non-empty, bounded set of signed reals has a supremum.
|- (((A (_ R. /\ -. A = (/)) /\ E.x(x e. R. /\ A.y(y e. R. -> (y e. A -> y <R x)))) -> E.x(x e. R. /\ A.y(y e. R. -> ((y e. A -> -. x <R y) /\ (y <R x -> E.z(z e. R. /\ (z e. A /\ y <R z)))))))
Syntax	cc 4026	Class of complex numbers.
class CC
Syntax	cr 4027	Class of real numbers.
class RR
Syntax	cc0 4028	Extend class notation to include the complex number 0.
class 0
Syntax	c1 4029	Extend class notation to include the complex number 1.
class 1
Syntax	ci 4030	Extend class notation to include the complex number i.
class i
Syntax	caddc 4031	Addition on complex numbers.
class +
Syntax	cmulc 4032	Multiplication on complex numbers. The token x. is a center dot.
class x.
Syntax	clt 4033	'Less than' predicate (defined over real subset of complex numbers).
class <
Definition	df-c 4034	Define the set of complex numbers.
|- CC = (R. X. R.)
Definition	df-0 4035	Define the complex number 0 (base 10).
|- 0 = <.0R, 0R>.
Definition	df-1 4036	Define the complex number 1 (base 10).
|- 1 = <.1R, 0R>.
Definition	df-i 4037	Define the complex (imaginary) number i.
|- i = <.0R, 1R>.
Definition	df-r 4038	Define the set of real numbers.
|- RR = (R. X. {0R})
Definition	df-plus 4039	Define addition over complex numbers.
|- + = {<.<.x, y>., z>. | ((x e. CC /\ y e. CC) /\ E.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = <.(w +R u), (v +R f)>.))}
Definition	df-mul 4040	Define multiplication over complex numbers.
|- x. = {<.<.x, y>., z>. | ((x e. CC /\ y e. CC) /\ E.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = <.((w .R u) +R (-1R .R (v .R f))), ((v .R u) +R (w .R f))>.))}
Definition	df-lt 4041	Define 'less than' on the real subset of complex numbers.
|- < = {<.x, y>. | ((x e. RR /\ y e. RR) /\ E.zE.w((x = <.z, 0R>. /\ y = <.w, 0R>.) /\ z <R w))}
Theorem	opelcn 4042	Ordered pair membership in the class of complex numbers.
|- B e. V   =>   |- (<.A, B>. e. CC <-> (A e. R. /\ B e. R.))
Theorem	opelreal 4043	Ordered pair membership in class of real subset of complex numbers.
|- (<.A, 0R>. e. RR <-> A e. R.)
Theorem	elreal 4044	Membership in class of real numbers.
|- (A e. RR <-> E.x(x e. R. /\ <.x, 0R>. = A))
Theorem	0ncn 4045	The empty set is not a complex number.
|- -. (/) e. CC
Theorem	ltrelre 4046	'Less than' is a relation on real numbers.
|- < (_ (RR X. RR)
Theorem	addcnsr 4047	Addition of complex numbers in terms of signed reals.
|- (((A e. R. /\ B e. R.) /\ (C e. R. /\ D e. R.)) -> (<.A, B>. + <.C, D>.) = <.(A +R C), (B +R D)>.)
Theorem	mulcnsr 4048	Multiplication of complex numbers in terms of signed reals.
|- (((A e. R. /\ B e. R.) /\ (C e. R. /\ D e. R.)) -> (<.A, B>. x. <.C, D>.) = <.((A .R C) +R (-1R .R (B .R D))), ((B .R C) +R (A .R D))>.)
Theorem	eqresr 4049	Equality of real numbers in terms of intermediate signed reals.
|- A e. V   =>   |- (<.A, 0R>. = <.B, 0R>. <-> A = B)
Theorem	addresr 4050	Addition of real numbers in terms of intermediate signed reals.
|- ((A e. R. /\ B e. R.) -> (<.A, 0R>. + <.B, 0R>.) = <.(A +R B), 0R>.)
Theorem	mulresr 4051	Multiplication of real numbers in terms of intermediate signed reals.
|- B e. V   =>   |- ((A e. R. /\ B e. R.) -> (<.A, 0R>. x. <.B, <