mmtheorems53 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 5201-5300 - Page 53 of 107

Type	Label	Description
Statement
Theorem	addgt0sr 5201	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 5202	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 5203	The square of a nonzero signed real is positive.
|- A e. V   =>   |- (A e. R. -> (-. A = 0R -> 0R <R (A .R A)))
Theorem	recexsr 5204	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 5205	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 5206	Mapping from positive signed reals to positive reals.
|- A e. V   =>   |- (0R <R [<.(A +P. 1P), 1P>.] ~R <-> A e. P.)
Theorem	ltpsrpr 5207	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 5208	Equivalence for positive signed real.
|- A e. V   =>   |- (0R <R A <-> E.x(x e. P. /\ [<.(x +P. 1P), 1P>.] ~R = A))
Theorem	suppsrlem 5209	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 5210	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 5211	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 5212	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 5213	Lemma for supremum theorem.
Theorem	supsrlem2 5214	Lemma for supremum theorem.
Theorem	supsrlem3 5215	Lemma for supremum theorem.
Theorem	supsrlem4 5216	Lemma for supremum theorem.
Theorem	supsrlem5 5217	Lemma for supremum theorem.
Theorem	supsrlem6 5218	Lemma for supremum theorem.
Theorem	supsr 5219	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 5220	Class of complex numbers.
class CC
Syntax	cr 5221	Class of real numbers.
class RR
Syntax	cc0 5222	Extend class notation to include the complex number 0.
class 0
Syntax	c1 5223	Extend class notation to include the complex number 1.
class 1
Syntax	ci 5224	Extend class notation to include the complex number i.
class i
Syntax	caddc 5225	Addition on complex numbers.
class +
Syntax	cltrr 5226	'Less than' predicate (defined over real subset of complex numbers).
class <R
Syntax	cmul 5227	Multiplication on complex numbers. The token x. is a center dot.
class x.
Definition	df-c 5228	Define the set of complex numbers. The 25 axioms for complex numbers start at axcnex 5255.
|- CC = (R. X. R.)
Definition	df-0 5229	Define the complex number 0 (base 10).
|- 0 = <.0R, 0R>.
Definition	df-1 5230	Define the complex number 1 (base 10).
|- 1 = <.1R, 0R>.
Definition	df-i 5231	Define the complex number i (the imaginary unit).
|- i = <.0R, 1R>.
Definition	df-r 5232	Define the set of real numbers.
|- RR = (R. X. {0R})
Definition	df-plus 5233	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 5234	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 5235	Define 'less than' on the real subset of complex numbers.
|- <R = {<.x, y>. | ((x e. RR /\ y e. RR) /\ E.zE.w((x = <.z, 0R>. /\ y = <.w, 0R>.) /\ z <R w))}
Theorem	opelcn 5236	Ordered pair membership in the class of complex numbers.
|- B e. V   =>   |- (<.A, B>. e. CC <-> (A e. R. /\ B e. R.))
Theorem	opelreal 5237	Ordered pair membership in class of real subset of complex numbers.
|- (<.A, 0R>. e. RR <-> A e. R.)
Theorem	elreal 5238	Membership in class of real numbers.
|- (A e. RR <-> E.x(x e. R. /\ <.x, 0R>. = A))
Theorem	0ncn 5239	The empty set is not a complex number. Note: do not use this after the real number axioms are developed, since it is a construction-dependent property.
|- -. (/) e. CC
Theorem	ltrelre 5240	'Less than' is a relation on real numbers.
|- <R (_ (RR X. RR)
Theorem	addcnsr 5241	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 5242	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 5243	Equality of real numbers in terms of intermediate signed reals.
|- A e. V   =>   |- (<.A, 0R>. = <.B, 0R>. <-> A = B)
Theorem	addresr 5244	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 5245	Multiplication of real numbers in terms of intermediate signed reals.
|- B e. V   =>   |- ((A e. R. /\ B e. R.) -> (<.A, 0R>. x. <.B, 0R>.) = <.(A .R B), 0R>.)
Theorem	ltresr 5246	Ordering of real subset of complex numbers in terms of signed reals.
|- A e. V   &   |- B e. V   =>   |- (<.A, 0R>. <R <.B, 0R>. <-> A <R B)
Theorem	suprelem 5247	Mapping of non-empty subset from signed reals to reals.
|- B = {w | <.w, 0R>. e. A}   =>   |- ((A (_ RR /\ -. A = (/)) -> (B (_ R. /\ -. B = (/)))
Theorem	supre 5248	A non-empty, bounded-above set of reals has a supremum.
|- B = {w | <.w, 0R>. e. A}   =>   |- (((A (_ RR /\ -. A = (/)) /\ E.x(x e. RR /\ A.y(y e. RR -> (y e. A -> y <R x)))) -> E.x(x e. RR /\ A.y(y e. RR -> ((y e. A -> -. x <R y) /\ (y <R x -> E.z(z e. RR /\ (z e. A /\ y <R z)))))))
Theorem	ltsor 5249	'Less than' is a strict ordering on real subset of complex numbers. Note: use ltso 5500 and not this one after the complex number postulates are derived, in order to maintain a "clean" derivation of complex number theorems directly from postulates. The artificial right conjunct is intended to help discourage its accidental use in place of ltso 5500.
|- ( <R Or RR /\ RR = RR)
Theorem	dfcnqs 5250	Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in CC from those in R.. The trick involves qsid 4299, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) acts as an identity divisor for the quotient set operation. This lets us "pretend" that CC is a quotient set, even though it is not (compare df-c 5228), and allows us to reuse some of the equivalence class lemmas we developed for the transition from positive reals to signed reals, etc.
|- CC = ((R. X. R.)/.`'E)
Theorem	addcnsrec 5251	Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 5250 and mulcnsrec 5252.
|- (((A e. R. /\ B e. R.) /\ (C e. R. /\ D e. R.)) -> ([<.A, B>.]`'E + [<.C, D>.]`'E) = [<.(A +R C), (B +R D)>.]`'E)
Theorem	mulcnsrec 5252	Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecid 4298, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set