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	pn0sr 5201	A signed real plus its negative is zero.
|- (A e. R. -> (A +R (A .R -1R)) = 0R)
Theorem	negexsr 5202	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 5203	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 5204	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 5205	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 5206	The square of a nonzero signed real is positive.
|- A e. V   =>   |- (A e. R. -> (-. A = 0R -> 0R <R (A .R A)))
Theorem	recexsr 5207	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 5208	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 5209	Mapping from positive signed reals to positive reals.
|- A e. V   =>   |- (0R <R [<.(A +P. 1P), 1P>.] ~R <-> A e. P.)
Theorem	ltpsrpr 5210	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 5211	Equivalence for positive signed real.
|- A e. V   =>   |- (0R <R A <-> E.x(x e. P. /\ [<.(x +P. 1P), 1P>.] ~R = A))
Theorem	suppsrlem 5212	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 5213	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 5214	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 5215	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 5216	Lemma for supremum theorem.
Theorem	supsrlem2 5217	Lemma for supremum theorem.
Theorem	supsrlem3 5218	Lemma for supremum theorem.
Theorem	supsrlem4 5219	Lemma for supremum theorem.
Theorem	supsrlem5 5220	Lemma for supremum theorem.
Theorem	supsrlem6 5221	Lemma for supremum theorem.
Theorem	supsr 5222	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 5223	Class of complex numbers.
class CC
Syntax	cr 5224	Class of real numbers.
class RR
Syntax	cc0 5225	Extend class notation to include the complex number 0.
class 0
Syntax	c1 5226	Extend class notation to include the complex number 1.
class 1
Syntax	ci 5227	Extend class notation to include the complex number i.
class i
Syntax	caddc 5228	Addition on complex numbers.
class +
Syntax	cltrr 5229	'Less than' predicate (defined over real subset of complex numbers).
class <R
Syntax	cmul 5230	Multiplication on complex numbers. The token x. is a center dot.
class x.
Definition	df-c 5231	Define the set of complex numbers. The 25 axioms for complex numbers start at axcnex 5258.
|- CC = (R. X. R.)
Definition	df-0 5232	Define the complex number 0 (base 10).
|- 0 = <.0R, 0R>.
Definition	df-1 5233	Define the complex number 1 (base 10).
|- 1 = <.1R, 0R>.
Definition	df-i 5234	Define the complex number i (the imaginary unit).
|- i = <.0R, 1R>.
Definition	df-r 5235	Define the set of real numbers.
|- RR = (R. X. {0R})
Definition	df-plus 5236	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 5237	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 5238	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 5239	Ordered pair membership in the class of complex numbers.
|- B e. V   =>   |- (<.A, B>. e. CC <-> (A e. R. /\ B e. R.))
Theorem	opelreal 5240	Ordered pair membership in class of real subset of complex numbers.
|- (<.A, 0R>. e. RR <-> A e. R.)
Theorem	elreal 5241	Membership in class of real numbers.
|- (A e. RR <-> E.x(x e. R. /\ <.x, 0R>. = A))
Theorem	0ncn 5242	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 5243	'Less than' is a relation on real numbers.
|- <R (_ (RR X. RR)
Theorem	addcnsr 5244	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 5245	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 5246	Equality of real numbers in terms of intermediate signed reals.
|- A e. V   =>   |- (<.A, 0R>. = <.B, 0R>. <-> A = B)
Theorem	addresr 5247	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 5248	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 5249	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 5250	Mapping of non-empty subset from signed reals to reals.
|- B = {w | <.w, 0R>. e. A}   =>   |- ((A (_ RR /\ -. A = (/)) -> (B (_ R. /\ -. B = (/)))
Theorem	supre 5251	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 5252	'Less than' is a strict ordering on real subset of complex numbers. Note: use ltso 5503 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 5503.
|- ( <R Or RR /\ RR = RR)
Theorem	dfcnqs 5253	Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in CC from those in R.. The trick involves qsid 4301, 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 5231), 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.)