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Statement List for Metamath Proof Explorer - 5201-5300 - Page 53 of 107
TypeLabelDescription
Statement
 
Theoremmappsrpr 5201 Mapping from positive signed reals to positive reals.
|- A e. V   =>   |- (0R <R [<.(A +P. 1P), 1P>.] ~R <-> A e. P.)
 
Theoremltpsrpr 5202 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)
 
Theoremmap2psrpr 5203 Equivalence for positive signed real.
|- A e. V   =>   |- (0R <R A <-> E.x(x e. P. /\ [<.(x +P. 1P), 1P>.] ~R = A))
 
Theoremsuppsrlem 5204 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 = (/)))
 
Theoremsuppsr 5205 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)))))))
 
Theoremsuppsr2 5206 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)))))))
 
Theoremsuppsr3 5207 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)))))))
 
Theoremsupsrlem1 5208 Lemma for supremum theorem.
 
Theoremsupsrlem2 5209 Lemma for supremum theorem.
 
Theoremsupsrlem3 5210 Lemma for supremum theorem.
 
Theoremsupsrlem4 5211 Lemma for supremum theorem.
 
Theoremsupsrlem5 5212 Lemma for supremum theorem.
 
Theoremsupsrlem6 5213 Lemma for supremum theorem.
 
Theoremsupsr 5214 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)))))))
 
Syntaxcc 5215 Class of complex numbers.
class CC
 
Syntaxcr 5216 Class of real numbers.
class RR
 
Syntaxcc0 5217 Extend class notation to include the complex number 0.
class 0
 
Syntaxc1 5218 Extend class notation to include the complex number 1.
class 1
 
Syntaxci 5219 Extend class notation to include the complex number i.
class i
 
Syntaxcaddc 5220 Addition on complex numbers.
class +
 
Syntaxcltrr 5221 'Less than' predicate (defined over real subset of complex numbers).
class <R
 
Syntaxcmul 5222 Multiplication on complex numbers. The token x. is a center dot.
class x.
 
Definitiondf-c 5223 Define the set of complex numbers. The 25 axioms for complex numbers start at axcnex 5250.
|- CC = (R. X. R.)
 
Definitiondf-0 5224 Define the complex number 0 (base 10).
|- 0 = <.0R, 0R>.
 
Definitiondf-1 5225 Define the complex number 1 (base 10).
|- 1 = <.1R, 0R>.
 
Definitiondf-i 5226 Define the complex number i (the imaginary unit).
|- i = <.0R, 1R>.
 
Definitiondf-r 5227 Define the set of real numbers.
|- RR = (R. X. {0R})
 
Definitiondf-plus 5228 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)>.))}
 
Definitiondf-mul 5229 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))>.))}
 
Definitiondf-lt 5230 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))}
 
Theoremopelcn 5231 Ordered pair membership in the class of complex numbers.
|- B e. V   =>   |- (<.A, B>. e. CC <-> (A e. R. /\ B e. R.))
 
Theoremopelreal 5232 Ordered pair membership in class of real subset of complex numbers.
|- (<.A, 0R>. e. RR <-> A e. R.)
 
Theoremelreal 5233 Membership in class of real numbers.
|- (A e. RR <-> E.x(x e. R. /\ <.x, 0R>. = A))
 
Theorem0ncn 5234 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
 
Theoremltrelre 5235 'Less than' is a relation on real numbers.
|- <R (_ (RR X. RR)
 
Theoremaddcnsr 5236 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)>.)
 
Theoremmulcnsr 5237 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))>.)
 
Theoremeqresr 5238 Equality of real numbers in terms of intermediate signed reals.
|- A e. V   =>   |- (<.A, 0R>. = <.B, 0R>. <-> A = B)
 
Theoremaddresr 5239 Addition of real numbers in terms of intermediate signed reals.
|- ((A e. R. /\ B e. R.) -> (<.A, 0R>. + <.B, 0R>.) = <.(A +R B), 0R>.)
 
Theoremmulresr 5240 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>.)
 
Theoremltresr 5241 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)
 
Theoremsuprelem 5242 Mapping of non-empty subset from signed reals to reals.
|- B = {w | <.w, 0R>. e. A}   =>   |- ((A (_ RR /\ -. A = (/)) -> (B (_ R. /\ -. B = (/)))
 
Theoremsupre 5243 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)))))))
 
Theoremltsor 5244 'Less than' is a strict ordering on real subset of complex numbers. Note: use ltso 5495 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 5495.
|- ( <R Or RR /\ RR = RR)
 
Theoremdfcnqs 5245 Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in CC from those in R.. The trick involves qsid 4294, 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 5223), 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)
 
Theoremaddcnsrec 5246 Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 5245 and mulcnsrec 5247.
|- (((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)
 
Theoremmulcnsrec 5247 Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecid 4293, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 5245.

Note: This is the last lemma (from which the axioms will be derived) in the construction of real and complex numbers. The construction starts at cnpi 4955.

|- (((A e. R. /\ B e. R.) /\ (C e. R. /\ D e. R.)) -> ([<.A, B>.]`'E x. [<.C, D>.]`'E) = [<.((A .R C) +R (-1R .R (B .R D))), ((B .R C) +R (A .R D))>.]`'E)
 
Real and complex number postulates
 
Theoremaxaddopr 5248 Addition is an operation on the complex numbers. This theorem can be used as an alternate axiom for complex numbers in place of the less specific axaddcl 5254.
|- + :(CC X. CC)-->CC
 
Theoremaxmulopr 5249 Multiplication is an operation on the complex numbers. This theorem can be used as an alternate axiom for complex numbers in place of the less specific axmulcl 5256.
|- x. :(CC X. CC)-->CC
 
Theoremaxcnex 5250 The class of complex numbers is a set, i.e. it is a member of the universe of sets V (see isset 1811). Axiom 1 of 25 for real and complex numbers, derived from ZF set theory.
|- CC e. V
 
Theoremaxresscn 5251 The real numbers are a subset of the complex numbers. Axiom 2 of 25 for real and complex numbers, derived from ZF set theory.
|- RR (_ CC
 
Theoremax1cn 5252 1 is a complex number. Axiom 3 of 25 for real and complex numbers, derived from ZF set theory.
|- 1 e. CC
 
Theoremaxicn 5253 i is a complex number. Axiom 4 of 25 for real and complex numbers, derived from ZF set theory.
|- i e. CC
 
Theoremaxaddcl 5254 Closure law for addition of complex numbers. Axiom 5 of 25 for real and complex numbers, derived from ZF set theory.
|- ((A e. CC /\ B e. CC) -> (A + B) e. CC