mmtheorems55 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 5401-5500 - Page 55 of 123

Type	Label	Description
Statement
Definition	df-lt 5401	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 5402	Ordered pair membership in the class of complex numbers.
|- B e. V   =>   |- (<.A, B>. e. CC <-> (A e. R. /\ B e. R.))
Theorem	opelreal 5403	Ordered pair membership in class of real subset of complex numbers.
|- (<.A, 0R>. e. RR <-> A e. R.)
Theorem	elreal 5404	Membership in class of real numbers.
|- (A e. RR <-> E.x(x e. R. /\ <.x, 0R>. = A))
Theorem	0ncn 5405	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 5406	'Less than' is a relation on real numbers.
|- <R (_ (RR X. RR)
Theorem	addcnsr 5407	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 5408	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 5409	Equality of real numbers in terms of intermediate signed reals.
|- A e. V   =>   |- (<.A, 0R>. = <.B, 0R>. <-> A = B)
Theorem	addresr 5410	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 5411	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 5412	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 5413	Mapping of non-empty subset from signed reals to reals.
|- B = {w | <.w, 0R>. e. A}   =>   |- ((A (_ RR /\ -. A = (/)) -> (B (_ R. /\ -. B = (/)))
Theorem	supre 5414	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 5415	'Less than' is a strict ordering on real subset of complex numbers. Note: use ltso 5666 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 5666.
|- ( <R Or RR /\ RR = RR)
Theorem	dfcnqs 5416	Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in CC from those in R.. The trick involves qsid 4442, 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 5394), 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 5417	Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 5416 and mulcnsrec 5418.
|- (((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 5418	Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecid 4441, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 5416. 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 5126.
|- (((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
Theorem	axaddopr 5419	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 5425.
|- + :(CC X. CC)-->CC
Theorem	axmulopr 5420	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 5427.
|- x. :(CC X. CC)-->CC
Theorem	axcnex 5421	The class of complex numbers is a set, i.e. it is a member of the universe of sets V (see isset 1860). Axiom 1 of 25 for real and complex numbers, derived from ZF set theory.
|- CC e. V
Theorem	axresscn 5422	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
Theorem	ax1cn 5423	1 is a complex number. Axiom 3 of 25 for real and complex numbers, derived from ZF set theory.
|- 1 e. CC
Theorem	axicn 5424	i is a complex number. Axiom 4 of 25 for real and complex numbers, derived from ZF set theory.
|- i e. CC
Theorem	axaddcl 5425	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)
Theorem	axaddrcl 5426	Closure law for addition in the real subfield of complex numbers. Axiom 6 of 25 for real and complex numbers, derived from ZF set theory.
|- ((A e. RR /\ B e. RR) -> (A + B) e. RR)
Theorem	axmulcl 5427	Closure law for multiplication of complex numbers. Axiom 7 of 25 for real and complex numbers, derived from ZF set theory.
|- ((A e. CC /\ B e. CC) -> (A x. B) e. CC)
Theorem	axmulrcl 5428	Closure law for multiplication in the real subfield of complex numbers. Axiom 8 of 25 for real and complex numbers, derived from ZF set theory.
|- ((A e. RR /\ B e. RR) -> (A x. B) e. RR)
Theorem	axaddcom 5429	Addition of complex numbers is commutative. Axiom 9 of 25 for real and complex numbers, derived from ZF set theory.
|- ((A e. CC /\ B e. CC) -> (A + B) = (B + A))
Theorem	axmulcom 5430	Multiplication of complex numbers is commutative. Axiom 10 of 25 for real and complex numbers, derived from ZF set theory.
|- ((A e. CC /\ B e. CC) -> (A x. B) = (B x. A))
Theorem	axaddass 5431	Addition of complex numbers is associative. This theorem transfers the associative laws for the real and imaginary signed real components of complex number pairs, to complex number addition itself. Axiom 11 of 25 for real and complex numbers, derived from ZF set theory.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A + B) + C) = (A + (B + C)))
Theorem	axmulass 5432	Multiplication of complex numbers is associative. Axiom 12 of 25 for real and complex numbers, derived from ZF set theory.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A x. B) x. C) = (A x. (B x. C)))
Theorem	axdistr 5433	Distributive law for complex numbers. Axiom 13 of 25 for real and complex numbers, derived from ZF set theory.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> (A x. (B + C)) = ((A x. B) + (A x. C)))
Theorem	ax1ne0 5434	1 and 0 are distinct. Axiom 14 of 25 for real and complex numbers, derived from ZF set theory.
|- 1 =/= 0
Theorem	ax0id 5435	0 is an identity element for addition. Axiom 15 of 25 for real and complex numbers, derived from ZF set theory.
|- (A e. CC -> (A + 0) = A)
Theorem	ax1id 5436	1 is an identity element for multiplication. Axiom 16 of 25 for real and complex numbers, derived from ZF set theory.
|- (A e. CC -> (A x. 1) = A)
Theorem	axrnegex 5437	Existence of negative of real number. Axiom 17 of 25 for real and complex numbers, derived from ZF set theory.
|- (A e. RR -> E.x e. RR (A + x) = 0)
Theorem	axrrecex 5438	Existence of reciprocal of nonzero real number. Axiom 18 of 25 for real and complex numbers, derived from ZF set theory.
|- ((A e. RR /\ A =/= 0) -> E.x e. RR (A x. x) = 1)
Theorem	axi2m1 5439	i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom 19 of 25 for real and complex numbers, derived from ZF set theory.
|- ((i x. i) + 1) = 0
Theorem	axcnre 5440	A complex number can be expressed in terms of two reals. Definition 10-1.1(v) of [Gleason] p. 130. Axiom 20 of 25 for real and complex numbers, derived from ZF set theory.
|- (A e. CC -> E.x e. RR E.y e. RR A = (x + (i x. y)))
Theorem	pre-axlttri 5441	Ordering on reals satisfies strict trichotomy. Axiom 21 of 25 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axlttri 5657.
|- ((A e. RR /\ B e. RR) -> (A <R B <-> -. (A = B \/ B <R A)))
Theorem	pre-axlttrn 5442	Ordering on reals is transitive. Axiom 22 of 25 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axlttrn 5658.
|- ((A e. RR /\ B e. RR /\ C e. RR) -> ((A <R B /\ B <R C) -> A <R C))
Theorem	pre-axltadd 5443	Ordering property of addition on reals. Axiom 23 of 25 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axltadd 5659.
|- ((A e. RR /\ B e. RR /\ C e. RR) -> (A <R B -> (C + A) <R (C + B)))
Theorem	pre-axmulgt0 5444	The product of two positive reals is positive. Axiom 24 of 25 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axmulgt0 5660.
|- ((A e. RR /\ B e. RR) -> ((0 <R A /\ 0 <R B) -> 0 <R (A x. B)))
Theorem	pre-axsup 5445	A non-empty, bounded-above set of reals has a supremum. Axiom 25 of 25 for real and complex numbers, derived from ZF set theory. Note: The more general version with ordering on extended reals is axsup 5661.
|- ((A (_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A y <R x) -> E.x e. RR (A.y e. A -. x <R y /\ A.y e. RR (y <R x -> E.z e. A y <R z)))
Real and complex numbers - basic operations
Syntax	cmin 5446	Extend class notation to include subtraction.
class -
Syntax	cneg 5447	Extend class notation to include unary minus. The symbol -u is not a class by itself but part of a compound class definition. We do this rather than making it a formal function since it is so commonly used. Note: We use different symbols for unary minus (-u) and subtraction cmin 5446 (-) to prevent syntax ambiguity. For example, looking at the syntax definition co 4021, if we used the same symbol then "( - A - B)" could mean either "- A" minus "B", or it could represent the (meaningless) operation of classes "-" and "- B" connected with "operation" "A". On the other hand, "(-uA - B)" is unambiguous.
class -uA
Syntax	cdiv 5448	Extend class notation to include division.
class /
Syntax	cle 5449	Extend wff notation to include the 'less than or equal to' relation.
class <_
Syntax	cn 5450	Extend class notation to include the class of positive integers.
class NN
Syntax	cn0 5451	Extend class notation to include the class of nonnegative integers.
class NN0
Syntax	cz 5452	Extend class notation to include the class of integers.
class ZZ
Syntax	cq 5453	Extend class notation to include the class of rationals.
class QQ
Syntax	crp 5454	Extend class notation to include the class of positive reals.
class RR+
Some deductions from the field axioms for complex numbers
Theorem	addcl 5455	Alias for axaddcl 5425, for naming consistency with addcli 5474.
|- ((A e. CC /\ B e. CC) -> (A + B) e. CC)
Theorem	readdcl 5456	Alias for axaddrcl 5426, for naming consistency with readdcli 5488.
|- ((A e. RR /\ B e. RR) -> (A + B) e. RR)
Theorem	mulcl 5457	Alias for axmulcl 5427, for naming consistency with mulcli 5475.
|- ((A e. CC /\ B e. CC) -> (A x. B) e. CC)
Theorem	remulcl 5458	Alias for axmulrcl 5428, for naming consistency with remulcli 5489.
|- ((A e. RR /\ B e. RR) -> (A x. B) e. RR)
Theorem	addcom 5459	Alias for axaddcom 5429, for naming consistency with addcomi 5476.
|- ((A e. CC /\ B e. CC) -> (A + B) = (B + A))
Theorem	mulcom 5460	Alias for axmulcom 5430, for naming consistency with mulcomi 5477.
|- ((A e. CC /\ B e. CC) -> (A x. B) = (B x. A))
Theorem	addass 5461	Alias for axaddass 5431, for naming consistency with addassi 5478.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A + B) + C) = (A + (B + C)))
Theorem	mulass 5462	Alias for axmulass 5432, for naming consistency with mulassi 5479.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A x. B) x. C) = (A x. (B x. C)))
Theorem	adddi 5463	Alias for axdistr 5433, for naming consistency with adddii 5480.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> (A x. (B + C)) = ((A x. B) + (A x. C)))
Theorem	addid1 5464	Alias for ax0id 5435, for naming consistency with addid1i 5484.
|- (A e. CC -> (A + 0) = A)
Theorem	mulid1 5465	Alias for ax1id 5436, for naming consistency with mulid1i 5486.
|- (A e. CC -> (A x. 1) = A)
Theorem	reex 5466	The set of real numbers exists.
|- RR e. V
Theorem	recn 5467	A real number is a complex number.
|- (A e. RR -> A e. CC)
Theorem	recni 5468	A real number is a complex number.
|- A e. RR   =>   |- A e. CC
Theorem	recnd 5469	Deduction from real number to complex number.
|- (ph -> A e. RR)   =>   |- (ph -> A e. CC)
Theorem	elimne0 5470	Hypothesis for weak deduction theorem to eliminate A =/= 0.
|- if(A =/= 0, A, 1) =/= 0
Theorem	addex 5471	The addition operation is a set.
|- + e. V
Theorem	mulex 5472	The multiplication operation is a set.
|- x. e. V
Theorem	adddir 5473	Distributive law for complex numbers.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A + B) x. C) = ((A x. C) + (B x. C)))
Theorem	addcli 5474	Closure law for addition.
|- A e. CC   &   |- B e. CC   =>   |- (A + B) e. CC
Theorem	mulcli 5475	Closure law for multiplication.
|- A e. CC   &   |- B e. CC   =>   |- (A x. B) e. CC
Theorem	addcomi 5476	Commutative law for addition.
|- A e. CC   &   |- B e. CC   =>   |- (A + B) = (B + A)
Theorem	mulcomi 5477	Commutative law for multiplication.
|- A e. CC   &   |- B e. CC   =>   |- (A x. B) = (B x. A)
Theorem	addassi 5478	Associative law for addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A + B) + C) = (A + (B + C))
Theorem	mulassi 5479	Associative law for multiplication.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A x. B) x. C) = (A x. (B x. C))
Theorem	adddii 5480	Distributive law.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- (A x. (B + C)) = ((A x. B) + (A x. C))
Theorem	adddiri 5481	Distributive law.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A + B) x. C) = ((A x. C) + (B x. C))
Theorem	0cn 5482	0 is a complex number.
|- 0 e. CC
Theorem	addid2 5483	Identity law for addition.
|- (A e. CC -> (0 + A) = A)
Theorem	addid1i 5484	Identity law for addition.
|- A e. CC   =>   |- (A + 0) = A
Theorem	addid2i 5485	Identity law for addition.
|- A e. CC   =>   |- (0 + A) = A
Theorem	mulid1i 5486	Identity law for multiplication.
|- A e. CC   =>   |- (A x. 1) = A
Theorem	mulid2i 5487	Identity law for multiplication.
|- A e. CC   =>   |- (1 x. A) = A
Theorem	readdcli 5488	Closure law for addition of reals.
|- A e. RR   &   |- B e. RR   =>   |- (A + B) e. RR
Theorem	remulcli 5489	Closure law for multiplication of reals.
|- A e. RR   &   |- B e. RR   =>   |- (A x. B) e. RR
Addition
Theorem	add12 5490	Commutative/associative law that swaps the first two terms in a triple sum.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> (A + (B + C)) = (B + (A + C)))
Theorem	add23 5491	Commutative/associative law that swaps the last two terms in a triple sum.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A + B) + C) = ((A + C) + B))
Theorem	add4 5492	Rearrangement of 4 terms in a sum.
|- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> ((A + B) + (C + D)) = ((A + C) + (B + D)))
Theorem	add42 5493	Rearrangement of 4 terms in a sum.
|- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> ((A + B) + (C + D)) = ((A + C) + (D + B)))
Theorem	add12i 5494	Commutative/associative law that swaps the first two terms in a triple sum.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- (A + (B + C)) = (B + (A + C))
Theorem	add23i 5495	Commutative/associative law that swaps the last two terms in a triple sum.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A + B) + C) = ((A + C) + B)
Theorem	add4i 5496	Rearrangement of 4 terms in a sum.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   &   |- D e. CC   =>   |- ((A + B) + (C + D)) = ((A + C) + (B + D))
Theorem	add42i 5497	Rearrangement of 4 terms in a sum.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   &   |- D e. CC   =>   |- ((A + B) + (C + D)) = ((A + C) + (D + B))
Theorem	peano2cn 5498	A theorem for complex numbers analogous the second Peano postulate peano2nn 6080.
|- (A e. CC -> (A + 1) e. CC)
Subtraction
Theorem	cnegexlem1 5499	Lemma for cnegex 5502.
Theorem	cnegexlem2 5500	Lemma for cnegex 5502.