mmtheorems53 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 5201-5300 - Page 53 of 105

Type	Label	Description
Statement
Theorem	axmulass 5201	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 5202	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 5203	1 and 0 are distinct. Axiom 14 of 25 for real and complex numbers, derived from ZF set theory.
|- 1 =/= 0
Theorem	ax0id 5204	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 5205	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 5206	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 5207	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 5208	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 5209	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 5210	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 5426.
|- ((A e. RR /\ B e. RR) -> (A <R B <-> -. (A = B \/ B <R A)))
Theorem	pre-axlttrn 5211	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 5427.
|- ((A e. RR /\ B e. RR /\ C e. RR) -> ((A <R B /\ B <R C) -> A <R C))
Theorem	pre-axltadd 5212	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 5428.
|- ((A e. RR /\ B e. RR /\ C e. RR) -> (A <R B -> (C + A) <R (C + B)))
Theorem	pre-axmulgt0 5213	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 5429.
|- ((A e. RR /\ B e. RR) -> ((0 <R A /\ 0 <R B) -> 0 <R (A x. B)))
Theorem	pre-axsup 5214	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 5430.
|- ((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 5215	Extend class notation to include subtraction.
class -
Syntax	cneg 5216	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 a different symbols for unary minus (-u) and subtraction cmin 5215 (-) to prevent syntax ambiguity. For example, looking at the syntax definition co 3902, 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 5217	Extend class notation to include division.
class /
Syntax	cle 5218	Extend wff notation to include the 'less than or equal to' relation.
class <_
Syntax	cn 5219	Extend class notation to include the class of positive integers.
class NN
Syntax	cn0 5220	Extend class notation to include the class of nonnegative integers.
class NN0
Syntax	cz 5221	Extend class notation to include the class of integers.
class ZZ
Syntax	cq 5222	Extend class notation to include the class of rationals.
class QQ
Syntax	crp 5223	Extend class notation to include the class of positive reals.
class RR+
Some deductions from the field axioms for complex numbers
Theorem	addclt 5224	Alias for axaddcl 5194, for naming consistency with addcl 5243.
|- ((A e. CC /\ B e. CC) -> (A + B) e. CC)
Theorem	readdclt 5225	Alias for axaddrcl 5195, for naming consistency with readdcl 5257.
|- ((A e. RR /\ B e. RR) -> (A + B) e. RR)
Theorem	mulclt 5226	Alias for axmulcl 5196, for naming consistency with mulcl 5244.
|- ((A e. CC /\ B e. CC) -> (A x. B) e. CC)
Theorem	remulclt 5227	Alias for axmulrcl 5197, for naming consistency with remulcl 5258.
|- ((A e. RR /\ B e. RR) -> (A x. B) e. RR)
Theorem	addcomt 5228	Alias for axaddcom 5198, for naming consistency with addcom 5245.
|- ((A e. CC /\ B e. CC) -> (A + B) = (B + A))
Theorem	mulcomt 5229	Alias for axmulcom 5199, for naming consistency with mulcom 5246.
|- ((A e. CC /\ B e. CC) -> (A x. B) = (B x. A))
Theorem	addasst 5230	Alias for axaddass 5200, for naming consistency with addass 5247.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A + B) + C) = (A + (B + C)))
Theorem	mulasst 5231	Alias for axmulass 5201, for naming consistency with mulass 5248.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A x. B) x. C) = (A x. (B x. C)))
Theorem	adddit 5232	Alias for axdistr 5202, for naming consistency with adddi 5249.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> (A x. (B + C)) = ((A x. B) + (A x. C)))
Theorem	addid1t 5233	Alias for ax0id 5204, for naming consistency with addid1 5253.
|- (A e. CC -> (A + 0) = A)
Theorem	mulid1t 5234	Alias for ax1id 5205, for naming consistency with mulid1 5255.
|- (A e. CC -> (A x. 1) = A)
Theorem	reex 5235	The set of real numbers exists.
|- RR e. V
Theorem	recnt 5236	A real number is a complex number.
|- (A e. RR -> A e. CC)
Theorem	recn 5237	A real number is a complex number.
|- A e. RR   =>   |- A e. CC
Theorem	recnd 5238	Deduction from real number to complex number.
|- (ph -> A e. RR)   =>   |- (ph -> A e. CC)
Theorem	elimne0 5239	Hypothesis for weak deduction theorem to eliminate A =/= 0.
|- if(A =/= 0, A, 1) =/= 0
Theorem	addex 5240	The addition operation is a set.
|- + e. V
Theorem	mulex 5241	The multiplication operation is a set.
|- x. e. V
Theorem	adddirt 5242	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	addcl 5243	Closure law for addition.
|- A e. CC   &   |- B e. CC   =>   |- (A + B) e. CC
Theorem	mulcl 5244	Closure law for multiplication.
|- A e. CC   &   |- B e. CC   =>   |- (A x. B) e. CC
Theorem	addcom 5245	Commutative law for addition.
|- A e. CC   &   |- B e. CC   =>   |- (A + B) = (B + A)
Theorem	mulcom 5246	Commutative law for multiplication.
|- A e. CC   &   |- B e. CC   =>   |- (A x. B) = (B x. A)
Theorem	addass 5247	Associative law for addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A + B) + C) = (A + (B + C))
Theorem	mulass 5248	Associative law for multiplication.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A x. B) x. C) = (A x. (B x. C))
Theorem	adddi 5249	Distributive law.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- (A x. (B + C)) = ((A x. B) + (A x. C))
Theorem	adddir 5250	Distributive law.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A + B) x. C) = ((A x. C) + (B x. C))
Theorem	0cn 5251	0 is a complex number.
|- 0 e. CC
Theorem	addid2t 5252	Identity law for addition.
|- (A e. CC -> (0 + A) = A)
Theorem	addid1 5253	Identity law for addition.
|- A e. CC   =>   |- (A + 0) = A
Theorem	addid2 5254	Identity law for addition.
|- A e. CC   =>   |- (0 + A) = A
Theorem	mulid1 5255	Identity law for multiplication.
|- A e. CC   =>   |- (A x. 1) = A
Theorem	mulid2 5256	Identity law for multiplication.
|- A e. CC   =>   |- (1 x. A) = A
Theorem	readdcl 5257	Closure law for addition of reals.
|- A e. RR   &   |- B e. RR   =>   |- (A + B) e. RR
Theorem	remulcl 5258	Closure law for multiplication of reals.
|- A e. RR   &   |- B e. RR   =>   |- (A x. B) e. RR
Addition
Theorem	add12t 5259	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	add23t 5260	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	add4t 5261	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	add42t 5262	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	add12 5263	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 5264	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 5265	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 5266	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 5267