Metamath Proof Explorer - mmtheorems41

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Statement List for Metamath Proof Explorer - 4001-4100 - Page 41 of 58

Type	Label	Description
Statement
Theorem	1idsr 4001	1 is an identity element for multiplication.
⊢ (A ∈ R → (A ·R 1R) = A)
Theorem	00sr 4002	A signed real times 0 is 0.
⊢ (A ∈ R → (A ·R 0R) = 0R)
Theorem	ltasr 4003	Ordering property of addition.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ (C ∈ R → (A <R B ↔ (C +R A) <R (C +R B)))
Theorem	pn0sr 4004	A signed real plus its negative is zero.
⊢ (A ∈ R → (A +R (A ·R -1R)) = 0R)
Theorem	negexsr 4005	Existence of negative signed real. Part of Proposition 9-4.3 of [Gleason] p. 126.
⊢ (A ∈ R → ∃x(x ∈ R ∧ (A +R x) = 0R))
Theorem	recexsrlem 4006	The reciprocal of a positive signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126.
⊢ A ∈ V    ⇒   ⊢ (0R <R A → ∃x(x ∈ R ∧ (A ·R x) = 1R))
Theorem	addgt0sr 4007	The sum of two positive signed reals is positive.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ ((0R <R A ∧ 0R <R B) → 0R <R (A +R B))
Theorem	mulgt0sr 4008	The product of two positive signed reals is positive.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ ((0R <R A ∧ 0R <R B) → 0R <R (A ·R B))
Theorem	sqgt0sr 4009	The square of a nonzero signed real is positive.
⊢ A ∈ V    ⇒   ⊢ (A ∈ R → (¬ A = 0R → 0R <R (A ·R A)))
Theorem	recexsr 4010	The reciprocal of a nonzero signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126.
⊢ A ∈ V    ⇒   ⊢ (A ∈ R → (¬ A = 0R → ∃x(x ∈ R ∧ (A ·R x) = 1R)))
Theorem	ssgt0sr 4011	The sum of squares of signed reals is positive if one is nonzero.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ ((A ∈ R ∧ B ∈ R) → (¬ (A = 0R ∧ B = 0R) → 0R <R ((A ·R A) +R (B ·R B))))
Theorem	mappsrpr 4012	Mapping from positive signed reals to positive reals.
⊢ A ∈ V    ⇒   ⊢ (0R <R [⟨(A +P 1P), 1P⟩] ~R ↔ A ∈ P)
Theorem	ltpsrpr 4013	Mapping of order from positive signed reals to positive reals.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ ([⟨(A +P 1P), 1P⟩] ~R <R [⟨(B +P 1P), 1P⟩] ~R ↔ A<P B)
Theorem	map2psrpr 4014	Equivalence for positive signed real.
⊢ A ∈ V    ⇒   ⊢ (0R <R A ↔ ∃x(x ∈ P ∧ [⟨(x +P 1P), 1P⟩] ~R = A))
Theorem	suppsrlem 4015	Mapping of non-empty subset from positive reals to positive signed reals.
⊢ B = {w∣[⟨(w +P 1P), 1P⟩] ~R ∈ A}    ⇒   ⊢ ((∀x(x ∈ A → 0R <R x) ∧ ¬ A = ∅) → (B ⊆ P ∧ ¬ B = ∅))
Theorem	suppsr 4016	A non-empty, bounded set of positive signed reals has a supremum.
⊢ B = {w∣[⟨(w +P 1P), 1P⟩] ~R ∈ A}    ⇒   ⊢ (((∀x(x ∈ A → 0R <R x) ∧ ¬ A = ∅) ∧ ∃x(0R <R x ∧ ∀y(0R <R y → (y ∈ A → y <R x)))) → ∃x(0R <R x ∧ ∀y(0R <R y → ((y ∈ A → ¬ x <R y) ∧ (y <R x → ∃z(0R <R z ∧ (z ∈ A ∧ y <R z)))))))
Theorem	suppsr2 4017	A non-empty, bounded set of positive signed reals has a supremum. (Converts quantifier restrictions to all reals.)
⊢ (((∀x(x ∈ A → 0R <R x) ∧ ¬ A = ∅) ∧ ∃x(x ∈ R ∧ ∀y(y ∈ R → (y ∈ A → y <R x)))) → ∃x(x ∈ R ∧ ∀y(y ∈ R → ((y ∈ A → ¬ x <R y) ∧ (y <R x → ∃z(z ∈ R ∧ (z ∈ A ∧ y <R z)))))))
Theorem	suppsr3 4018	A non-empty, bounded set with at least one positive real has a supremum.
⊢ B = {y∣(y ∈ A ∧ 0R <R y)}    ⇒   ⊢ ((∃y(y ∈ A ∧ 0R <R y) ∧ ∃x(x ∈ R ∧ ∀y(y ∈ R → (y ∈ A → y <R x)))) → ∃x(x ∈ R ∧ ∀y(y ∈ R → ((y ∈ A → ¬ x <R y) ∧ (y <R x → ∃z(z ∈ R ∧ (z ∈ A ∧ y <R z)))))))
Theorem	supsrlem1 4019	Lemma for supremum theorem.
⊢ C ∈ R    ⇒   ⊢ ((C +R (A +R -1R)) ∈ R ↔ A ∈ R)
Theorem	supsrlem2 4020	Lemma for supremum theorem.
⊢ C ∈ R    ⇒   ⊢ (A ∈ R ↔ ∃x(x ∈ R ∧ (C +R (x +R -1R)) = A))
Theorem	supsrlem3 4021	Lemma for supremum theorem.
⊢ C ∈ R    &   ⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ ((C +R (A +R -1R)) <R (C +R (B +R -1R)) ↔ A <R B)
Theorem	supsrlem4 4022	Lemma for supremum theorem.
⊢ C ∈ R    &   ⊢ B = {w∣(C +R (w +R -1R)) ∈ A}    &   ⊢ D ∈ V    ⇒   ⊢ (D ∈ B ↔ (C +R (D +R -1R)) ∈ A)
Theorem	supsrlem5 4023	Lemma for supremum theorem.
⊢ C ∈ R    &   ⊢ B = {w∣(C +R (w +R -1R)) ∈ A}    ⇒   ⊢ (C ∈ A → ∃y(y ∈ B ∧ 0R <R y))
Theorem	supsrlem6 4024	Lemma for supremum theorem.
⊢ C ∈ R    &   ⊢ B = {w∣(C +R (w +R -1R)) ∈ A}    ⇒   ⊢ ((C ∈ A ∧ ∃x(x ∈ R ∧ ∀y(y ∈ R → (y ∈ A → y <R x)))) → ∃x(x ∈ R ∧ ∀y(y ∈ R → ((y ∈ A → ¬ x <R y) ∧ (y <R x → ∃z(z ∈ R ∧ (z ∈ A ∧ y <R z)))))))
Theorem	supsr 4025	A non-empty, bounded set of signed reals has a supremum.
⊢ (((A ⊆ R ∧ ¬ A = ∅) ∧ ∃x(x ∈ R ∧ ∀y(y ∈ R → (y ∈ A → y <R x)))) → ∃x(x ∈ R ∧ ∀y(y ∈ R → ((y ∈ A → ¬ x <R y) ∧ (y <R x → ∃z(z ∈ R ∧ (z ∈ A ∧ y <R z)))))))
Syntax	cc 4026	Class of complex numbers.
class ℂ
Syntax	cr 4027	Class of real numbers.
class ℝ
Syntax	cc0 4028	Extend class notation to include the complex number 0.
class 0
Syntax	c1 4029	Extend class notation to include the complex number 1.
class 1
Syntax	ci 4030	Extend class notation to include the complex number i.
class i
Syntax	caddc 4031	Addition on complex numbers.
class +
Syntax	cmulc 4032	Multiplication on complex numbers. The token · is a center dot.
class ·
Syntax	clt 4033	'Less than' predicate (defined over real subset of complex numbers).
class <
Definition	df-c 4034	Define the set of complex numbers.
⊢ ℂ = (R × R)
Definition	df-0 4035	Define the complex number 0 (base 10).
⊢ 0 = ⟨0R, 0R⟩
Definition	df-1 4036	Define the complex number 1 (base 10).
⊢ 1 = ⟨1R, 0R⟩
Definition	df-i 4037	Define the complex (imaginary) number i.
⊢ i = ⟨0R, 1R⟩
Definition	df-r 4038	Define the set of real numbers.
⊢ ℝ = (R × {0R})
Definition	df-plus 4039	Define addition over complex numbers.
⊢ + = {⟨⟨x, y⟩, z⟩∣((x ∈ ℂ ∧ y ∈ ℂ) ∧ ∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = ⟨(w +R u), (v +R f)⟩))}
Definition	df-mul 4040	Define multiplication over complex numbers.
⊢ · = {⟨⟨x, y⟩, z⟩∣((x ∈ ℂ ∧ y ∈ ℂ) ∧ ∃w∃v∃u∃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 4041	Define 'less than' on the real subset of complex numbers.
⊢ < = {⟨x, y⟩∣((x ∈ ℝ ∧ y ∈ ℝ) ∧ ∃z∃w((x = ⟨z, 0R⟩ ∧ y = ⟨w, 0R⟩) ∧ z <R w))}
Theorem	opelcn 4042	Ordered pair membership in the class of complex numbers.
⊢ B ∈ V    ⇒   ⊢ (⟨A, B⟩ ∈ ℂ ↔ (A ∈ R ∧ B ∈ R))
Theorem	opelreal 4043	Ordered pair membership in class of real subset of complex numbers.
⊢ (⟨A, 0R⟩ ∈ ℝ ↔ A ∈ R)
Theorem	elreal 4044	Membership in class of real numbers.
⊢ (A ∈ ℝ ↔ ∃x(x ∈ R ∧ ⟨x, 0R⟩ = A))
Theorem	0ncn 4045	The empty set is not a complex number.
⊢ ¬ ∅ ∈ ℂ
Theorem	ltrelre 4046	'Less than' is a relation on real numbers.
⊢ < ⊆ (ℝ × ℝ)
Theorem	addcnsr 4047	Addition of complex numbers in terms of signed reals.
⊢ (((A ∈ R ∧ B ∈ R) ∧ (C ∈ R ∧ D ∈ R)) → (⟨A, B⟩ + ⟨C, D⟩) = ⟨(A +R C), (B +R D)⟩)
Theorem	mulcnsr 4048	Multiplication of complex numbers in terms of signed reals.
⊢ (((A ∈ R ∧ B ∈ R) ∧ (C ∈ R ∧ D ∈ R)) → (⟨A, B⟩ · ⟨C, D⟩) = ⟨((A ·R C) +R (-1R ·R (B ·R D))), ((B ·R C) +R (A ·R D))⟩)
Theorem	eqresr 4049	Equality of real numbers in terms of intermediate signed reals.
⊢ A ∈ V    ⇒   ⊢ (⟨A, 0R⟩ = ⟨B, 0R⟩ ↔ A = B)
Theorem	addresr 4050	Addition of real numbers in terms of intermediate signed reals.
⊢ ((A ∈ R ∧ B ∈ R) → (⟨A, 0R⟩ + ⟨B, 0R⟩) = ⟨(A +R B), 0R⟩)
Theorem	mulresr 4051	Multiplication of real numbers in terms of intermediate signed reals.
⊢ B ∈ V    ⇒   ⊢ ((A ∈ R ∧ B ∈ R) → (⟨A, 0R⟩ · ⟨B, 0R⟩) = ⟨(A ·R B), 0R⟩)
Theorem	ltresr 4052	Ordering of real subset of complex numbers in terms of signed reals.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ (⟨A, 0R⟩ < ⟨B, 0R⟩ ↔ A <R B)
Theorem	suprelem 4053	Mapping of non-empty subset from signed reals to reals.
⊢ B = {w∣⟨w, 0R⟩ ∈ A}    ⇒   ⊢ ((A ⊆ ℝ ∧ ¬ A = ∅) → (B ⊆ R ∧ ¬ B = ∅))
Theorem	supre 4054	A non-empty, bounded-above set of reals has a supremum.
⊢ B = {w∣⟨w, 0R⟩ ∈ A}    ⇒   ⊢ (((A ⊆ ℝ ∧ ¬ A = ∅) ∧ ∃x(x ∈ ℝ ∧ ∀y(y ∈ ℝ → (y ∈ A → y < x)))) → ∃x(x ∈ ℝ ∧ ∀y(y ∈ ℝ → ((y ∈ A → ¬ x < y) ∧ (y < x → ∃z(z ∈ ℝ ∧ (z ∈ A ∧ y < z)))))))
Theorem	ltsor 4055	'Less than' is a strict ordering on real subset of complex numbers. Note: use ltso 4279 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.
⊢ < Or ℝ
Theorem	dfcnqs 4056	Technical trick to permit reuse of some of our previous equivalence class lemmas to prove arithmetic operation laws in ℂ from those in R. Note that converse epsilon is not an equivalence relation, but the lemmas we reuse do not require it.
⊢ ℂ = ((R × R) / ◡E)
Theorem	addcnsrec 4057	Technical trick to permit re-use of some equivalence class lemmas for operation laws.
⊢ (((A ∈ R ∧ B ∈ R) ∧ (C ∈ R ∧ D ∈ R)) → ([⟨A, B⟩]◡E + [⟨C, D⟩]◡E) = [⟨(A +R C), (B +R D)⟩]◡E)
Theorem	mulcnsrec 4058	Technical trick to permit re-use of some equivalence class lemmas for operation laws.
⊢ (((A ∈ R ∧ B ∈ R) ∧ (C ∈ R ∧ D ∈ R)) → ([⟨A, B⟩]◡E · [⟨C, D⟩]◡E) = [⟨((A ·R C) +R (-1R ·R (B ·R D))), ((B ·R C) +R (A ·R D))⟩]◡E)
Theorem	axaddex 4059	The addition operation is a set. Supplemental axiom for complex numbers (needed by seqsuc 4671). (It is not clear if this should be added to the "official" list of complex number axioms. df-seq 4661 is an artificial construct that in principle could be defined differently so as not to require this axiom. Currently we do not include this axiom on the Metamath Proof Explorer page with the Axioms for Complex Numbers.)
⊢ + ∈ V
Theorem	axmulex 4060	The multiplication operation is a set. Supplemental axiom for complex numbers (needed by seqsuc 4671). (It is not clear if this should be added to the "official" list of complex number axioms. df-seq 4661 is an artificial construct that in principle could be defined differently so as not to require this axiom. Currently we do not include this axiom on the Metamath Proof Explorer page with the Axioms for Complex Numbers.)
⊢ · ∈ V
Theorem	axcnex 4061	The class of complex numbers is a set, i.e. it is a member of the universe of sets V (see isset 1351). One of the 28 axioms for real and complex numbers, derived from ZF set theory.
⊢ ℂ ∈ V
Theorem	axresscn 4062	The real numbers are a subset of the complex numbers. One of the 28 axioms for real and complex numbers, derived from ZF set theory.
⊢ ℝ ⊆ ℂ
Theorem	ax0re 4063	0 is a real number. One of the 28 axioms for real and complex numbers, derived from ZF set theory.
⊢ 0 ∈ ℝ
Theorem	ax1re 4064	1 is a real number. One of the 28 axioms for real and complex numbers, derived from ZF set theory.
⊢ 1 ∈ ℝ
Theorem	axicn 4065	i is a complex number. One of the 28 axioms for real and complex numbers, derived from ZF set theory.
⊢ i ∈ ℂ
Theorem	axaddcl 4066	Closure law for addition of complex numbers. One of the 28 axioms for real and complex numbers, derived from ZF set theory.
⊢ ((A ∈ ℂ ∧ B ∈ ℂ) → (A + B) ∈ ℂ)
Theorem	axaddrcl 4067	Closure law for addition in the real subfield of complex numbers. One of the 28 axioms for real and complex numbers, derived from ZF set theory.
⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → (A + B) ∈ ℝ)
Theorem	axmulcl 4068	Closure law for multiplication of complex numbers. One of the 28 axioms for real and complex numbers, derived from ZF set theory.
⊢ ((A ∈ ℂ ∧ B ∈ ℂ) → (A · B) ∈ ℂ)
Theorem	axmulrcl 4069	Closure law for multiplication in the real subfield of complex numbers. One of the 28 axioms for real and complex numbers, derived from ZF set theory.
⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → (A · B) ∈ ℝ)
Theorem	axaddcom 4070	Addition of complex numbers is commutative. One of the 28 axioms for real and complex numbers, derived from ZF set theory.
⊢ ((A ∈ ℂ ∧ B ∈ ℂ) → (A + B) = (B + A))
Theorem	axmulcom 4071	Multiplication of complex numbers is commutative. One of the 28 axioms for real and complex numbers, derived from ZF set theory.
⊢ ((A ∈ ℂ ∧ B ∈ ℂ) → (A · B) = (B · A))
Theorem	axaddass 4072	Addition of complex numbers is associative. One of the 28 axioms for real and complex numbers, derived from ZF set theory.
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ) → ((A + B) + C) = (A + (B + C)))
Theorem	axmulass 4073	Multiplication of complex numbers is associative. One of the 28 axioms for real and complex numbers, derived from ZF set theory.
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ) → ((A · B) · C) = (A · (B · C)))
Theorem	axdistr 4074	Distributive law for complex numbers. One of the 28 axioms for real and complex numbers, derived from ZF set theory.
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ) → (A · (B + C)) = ((A · B) + (A · C)))
Theorem	ax1ne0 4075	1 and 0 are distinct. One of the 28 axioms for real and complex numbers, derived from ZF set theory.
⊢ 1 ≠ 0
Theorem	ax0id 4076	0 is an identity element for addition. One of the 28 axioms for real and complex numbers, derived from ZF set theory.
⊢ (A ∈ ℂ → (A + 0) = A)
Theorem	ax1id 4077	1 is an identity element for multiplication. One of the 28 axioms for real and complex numbers, derived from ZF set theory.
⊢ (A ∈ ℂ → (A · 1) = A)
Theorem	axnegex 4078	Existence of negative of complex number. One of the 28 axioms for real and complex numbers, derived from ZF set theory.
⊢ (A ∈ ℂ → ∃x ∈ ℂ (A + x) = 0)
Theorem	axrecex 4079	Existence of reciprocal of nonzero complex number. One of the 28 axioms for real and complex numbers, derived from ZF set theory.
⊢ ((A ∈ ℂ ∧ A ≠ 0) → ∃x ∈ ℂ (A · x) = 1)
Theorem	axrnegex 4080	Existence of negative of real number. One of the 28 axioms for real and complex numbers, derived from ZF set theory.
⊢ (A ∈ ℝ → ∃x ∈ ℝ (A + x) = 0)
Theorem	axrrecex 4081	Existence of reciprocal of nonzero real number. One of the 28 axioms for real and complex numbers, derived from ZF set theory.
⊢ ((A ∈ ℝ ∧ A ≠ 0) → ∃x ∈ ℝ (A · x) = 1)
Theorem	axi2m1 4082	i-squared equals -1 (expressed as i-squared plus 1 is 0). One of the 28 axioms for real and complex numbers, derived from ZF set theory.
⊢ ((i · i) + 1) = 0
Theorem	axlttri 4083	Ordering on reals satisfies strict trichotomy. One of the 28 axioms for real and complex numbers, derived from ZF set theory.
⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → (A < B ↔ ¬ (A = B ∨ B < A)))
Theorem	axlttrn 4084	Ordering on reals is transitive. One of the 28 axioms for real and complex numbers, derived from ZF set theory.
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ C ∈ ℝ) → ((A < B ∧ B < C) → A < C))
Theorem	axltadd 4085	Ordering property of addition on reals. One of the 28 axioms for real and complex numbers, derived from ZF set theory.
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ C ∈ ℝ) → (A < B → (C + A) < (C + B)))
Theorem	axmulgt0 4086	The product of two positive reals is positive. One of the 28 axioms for real and complex numbers, derived from ZF set theory.
⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → ((0 < A ∧ 0 < B) → 0 < (A · B)))
Theorem	axcnre 4087	A complex number can be expressed in terms of two reals. Definition 10-1.1(v) of [Gleason] p. 130. One of the 28 axioms for real and complex numbers, derived from ZF set theory.
⊢ (A ∈ ℂ → ∃x ∈ ℝ ∃y ∈ ℝ A = (x + (y · i)))
Theorem	axsup 4088	A non-empty, bounded-above set of reals has a supremum. One of the 28 axioms for real and complex numbers, derived from ZF set theory.
⊢ ((A ⊆ ℝ ∧ A ≠ ∅ ∧ ∃x ∈ ℝ ∀y ∈ A y < x) → ∃x ∈ ℝ (∀y ∈ A ¬ x < y ∧ ∀y ∈ ℝ (y < x → ∃z ∈ A y < z)))
Syntax	cmin 4089	Extend class notation to include subtraction.
class −
Syntax	cneg 4090	Extend class notation to include unary minus. The symbol - 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 (-) and subtraction cmin 4089 (−) to prevent syntax ambiguity. For example, looking at the syntax definition co 3001, 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, "(-A − B)" is unambiguous.
class -A
Syntax	cdiv 4091	Extend class notation to include division.
class /
Syntax	cle 4092	Extend wff notation to include the 'less than or equal to' relation.
class ≤
Syntax	cn 4093	Extend class notation to include the class of positive integers.
class ℕ
Syntax	cn0 4094	Extend class notation to include the class of nonnegative integers.
class ℕ0
Syntax	cz 4095	Extend class notation to include the class of integers.
class ℤ
Syntax	cq 4096	Extend class notation to include the class of rationals.
class ℚ
Theorem	recnt 4097	A real number is a complex number.
⊢ (A ∈ ℝ → A ∈ ℂ)
Theorem	recn 4098	A real number is a complex number.
⊢ A ∈ ℝ    ⇒   ⊢ A ∈ ℂ
Theorem	recnd 4099	Deduction from real number to complex number.
⊢ (φ → A ∈ ℝ)    ⇒   ⊢ (φ → A ∈ ℂ)
Theorem	0cn 4100	0 is a complex number.
⊢ 0 ∈ ℂ