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 ∈ 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 ∈ R → ∃x(x ∈ 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 ∈ V    ⇒   ⊢ (0R <R A → ∃x(x ∈ R ⋀ (A ·R x) = 1R))
Theorem	addgt0sr 5204	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 5205	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 5206	The square of a nonzero signed real is positive.
⊢ A ∈ V    ⇒   ⊢ (A ∈ 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 ∈ V    ⇒   ⊢ (A ∈ R → (¬ A = 0R → ∃x(x ∈ R ⋀ (A ·R x) = 1R)))
Theorem	ssgt0sr 5208	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 5209	Mapping from positive signed reals to positive reals.
⊢ A ∈ V    ⇒   ⊢ (0R <R [⟨(A +P 1P), 1P⟩] ~R ↔ A ∈ P)
Theorem	ltpsrpr 5210	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 5211	Equivalence for positive signed real.
⊢ A ∈ V    ⇒   ⊢ (0R <R A ↔ ∃x(x ∈ 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 ∈ A}    ⇒   ⊢ ((∀x(x ∈ 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 ∈ 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 5214	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 5215	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 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 = ∅) ⋀ ∃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 5223	Class of complex numbers.
class ℂ
Syntax	cr 5224	Class of real numbers.
class ℝ
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 · is a center dot.
class ·
Definition	df-c 5231	Define the set of complex numbers. The 25 axioms for complex numbers start at axcnex 5258.
⊢ ℂ = (R × 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.
⊢ ℝ = (R × {0R})
Definition	df-plus 5236	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 5237	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 5238	Define 'less than' on the real subset of complex numbers.
⊢ <ℝ = {⟨x, y⟩∣((x ∈ ℝ ⋀ y ∈ ℝ) ⋀ ∃z∃w((x = ⟨z, 0R⟩ ⋀ y = ⟨w, 0R⟩) ⋀ z <R w))}
Theorem	opelcn 5239	Ordered pair membership in the class of complex numbers.
⊢ B ∈ V    ⇒   ⊢ (⟨A, B⟩ ∈ ℂ ↔ (A ∈ R ⋀ B ∈ R))
Theorem	opelreal 5240	Ordered pair membership in class of real subset of complex numbers.
⊢ (⟨A, 0R⟩ ∈ ℝ ↔ A ∈ R)
Theorem	elreal 5241	Membership in class of real numbers.
⊢ (A ∈ ℝ ↔ ∃x(x ∈ 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.
⊢ ¬ ∅ ∈ ℂ
Theorem	ltrelre 5243	'Less than' is a relation on real numbers.
⊢ <ℝ ⊆ (ℝ × ℝ)
Theorem	addcnsr 5244	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 5245	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 5246	Equality of real numbers in terms of intermediate signed reals.
⊢ A ∈ V    ⇒   ⊢ (⟨A, 0R⟩ = ⟨B, 0R⟩ ↔ A = B)
Theorem	addresr 5247	Addition of real numbers in terms of intermediate signed reals.
⊢ ((A ∈ R ⋀ B ∈ R) → (⟨A, 0R⟩ + ⟨B, 0R⟩) = ⟨(A +R B), 0R⟩)
Theorem	mulresr 5248	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 5249	Ordering of real subset of complex numbers in terms of signed reals.
⊢ A ∈ V    &   ⊢ B ∈ 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⟩ ∈ A}    ⇒   ⊢ ((A ⊆ ℝ ⋀ ¬ A = ∅) → (B ⊆ R ⋀ ¬ B = ∅))
Theorem	supre 5251	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 <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	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 ℝ ⋀ ℝ = ℝ)
Theorem	dfcnqs 5253	Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in ℂ 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 ℂ 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.
⊢ ℂ = ((R × R) / ◡E)
Theorem	addcnsrec 5254	Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 5253 and mulcnsrec 5255.
⊢ (((A ∈ R ⋀ B ∈ R) ⋀ (C ∈ R ⋀ D ∈ R)) → ([⟨A, B⟩]◡E + [⟨C, D⟩]◡E) = [⟨(A +R C), (B +R D)⟩]◡E)
Theorem	mulcnsrec 5255	Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecid 4300, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 5253. 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 4963.
⊢ (((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)
Real and complex number postulates
Theorem	axaddopr 5256	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 5262.
⊢ + :(ℂ × ℂ)–→ℂ
Theorem	axmulopr 5257	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 5264.
⊢ · :(ℂ × ℂ)–→ℂ
Theorem	axcnex 5258	The class of complex numbers is a set, i.e. it is a member of the universe of sets V (see isset 1810). Axiom 1 of 25 for real and complex numbers, derived from ZF set theory.
⊢ ℂ ∈ V
Theorem	axresscn 5259	The real numbers are a subset of the complex numbers. Axiom 2 of 25 for real and complex numbers, derived from ZF set theory.
⊢ ℝ ⊆ ℂ
Theorem	ax1cn 5260	1 is a complex number. Axiom 3 of 25 for real and complex numbers, derived from ZF set theory.
⊢ 1 ∈ ℂ
Theorem	axicn 5261	i is a complex number. Axiom 4 of 25 for real and complex numbers, derived from ZF set theory.
⊢ i ∈ ℂ
Theorem	axaddcl 5262	Closure law for addition of complex numbers. Axiom 5 of 25 for real and complex numbers, derived from ZF set theory.
⊢ ((A ∈ ℂ ⋀ B ∈ ℂ) → (A + B) ∈ ℂ)
Theorem	axaddrcl 5263	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 ∈ ℝ ⋀ B ∈ ℝ) → (A + B) ∈ ℝ)
Theorem	axmulcl 5264	Closure law for multiplication of complex numbers. Axiom 7 of 25 for real and complex numbers, derived from ZF set theory.
⊢ ((A ∈ ℂ ⋀ B ∈ ℂ) → (A · B) ∈ ℂ)
Theorem	axmulrcl 5265	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 ∈ ℝ ⋀ B ∈ ℝ) → (A · B) ∈ ℝ)
Theorem	axaddcom 5266	Addition of complex numbers is commutative. Axiom 9 of 25 for real and complex numbers, derived from ZF set theory.
⊢ ((A ∈ ℂ ⋀ B ∈ ℂ) → (A + B) = (B + A))
Theorem	axmulcom 5267	Multiplication of complex numbers is commutative. Axiom 10 of 25 for real and complex numbers, derived from ZF set theory.
⊢ ((A ∈ ℂ ⋀ B ∈ ℂ) → (A · B) = (B · A))
Theorem	axaddass 5268	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 ∈ ℂ ⋀ B ∈ ℂ ⋀ C ∈ ℂ) → ((A + B) + C) = (A + (B + C)))
Theorem	axmulass 5269	Multiplication of complex numbers is associative. Axiom 12 of 25 for real and complex numbers, derived from ZF set theory.
⊢ ((A ∈ ℂ ⋀ B ∈ ℂ ⋀ C ∈ ℂ) → ((A · B) · C) = (A · (B · C)))
Theorem	axdistr 5270	Distributive law for complex numbers. Axiom 13 of 25 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 5271	1 and 0 are distinct. Axiom 14 of 25 for real and complex numbers, derived from ZF set theory.
⊢ 1 ≠ 0
Theorem	ax0id 5272	0 is an identity element for addition. Axiom 15 of 25 for real and complex numbers, derived from ZF set theory.
⊢ (A ∈ ℂ → (A + 0) = A)
Theorem	ax1id 5273	1 is an identity element for multiplication. Axiom 16 of 25 for real and complex numbers, derived from ZF set theory.
⊢ (A ∈ ℂ → (A · 1) = A)
Theorem	axrnegex 5274	Existence of negative of real number. Axiom 17 of 25 for real and complex numbers, derived from ZF set theory.
⊢ (A ∈ ℝ → ∃x ∈ ℝ (A + x) = 0)
Theorem	axrrecex 5275	Existence of reciprocal of nonzero real number. Axiom 18 of 25 for real and complex numbers, derived from ZF set theory.
⊢ ((A ∈ ℝ ⋀ A ≠ 0) → ∃x ∈ ℝ (A · x) = 1)
Theorem	axi2m1 5276	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 · i) + 1) = 0
Theorem	axcnre 5277	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 ∈ ℂ → ∃x ∈ ℝ ∃y ∈ ℝ A = (x + (i · y)))
Theorem	pre-axlttri 5278	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 5494.
⊢ ((A ∈ ℝ ⋀ B ∈ ℝ) → (A <ℝ B ↔ ¬ (A = B ⋁ B <ℝ A)))
Theorem	pre-axlttrn 5279	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 5495.
⊢ ((A ∈ ℝ ⋀ B ∈ ℝ ⋀ C ∈ ℝ) → ((A <ℝ B ⋀ B <ℝ C) → A <ℝ C))
Theorem	pre-axltadd 5280	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 5496.
⊢ ((A ∈ ℝ ⋀ B ∈ ℝ ⋀ C ∈ ℝ) → (A <ℝ B → (C + A) <ℝ (C + B)))
Theorem	pre-axmulgt0 5281	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 5497.
⊢ ((A ∈ ℝ ⋀ B ∈ ℝ) → ((0 <ℝ A ⋀ 0 <ℝ B) → 0 <ℝ (A · B)))
Theorem	pre-axsup 5282	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 5498.
⊢ ((A ⊆ ℝ ⋀ A ≠ ∅ ⋀ ∃x ∈ ℝ ∀y ∈ A y <ℝ x) → ∃x ∈ ℝ (∀y ∈ A ¬ x <ℝ y ⋀ ∀y ∈ ℝ (y <ℝ x → ∃z ∈ A y <ℝ z)))
Real and complex numbers - basic operations
Syntax	cmin 5283	Extend class notation to include subtraction.
class −
Syntax	cneg 5284	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 5283 (−) to prevent syntax ambiguity. For example, looking at the syntax definition co 3964, 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 5285	Extend class notation to include division.
class /
Syntax	cle 5286	Extend wff notation to include the 'less than or equal to' relation.
class ≤
Syntax	cn 5287	Extend class notation to include the class of positive integers.
class ℕ
Syntax	cn0 5288	Extend class notation to include the class of nonnegative integers.
class ℕ0
Syntax	cz 5289	Extend class notation to include the class of integers.
class ℤ
Syntax	cq 5290	Extend class notation to include the class of rationals.
class ℚ
Syntax	crp 5291	Extend class notation to include the class of positive reals.
class ℝ+
Some deductions from the field axioms for complex numbers
Theorem	addclt 5292	Alias for axaddcl 5262, for naming consistency with addcl 5311.
⊢ ((A ∈ ℂ ⋀ B ∈ ℂ) → (A + B) ∈ ℂ)
Theorem	readdclt 5293	Alias for axaddrcl 5263, for naming consistency with readdcl 5325.
⊢ ((A ∈ ℝ ⋀ B ∈ ℝ) → (A + B) ∈ ℝ)
Theorem	mulclt 5294	Alias for axmulcl 5264, for naming consistency with mulcl 5312.
⊢ ((A ∈ ℂ ⋀ B ∈ ℂ) → (A · B) ∈ ℂ)
Theorem	remulclt 5295	Alias for axmulrcl 5265, for naming consistency with remulcl 5326.
⊢ ((A ∈ ℝ ⋀ B ∈ ℝ) → (A · B) ∈ ℝ)
Theorem	addcomt 5296	Alias for axaddcom 5266, for naming consistency with addcom 5313.
⊢ ((A ∈ ℂ ⋀ B ∈ ℂ) → (A + B) = (B + A))
Theorem	mulcomt 5297	Alias for axmulcom 5267, for naming consistency with mulcom 5314.
⊢ ((A ∈ ℂ ⋀ B ∈ ℂ) → (A · B) = (B · A))
Theorem	addasst 5298	Alias for axaddass 5268, for naming consistency with addass 5315.
⊢ ((A ∈ ℂ ⋀ B ∈ ℂ ⋀ C ∈ ℂ) → ((A + B) + C) = (A + (B + C)))
Theorem	mulasst 5299	Alias for axmulass 5269, for naming consistency with mulass 5316.
⊢ ((A ∈ ℂ ⋀ B ∈ ℂ ⋀ C ∈ ℂ) → ((A · B) · C) = (A · (B · C)))
Theorem	adddit 5300	Alias for axdistr 5270, for naming consistency with adddi 5317.
⊢ ((A ∈ ℂ ⋀ B ∈ ℂ ⋀ C ∈ ℂ) → (A · (B + C)) = ((A · B) + (A · C)))