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Theorem List for Intuitionistic Logic Explorer - 9001-9100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremexpnegapd 9001 Value of a complex number raised to a negative power. (Contributed by Jim Kingdon, 12-Jun-2020.)
(φA ℂ)    &   (φA # 0)    &   (φ𝑁 ℤ)       (φ → (A↑-𝑁) = (1 / (A𝑁)))
 
Theoremexprecapd 9002 Nonnegative integer exponentiation of a reciprocal. (Contributed by Jim Kingdon, 12-Jun-2020.)
(φA ℂ)    &   (φA # 0)    &   (φ𝑁 ℤ)       (φ → ((1 / A)↑𝑁) = (1 / (A𝑁)))
 
Theoremexpp1zapd 9003 Value of a nonzero complex number raised to an integer power plus one. (Contributed by Jim Kingdon, 12-Jun-2020.)
(φA ℂ)    &   (φA # 0)    &   (φ𝑁 ℤ)       (φ → (A↑(𝑁 + 1)) = ((A𝑁) · A))
 
Theoremexpm1apd 9004 Value of a complex number raised to an integer power minus one. (Contributed by Jim Kingdon, 12-Jun-2020.)
(φA ℂ)    &   (φA # 0)    &   (φ𝑁 ℤ)       (φ → (A↑(𝑁 − 1)) = ((A𝑁) / A))
 
Theoremexpsubapd 9005 Exponent subtraction law for nonnegative integer exponentiation. (Contributed by Jim Kingdon, 12-Jun-2020.)
(φA ℂ)    &   (φA # 0)    &   (φ𝑁 ℤ)    &   (φ𝑀 ℤ)       (φ → (A↑(𝑀𝑁)) = ((A𝑀) / (A𝑁)))
 
Theoremsqmuld 9006 Distribution of square over multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → ((A · B)↑2) = ((A↑2) · (B↑2)))
 
Theoremsqdivapd 9007 Distribution of square over division. (Contributed by Jim Kingdon, 13-Jun-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)       (φ → ((A / B)↑2) = ((A↑2) / (B↑2)))
 
Theoremexpdivapd 9008 Nonnegative integer exponentiation of a quotient. (Contributed by Jim Kingdon, 13-Jun-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)    &   (φ𝑁 0)       (φ → ((A / B)↑𝑁) = ((A𝑁) / (B𝑁)))
 
Theoremmulexpd 9009 Positive integer exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝑁 0)       (φ → ((A · B)↑𝑁) = ((A𝑁) · (B𝑁)))
 
Theorem0expd 9010 Value of zero raised to a positive integer power. (Contributed by Mario Carneiro, 28-May-2016.)
(φ𝑁 ℕ)       (φ → (0↑𝑁) = 0)
 
Theoremreexpcld 9011 Closure of exponentiation of reals. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φ𝑁 0)       (φ → (A𝑁) ℝ)
 
Theoremexpge0d 9012 Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φ𝑁 0)    &   (φ → 0 ≤ A)       (φ → 0 ≤ (A𝑁))
 
Theoremexpge1d 9013 Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φ𝑁 0)    &   (φ → 1 ≤ A)       (φ → 1 ≤ (A𝑁))
 
Theoremnnsqcld 9014 The naturals are closed under squaring. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℕ)       (φ → (A↑2) ℕ)
 
Theoremnnexpcld 9015 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℕ)    &   (φ𝑁 0)       (φ → (A𝑁) ℕ)
 
Theoremnn0expcld 9016 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.)
(φA 0)    &   (φ𝑁 0)       (φ → (A𝑁) 0)
 
Theoremrpexpcld 9017 Closure law for exponentiation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)    &   (φ𝑁 ℤ)       (φ → (A𝑁) +)
 
Theoremreexpclzapd 9018 Closure of exponentiation of reals. (Contributed by Jim Kingdon, 13-Jun-2020.)
(φA ℝ)    &   (φA # 0)    &   (φ𝑁 ℤ)       (φ → (A𝑁) ℝ)
 
Theoremresqcld 9019 Closure of square in reals. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)       (φ → (A↑2) ℝ)
 
Theoremsqge0d 9020 A square of a real is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)       (φ → 0 ≤ (A↑2))
 
Theoremsqgt0apd 9021 The square of a real apart from zero is positive. (Contributed by Jim Kingdon, 13-Jun-2020.)
(φA ℝ)    &   (φA # 0)       (φ → 0 < (A↑2))
 
Theoremleexp2ad 9022 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φ → 1 ≤ A)    &   (φ𝑁 (ℤ𝑀))       (φ → (A𝑀) ≤ (A𝑁))
 
Theoremleexp2rd 9023 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φ𝑀 0)    &   (φ𝑁 (ℤ𝑀))    &   (φ → 0 ≤ A)    &   (φA ≤ 1)       (φ → (A𝑁) ≤ (A𝑀))
 
Theoremlt2sqd 9024 The square function on nonnegative reals is strictly monotonic. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ → 0 ≤ A)    &   (φ → 0 ≤ B)       (φ → (A < B ↔ (A↑2) < (B↑2)))
 
Theoremle2sqd 9025 The square function on nonnegative reals is monotonic. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ → 0 ≤ A)    &   (φ → 0 ≤ B)       (φ → (AB ↔ (A↑2) ≤ (B↑2)))
 
Theoremsq11d 9026 The square function is one-to-one for nonnegative reals. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ → 0 ≤ A)    &   (φ → 0 ≤ B)    &   (φ → (A↑2) = (B↑2))       (φA = B)
 
3.6  Elementary real and complex functions
 
3.6.1  Real and imaginary parts; conjugate
 
Syntaxccj 9027 Extend class notation to include complex conjugate function.
class
 
Syntaxcre 9028 Extend class notation to include real part of a complex number.
class
 
Syntaxcim 9029 Extend class notation to include imaginary part of a complex number.
class
 
Definitiondf-cj 9030* Define the complex conjugate function. See cjcli 9101 for its closure and cjval 9033 for its value. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
∗ = (x ℂ ↦ (y ℂ ((x + y) (i · (xy)) ℝ)))
 
Definitiondf-re 9031 Define a function whose value is the real part of a complex number. See reval 9037 for its value, recli 9099 for its closure, and replim 9047 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
ℜ = (x ℂ ↦ ((x + (∗‘x)) / 2))
 
Definitiondf-im 9032 Define a function whose value is the imaginary part of a complex number. See imval 9038 for its value, imcli 9100 for its closure, and replim 9047 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
ℑ = (x ℂ ↦ (ℜ‘(x / i)))
 
Theoremcjval 9033* The value of the conjugate of a complex number. (Contributed by Mario Carneiro, 6-Nov-2013.)
(A ℂ → (∗‘A) = (x ℂ ((A + x) (i · (Ax)) ℝ)))
 
Theoremcjth 9034 The defining property of the complex conjugate. (Contributed by Mario Carneiro, 6-Nov-2013.)
(A ℂ → ((A + (∗‘A)) (i · (A − (∗‘A))) ℝ))
 
Theoremcjf 9035 Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)
∗:ℂ⟶ℂ
 
Theoremcjcl 9036 The conjugate of a complex number is a complex number (closure law). (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
(A ℂ → (∗‘A) ℂ)
 
Theoremreval 9037 The value of the real part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
(A ℂ → (ℜ‘A) = ((A + (∗‘A)) / 2))
 
Theoremimval 9038 The value of the imaginary part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
(A ℂ → (ℑ‘A) = (ℜ‘(A / i)))
 
Theoremimre 9039 The imaginary part of a complex number in terms of the real part function. (Contributed by NM, 12-May-2005.) (Revised by Mario Carneiro, 6-Nov-2013.)
(A ℂ → (ℑ‘A) = (ℜ‘(-i · A)))
 
Theoremreim 9040 The real part of a complex number in terms of the imaginary part function. (Contributed by Mario Carneiro, 31-Mar-2015.)
(A ℂ → (ℜ‘A) = (ℑ‘(i · A)))
 
Theoremrecl 9041 The real part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
(A ℂ → (ℜ‘A) ℝ)
 
Theoremimcl 9042 The imaginary part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
(A ℂ → (ℑ‘A) ℝ)
 
Theoremref 9043 Domain and codomain of the real part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)
ℜ:ℂ⟶ℝ
 
Theoremimf 9044 Domain and codomain of the imaginary part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)
ℑ:ℂ⟶ℝ
 
Theoremcrre 9045 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 12-May-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
((A B ℝ) → (ℜ‘(A + (i · B))) = A)
 
Theoremcrim 9046 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 12-May-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
((A B ℝ) → (ℑ‘(A + (i · B))) = B)
 
Theoremreplim 9047 Reconstruct a complex number from its real and imaginary parts. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 7-Nov-2013.)
(A ℂ → A = ((ℜ‘A) + (i · (ℑ‘A))))
 
Theoremremim 9048 Value of the conjugate of a complex number. The value is the real part minus i times the imaginary part. Definition 10-3.2 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 7-Nov-2013.)
(A ℂ → (∗‘A) = ((ℜ‘A) − (i · (ℑ‘A))))
 
Theoremreim0 9049 The imaginary part of a real number is 0. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
(A ℝ → (ℑ‘A) = 0)
 
Theoremreim0b 9050 A number is real iff its imaginary part is 0. (Contributed by NM, 26-Sep-2005.)
(A ℂ → (A ℝ ↔ (ℑ‘A) = 0))
 
Theoremrereb 9051 A number is real iff it equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 20-Aug-2008.)
(A ℂ → (A ℝ ↔ (ℜ‘A) = A))
 
Theoremmulreap 9052 A product with a real multiplier apart from zero is real iff the multiplicand is real. (Contributed by Jim Kingdon, 14-Jun-2020.)
((A B B # 0) → (A ℝ ↔ (B · A) ℝ))
 
Theoremrere 9053 A real number equals its real part. One direction of Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Paul Chapman, 7-Sep-2007.)
(A ℝ → (ℜ‘A) = A)
 
Theoremcjreb 9054 A number is real iff it equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 2-Jul-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
(A ℂ → (A ℝ ↔ (∗‘A) = A))
 
Theoremrecj 9055 Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014.)
(A ℂ → (ℜ‘(∗‘A)) = (ℜ‘A))
 
Theoremreneg 9056 Real part of negative. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
(A ℂ → (ℜ‘-A) = -(ℜ‘A))
 
Theoremreadd 9057 Real part distributes over addition. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
((A B ℂ) → (ℜ‘(A + B)) = ((ℜ‘A) + (ℜ‘B)))
 
Theoremresub 9058 Real part distributes over subtraction. (Contributed by NM, 17-Mar-2005.)
((A B ℂ) → (ℜ‘(AB)) = ((ℜ‘A) − (ℜ‘B)))
 
Theoremremullem 9059 Lemma for remul 9060, immul 9067, and cjmul 9073. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
((A B ℂ) → ((ℜ‘(A · B)) = (((ℜ‘A) · (ℜ‘B)) − ((ℑ‘A) · (ℑ‘B))) (ℑ‘(A · B)) = (((ℜ‘A) · (ℑ‘B)) + ((ℑ‘A) · (ℜ‘B))) (∗‘(A · B)) = ((∗‘A) · (∗‘B))))
 
Theoremremul 9060 Real part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
((A B ℂ) → (ℜ‘(A · B)) = (((ℜ‘A) · (ℜ‘B)) − ((ℑ‘A) · (ℑ‘B))))
 
Theoremremul2 9061 Real part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
((A B ℂ) → (ℜ‘(A · B)) = (A · (ℜ‘B)))
 
Theoremredivap 9062 Real part of a division. Related to remul2 9061. (Contributed by Jim Kingdon, 14-Jun-2020.)
((A B B # 0) → (ℜ‘(A / B)) = ((ℜ‘A) / B))
 
Theoremimcj 9063 Imaginary part of a complex conjugate. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
(A ℂ → (ℑ‘(∗‘A)) = -(ℑ‘A))
 
Theoremimneg 9064 The imaginary part of a negative number. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
(A ℂ → (ℑ‘-A) = -(ℑ‘A))
 
Theoremimadd 9065 Imaginary part distributes over addition. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
((A B ℂ) → (ℑ‘(A + B)) = ((ℑ‘A) + (ℑ‘B)))
 
Theoremimsub 9066 Imaginary part distributes over subtraction. (Contributed by NM, 18-Mar-2005.)
((A B ℂ) → (ℑ‘(AB)) = ((ℑ‘A) − (ℑ‘B)))
 
Theoremimmul 9067 Imaginary part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
((A B ℂ) → (ℑ‘(A · B)) = (((ℜ‘A) · (ℑ‘B)) + ((ℑ‘A) · (ℜ‘B))))
 
Theoremimmul2 9068 Imaginary part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
((A B ℂ) → (ℑ‘(A · B)) = (A · (ℑ‘B)))
 
Theoremimdivap 9069 Imaginary part of a division. Related to immul2 9068. (Contributed by Jim Kingdon, 14-Jun-2020.)
((A B B # 0) → (ℑ‘(A / B)) = ((ℑ‘A) / B))
 
Theoremcjre 9070 A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 8-Oct-1999.)
(A ℝ → (∗‘A) = A)
 
Theoremcjcj 9071 The conjugate of the conjugate is the original complex number. Proposition 10-3.4(e) of [Gleason] p. 133. (Contributed by NM, 29-Jul-1999.) (Proof shortened by Mario Carneiro, 14-Jul-2014.)
(A ℂ → (∗‘(∗‘A)) = A)
 
Theoremcjadd 9072 Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by NM, 31-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
((A B ℂ) → (∗‘(A + B)) = ((∗‘A) + (∗‘B)))
 
Theoremcjmul 9073 Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by NM, 29-Jul-1999.) (Proof shortened by Mario Carneiro, 14-Jul-2014.)
((A B ℂ) → (∗‘(A · B)) = ((∗‘A) · (∗‘B)))
 
Theoremipcnval 9074 Standard inner product on complex numbers. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
((A B ℂ) → (ℜ‘(A · (∗‘B))) = (((ℜ‘A) · (ℜ‘B)) + ((ℑ‘A) · (ℑ‘B))))
 
Theoremcjmulrcl 9075 A complex number times its conjugate is real. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
(A ℂ → (A · (∗‘A)) ℝ)
 
Theoremcjmulval 9076 A complex number times its conjugate. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 14-Jul-2014.)
(A ℂ → (A · (∗‘A)) = (((ℜ‘A)↑2) + ((ℑ‘A)↑2)))
 
Theoremcjmulge0 9077 A complex number times its conjugate is nonnegative. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
(A ℂ → 0 ≤ (A · (∗‘A)))
 
Theoremcjneg 9078 Complex conjugate of negative. (Contributed by NM, 27-Feb-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
(A ℂ → (∗‘-A) = -(∗‘A))
 
Theoremaddcj 9079 A number plus its conjugate is twice its real part. Compare Proposition 10-3.4(h) of [Gleason] p. 133. (Contributed by NM, 21-Jan-2007.) (Revised by Mario Carneiro, 14-Jul-2014.)
(A ℂ → (A + (∗‘A)) = (2 · (ℜ‘A)))
 
Theoremcjsub 9080 Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.)
((A B ℂ) → (∗‘(AB)) = ((∗‘A) − (∗‘B)))
 
Theoremcjexp 9081 Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.)
((A 𝑁 0) → (∗‘(A𝑁)) = ((∗‘A)↑𝑁))
 
Theoremimval2 9082 The imaginary part of a number in terms of complex conjugate. (Contributed by NM, 30-Apr-2005.)
(A ℂ → (ℑ‘A) = ((A − (∗‘A)) / (2 · i)))
 
Theoremre0 9083 The real part of zero. (Contributed by NM, 27-Jul-1999.)
(ℜ‘0) = 0
 
Theoremim0 9084 The imaginary part of zero. (Contributed by NM, 27-Jul-1999.)
(ℑ‘0) = 0
 
Theoremre1 9085 The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
(ℜ‘1) = 1
 
Theoremim1 9086 The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
(ℑ‘1) = 0
 
Theoremrei 9087 The real part of i. (Contributed by Scott Fenton, 9-Jun-2006.)
(ℜ‘i) = 0
 
Theoremimi 9088 The imaginary part of i. (Contributed by Scott Fenton, 9-Jun-2006.)
(ℑ‘i) = 1
 
Theoremcj0 9089 The conjugate of zero. (Contributed by NM, 27-Jul-1999.)
(∗‘0) = 0
 
Theoremcji 9090 The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.)
(∗‘i) = -i
 
Theoremcjreim 9091 The conjugate of a representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.)
((A B ℝ) → (∗‘(A + (i · B))) = (A − (i · B)))
 
Theoremcjreim2 9092 The conjugate of the representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)
((A B ℝ) → (∗‘(A − (i · B))) = (A + (i · B)))
 
Theoremcj11 9093 Complex conjugate is a one-to-one function. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Eric Schmidt, 2-Jul-2009.)
((A B ℂ) → ((∗‘A) = (∗‘B) ↔ A = B))
 
Theoremcjap 9094 Complex conjugate and apartness. (Contributed by Jim Kingdon, 14-Jun-2020.)
((A B ℂ) → ((∗‘A) # (∗‘B) ↔ A # B))
 
Theoremcjap0 9095 A number is apart from zero iff its complex conjugate is apart from zero. (Contributed by Jim Kingdon, 14-Jun-2020.)
(A ℂ → (A # 0 ↔ (∗‘A) # 0))
 
Theoremcjne0 9096 A number is nonzero iff its complex conjugate is nonzero. (Contributed by NM, 29-Apr-2005.)
(A ℂ → (A ≠ 0 ↔ (∗‘A) ≠ 0))
 
Theoremcjdivap 9097 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
((A B B # 0) → (∗‘(A / B)) = ((∗‘A) / (∗‘B)))
 
Theoremcnrecnv 9098* The inverse to the canonical bijection from (ℝ × ℝ) to from cnref1o 8317. (Contributed by Mario Carneiro, 25-Aug-2014.)
𝐹 = (x ℝ, y ℝ ↦ (x + (i · y)))       𝐹 = (z ℂ ↦ ⟨(ℜ‘z), (ℑ‘z)⟩)
 
Theoremrecli 9099 The real part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
A        (ℜ‘A)
 
Theoremimcli 9100 The imaginary part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
A        (ℑ‘A)
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