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Theorem List for Intuitionistic Logic Explorer - 9001-9100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremneg1sqe1 9001 -1 squared is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(-1↑2) = 1
 
Theoremsq2 9002 The square of 2 is 4. (Contributed by NM, 22-Aug-1999.)
(2↑2) = 4
 
Theoremsq3 9003 The square of 3 is 9. (Contributed by NM, 26-Apr-2006.)
(3↑2) = 9
 
Theoremcu2 9004 The cube of 2 is 8. (Contributed by NM, 2-Aug-2004.)
(2↑3) = 8
 
Theoremirec 9005 The reciprocal of i. (Contributed by NM, 11-Oct-1999.)
(1 / i) = -i
 
Theoremi2 9006 i squared. (Contributed by NM, 6-May-1999.)
(i↑2) = -1
 
Theoremi3 9007 i cubed. (Contributed by NM, 31-Jan-2007.)
(i↑3) = -i
 
Theoremi4 9008 i to the fourth power. (Contributed by NM, 31-Jan-2007.)
(i↑4) = 1
 
Theoremnnlesq 9009 A positive integer is less than or equal to its square. (Contributed by NM, 15-Sep-1999.) (Revised by Mario Carneiro, 12-Sep-2015.)
(𝑁 ℕ → 𝑁 ≤ (𝑁↑2))
 
Theoremexpnass 9010 A counterexample showing that exponentiation is not associative. (Contributed by Stefan Allan and Gérard Lang, 21-Sep-2010.)
((3↑3)↑3) < (3↑(3↑3))
 
Theoremsubsq 9011 Factor the difference of two squares. (Contributed by NM, 21-Feb-2008.)
((A B ℂ) → ((A↑2) − (B↑2)) = ((A + B) · (AB)))
 
Theoremsubsq2 9012 Express the difference of the squares of two numbers as a polynomial in the difference of the numbers. (Contributed by NM, 21-Feb-2008.)
((A B ℂ) → ((A↑2) − (B↑2)) = (((AB)↑2) + ((2 · B) · (AB))))
 
Theorembinom2i 9013 The square of a binomial. (Contributed by NM, 11-Aug-1999.)
A     &   B        ((A + B)↑2) = (((A↑2) + (2 · (A · B))) + (B↑2))
 
Theoremsubsqi 9014 Factor the difference of two squares. (Contributed by NM, 7-Feb-2005.)
A     &   B        ((A↑2) − (B↑2)) = ((A + B) · (AB))
 
Theorembinom2 9015 The square of a binomial. (Contributed by FL, 10-Dec-2006.)
((A B ℂ) → ((A + B)↑2) = (((A↑2) + (2 · (A · B))) + (B↑2)))
 
Theorembinom21 9016 Special case of binom2 9015 where B = 1. (Contributed by Scott Fenton, 11-May-2014.)
(A ℂ → ((A + 1)↑2) = (((A↑2) + (2 · A)) + 1))
 
Theorembinom2sub 9017 Expand the square of a subtraction. (Contributed by Scott Fenton, 10-Jun-2013.)
((A B ℂ) → ((AB)↑2) = (((A↑2) − (2 · (A · B))) + (B↑2)))
 
Theorembinom2subi 9018 Expand the square of a subtraction. (Contributed by Scott Fenton, 13-Jun-2013.)
A     &   B        ((AB)↑2) = (((A↑2) − (2 · (A · B))) + (B↑2))
 
Theorembinom3 9019 The cube of a binomial. (Contributed by Mario Carneiro, 24-Apr-2015.)
((A B ℂ) → ((A + B)↑3) = (((A↑3) + (3 · ((A↑2) · B))) + ((3 · (A · (B↑2))) + (B↑3))))
 
Theoremzesq 9020 An integer is even iff its square is even. (Contributed by Mario Carneiro, 12-Sep-2015.)
(𝑁 ℤ → ((𝑁 / 2) ℤ ↔ ((𝑁↑2) / 2) ℤ))
 
Theoremnnesq 9021 A positive integer is even iff its square is even. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 12-Sep-2015.)
(𝑁 ℕ → ((𝑁 / 2) ℕ ↔ ((𝑁↑2) / 2) ℕ))
 
Theorembernneq 9022 Bernoulli's inequality, due to Johan Bernoulli (1667-1748). (Contributed by NM, 21-Feb-2005.)
((A 𝑁 0 -1 ≤ A) → (1 + (A · 𝑁)) ≤ ((1 + A)↑𝑁))
 
Theorembernneq2 9023 Variation of Bernoulli's inequality bernneq 9022. (Contributed by NM, 18-Oct-2007.)
((A 𝑁 0 0 ≤ A) → (((A − 1) · 𝑁) + 1) ≤ (A𝑁))
 
Theorembernneq3 9024 A corollary of bernneq 9022. (Contributed by Mario Carneiro, 11-Mar-2014.)
((𝑃 (ℤ‘2) 𝑁 0) → 𝑁 < (𝑃𝑁))
 
Theoremexpnbnd 9025* Exponentiation with a mantissa greater than 1 has no upper bound. (Contributed by NM, 20-Oct-2007.)
((A B 1 < B) → 𝑘 A < (B𝑘))
 
Theoremexpnlbnd 9026* The reciprocal of exponentiation with a mantissa greater than 1 has no lower bound. (Contributed by NM, 18-Jul-2008.)
((A + B 1 < B) → 𝑘 ℕ (1 / (B𝑘)) < A)
 
Theoremexpnlbnd2 9027* The reciprocal of exponentiation with a mantissa greater than 1 has no lower bound. (Contributed by NM, 18-Jul-2008.) (Proof shortened by Mario Carneiro, 5-Jun-2014.)
((A + B 1 < B) → 𝑗 𝑘 (ℤ𝑗)(1 / (B𝑘)) < A)
 
Theoremexp0d 9028 Value of a complex number raised to the 0th power. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℂ)       (φ → (A↑0) = 1)
 
Theoremexp1d 9029 Value of a complex number raised to the first power. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℂ)       (φ → (A↑1) = A)
 
Theoremexpeq0d 9030 Positive integer exponentiation is 0 iff its mantissa is 0. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℂ)    &   (φ𝑁 ℕ)    &   (φ → (A𝑁) = 0)       (φA = 0)
 
Theoremsqvald 9031 Value of square. Inference version. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℂ)       (φ → (A↑2) = (A · A))
 
Theoremsqcld 9032 Closure of square. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℂ)       (φ → (A↑2) ℂ)
 
Theoremsqeq0d 9033 A number is zero iff its square is zero. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℂ)    &   (φ → (A↑2) = 0)       (φA = 0)
 
Theoremexpcld 9034 Closure law for nonnegative integer exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℂ)    &   (φ𝑁 0)       (φ → (A𝑁) ℂ)
 
Theoremexpp1d 9035 Value of a complex number raised to a nonnegative integer power plus one. Part of Definition 10-4.1 of [Gleason] p. 134. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℂ)    &   (φ𝑁 0)       (φ → (A↑(𝑁 + 1)) = ((A𝑁) · A))
 
Theoremexpaddd 9036 Sum of exponents law for nonnegative integer exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℂ)    &   (φ𝑁 0)    &   (φ𝑀 0)       (φ → (A↑(𝑀 + 𝑁)) = ((A𝑀) · (A𝑁)))
 
Theoremexpmuld 9037 Product of exponents law for positive integer exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℂ)    &   (φ𝑁 0)    &   (φ𝑀 0)       (φ → (A↑(𝑀 · 𝑁)) = ((A𝑀)↑𝑁))
 
Theoremsqrecapd 9038 Square of reciprocal. (Contributed by Jim Kingdon, 12-Jun-2020.)
(φA ℂ)    &   (φA # 0)       (φ → ((1 / A)↑2) = (1 / (A↑2)))
 
Theoremexpclzapd 9039 Closure law for integer exponentiation. (Contributed by Jim Kingdon, 12-Jun-2020.)
(φA ℂ)    &   (φA # 0)    &   (φ𝑁 ℤ)       (φ → (A𝑁) ℂ)
 
Theoremexpap0d 9040 Nonnegative integer exponentiation is nonzero if its mantissa is nonzero. (Contributed by Jim Kingdon, 12-Jun-2020.)
(φA ℂ)    &   (φA # 0)    &   (φ𝑁 ℤ)       (φ → (A𝑁) # 0)
 
Theoremexpnegapd 9041 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 9042 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 9043 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 9044 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 9045 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 9046 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 9047 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 9048 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 9049 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 9050 Value of zero raised to a positive integer power. (Contributed by Mario Carneiro, 28-May-2016.)
(φ𝑁 ℕ)       (φ → (0↑𝑁) = 0)
 
Theoremreexpcld 9051 Closure of exponentiation of reals. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φ𝑁 0)       (φ → (A𝑁) ℝ)
 
Theoremexpge0d 9052 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 9053 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 9054 The naturals are closed under squaring. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℕ)       (φ → (A↑2) ℕ)
 
Theoremnnexpcld 9055 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℕ)    &   (φ𝑁 0)       (φ → (A𝑁) ℕ)
 
Theoremnn0expcld 9056 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.)
(φA 0)    &   (φ𝑁 0)       (φ → (A𝑁) 0)
 
Theoremrpexpcld 9057 Closure law for exponentiation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
(φA +)    &   (φ𝑁 ℤ)       (φ → (A𝑁) +)
 
Theoremreexpclzapd 9058 Closure of exponentiation of reals. (Contributed by Jim Kingdon, 13-Jun-2020.)
(φA ℝ)    &   (φA # 0)    &   (φ𝑁 ℤ)       (φ → (A𝑁) ℝ)
 
Theoremresqcld 9059 Closure of square in reals. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)       (φ → (A↑2) ℝ)
 
Theoremsqge0d 9060 A square of a real is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)       (φ → 0 ≤ (A↑2))
 
Theoremsqgt0apd 9061 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 9062 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)    &   (φ → 1 ≤ A)    &   (φ𝑁 (ℤ𝑀))       (φ → (A𝑀) ≤ (A𝑁))
 
Theoremleexp2rd 9063 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 9064 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 9065 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 9066 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 9067 Extend class notation to include complex conjugate function.
class
 
Syntaxcre 9068 Extend class notation to include real part of a complex number.
class
 
Syntaxcim 9069 Extend class notation to include imaginary part of a complex number.
class
 
Definitiondf-cj 9070* Define the complex conjugate function. See cjcli 9141 for its closure and cjval 9073 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 9071 Define a function whose value is the real part of a complex number. See reval 9077 for its value, recli 9139 for its closure, and replim 9087 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
ℜ = (x ℂ ↦ ((x + (∗‘x)) / 2))
 
Definitiondf-im 9072 Define a function whose value is the imaginary part of a complex number. See imval 9078 for its value, imcli 9140 for its closure, and replim 9087 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
ℑ = (x ℂ ↦ (ℜ‘(x / i)))
 
Theoremcjval 9073* 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 9074 The defining property of the complex conjugate. (Contributed by Mario Carneiro, 6-Nov-2013.)
(A ℂ → ((A + (∗‘A)) (i · (A − (∗‘A))) ℝ))
 
Theoremcjf 9075 Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)
∗:ℂ⟶ℂ
 
Theoremcjcl 9076 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 9077 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 9078 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 9079 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 9080 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 9081 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 9082 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 9083 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 9084 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 9085 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 9086 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 9087 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 9088 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 9089 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 9090 A number is real iff its imaginary part is 0. (Contributed by NM, 26-Sep-2005.)
(A ℂ → (A ℝ ↔ (ℑ‘A) = 0))
 
Theoremrereb 9091 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 9092 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 9093 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 9094 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 9095 Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014.)
(A ℂ → (ℜ‘(∗‘A)) = (ℜ‘A))
 
Theoremreneg 9096 Real part of negative. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
(A ℂ → (ℜ‘-A) = -(ℜ‘A))
 
Theoremreadd 9097 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 9098 Real part distributes over subtraction. (Contributed by NM, 17-Mar-2005.)
((A B ℂ) → (ℜ‘(AB)) = ((ℜ‘A) − (ℜ‘B)))
 
Theoremremullem 9099 Lemma for remul 9100, immul 9107, and cjmul 9113. (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 9100 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))))
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