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Theorem List for Intuitionistic Logic Explorer - 9101-9200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremremul2 9101 Real part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
((A B ℂ) → (ℜ‘(A · B)) = (A · (ℜ‘B)))
 
Theoremredivap 9102 Real part of a division. Related to remul2 9101. (Contributed by Jim Kingdon, 14-Jun-2020.)
((A B B # 0) → (ℜ‘(A / B)) = ((ℜ‘A) / B))
 
Theoremimcj 9103 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 9104 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 9105 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 9106 Imaginary part distributes over subtraction. (Contributed by NM, 18-Mar-2005.)
((A B ℂ) → (ℑ‘(AB)) = ((ℑ‘A) − (ℑ‘B)))
 
Theoremimmul 9107 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 9108 Imaginary part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
((A B ℂ) → (ℑ‘(A · B)) = (A · (ℑ‘B)))
 
Theoremimdivap 9109 Imaginary part of a division. Related to immul2 9108. (Contributed by Jim Kingdon, 14-Jun-2020.)
((A B B # 0) → (ℑ‘(A / B)) = ((ℑ‘A) / B))
 
Theoremcjre 9110 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 9111 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 9112 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 9113 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 9114 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 9115 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 9116 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 9117 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 9118 Complex conjugate of negative. (Contributed by NM, 27-Feb-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
(A ℂ → (∗‘-A) = -(∗‘A))
 
Theoremaddcj 9119 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 9120 Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.)
((A B ℂ) → (∗‘(AB)) = ((∗‘A) − (∗‘B)))
 
Theoremcjexp 9121 Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.)
((A 𝑁 0) → (∗‘(A𝑁)) = ((∗‘A)↑𝑁))
 
Theoremimval2 9122 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 9123 The real part of zero. (Contributed by NM, 27-Jul-1999.)
(ℜ‘0) = 0
 
Theoremim0 9124 The imaginary part of zero. (Contributed by NM, 27-Jul-1999.)
(ℑ‘0) = 0
 
Theoremre1 9125 The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
(ℜ‘1) = 1
 
Theoremim1 9126 The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
(ℑ‘1) = 0
 
Theoremrei 9127 The real part of i. (Contributed by Scott Fenton, 9-Jun-2006.)
(ℜ‘i) = 0
 
Theoremimi 9128 The imaginary part of i. (Contributed by Scott Fenton, 9-Jun-2006.)
(ℑ‘i) = 1
 
Theoremcj0 9129 The conjugate of zero. (Contributed by NM, 27-Jul-1999.)
(∗‘0) = 0
 
Theoremcji 9130 The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.)
(∗‘i) = -i
 
Theoremcjreim 9131 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 9132 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 9133 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 9134 Complex conjugate and apartness. (Contributed by Jim Kingdon, 14-Jun-2020.)
((A B ℂ) → ((∗‘A) # (∗‘B) ↔ A # B))
 
Theoremcjap0 9135 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 9136 A number is nonzero iff its complex conjugate is nonzero. (Contributed by NM, 29-Apr-2005.)
(A ℂ → (A ≠ 0 ↔ (∗‘A) ≠ 0))
 
Theoremcjdivap 9137 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
((A B B # 0) → (∗‘(A / B)) = ((∗‘A) / (∗‘B)))
 
Theoremcnrecnv 9138* The inverse to the canonical bijection from (ℝ × ℝ) to from cnref1o 8357. (Contributed by Mario Carneiro, 25-Aug-2014.)
𝐹 = (x ℝ, y ℝ ↦ (x + (i · y)))       𝐹 = (z ℂ ↦ ⟨(ℜ‘z), (ℑ‘z)⟩)
 
Theoremrecli 9139 The real part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
A        (ℜ‘A)
 
Theoremimcli 9140 The imaginary part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
A        (ℑ‘A)
 
Theoremcjcli 9141 Closure law for complex conjugate. (Contributed by NM, 11-May-1999.)
A        (∗‘A)
 
Theoremreplimi 9142 Construct a complex number from its real and imaginary parts. (Contributed by NM, 1-Oct-1999.)
A        A = ((ℜ‘A) + (i · (ℑ‘A)))
 
Theoremcjcji 9143 The conjugate of the conjugate is the original complex number. Proposition 10-3.4(e) of [Gleason] p. 133. (Contributed by NM, 11-May-1999.)
A        (∗‘(∗‘A)) = A
 
Theoremreim0bi 9144 A number is real iff its imaginary part is 0. (Contributed by NM, 29-May-1999.)
A        (A ℝ ↔ (ℑ‘A) = 0)
 
Theoremrerebi 9145 A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 27-Oct-1999.)
A        (A ℝ ↔ (ℜ‘A) = A)
 
Theoremcjrebi 9146 A number is real iff it equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 11-Oct-1999.)
A        (A ℝ ↔ (∗‘A) = A)
 
Theoremrecji 9147 Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
A        (ℜ‘(∗‘A)) = (ℜ‘A)
 
Theoremimcji 9148 Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
A        (ℑ‘(∗‘A)) = -(ℑ‘A)
 
Theoremcjmulrcli 9149 A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.)
A        (A · (∗‘A))
 
Theoremcjmulvali 9150 A complex number times its conjugate. (Contributed by NM, 2-Oct-1999.)
A        (A · (∗‘A)) = (((ℜ‘A)↑2) + ((ℑ‘A)↑2))
 
Theoremcjmulge0i 9151 A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.)
A        0 ≤ (A · (∗‘A))
 
Theoremrenegi 9152 Real part of negative. (Contributed by NM, 2-Aug-1999.)
A        (ℜ‘-A) = -(ℜ‘A)
 
Theoremimnegi 9153 Imaginary part of negative. (Contributed by NM, 2-Aug-1999.)
A        (ℑ‘-A) = -(ℑ‘A)
 
Theoremcjnegi 9154 Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.)
A        (∗‘-A) = -(∗‘A)
 
Theoremaddcji 9155 A number plus its conjugate is twice its real part. Compare Proposition 10-3.4(h) of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)
A        (A + (∗‘A)) = (2 · (ℜ‘A))
 
Theoremreaddi 9156 Real part distributes over addition. (Contributed by NM, 28-Jul-1999.)
A     &   B        (ℜ‘(A + B)) = ((ℜ‘A) + (ℜ‘B))
 
Theoremimaddi 9157 Imaginary part distributes over addition. (Contributed by NM, 28-Jul-1999.)
A     &   B        (ℑ‘(A + B)) = ((ℑ‘A) + (ℑ‘B))
 
Theoremremuli 9158 Real part of a product. (Contributed by NM, 28-Jul-1999.)
A     &   B        (ℜ‘(A · B)) = (((ℜ‘A) · (ℜ‘B)) − ((ℑ‘A) · (ℑ‘B)))
 
Theoremimmuli 9159 Imaginary part of a product. (Contributed by NM, 28-Jul-1999.)
A     &   B        (ℑ‘(A · B)) = (((ℜ‘A) · (ℑ‘B)) + ((ℑ‘A) · (ℜ‘B)))
 
Theoremcjaddi 9160 Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)
A     &   B        (∗‘(A + B)) = ((∗‘A) + (∗‘B))
 
Theoremcjmuli 9161 Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)
A     &   B        (∗‘(A · B)) = ((∗‘A) · (∗‘B))
 
Theoremipcni 9162 Standard inner product on complex numbers. (Contributed by NM, 2-Oct-1999.)
A     &   B        (ℜ‘(A · (∗‘B))) = (((ℜ‘A) · (ℜ‘B)) + ((ℑ‘A) · (ℑ‘B)))
 
Theoremcjdivapi 9163 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
A     &   B        (B # 0 → (∗‘(A / B)) = ((∗‘A) / (∗‘B)))
 
Theoremcrrei 9164 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.)
A     &   B        (ℜ‘(A + (i · B))) = A
 
Theoremcrimi 9165 The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.)
A     &   B        (ℑ‘(A + (i · B))) = B
 
Theoremrecld 9166 The real part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.)
(φA ℂ)       (φ → (ℜ‘A) ℝ)
 
Theoremimcld 9167 The imaginary part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.)
(φA ℂ)       (φ → (ℑ‘A) ℝ)
 
Theoremcjcld 9168 Closure law for complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
(φA ℂ)       (φ → (∗‘A) ℂ)
 
Theoremreplimd 9169 Construct a complex number from its real and imaginary parts. (Contributed by Mario Carneiro, 29-May-2016.)
(φA ℂ)       (φA = ((ℜ‘A) + (i · (ℑ‘A))))
 
Theoremremimd 9170 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 Mario Carneiro, 29-May-2016.)
(φA ℂ)       (φ → (∗‘A) = ((ℜ‘A) − (i · (ℑ‘A))))
 
Theoremcjcjd 9171 The conjugate of the conjugate is the original complex number. Proposition 10-3.4(e) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
(φA ℂ)       (φ → (∗‘(∗‘A)) = A)
 
Theoremreim0bd 9172 A number is real iff its imaginary part is 0. (Contributed by Mario Carneiro, 29-May-2016.)
(φA ℂ)    &   (φ → (ℑ‘A) = 0)       (φA ℝ)
 
Theoremrerebd 9173 A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
(φA ℂ)    &   (φ → (ℜ‘A) = A)       (φA ℝ)
 
Theoremcjrebd 9174 A number is real iff it equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
(φA ℂ)    &   (φ → (∗‘A) = A)       (φA ℝ)
 
Theoremcjne0d 9175 A number is nonzero iff its complex conjugate is nonzero. (Contributed by Mario Carneiro, 29-May-2016.)
(φA ℂ)    &   (φA ≠ 0)       (φ → (∗‘A) ≠ 0)
 
Theoremrecjd 9176 Real part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
(φA ℂ)       (φ → (ℜ‘(∗‘A)) = (ℜ‘A))
 
Theoremimcjd 9177 Imaginary part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
(φA ℂ)       (φ → (ℑ‘(∗‘A)) = -(ℑ‘A))
 
Theoremcjmulrcld 9178 A complex number times its conjugate is real. (Contributed by Mario Carneiro, 29-May-2016.)
(φA ℂ)       (φ → (A · (∗‘A)) ℝ)
 
Theoremcjmulvald 9179 A complex number times its conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
(φA ℂ)       (φ → (A · (∗‘A)) = (((ℜ‘A)↑2) + ((ℑ‘A)↑2)))
 
Theoremcjmulge0d 9180 A complex number times its conjugate is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
(φA ℂ)       (φ → 0 ≤ (A · (∗‘A)))
 
Theoremrenegd 9181 Real part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
(φA ℂ)       (φ → (ℜ‘-A) = -(ℜ‘A))
 
Theoremimnegd 9182 Imaginary part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
(φA ℂ)       (φ → (ℑ‘-A) = -(ℑ‘A))
 
Theoremcjnegd 9183 Complex conjugate of negative. (Contributed by Mario Carneiro, 29-May-2016.)
(φA ℂ)       (φ → (∗‘-A) = -(∗‘A))
 
Theoremaddcjd 9184 A number plus its conjugate is twice its real part. Compare Proposition 10-3.4(h) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
(φA ℂ)       (φ → (A + (∗‘A)) = (2 · (ℜ‘A)))
 
Theoremcjexpd 9185 Complex conjugate of positive integer exponentiation. (Contributed by Mario Carneiro, 29-May-2016.)
(φA ℂ)    &   (φ𝑁 0)       (φ → (∗‘(A𝑁)) = ((∗‘A)↑𝑁))
 
Theoremreaddd 9186 Real part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → (ℜ‘(A + B)) = ((ℜ‘A) + (ℜ‘B)))
 
Theoremimaddd 9187 Imaginary part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → (ℑ‘(A + B)) = ((ℑ‘A) + (ℑ‘B)))
 
Theoremresubd 9188 Real part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → (ℜ‘(AB)) = ((ℜ‘A) − (ℜ‘B)))
 
Theoremimsubd 9189 Imaginary part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → (ℑ‘(AB)) = ((ℑ‘A) − (ℑ‘B)))
 
Theoremremuld 9190 Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → (ℜ‘(A · B)) = (((ℜ‘A) · (ℜ‘B)) − ((ℑ‘A) · (ℑ‘B))))
 
Theoremimmuld 9191 Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → (ℑ‘(A · B)) = (((ℜ‘A) · (ℑ‘B)) + ((ℑ‘A) · (ℜ‘B))))
 
Theoremcjaddd 9192 Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → (∗‘(A + B)) = ((∗‘A) + (∗‘B)))
 
Theoremcjmuld 9193 Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → (∗‘(A · B)) = ((∗‘A) · (∗‘B)))
 
Theoremipcnd 9194 Standard inner product on complex numbers. (Contributed by Mario Carneiro, 29-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → (ℜ‘(A · (∗‘B))) = (((ℜ‘A) · (ℜ‘B)) + ((ℑ‘A) · (ℑ‘B))))
 
Theoremcjdivapd 9195 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 15-Jun-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)       (φ → (∗‘(A / B)) = ((∗‘A) / (∗‘B)))
 
Theoremrered 9196 A real number equals its real part. One direction of Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
(φA ℝ)       (φ → (ℜ‘A) = A)
 
Theoremreim0d 9197 The imaginary part of a real number is 0. (Contributed by Mario Carneiro, 29-May-2016.)
(φA ℝ)       (φ → (ℑ‘A) = 0)
 
Theoremcjred 9198 A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
(φA ℝ)       (φ → (∗‘A) = A)
 
Theoremremul2d 9199 Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
(φA ℝ)    &   (φB ℂ)       (φ → (ℜ‘(A · B)) = (A · (ℜ‘B)))
 
Theoremimmul2d 9200 Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
(φA ℝ)    &   (φB ℂ)       (φ → (ℑ‘(A · B)) = (A · (ℑ‘B)))
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