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Theorem List for Intuitionistic Logic Explorer - 10501-10600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremshftvalg 10501 Value of a sequence shifted by 𝐴. (Contributed by Scott Fenton, 16-Dec-2017.)
((𝐹𝑉𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴)‘𝐵) = (𝐹‘(𝐵𝐴)))

Theoremshftval4g 10502 Value of a sequence shifted by -𝐴. (Contributed by Jim Kingdon, 19-Aug-2021.)
((𝐹𝑉𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift -𝐴)‘𝐵) = (𝐹‘(𝐴 + 𝐵)))

Theoremseq3shft 10503* Shifting the index set of a sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 17-Oct-2022.)
(𝜑𝐹𝑉)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   ((𝜑𝑥 ∈ (ℤ‘(𝑀𝑁))) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → seq𝑀( + , (𝐹 shift 𝑁)) = (seq(𝑀𝑁)( + , 𝐹) shift 𝑁))

4.7.2  Real and imaginary parts; conjugate

Syntaxccj 10504 Extend class notation to include complex conjugate function.
class

Syntaxcre 10505 Extend class notation to include real part of a complex number.
class

Syntaxcim 10506 Extend class notation to include imaginary part of a complex number.
class

Definitiondf-cj 10507* Define the complex conjugate function. See cjcli 10578 for its closure and cjval 10510 for its value. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
∗ = (𝑥 ∈ ℂ ↦ (𝑦 ∈ ℂ ((𝑥 + 𝑦) ∈ ℝ ∧ (i · (𝑥𝑦)) ∈ ℝ)))

Definitiondf-re 10508 Define a function whose value is the real part of a complex number. See reval 10514 for its value, recli 10576 for its closure, and replim 10524 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
ℜ = (𝑥 ∈ ℂ ↦ ((𝑥 + (∗‘𝑥)) / 2))

Definitiondf-im 10509 Define a function whose value is the imaginary part of a complex number. See imval 10515 for its value, imcli 10577 for its closure, and replim 10524 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
ℑ = (𝑥 ∈ ℂ ↦ (ℜ‘(𝑥 / i)))

Theoremcjval 10510* The value of the conjugate of a complex number. (Contributed by Mario Carneiro, 6-Nov-2013.)
(𝐴 ∈ ℂ → (∗‘𝐴) = (𝑥 ∈ ℂ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴𝑥)) ∈ ℝ)))

Theoremcjth 10511 The defining property of the complex conjugate. (Contributed by Mario Carneiro, 6-Nov-2013.)
(𝐴 ∈ ℂ → ((𝐴 + (∗‘𝐴)) ∈ ℝ ∧ (i · (𝐴 − (∗‘𝐴))) ∈ ℝ))

Theoremcjf 10512 Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)
∗:ℂ⟶ℂ

Theoremcjcl 10513 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 10514 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 10515 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 10516 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 10517 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 10518 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 10519 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 10520 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 10521 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 10522 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 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (ℜ‘(𝐴 + (i · 𝐵))) = 𝐴)

Theoremcrim 10523 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 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (ℑ‘(𝐴 + (i · 𝐵))) = 𝐵)

Theoremreplim 10524 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 10525 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 10526 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 10527 A number is real iff its imaginary part is 0. (Contributed by NM, 26-Sep-2005.)
(𝐴 ∈ ℂ → (𝐴 ∈ ℝ ↔ (ℑ‘𝐴) = 0))

Theoremrereb 10528 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 10529 A product with a real multiplier apart from zero is real iff the multiplicand is real. (Contributed by Jim Kingdon, 14-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 # 0) → (𝐴 ∈ ℝ ↔ (𝐵 · 𝐴) ∈ ℝ))

Theoremrere 10530 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 10531 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 10532 Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ ℂ → (ℜ‘(∗‘𝐴)) = (ℜ‘𝐴))

Theoremreneg 10533 Real part of negative. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ ℂ → (ℜ‘-𝐴) = -(ℜ‘𝐴))

Theoremreadd 10534 Real part distributes over addition. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 + 𝐵)) = ((ℜ‘𝐴) + (ℜ‘𝐵)))

Theoremresub 10535 Real part distributes over subtraction. (Contributed by NM, 17-Mar-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴𝐵)) = ((ℜ‘𝐴) − (ℜ‘𝐵)))

Theoremremullem 10536 Lemma for remul 10537, immul 10544, and cjmul 10550. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
((𝐴 ∈ ℂ ∧ 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 10537 Real part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 · 𝐵)) = (((ℜ‘𝐴) · (ℜ‘𝐵)) − ((ℑ‘𝐴) · (ℑ‘𝐵))))

Theoremremul2 10538 Real part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 · 𝐵)) = (𝐴 · (ℜ‘𝐵)))

Theoremredivap 10539 Real part of a division. Related to remul2 10538. (Contributed by Jim Kingdon, 14-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 # 0) → (ℜ‘(𝐴 / 𝐵)) = ((ℜ‘𝐴) / 𝐵))

Theoremimcj 10540 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 10541 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 10542 Imaginary part distributes over addition. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴 + 𝐵)) = ((ℑ‘𝐴) + (ℑ‘𝐵)))

Theoremimsub 10543 Imaginary part distributes over subtraction. (Contributed by NM, 18-Mar-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴𝐵)) = ((ℑ‘𝐴) − (ℑ‘𝐵)))

Theoremimmul 10544 Imaginary part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴 · 𝐵)) = (((ℜ‘𝐴) · (ℑ‘𝐵)) + ((ℑ‘𝐴) · (ℜ‘𝐵))))

Theoremimmul2 10545 Imaginary part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴 · 𝐵)) = (𝐴 · (ℑ‘𝐵)))

Theoremimdivap 10546 Imaginary part of a division. Related to immul2 10545. (Contributed by Jim Kingdon, 14-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 # 0) → (ℑ‘(𝐴 / 𝐵)) = ((ℑ‘𝐴) / 𝐵))

Theoremcjre 10547 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 10548 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 10549 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 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∗‘(𝐴 + 𝐵)) = ((∗‘𝐴) + (∗‘𝐵)))

Theoremcjmul 10550 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 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∗‘(𝐴 · 𝐵)) = ((∗‘𝐴) · (∗‘𝐵)))

Theoremipcnval 10551 Standard inner product on complex numbers. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 · (∗‘𝐵))) = (((ℜ‘𝐴) · (ℜ‘𝐵)) + ((ℑ‘𝐴) · (ℑ‘𝐵))))

Theoremcjmulrcl 10552 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 10553 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 10554 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 10555 Complex conjugate of negative. (Contributed by NM, 27-Feb-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ ℂ → (∗‘-𝐴) = -(∗‘𝐴))

Theoremaddcj 10556 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 10557 Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∗‘(𝐴𝐵)) = ((∗‘𝐴) − (∗‘𝐵)))

Theoremcjexp 10558 Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (∗‘(𝐴𝑁)) = ((∗‘𝐴)↑𝑁))

Theoremimval2 10559 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 10560 The real part of zero. (Contributed by NM, 27-Jul-1999.)
(ℜ‘0) = 0

Theoremim0 10561 The imaginary part of zero. (Contributed by NM, 27-Jul-1999.)
(ℑ‘0) = 0

Theoremre1 10562 The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
(ℜ‘1) = 1

Theoremim1 10563 The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
(ℑ‘1) = 0

Theoremrei 10564 The real part of i. (Contributed by Scott Fenton, 9-Jun-2006.)
(ℜ‘i) = 0

Theoremimi 10565 The imaginary part of i. (Contributed by Scott Fenton, 9-Jun-2006.)
(ℑ‘i) = 1

Theoremcj0 10566 The conjugate of zero. (Contributed by NM, 27-Jul-1999.)
(∗‘0) = 0

Theoremcji 10567 The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.)
(∗‘i) = -i

Theoremcjreim 10568 The conjugate of a representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (∗‘(𝐴 + (i · 𝐵))) = (𝐴 − (i · 𝐵)))

Theoremcjreim2 10569 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 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (∗‘(𝐴 − (i · 𝐵))) = (𝐴 + (i · 𝐵)))

Theoremcj11 10570 Complex conjugate is a one-to-one function. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Eric Schmidt, 2-Jul-2009.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((∗‘𝐴) = (∗‘𝐵) ↔ 𝐴 = 𝐵))

Theoremcjap 10571 Complex conjugate and apartness. (Contributed by Jim Kingdon, 14-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((∗‘𝐴) # (∗‘𝐵) ↔ 𝐴 # 𝐵))

Theoremcjap0 10572 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 10573 A number is nonzero iff its complex conjugate is nonzero. Also see cjap0 10572 which is similar but for apartness. (Contributed by NM, 29-Apr-2005.)
(𝐴 ∈ ℂ → (𝐴 ≠ 0 ↔ (∗‘𝐴) ≠ 0))

Theoremcjdivap 10574 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (∗‘(𝐴 / 𝐵)) = ((∗‘𝐴) / (∗‘𝐵)))

Theoremcnrecnv 10575* The inverse to the canonical bijection from (ℝ × ℝ) to from cnref1o 9342. (Contributed by Mario Carneiro, 25-Aug-2014.)
𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))       𝐹 = (𝑧 ∈ ℂ ↦ ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩)

Theoremrecli 10576 The real part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
𝐴 ∈ ℂ       (ℜ‘𝐴) ∈ ℝ

Theoremimcli 10577 The imaginary part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
𝐴 ∈ ℂ       (ℑ‘𝐴) ∈ ℝ

Theoremcjcli 10578 Closure law for complex conjugate. (Contributed by NM, 11-May-1999.)
𝐴 ∈ ℂ       (∗‘𝐴) ∈ ℂ

Theoremreplimi 10579 Construct a complex number from its real and imaginary parts. (Contributed by NM, 1-Oct-1999.)
𝐴 ∈ ℂ       𝐴 = ((ℜ‘𝐴) + (i · (ℑ‘𝐴)))

Theoremcjcji 10580 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 ∈ ℂ       (∗‘(∗‘𝐴)) = 𝐴

Theoremreim0bi 10581 A number is real iff its imaginary part is 0. (Contributed by NM, 29-May-1999.)
𝐴 ∈ ℂ       (𝐴 ∈ ℝ ↔ (ℑ‘𝐴) = 0)

Theoremrerebi 10582 A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 27-Oct-1999.)
𝐴 ∈ ℂ       (𝐴 ∈ ℝ ↔ (ℜ‘𝐴) = 𝐴)

Theoremcjrebi 10583 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 ∈ ℂ       (𝐴 ∈ ℝ ↔ (∗‘𝐴) = 𝐴)

Theoremrecji 10584 Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ ℂ       (ℜ‘(∗‘𝐴)) = (ℜ‘𝐴)

Theoremimcji 10585 Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ ℂ       (ℑ‘(∗‘𝐴)) = -(ℑ‘𝐴)

Theoremcjmulrcli 10586 A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.)
𝐴 ∈ ℂ       (𝐴 · (∗‘𝐴)) ∈ ℝ

Theoremcjmulvali 10587 A complex number times its conjugate. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ ℂ       (𝐴 · (∗‘𝐴)) = (((ℜ‘𝐴)↑2) + ((ℑ‘𝐴)↑2))

Theoremcjmulge0i 10588 A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.)
𝐴 ∈ ℂ       0 ≤ (𝐴 · (∗‘𝐴))

Theoremrenegi 10589 Real part of negative. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℂ       (ℜ‘-𝐴) = -(ℜ‘𝐴)

Theoremimnegi 10590 Imaginary part of negative. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℂ       (ℑ‘-𝐴) = -(ℑ‘𝐴)

Theoremcjnegi 10591 Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℂ       (∗‘-𝐴) = -(∗‘𝐴)

Theoremaddcji 10592 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 ∈ ℂ       (𝐴 + (∗‘𝐴)) = (2 · (ℜ‘𝐴))

𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (ℜ‘(𝐴 + 𝐵)) = ((ℜ‘𝐴) + (ℜ‘𝐵))

𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (ℑ‘(𝐴 + 𝐵)) = ((ℑ‘𝐴) + (ℑ‘𝐵))

Theoremremuli 10595 Real part of a product. (Contributed by NM, 28-Jul-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (ℜ‘(𝐴 · 𝐵)) = (((ℜ‘𝐴) · (ℜ‘𝐵)) − ((ℑ‘𝐴) · (ℑ‘𝐵)))

Theoremimmuli 10596 Imaginary part of a product. (Contributed by NM, 28-Jul-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (ℑ‘(𝐴 · 𝐵)) = (((ℜ‘𝐴) · (ℑ‘𝐵)) + ((ℑ‘𝐴) · (ℜ‘𝐵)))

Theoremcjaddi 10597 Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (∗‘(𝐴 + 𝐵)) = ((∗‘𝐴) + (∗‘𝐵))

Theoremcjmuli 10598 Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (∗‘(𝐴 · 𝐵)) = ((∗‘𝐴) · (∗‘𝐵))

Theoremipcni 10599 Standard inner product on complex numbers. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (ℜ‘(𝐴 · (∗‘𝐵))) = (((ℜ‘𝐴) · (ℜ‘𝐵)) + ((ℑ‘𝐴) · (ℑ‘𝐵)))

Theoremcjdivapi 10600 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 # 0 → (∗‘(𝐴 / 𝐵)) = ((∗‘𝐴) / (∗‘𝐵)))

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