Theorem List for Intuitionistic Logic Explorer - 10801-10900 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | shftfval 10801* |
The value of the sequence shifter operation is a function on ℂ.
𝐴 is ordinarily an integer.
(Contributed by NM, 20-Jul-2005.)
(Revised by Mario Carneiro, 3-Nov-2013.)
|
⊢ 𝐹 ∈ V ⇒ ⊢ (𝐴 ∈ ℂ → (𝐹 shift 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)}) |
|
Theorem | shftdm 10802* |
Domain of a relation shifted by 𝐴. The set on the right is more
commonly notated as (dom 𝐹 + 𝐴) (meaning add 𝐴 to every
element of dom 𝐹). (Contributed by Mario Carneiro,
3-Nov-2013.)
|
⊢ 𝐹 ∈ V ⇒ ⊢ (𝐴 ∈ ℂ → dom (𝐹 shift 𝐴) = {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ dom 𝐹}) |
|
Theorem | shftfib 10803 |
Value of a fiber of the relation 𝐹. (Contributed by Mario
Carneiro, 4-Nov-2013.)
|
⊢ 𝐹 ∈ V ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴) “ {𝐵}) = (𝐹 “ {(𝐵 − 𝐴)})) |
|
Theorem | shftfn 10804* |
Functionality and domain of a sequence shifted by 𝐴. (Contributed
by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
|
⊢ 𝐹 ∈ V ⇒ ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ∈ ℂ) → (𝐹 shift 𝐴) Fn {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵}) |
|
Theorem | shftval 10805 |
Value of a sequence shifted by 𝐴. (Contributed by NM,
20-Jul-2005.) (Revised by Mario Carneiro, 4-Nov-2013.)
|
⊢ 𝐹 ∈ V ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴)‘𝐵) = (𝐹‘(𝐵 − 𝐴))) |
|
Theorem | shftval2 10806 |
Value of a sequence shifted by 𝐴 − 𝐵. (Contributed by NM,
20-Jul-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
|
⊢ 𝐹 ∈ V ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐹 shift (𝐴 − 𝐵))‘(𝐴 + 𝐶)) = (𝐹‘(𝐵 + 𝐶))) |
|
Theorem | shftval3 10807 |
Value of a sequence shifted by 𝐴 − 𝐵. (Contributed by NM,
20-Jul-2005.)
|
⊢ 𝐹 ∈ V ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift (𝐴 − 𝐵))‘𝐴) = (𝐹‘𝐵)) |
|
Theorem | shftval4 10808 |
Value of a sequence shifted by -𝐴. (Contributed by NM,
18-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
|
⊢ 𝐹 ∈ V ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift -𝐴)‘𝐵) = (𝐹‘(𝐴 + 𝐵))) |
|
Theorem | shftval5 10809 |
Value of a shifted sequence. (Contributed by NM, 19-Aug-2005.)
(Revised by Mario Carneiro, 5-Nov-2013.)
|
⊢ 𝐹 ∈ V ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴)‘(𝐵 + 𝐴)) = (𝐹‘𝐵)) |
|
Theorem | shftf 10810* |
Functionality of a shifted sequence. (Contributed by NM, 19-Aug-2005.)
(Revised by Mario Carneiro, 5-Nov-2013.)
|
⊢ 𝐹 ∈ V ⇒ ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐴 ∈ ℂ) → (𝐹 shift 𝐴):{𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵}⟶𝐶) |
|
Theorem | 2shfti 10811 |
Composite shift operations. (Contributed by NM, 19-Aug-2005.) (Revised
by Mario Carneiro, 5-Nov-2013.)
|
⊢ 𝐹 ∈ V ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴) shift 𝐵) = (𝐹 shift (𝐴 + 𝐵))) |
|
Theorem | shftidt2 10812 |
Identity law for the shift operation. (Contributed by Mario Carneiro,
5-Nov-2013.)
|
⊢ 𝐹 ∈ V ⇒ ⊢ (𝐹 shift 0) = (𝐹 ↾ ℂ) |
|
Theorem | shftidt 10813 |
Identity law for the shift operation. (Contributed by NM, 19-Aug-2005.)
(Revised by Mario Carneiro, 5-Nov-2013.)
|
⊢ 𝐹 ∈ V ⇒ ⊢ (𝐴 ∈ ℂ → ((𝐹 shift 0)‘𝐴) = (𝐹‘𝐴)) |
|
Theorem | shftcan1 10814 |
Cancellation law for the shift operation. (Contributed by NM,
4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
|
⊢ 𝐹 ∈ V ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐹 shift 𝐴) shift -𝐴)‘𝐵) = (𝐹‘𝐵)) |
|
Theorem | shftcan2 10815 |
Cancellation law for the shift operation. (Contributed by NM,
4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
|
⊢ 𝐹 ∈ V ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐹 shift -𝐴) shift 𝐴)‘𝐵) = (𝐹‘𝐵)) |
|
Theorem | shftvalg 10816 |
Value of a sequence shifted by 𝐴. (Contributed by Scott Fenton,
16-Dec-2017.)
|
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴)‘𝐵) = (𝐹‘(𝐵 − 𝐴))) |
|
Theorem | shftval4g 10817 |
Value of a sequence shifted by -𝐴. (Contributed by Jim Kingdon,
19-Aug-2021.)
|
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift -𝐴)‘𝐵) = (𝐹‘(𝐴 + 𝐵))) |
|
Theorem | seq3shft 10818* |
Shifting the index set of a sequence. (Contributed by NM, 17-Mar-2005.)
(Revised by Jim Kingdon, 17-Oct-2022.)
|
⊢ (𝜑 → 𝐹 ∈ 𝑉)
& ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(𝑀 − 𝑁))) → (𝐹‘𝑥) ∈ 𝑆)
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) ⇒ ⊢ (𝜑 → seq𝑀( + , (𝐹 shift 𝑁)) = (seq(𝑀 − 𝑁)( + , 𝐹) shift 𝑁)) |
|
4.7.2 Real and imaginary parts;
conjugate
|
|
Syntax | ccj 10819 |
Extend class notation to include complex conjugate function.
|
class ∗ |
|
Syntax | cre 10820 |
Extend class notation to include real part of a complex number.
|
class ℜ |
|
Syntax | cim 10821 |
Extend class notation to include imaginary part of a complex number.
|
class ℑ |
|
Definition | df-cj 10822* |
Define the complex conjugate function. See cjcli 10893 for its closure and
cjval 10825 for its value. (Contributed by NM,
9-May-1999.) (Revised by
Mario Carneiro, 6-Nov-2013.)
|
⊢ ∗ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑥 + 𝑦) ∈ ℝ ∧ (i · (𝑥 − 𝑦)) ∈ ℝ))) |
|
Definition | df-re 10823 |
Define a function whose value is the real part of a complex number. See
reval 10829 for its value, recli 10891 for its closure, and replim 10839 for its use
in decomposing a complex number. (Contributed by NM, 9-May-1999.)
|
⊢ ℜ = (𝑥 ∈ ℂ ↦ ((𝑥 + (∗‘𝑥)) / 2)) |
|
Definition | df-im 10824 |
Define a function whose value is the imaginary part of a complex number.
See imval 10830 for its value, imcli 10892 for its closure, and replim 10839 for its
use in decomposing a complex number. (Contributed by NM,
9-May-1999.)
|
⊢ ℑ = (𝑥 ∈ ℂ ↦ (ℜ‘(𝑥 / i))) |
|
Theorem | cjval 10825* |
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
· (𝐴 − 𝑥)) ∈
ℝ))) |
|
Theorem | cjth 10826 |
The defining property of the complex conjugate. (Contributed by Mario
Carneiro, 6-Nov-2013.)
|
⊢ (𝐴 ∈ ℂ → ((𝐴 + (∗‘𝐴)) ∈ ℝ ∧ (i · (𝐴 − (∗‘𝐴))) ∈
ℝ)) |
|
Theorem | cjf 10827 |
Domain and codomain of the conjugate function. (Contributed by Mario
Carneiro, 6-Nov-2013.)
|
⊢
∗:ℂ⟶ℂ |
|
Theorem | cjcl 10828 |
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) |
|
Theorem | reval 10829 |
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)) |
|
Theorem | imval 10830 |
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))) |
|
Theorem | imre 10831 |
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))) |
|
Theorem | reim 10832 |
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))) |
|
Theorem | recl 10833 |
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) |
|
Theorem | imcl 10834 |
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) |
|
Theorem | ref 10835 |
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 |
|
Theorem | imf 10836 |
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 |
|
Theorem | crre 10837 |
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) |
|
Theorem | crim 10838 |
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) |
|
Theorem | replim 10839 |
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)))) |
|
Theorem | remim 10840 |
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)))) |
|
Theorem | reim0 10841 |
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) |
|
Theorem | reim0b 10842 |
A number is real iff its imaginary part is 0. (Contributed by NM,
26-Sep-2005.)
|
⊢ (𝐴 ∈ ℂ → (𝐴 ∈ ℝ ↔ (ℑ‘𝐴) = 0)) |
|
Theorem | rereb 10843 |
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)) |
|
Theorem | mulreap 10844 |
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)) |
|
Theorem | rere 10845 |
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) |
|
Theorem | cjreb 10846 |
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)) |
|
Theorem | recj 10847 |
Real part of a complex conjugate. (Contributed by Mario Carneiro,
14-Jul-2014.)
|
⊢ (𝐴 ∈ ℂ →
(ℜ‘(∗‘𝐴)) = (ℜ‘𝐴)) |
|
Theorem | reneg 10848 |
Real part of negative. (Contributed by NM, 17-Mar-2005.) (Revised by
Mario Carneiro, 14-Jul-2014.)
|
⊢ (𝐴 ∈ ℂ → (ℜ‘-𝐴) = -(ℜ‘𝐴)) |
|
Theorem | readd 10849 |
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))) |
|
Theorem | resub 10850 |
Real part distributes over subtraction. (Contributed by NM,
17-Mar-2005.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) →
(ℜ‘(𝐴 −
𝐵)) = ((ℜ‘𝐴) − (ℜ‘𝐵))) |
|
Theorem | remullem 10851 |
Lemma for remul 10852, immul 10859, and cjmul 10865. (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)))) |
|
Theorem | remul 10852 |
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)))) |
|
Theorem | remul2 10853 |
Real part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) →
(ℜ‘(𝐴 ·
𝐵)) = (𝐴 · (ℜ‘𝐵))) |
|
Theorem | redivap 10854 |
Real part of a division. Related to remul2 10853. (Contributed by Jim
Kingdon, 14-Jun-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 # 0) → (ℜ‘(𝐴 / 𝐵)) = ((ℜ‘𝐴) / 𝐵)) |
|
Theorem | imcj 10855 |
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)) |
|
Theorem | imneg 10856 |
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)) |
|
Theorem | imadd 10857 |
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))) |
|
Theorem | imsub 10858 |
Imaginary part distributes over subtraction. (Contributed by NM,
18-Mar-2005.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) →
(ℑ‘(𝐴 −
𝐵)) =
((ℑ‘𝐴) −
(ℑ‘𝐵))) |
|
Theorem | immul 10859 |
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)))) |
|
Theorem | immul2 10860 |
Imaginary part of a product. (Contributed by Mario Carneiro,
2-Aug-2014.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) →
(ℑ‘(𝐴 ·
𝐵)) = (𝐴 · (ℑ‘𝐵))) |
|
Theorem | imdivap 10861 |
Imaginary part of a division. Related to immul2 10860. (Contributed by Jim
Kingdon, 14-Jun-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 # 0) → (ℑ‘(𝐴 / 𝐵)) = ((ℑ‘𝐴) / 𝐵)) |
|
Theorem | cjre 10862 |
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) |
|
Theorem | cjcj 10863 |
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) |
|
Theorem | cjadd 10864 |
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))) |
|
Theorem | cjmul 10865 |
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))) |
|
Theorem | ipcnval 10866 |
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)))) |
|
Theorem | cjmulrcl 10867 |
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) |
|
Theorem | cjmulval 10868 |
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))) |
|
Theorem | cjmulge0 10869 |
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))) |
|
Theorem | cjneg 10870 |
Complex conjugate of negative. (Contributed by NM, 27-Feb-2005.)
(Revised by Mario Carneiro, 14-Jul-2014.)
|
⊢ (𝐴 ∈ ℂ →
(∗‘-𝐴) =
-(∗‘𝐴)) |
|
Theorem | addcj 10871 |
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))) |
|
Theorem | cjsub 10872 |
Complex conjugate distributes over subtraction. (Contributed by NM,
28-Apr-2005.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) →
(∗‘(𝐴 −
𝐵)) =
((∗‘𝐴)
− (∗‘𝐵))) |
|
Theorem | cjexp 10873 |
Complex conjugate of positive integer exponentiation. (Contributed by
NM, 7-Jun-2006.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) →
(∗‘(𝐴↑𝑁)) = ((∗‘𝐴)↑𝑁)) |
|
Theorem | imval2 10874 |
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))) |
|
Theorem | re0 10875 |
The real part of zero. (Contributed by NM, 27-Jul-1999.)
|
⊢ (ℜ‘0) = 0 |
|
Theorem | im0 10876 |
The imaginary part of zero. (Contributed by NM, 27-Jul-1999.)
|
⊢ (ℑ‘0) = 0 |
|
Theorem | re1 10877 |
The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
|
⊢ (ℜ‘1) = 1 |
|
Theorem | im1 10878 |
The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
|
⊢ (ℑ‘1) = 0 |
|
Theorem | rei 10879 |
The real part of i. (Contributed by Scott Fenton,
9-Jun-2006.)
|
⊢ (ℜ‘i) = 0 |
|
Theorem | imi 10880 |
The imaginary part of i. (Contributed by Scott Fenton,
9-Jun-2006.)
|
⊢ (ℑ‘i) = 1 |
|
Theorem | cj0 10881 |
The conjugate of zero. (Contributed by NM, 27-Jul-1999.)
|
⊢ (∗‘0) = 0 |
|
Theorem | cji 10882 |
The complex conjugate of the imaginary unit. (Contributed by NM,
26-Mar-2005.)
|
⊢ (∗‘i) = -i |
|
Theorem | cjreim 10883 |
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))) |
|
Theorem | cjreim2 10884 |
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))) |
|
Theorem | cj11 10885 |
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)) |
|
Theorem | cjap 10886 |
Complex conjugate and apartness. (Contributed by Jim Kingdon,
14-Jun-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) →
((∗‘𝐴) #
(∗‘𝐵) ↔
𝐴 # 𝐵)) |
|
Theorem | cjap0 10887 |
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)) |
|
Theorem | cjne0 10888 |
A number is nonzero iff its complex conjugate is nonzero. Also see
cjap0 10887 which is similar but for apartness.
(Contributed by NM,
29-Apr-2005.)
|
⊢ (𝐴 ∈ ℂ → (𝐴 ≠ 0 ↔ (∗‘𝐴) ≠ 0)) |
|
Theorem | cjdivap 10889 |
Complex conjugate distributes over division. (Contributed by Jim Kingdon,
14-Jun-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (∗‘(𝐴 / 𝐵)) = ((∗‘𝐴) / (∗‘𝐵))) |
|
Theorem | cnrecnv 10890* |
The inverse to the canonical bijection from (ℝ ×
ℝ) to ℂ
from cnref1o 9626. (Contributed by Mario Carneiro,
25-Aug-2014.)
|
⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦))) ⇒ ⊢ ◡𝐹 = (𝑧 ∈ ℂ ↦
〈(ℜ‘𝑧),
(ℑ‘𝑧)〉) |
|
Theorem | recli 10891 |
The real part of a complex number is real (closure law). (Contributed
by NM, 11-May-1999.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ (ℜ‘𝐴) ∈ ℝ |
|
Theorem | imcli 10892 |
The imaginary part of a complex number is real (closure law).
(Contributed by NM, 11-May-1999.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ (ℑ‘𝐴) ∈
ℝ |
|
Theorem | cjcli 10893 |
Closure law for complex conjugate. (Contributed by NM, 11-May-1999.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ (∗‘𝐴) ∈
ℂ |
|
Theorem | replimi 10894 |
Construct a complex number from its real and imaginary parts.
(Contributed by NM, 1-Oct-1999.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ 𝐴 = ((ℜ‘𝐴) + (i · (ℑ‘𝐴))) |
|
Theorem | cjcji 10895 |
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 |
|
Theorem | reim0bi 10896 |
A number is real iff its imaginary part is 0. (Contributed by NM,
29-May-1999.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ (𝐴 ∈ ℝ ↔ (ℑ‘𝐴) = 0) |
|
Theorem | rerebi 10897 |
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) |
|
Theorem | cjrebi 10898 |
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) |
|
Theorem | recji 10899 |
Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢
(ℜ‘(∗‘𝐴)) = (ℜ‘𝐴) |
|
Theorem | imcji 10900 |
Imaginary part of a complex conjugate. (Contributed by NM,
2-Oct-1999.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢
(ℑ‘(∗‘𝐴)) = -(ℑ‘𝐴) |