![]() |
Metamath
Proof Explorer Theorem List (p. 146 of 454) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | ![]() (1-28701) |
![]() (28702-30224) |
![]() (30225-45333) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cjexp 14501 | Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (∗‘(𝐴↑𝑁)) = ((∗‘𝐴)↑𝑁)) | ||
Theorem | imval2 14502 | 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 14503 | The real part of zero. (Contributed by NM, 27-Jul-1999.) |
⊢ (ℜ‘0) = 0 | ||
Theorem | im0 14504 | The imaginary part of zero. (Contributed by NM, 27-Jul-1999.) |
⊢ (ℑ‘0) = 0 | ||
Theorem | re1 14505 | The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.) |
⊢ (ℜ‘1) = 1 | ||
Theorem | im1 14506 | The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.) |
⊢ (ℑ‘1) = 0 | ||
Theorem | rei 14507 | The real part of i. (Contributed by Scott Fenton, 9-Jun-2006.) |
⊢ (ℜ‘i) = 0 | ||
Theorem | imi 14508 | The imaginary part of i. (Contributed by Scott Fenton, 9-Jun-2006.) |
⊢ (ℑ‘i) = 1 | ||
Theorem | cj0 14509 | The conjugate of zero. (Contributed by NM, 27-Jul-1999.) |
⊢ (∗‘0) = 0 | ||
Theorem | cji 14510 | The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.) |
⊢ (∗‘i) = -i | ||
Theorem | cjreim 14511 | 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 14512 | 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 14513 | 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 | cjne0 14514 | A number is nonzero iff its complex conjugate is nonzero. (Contributed by NM, 29-Apr-2005.) |
⊢ (𝐴 ∈ ℂ → (𝐴 ≠ 0 ↔ (∗‘𝐴) ≠ 0)) | ||
Theorem | cjdiv 14515 | Complex conjugate distributes over division. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (∗‘(𝐴 / 𝐵)) = ((∗‘𝐴) / (∗‘𝐵))) | ||
Theorem | cnrecnv 14516* | The inverse to the canonical bijection from (ℝ × ℝ) to ℂ from cnref1o 12372. (Contributed by Mario Carneiro, 25-Aug-2014.) |
⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦))) ⇒ ⊢ ◡𝐹 = (𝑧 ∈ ℂ ↦ 〈(ℜ‘𝑧), (ℑ‘𝑧)〉) | ||
Theorem | sqeqd 14517 | A deduction for showing two numbers whose squares are equal are themselves equal. (Contributed by Mario Carneiro, 3-Apr-2015.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → (𝐴↑2) = (𝐵↑2)) & ⊢ (𝜑 → 0 ≤ (ℜ‘𝐴)) & ⊢ (𝜑 → 0 ≤ (ℜ‘𝐵)) & ⊢ ((𝜑 ∧ (ℜ‘𝐴) = 0 ∧ (ℜ‘𝐵) = 0) → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | recli 14518 | The real part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (ℜ‘𝐴) ∈ ℝ | ||
Theorem | imcli 14519 | The imaginary part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (ℑ‘𝐴) ∈ ℝ | ||
Theorem | cjcli 14520 | Closure law for complex conjugate. (Contributed by NM, 11-May-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (∗‘𝐴) ∈ ℂ | ||
Theorem | replimi 14521 | 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 14522 | 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 14523 | 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 14524 | 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 14525 | 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 14526 | Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (ℜ‘(∗‘𝐴)) = (ℜ‘𝐴) | ||
Theorem | imcji 14527 | Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (ℑ‘(∗‘𝐴)) = -(ℑ‘𝐴) | ||
Theorem | cjmulrcli 14528 | A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 · (∗‘𝐴)) ∈ ℝ | ||
Theorem | cjmulvali 14529 | A complex number times its conjugate. (Contributed by NM, 2-Oct-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 · (∗‘𝐴)) = (((ℜ‘𝐴)↑2) + ((ℑ‘𝐴)↑2)) | ||
Theorem | cjmulge0i 14530 | A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ 0 ≤ (𝐴 · (∗‘𝐴)) | ||
Theorem | renegi 14531 | Real part of negative. (Contributed by NM, 2-Aug-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (ℜ‘-𝐴) = -(ℜ‘𝐴) | ||
Theorem | imnegi 14532 | Imaginary part of negative. (Contributed by NM, 2-Aug-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (ℑ‘-𝐴) = -(ℑ‘𝐴) | ||
Theorem | cjnegi 14533 | Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (∗‘-𝐴) = -(∗‘𝐴) | ||
Theorem | addcji 14534 | 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)) | ||
Theorem | readdi 14535 | Real part distributes over addition. (Contributed by NM, 28-Jul-1999.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (ℜ‘(𝐴 + 𝐵)) = ((ℜ‘𝐴) + (ℜ‘𝐵)) | ||
Theorem | imaddi 14536 | Imaginary part distributes over addition. (Contributed by NM, 28-Jul-1999.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (ℑ‘(𝐴 + 𝐵)) = ((ℑ‘𝐴) + (ℑ‘𝐵)) | ||
Theorem | remuli 14537 | 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))) | ||
Theorem | immuli 14538 | 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))) | ||
Theorem | cjaddi 14539 | 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)) | ||
Theorem | cjmuli 14540 | 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)) | ||
Theorem | ipcni 14541 | 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))) | ||
Theorem | cjdivi 14542 | Complex conjugate distributes over division. (Contributed by NM, 29-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐵 ≠ 0 → (∗‘(𝐴 / 𝐵)) = ((∗‘𝐴) / (∗‘𝐵))) | ||
Theorem | crrei 14543 | The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (ℜ‘(𝐴 + (i · 𝐵))) = 𝐴 | ||
Theorem | crimi 14544 | The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (ℑ‘(𝐴 + (i · 𝐵))) = 𝐵 | ||
Theorem | recld 14545 | The real part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (ℜ‘𝐴) ∈ ℝ) | ||
Theorem | imcld 14546 | The imaginary part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (ℑ‘𝐴) ∈ ℝ) | ||
Theorem | cjcld 14547 | Closure law for complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (∗‘𝐴) ∈ ℂ) | ||
Theorem | replimd 14548 | Construct a complex number from its real and imaginary parts. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → 𝐴 = ((ℜ‘𝐴) + (i · (ℑ‘𝐴)))) | ||
Theorem | remimd 14549 | 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.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (∗‘𝐴) = ((ℜ‘𝐴) − (i · (ℑ‘𝐴)))) | ||
Theorem | cjcjd 14550 | 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.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (∗‘(∗‘𝐴)) = 𝐴) | ||
Theorem | reim0bd 14551 | A number is real iff its imaginary part is 0. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → (ℑ‘𝐴) = 0) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ) | ||
Theorem | rerebd 14552 | A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → (ℜ‘𝐴) = 𝐴) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ) | ||
Theorem | cjrebd 14553 | 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.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → (∗‘𝐴) = 𝐴) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ) | ||
Theorem | cjne0d 14554 | A number is nonzero iff its complex conjugate is nonzero. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) ⇒ ⊢ (𝜑 → (∗‘𝐴) ≠ 0) | ||
Theorem | recjd 14555 | Real part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (ℜ‘(∗‘𝐴)) = (ℜ‘𝐴)) | ||
Theorem | imcjd 14556 | Imaginary part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (ℑ‘(∗‘𝐴)) = -(ℑ‘𝐴)) | ||
Theorem | cjmulrcld 14557 | A complex number times its conjugate is real. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 · (∗‘𝐴)) ∈ ℝ) | ||
Theorem | cjmulvald 14558 | A complex number times its conjugate. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 · (∗‘𝐴)) = (((ℜ‘𝐴)↑2) + ((ℑ‘𝐴)↑2))) | ||
Theorem | cjmulge0d 14559 | A complex number times its conjugate is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → 0 ≤ (𝐴 · (∗‘𝐴))) | ||
Theorem | renegd 14560 | Real part of negative. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (ℜ‘-𝐴) = -(ℜ‘𝐴)) | ||
Theorem | imnegd 14561 | Imaginary part of negative. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (ℑ‘-𝐴) = -(ℑ‘𝐴)) | ||
Theorem | cjnegd 14562 | Complex conjugate of negative. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (∗‘-𝐴) = -(∗‘𝐴)) | ||
Theorem | addcjd 14563 | 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.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 + (∗‘𝐴)) = (2 · (ℜ‘𝐴))) | ||
Theorem | cjexpd 14564 | Complex conjugate of positive integer exponentiation. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (∗‘(𝐴↑𝑁)) = ((∗‘𝐴)↑𝑁)) | ||
Theorem | readdd 14565 | Real part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (ℜ‘(𝐴 + 𝐵)) = ((ℜ‘𝐴) + (ℜ‘𝐵))) | ||
Theorem | imaddd 14566 | Imaginary part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (ℑ‘(𝐴 + 𝐵)) = ((ℑ‘𝐴) + (ℑ‘𝐵))) | ||
Theorem | resubd 14567 | Real part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (ℜ‘(𝐴 − 𝐵)) = ((ℜ‘𝐴) − (ℜ‘𝐵))) | ||
Theorem | imsubd 14568 | Imaginary part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (ℑ‘(𝐴 − 𝐵)) = ((ℑ‘𝐴) − (ℑ‘𝐵))) | ||
Theorem | remuld 14569 | Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (ℜ‘(𝐴 · 𝐵)) = (((ℜ‘𝐴) · (ℜ‘𝐵)) − ((ℑ‘𝐴) · (ℑ‘𝐵)))) | ||
Theorem | immuld 14570 | Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (ℑ‘(𝐴 · 𝐵)) = (((ℜ‘𝐴) · (ℑ‘𝐵)) + ((ℑ‘𝐴) · (ℜ‘𝐵)))) | ||
Theorem | cjaddd 14571 | Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (∗‘(𝐴 + 𝐵)) = ((∗‘𝐴) + (∗‘𝐵))) | ||
Theorem | cjmuld 14572 | Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (∗‘(𝐴 · 𝐵)) = ((∗‘𝐴) · (∗‘𝐵))) | ||
Theorem | ipcnd 14573 | Standard inner product on complex numbers. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (ℜ‘(𝐴 · (∗‘𝐵))) = (((ℜ‘𝐴) · (ℜ‘𝐵)) + ((ℑ‘𝐴) · (ℑ‘𝐵)))) | ||
Theorem | cjdivd 14574 | Complex conjugate distributes over division. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ≠ 0) ⇒ ⊢ (𝜑 → (∗‘(𝐴 / 𝐵)) = ((∗‘𝐴) / (∗‘𝐵))) | ||
Theorem | rered 14575 | 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.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (ℜ‘𝐴) = 𝐴) | ||
Theorem | reim0d 14576 | The imaginary part of a real number is 0. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (ℑ‘𝐴) = 0) | ||
Theorem | cjred 14577 | A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (∗‘𝐴) = 𝐴) | ||
Theorem | remul2d 14578 | Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (ℜ‘(𝐴 · 𝐵)) = (𝐴 · (ℜ‘𝐵))) | ||
Theorem | immul2d 14579 | Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (ℑ‘(𝐴 · 𝐵)) = (𝐴 · (ℑ‘𝐵))) | ||
Theorem | redivd 14580 | Real part of a division. Related to remul2 14481. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) ⇒ ⊢ (𝜑 → (ℜ‘(𝐵 / 𝐴)) = ((ℜ‘𝐵) / 𝐴)) | ||
Theorem | imdivd 14581 | Imaginary part of a division. Related to remul2 14481. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) ⇒ ⊢ (𝜑 → (ℑ‘(𝐵 / 𝐴)) = ((ℑ‘𝐵) / 𝐴)) | ||
Theorem | crred 14582 | The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (ℜ‘(𝐴 + (i · 𝐵))) = 𝐴) | ||
Theorem | crimd 14583 | The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (ℑ‘(𝐴 + (i · 𝐵))) = 𝐵) | ||
Syntax | csqrt 14584 | Extend class notation to include square root of a complex number. |
class √ | ||
Syntax | cabs 14585 | Extend class notation to include a function for the absolute value (modulus) of a complex number. |
class abs | ||
Definition | df-sqrt 14586* |
Define a function whose value is the square root of a complex number.
For example, (√‘25) = 5 (ex-sqrt 28239).
Since (𝑦↑2) = 𝑥 iff (-𝑦↑2) = 𝑥, we ensure uniqueness by restricting the range to numbers with positive real part, or numbers with 0 real part and nonnegative imaginary part. A description can be found under "Principal square root of a complex number" at http://en.wikipedia.org/wiki/Square_root 28239. The square root symbol was introduced in 1525 by Christoff Rudolff. See sqrtcl 14713 for its closure, sqrtval 14588 for its value, sqrtth 14716 and sqsqrti 14727 for its relationship to squares, and sqrt11i 14736 for uniqueness. (Contributed by NM, 27-Jul-1999.) (Revised by Mario Carneiro, 8-Jul-2013.) |
⊢ √ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+))) | ||
Definition | df-abs 14587 | Define the function for the absolute value (modulus) of a complex number. See abscli 14747 for its closure and absval 14589 or absval2i 14749 for its value. For example, (abs‘-2) = 2 (ex-abs 28240). (Contributed by NM, 27-Jul-1999.) |
⊢ abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥)))) | ||
Theorem | sqrtval 14588* | Value of square root function. (Contributed by Mario Carneiro, 8-Jul-2013.) |
⊢ (𝐴 ∈ ℂ → (√‘𝐴) = (℩𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+))) | ||
Theorem | absval 14589 | The absolute value (modulus) of a complex number. Proposition 10-3.7(a) of [Gleason] p. 133. (Contributed by NM, 27-Jul-1999.) (Revised by Mario Carneiro, 7-Nov-2013.) |
⊢ (𝐴 ∈ ℂ → (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴)))) | ||
Theorem | rennim 14590 | A real number does not lie on the negative imaginary axis. (Contributed by Mario Carneiro, 8-Jul-2013.) |
⊢ (𝐴 ∈ ℝ → (i · 𝐴) ∉ ℝ+) | ||
Theorem | cnpart 14591 | The specification of restriction to the right half-plane partitions the complex plane without 0 into two disjoint pieces, which are related by a reflection about the origin (under the map 𝑥 ↦ -𝑥). (Contributed by Mario Carneiro, 8-Jul-2013.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((0 ≤ (ℜ‘𝐴) ∧ (i · 𝐴) ∉ ℝ+) ↔ ¬ (0 ≤ (ℜ‘-𝐴) ∧ (i · -𝐴) ∉ ℝ+))) | ||
Theorem | sqr0lem 14592 | Square root of zero. (Contributed by Mario Carneiro, 9-Jul-2013.) |
⊢ ((𝐴 ∈ ℂ ∧ ((𝐴↑2) = 0 ∧ 0 ≤ (ℜ‘𝐴) ∧ (i · 𝐴) ∉ ℝ+)) ↔ 𝐴 = 0) | ||
Theorem | sqrt0 14593 | Square root of zero. (Contributed by Mario Carneiro, 9-Jul-2013.) |
⊢ (√‘0) = 0 | ||
Theorem | sqrlem1 14594* | Lemma for 01sqrex 14601. (Contributed by Mario Carneiro, 10-Jul-2013.) |
⊢ 𝑆 = {𝑥 ∈ ℝ+ ∣ (𝑥↑2) ≤ 𝐴} & ⊢ 𝐵 = sup(𝑆, ℝ, < ) ⇒ ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → ∀𝑦 ∈ 𝑆 𝑦 ≤ 1) | ||
Theorem | sqrlem2 14595* | Lemma for 01sqrex 14601. (Contributed by Mario Carneiro, 10-Jul-2013.) |
⊢ 𝑆 = {𝑥 ∈ ℝ+ ∣ (𝑥↑2) ≤ 𝐴} & ⊢ 𝐵 = sup(𝑆, ℝ, < ) ⇒ ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → 𝐴 ∈ 𝑆) | ||
Theorem | sqrlem3 14596* | Lemma for 01sqrex 14601. (Contributed by Mario Carneiro, 10-Jul-2013.) |
⊢ 𝑆 = {𝑥 ∈ ℝ+ ∣ (𝑥↑2) ≤ 𝐴} & ⊢ 𝐵 = sup(𝑆, ℝ, < ) ⇒ ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → (𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧)) | ||
Theorem | sqrlem4 14597* | Lemma for 01sqrex 14601. (Contributed by Mario Carneiro, 10-Jul-2013.) |
⊢ 𝑆 = {𝑥 ∈ ℝ+ ∣ (𝑥↑2) ≤ 𝐴} & ⊢ 𝐵 = sup(𝑆, ℝ, < ) ⇒ ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → (𝐵 ∈ ℝ+ ∧ 𝐵 ≤ 1)) | ||
Theorem | sqrlem5 14598* | Lemma for 01sqrex 14601. (Contributed by Mario Carneiro, 10-Jul-2013.) |
⊢ 𝑆 = {𝑥 ∈ ℝ+ ∣ (𝑥↑2) ≤ 𝐴} & ⊢ 𝐵 = sup(𝑆, ℝ, < ) & ⊢ 𝑇 = {𝑦 ∣ ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑆 𝑦 = (𝑎 · 𝑏)} ⇒ ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → ((𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑣 ∈ ℝ ∀𝑢 ∈ 𝑇 𝑢 ≤ 𝑣) ∧ (𝐵↑2) = sup(𝑇, ℝ, < ))) | ||
Theorem | sqrlem6 14599* | Lemma for 01sqrex 14601. (Contributed by Mario Carneiro, 10-Jul-2013.) |
⊢ 𝑆 = {𝑥 ∈ ℝ+ ∣ (𝑥↑2) ≤ 𝐴} & ⊢ 𝐵 = sup(𝑆, ℝ, < ) & ⊢ 𝑇 = {𝑦 ∣ ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑆 𝑦 = (𝑎 · 𝑏)} ⇒ ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → (𝐵↑2) ≤ 𝐴) | ||
Theorem | sqrlem7 14600* | Lemma for 01sqrex 14601. (Contributed by Mario Carneiro, 10-Jul-2013.) (Proof shortened by AV, 9-Jul-2022.) |
⊢ 𝑆 = {𝑥 ∈ ℝ+ ∣ (𝑥↑2) ≤ 𝐴} & ⊢ 𝐵 = sup(𝑆, ℝ, < ) & ⊢ 𝑇 = {𝑦 ∣ ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑆 𝑦 = (𝑎 · 𝑏)} ⇒ ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → (𝐵↑2) = 𝐴) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |