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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | addcj 14501 | 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 14502 | Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∗‘(𝐴 − 𝐵)) = ((∗‘𝐴) − (∗‘𝐵))) | ||
Theorem | cjexp 14503 | Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (∗‘(𝐴↑𝑁)) = ((∗‘𝐴)↑𝑁)) | ||
Theorem | imval2 14504 | 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 14505 | The real part of zero. (Contributed by NM, 27-Jul-1999.) |
⊢ (ℜ‘0) = 0 | ||
Theorem | im0 14506 | The imaginary part of zero. (Contributed by NM, 27-Jul-1999.) |
⊢ (ℑ‘0) = 0 | ||
Theorem | re1 14507 | The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.) |
⊢ (ℜ‘1) = 1 | ||
Theorem | im1 14508 | The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.) |
⊢ (ℑ‘1) = 0 | ||
Theorem | rei 14509 | The real part of i. (Contributed by Scott Fenton, 9-Jun-2006.) |
⊢ (ℜ‘i) = 0 | ||
Theorem | imi 14510 | The imaginary part of i. (Contributed by Scott Fenton, 9-Jun-2006.) |
⊢ (ℑ‘i) = 1 | ||
Theorem | cj0 14511 | The conjugate of zero. (Contributed by NM, 27-Jul-1999.) |
⊢ (∗‘0) = 0 | ||
Theorem | cji 14512 | The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.) |
⊢ (∗‘i) = -i | ||
Theorem | cjreim 14513 | 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 14514 | 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 14515 | 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 14516 | A number is nonzero iff its complex conjugate is nonzero. (Contributed by NM, 29-Apr-2005.) |
⊢ (𝐴 ∈ ℂ → (𝐴 ≠ 0 ↔ (∗‘𝐴) ≠ 0)) | ||
Theorem | cjdiv 14517 | 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 14518* | The inverse to the canonical bijection from (ℝ × ℝ) to ℂ from cnref1o 12378. (Contributed by Mario Carneiro, 25-Aug-2014.) |
⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦))) ⇒ ⊢ ◡𝐹 = (𝑧 ∈ ℂ ↦ 〈(ℜ‘𝑧), (ℑ‘𝑧)〉) | ||
Theorem | sqeqd 14519 | 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 14520 | The real part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (ℜ‘𝐴) ∈ ℝ | ||
Theorem | imcli 14521 | The imaginary part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (ℑ‘𝐴) ∈ ℝ | ||
Theorem | cjcli 14522 | Closure law for complex conjugate. (Contributed by NM, 11-May-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (∗‘𝐴) ∈ ℂ | ||
Theorem | replimi 14523 | 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 14524 | 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 14525 | 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 14526 | 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 14527 | 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 14528 | Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (ℜ‘(∗‘𝐴)) = (ℜ‘𝐴) | ||
Theorem | imcji 14529 | Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (ℑ‘(∗‘𝐴)) = -(ℑ‘𝐴) | ||
Theorem | cjmulrcli 14530 | A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 · (∗‘𝐴)) ∈ ℝ | ||
Theorem | cjmulvali 14531 | A complex number times its conjugate. (Contributed by NM, 2-Oct-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 · (∗‘𝐴)) = (((ℜ‘𝐴)↑2) + ((ℑ‘𝐴)↑2)) | ||
Theorem | cjmulge0i 14532 | A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ 0 ≤ (𝐴 · (∗‘𝐴)) | ||
Theorem | renegi 14533 | Real part of negative. (Contributed by NM, 2-Aug-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (ℜ‘-𝐴) = -(ℜ‘𝐴) | ||
Theorem | imnegi 14534 | Imaginary part of negative. (Contributed by NM, 2-Aug-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (ℑ‘-𝐴) = -(ℑ‘𝐴) | ||
Theorem | cjnegi 14535 | Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (∗‘-𝐴) = -(∗‘𝐴) | ||
Theorem | addcji 14536 | 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 14537 | Real part distributes over addition. (Contributed by NM, 28-Jul-1999.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (ℜ‘(𝐴 + 𝐵)) = ((ℜ‘𝐴) + (ℜ‘𝐵)) | ||
Theorem | imaddi 14538 | Imaginary part distributes over addition. (Contributed by NM, 28-Jul-1999.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (ℑ‘(𝐴 + 𝐵)) = ((ℑ‘𝐴) + (ℑ‘𝐵)) | ||
Theorem | remuli 14539 | 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 14540 | 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 14541 | 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 14542 | 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 14543 | 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 14544 | 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 14545 | 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 14546 | 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 14547 | 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 14548 | 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 14549 | Closure law for complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (∗‘𝐴) ∈ ℂ) | ||
Theorem | replimd 14550 | 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 14551 | 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 14552 | 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 14553 | 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 14554 | 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 14555 | 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 14556 | 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 14557 | Real part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (ℜ‘(∗‘𝐴)) = (ℜ‘𝐴)) | ||
Theorem | imcjd 14558 | Imaginary part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (ℑ‘(∗‘𝐴)) = -(ℑ‘𝐴)) | ||
Theorem | cjmulrcld 14559 | A complex number times its conjugate is real. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 · (∗‘𝐴)) ∈ ℝ) | ||
Theorem | cjmulvald 14560 | 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 14561 | A complex number times its conjugate is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → 0 ≤ (𝐴 · (∗‘𝐴))) | ||
Theorem | renegd 14562 | Real part of negative. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (ℜ‘-𝐴) = -(ℜ‘𝐴)) | ||
Theorem | imnegd 14563 | Imaginary part of negative. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (ℑ‘-𝐴) = -(ℑ‘𝐴)) | ||
Theorem | cjnegd 14564 | Complex conjugate of negative. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (∗‘-𝐴) = -(∗‘𝐴)) | ||
Theorem | addcjd 14565 | 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 14566 | 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 14567 | 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 14568 | 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 14569 | 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 14570 | 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 14571 | 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 14572 | 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 14573 | 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 14574 | 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 14575 | 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 14576 | 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 14577 | 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 14578 | 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 14579 | 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 14580 | 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 14581 | 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 14582 | Real part of a division. Related to remul2 14483. (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 14583 | Imaginary part of a division. Related to remul2 14483. (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 14584 | 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 14585 | 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 14586 | Extend class notation to include square root of a complex number. |
class √ | ||
Syntax | cabs 14587 | Extend class notation to include a function for the absolute value (modulus) of a complex number. |
class abs | ||
Definition | df-sqrt 14588* |
Define a function whose value is the square root of a complex number.
For example, (√‘25) = 5 (ex-sqrt 28227).
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 28227. The square root symbol was introduced in 1525 by Christoff Rudolff. See sqrtcl 14715 for its closure, sqrtval 14590 for its value, sqrtth 14718 and sqsqrti 14729 for its relationship to squares, and sqrt11i 14738 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 14589 | Define the function for the absolute value (modulus) of a complex number. See abscli 14749 for its closure and absval 14591 or absval2i 14751 for its value. For example, (abs‘-2) = 2 (ex-abs 28228). (Contributed by NM, 27-Jul-1999.) |
⊢ abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥)))) | ||
Theorem | sqrtval 14590* | 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 14591 | 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 14592 | 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 14593 | 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 14594 | 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 14595 | Square root of zero. (Contributed by Mario Carneiro, 9-Jul-2013.) |
⊢ (√‘0) = 0 | ||
Theorem | sqrlem1 14596* | Lemma for 01sqrex 14603. (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 14597* | Lemma for 01sqrex 14603. (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 14598* | Lemma for 01sqrex 14603. (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 14599* | Lemma for 01sqrex 14603. (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 14600* | Lemma for 01sqrex 14603. (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, < ))) |
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