P. 152 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 15101-15200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cjsub 15101	Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∗‘(𝐴 − 𝐵)) = ((∗‘𝐴) − (∗‘𝐵)))
Theorem	cjexp 15102	Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (∗‘(𝐴↑𝑁)) = ((∗‘𝐴)↑𝑁))
Theorem	imval2 15103	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 15104	The real part of zero. (Contributed by NM, 27-Jul-1999.)
⊢ (ℜ‘0) = 0
Theorem	im0 15105	The imaginary part of zero. (Contributed by NM, 27-Jul-1999.)
⊢ (ℑ‘0) = 0
Theorem	re1 15106	The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
⊢ (ℜ‘1) = 1
Theorem	im1 15107	The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
⊢ (ℑ‘1) = 0
Theorem	rei 15108	The real part of i. (Contributed by Scott Fenton, 9-Jun-2006.)
⊢ (ℜ‘i) = 0
Theorem	imi 15109	The imaginary part of i. (Contributed by Scott Fenton, 9-Jun-2006.)
⊢ (ℑ‘i) = 1
Theorem	cj0 15110	The conjugate of zero. (Contributed by NM, 27-Jul-1999.)
⊢ (∗‘0) = 0
Theorem	cji 15111	The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.)
⊢ (∗‘i) = -i
Theorem	cjreim 15112	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 15113	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 15114	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 15115	A number is nonzero iff its complex conjugate is nonzero. (Contributed by NM, 29-Apr-2005.)
⊢ (𝐴 ∈ ℂ → (𝐴 ≠ 0 ↔ (∗‘𝐴) ≠ 0))
Theorem	cjdiv 15116	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 15117*	The inverse to the canonical bijection from (ℝ × ℝ) to ℂ from cnref1o 12974. (Contributed by Mario Carneiro, 25-Aug-2014.)
⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))    ⇒   ⊢ ◡𝐹 = (𝑧 ∈ ℂ ↦ ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩)
Theorem	sqeqd 15118	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 15119	The real part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (ℜ‘𝐴) ∈ ℝ
Theorem	imcli 15120	The imaginary part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (ℑ‘𝐴) ∈ ℝ
Theorem	cjcli 15121	Closure law for complex conjugate. (Contributed by NM, 11-May-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (∗‘𝐴) ∈ ℂ
Theorem	replimi 15122	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 15123	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 15124	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 15125	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 15126	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 15127	Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (ℜ‘(∗‘𝐴)) = (ℜ‘𝐴)
Theorem	imcji 15128	Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (ℑ‘(∗‘𝐴)) = -(ℑ‘𝐴)
Theorem	cjmulrcli 15129	A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 · (∗‘𝐴)) ∈ ℝ
Theorem	cjmulvali 15130	A complex number times its conjugate. (Contributed by NM, 2-Oct-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 · (∗‘𝐴)) = (((ℜ‘𝐴)↑2) + ((ℑ‘𝐴)↑2))
Theorem	cjmulge0i 15131	A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ 0 ≤ (𝐴 · (∗‘𝐴))
Theorem	renegi 15132	Real part of negative. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (ℜ‘-𝐴) = -(ℜ‘𝐴)
Theorem	imnegi 15133	Imaginary part of negative. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (ℑ‘-𝐴) = -(ℑ‘𝐴)
Theorem	cjnegi 15134	Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (∗‘-𝐴) = -(∗‘𝐴)
Theorem	addcji 15135	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 15136	Real part distributes over addition. (Contributed by NM, 28-Jul-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (ℜ‘(𝐴 + 𝐵)) = ((ℜ‘𝐴) + (ℜ‘𝐵))
Theorem	imaddi 15137	Imaginary part distributes over addition. (Contributed by NM, 28-Jul-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (ℑ‘(𝐴 + 𝐵)) = ((ℑ‘𝐴) + (ℑ‘𝐵))
Theorem	remuli 15138	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 15139	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 15140	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 15141	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 15142	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 15143	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 15144	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 15145	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 15146	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 15147	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 15148	Closure law for complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (∗‘𝐴) ∈ ℂ)
Theorem	replimd 15149	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 15150	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 15151	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 15152	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 15153	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 15154	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 15155	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 15156	Real part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (ℜ‘(∗‘𝐴)) = (ℜ‘𝐴))
Theorem	imcjd 15157	Imaginary part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (ℑ‘(∗‘𝐴)) = -(ℑ‘𝐴))
Theorem	cjmulrcld 15158	A complex number times its conjugate is real. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 · (∗‘𝐴)) ∈ ℝ)
Theorem	cjmulvald 15159	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 15160	A complex number times its conjugate is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → 0 ≤ (𝐴 · (∗‘𝐴)))
Theorem	renegd 15161	Real part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (ℜ‘-𝐴) = -(ℜ‘𝐴))
Theorem	imnegd 15162	Imaginary part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (ℑ‘-𝐴) = -(ℑ‘𝐴))
Theorem	cjnegd 15163	Complex conjugate of negative. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (∗‘-𝐴) = -(∗‘𝐴))
Theorem	addcjd 15164	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 15165	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 15166	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 15167	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 15168	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 15169	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 15170	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 15171	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 15172	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 15173	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 15174	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 15175	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 15176	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 15177	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 15178	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 15179	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 15180	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 15181	Real part of a division. Related to remul2 15082. (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 15182	Imaginary part of a division. Related to remul2 15082. (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 15183	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 15184	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)
5.9.4  Square root; absolute value
Syntax	csqrt 15185	Extend class notation to include square root of a complex number.
class √
Syntax	cabs 15186	Extend class notation to include a function for the absolute value (modulus) of a complex number.
class abs
Definition	df-sqrt 15187*	Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 29971). 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 29971. The square root symbol was introduced in 1525 by Christoff Rudolff. See sqrtcl 15313 for its closure, sqrtval 15189 for its value, sqrtth 15316 and sqsqrti 15327 for its relationship to squares, and sqrt11i 15336 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 15188	Define the function for the absolute value (modulus) of a complex number. See abscli 15347 for its closure and absval 15190 or absval2i 15349 for its value. For example, (abs‘-2) = 2 (ex-abs 29972). (Contributed by NM, 27-Jul-1999.)
⊢ abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥))))
Theorem	sqrtval 15189*	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 15190	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 15191	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 15192	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	sqrt0 15193	The square root of zero is zero. (Contributed by Mario Carneiro, 9-Jul-2013.)
⊢ (√‘0) = 0
Theorem	01sqrexlem1 15194*	Lemma for 01sqrex 15201. (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	01sqrexlem2 15195*	Lemma for 01sqrex 15201. (Contributed by Mario Carneiro, 10-Jul-2013.)
⊢ 𝑆 = {𝑥 ∈ ℝ+ ∣ (𝑥↑2) ≤ 𝐴}    &   ⊢ 𝐵 = sup(𝑆, ℝ, < )    ⇒   ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → 𝐴 ∈ 𝑆)
Theorem	01sqrexlem3 15196*	Lemma for 01sqrex 15201. (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	01sqrexlem4 15197*	Lemma for 01sqrex 15201. (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	01sqrexlem5 15198*	Lemma for 01sqrex 15201. (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	01sqrexlem6 15199*	Lemma for 01sqrex 15201. (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	01sqrexlem7 15200*	Lemma for 01sqrex 15201. (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)