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	sqeqd 15101	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 15102	The real part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (ℜ‘𝐴) ∈ ℝ
Theorem	imcli 15103	The imaginary part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (ℑ‘𝐴) ∈ ℝ
Theorem	cjcli 15104	Closure law for complex conjugate. (Contributed by NM, 11-May-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (∗‘𝐴) ∈ ℂ
Theorem	replimi 15105	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 15106	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 15107	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 15108	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 15109	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 15110	Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (ℜ‘(∗‘𝐴)) = (ℜ‘𝐴)
Theorem	imcji 15111	Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (ℑ‘(∗‘𝐴)) = -(ℑ‘𝐴)
Theorem	cjmulrcli 15112	A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 · (∗‘𝐴)) ∈ ℝ
Theorem	cjmulvali 15113	A complex number times its conjugate. (Contributed by NM, 2-Oct-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 · (∗‘𝐴)) = (((ℜ‘𝐴)↑2) + ((ℑ‘𝐴)↑2))
Theorem	cjmulge0i 15114	A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ 0 ≤ (𝐴 · (∗‘𝐴))
Theorem	renegi 15115	Real part of negative. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (ℜ‘-𝐴) = -(ℜ‘𝐴)
Theorem	imnegi 15116	Imaginary part of negative. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (ℑ‘-𝐴) = -(ℑ‘𝐴)
Theorem	cjnegi 15117	Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (∗‘-𝐴) = -(∗‘𝐴)
Theorem	addcji 15118	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 15119	Real part distributes over addition. (Contributed by NM, 28-Jul-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (ℜ‘(𝐴 + 𝐵)) = ((ℜ‘𝐴) + (ℜ‘𝐵))
Theorem	imaddi 15120	Imaginary part distributes over addition. (Contributed by NM, 28-Jul-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (ℑ‘(𝐴 + 𝐵)) = ((ℑ‘𝐴) + (ℑ‘𝐵))
Theorem	remuli 15121	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 15122	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 15123	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 15124	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 15125	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 15126	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 15127	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 15128	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 15129	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 15130	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 15131	Closure law for complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (∗‘𝐴) ∈ ℂ)
Theorem	replimd 15132	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 15133	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 15134	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 15135	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 15136	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 15137	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 15138	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 15139	Real part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (ℜ‘(∗‘𝐴)) = (ℜ‘𝐴))
Theorem	imcjd 15140	Imaginary part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (ℑ‘(∗‘𝐴)) = -(ℑ‘𝐴))
Theorem	cjmulrcld 15141	A complex number times its conjugate is real. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 · (∗‘𝐴)) ∈ ℝ)
Theorem	cjmulvald 15142	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 15143	A complex number times its conjugate is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → 0 ≤ (𝐴 · (∗‘𝐴)))
Theorem	renegd 15144	Real part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (ℜ‘-𝐴) = -(ℜ‘𝐴))
Theorem	imnegd 15145	Imaginary part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (ℑ‘-𝐴) = -(ℑ‘𝐴))
Theorem	cjnegd 15146	Complex conjugate of negative. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (∗‘-𝐴) = -(∗‘𝐴))
Theorem	addcjd 15147	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 15148	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 15149	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 15150	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 15151	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 15152	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 15153	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 15154	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 15155	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 15156	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 15157	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 15158	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 15159	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 15160	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 15161	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 15162	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 15163	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 15164	Real part of a division. Related to remul2 15065. (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 15165	Imaginary part of a division. Related to remul2 15065. (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 15166	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 15167	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 15168	Extend class notation to include square root of a complex number.
class √
Syntax	cabs 15169	Extend class notation to include a function for the absolute value (modulus) of a complex number.
class abs
Definition	df-sqrt 15170*	Define a function whose value is the square root of a complex number. For example, (√‘25) = 5 (ex-sqrt 30541). 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 30541. The square root symbol was introduced in 1525 by Christoff Rudolff. See sqrtcl 15297 for its closure, sqrtval 15172 for its value, sqrtth 15300 and sqsqrti 15311 for its relationship to squares, and sqrt11i 15320 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 15171	Define the function for the absolute value (modulus) of a complex number. See abscli 15331 for its closure and absval 15173 or absval2i 15333 for its value. For example, (abs‘-2) = 2 (ex-abs 30542). (Contributed by NM, 27-Jul-1999.)
⊢ abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥))))
Theorem	sqrtval 15172*	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 15173	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 15174	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 15175	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 15176	The square root of zero is zero. (Contributed by Mario Carneiro, 9-Jul-2013.)
⊢ (√‘0) = 0
Theorem	01sqrexlem1 15177*	Lemma for 01sqrex 15184. (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 15178*	Lemma for 01sqrex 15184. (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 15179*	Lemma for 01sqrex 15184. (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 15180*	Lemma for 01sqrex 15184. (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 15181*	Lemma for 01sqrex 15184. (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 15182*	Lemma for 01sqrex 15184. (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 15183*	Lemma for 01sqrex 15184. (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)
Theorem	01sqrex 15184*	Existence of a square root for reals in the interval (0, 1]. (Contributed by Mario Carneiro, 10-Jul-2013.)
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → ∃𝑥 ∈ ℝ+ (𝑥 ≤ 1 ∧ (𝑥↑2) = 𝐴))
Theorem	resqrex 15185*	Existence of a square root for positive reals. (Contributed by Mario Carneiro, 9-Jul-2013.)
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
Theorem	sqrmo 15186*	Uniqueness for the square root function. (Contributed by Mario Carneiro, 9-Jul-2013.) (Revised by NM, 17-Jun-2017.)
⊢ (𝐴 ∈ ℂ → ∃*𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+))
Theorem	resqreu 15187*	Existence and uniqueness for the real square root function. (Contributed by Mario Carneiro, 9-Jul-2013.)
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ∃!𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+))
Theorem	resqrtcl 15188	Closure of the square root function. (Contributed by Mario Carneiro, 9-Jul-2013.)
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (√‘𝐴) ∈ ℝ)
Theorem	resqrtthlem 15189	Lemma for resqrtth 15190. (Contributed by Mario Carneiro, 9-Jul-2013.)
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (((√‘𝐴)↑2) = 𝐴 ∧ 0 ≤ (ℜ‘(√‘𝐴)) ∧ (i · (√‘𝐴)) ∉ ℝ+))
Theorem	resqrtth 15190	Square root theorem over the reals. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 9-Jul-2013.)
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((√‘𝐴)↑2) = 𝐴)
Theorem	remsqsqrt 15191	Square of square root. (Contributed by Mario Carneiro, 10-Jul-2013.)
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((√‘𝐴) · (√‘𝐴)) = 𝐴)
Theorem	sqrtge0 15192	The square root function is nonnegative for nonnegative input. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 9-Jul-2013.)
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 0 ≤ (√‘𝐴))
Theorem	sqrtgt0 15193	The square root function is positive for positive input. (Contributed by Mario Carneiro, 10-Jul-2013.) (Revised by Mario Carneiro, 6-Sep-2013.)
⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 0 < (√‘𝐴))
Theorem	sqrtmul 15194	Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.) (Revised by Mario Carneiro, 29-May-2016.)
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (√‘(𝐴 · 𝐵)) = ((√‘𝐴) · (√‘𝐵)))
Theorem	sqrtle 15195	Square root is monotonic. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (𝐴 ≤ 𝐵 ↔ (√‘𝐴) ≤ (√‘𝐵)))
Theorem	sqrtlt 15196	Square root is strictly monotonic. Closed form of sqrtlti 15325. (Contributed by Scott Fenton, 17-Apr-2014.) (Proof shortened by Mario Carneiro, 29-May-2016.)
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (𝐴 < 𝐵 ↔ (√‘𝐴) < (√‘𝐵)))
Theorem	sqrt11 15197	The square root function is one-to-one. (Contributed by Scott Fenton, 11-Jun-2013.)
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((√‘𝐴) = (√‘𝐵) ↔ 𝐴 = 𝐵))
Theorem	sqrt00 15198	A square root is zero iff its argument is 0. (Contributed by NM, 27-Jul-1999.) (Proof shortened by Mario Carneiro, 29-May-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((√‘𝐴) = 0 ↔ 𝐴 = 0))
Theorem	rpsqrtcl 15199	The square root of a positive real is a positive real. (Contributed by NM, 22-Feb-2008.)
⊢ (𝐴 ∈ ℝ+ → (√‘𝐴) ∈ ℝ+)
Theorem	sqrtdiv 15200	Square root distributes over division. (Contributed by Mario Carneiro, 5-May-2016.)
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ+) → (√‘(𝐴 / 𝐵)) = ((√‘𝐴) / (√‘𝐵)))