P. 154 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 15301-15400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rddif 15301	The difference between a real number and its nearest integer is less than or equal to one half. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Sep-2015.)
⊢ (𝐴 ∈ ℝ → (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)) ≤ (1 / 2))
Theorem	absrdbnd 15302	Bound on the absolute value of a real number rounded to the nearest integer. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Sep-2015.)
⊢ (𝐴 ∈ ℝ → (abs‘(⌊‘(𝐴 + (1 / 2)))) ≤ ((⌊‘(abs‘𝐴)) + 1))
Theorem	fzomaxdiflem 15303	Lemma for fzomaxdif 15304. (Contributed by Stefan O'Rear, 6-Sep-2015.)
⊢ (((𝐴 ∈ (𝐶..^𝐷) ∧ 𝐵 ∈ (𝐶..^𝐷)) ∧ 𝐴 ≤ 𝐵) → (abs‘(𝐵 − 𝐴)) ∈ (0..^(𝐷 − 𝐶)))
Theorem	fzomaxdif 15304	A bound on the separation of two points in a half-open range. (Contributed by Stefan O'Rear, 6-Sep-2015.)
⊢ ((𝐴 ∈ (𝐶..^𝐷) ∧ 𝐵 ∈ (𝐶..^𝐷)) → (abs‘(𝐴 − 𝐵)) ∈ (0..^(𝐷 − 𝐶)))
Theorem	uzin2 15305	The upper integers are closed under intersection. (Contributed by Mario Carneiro, 24-Dec-2013.)
⊢ ((𝐴 ∈ ran ℤ≥ ∧ 𝐵 ∈ ran ℤ≥) → (𝐴 ∩ 𝐵) ∈ ran ℤ≥)
Theorem	rexanuz 15306*	Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 25-Dec-2013.)
⊢ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓) ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓))
Theorem	rexanre 15307*	Combine two different upper real properties into one. (Contributed by Mario Carneiro, 8-May-2016.)
⊢ (𝐴 ⊆ ℝ → (∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (𝜑 ∧ 𝜓)) ↔ (∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑) ∧ ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜓))))
Theorem	rexfiuz 15308*	Combine finitely many different upper integer properties into one. (Contributed by Mario Carneiro, 6-Jun-2014.)
⊢ (𝐴 ∈ Fin → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝐴 𝜑 ↔ ∀𝑛 ∈ 𝐴 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑))
Theorem	rexuz3 15309*	Restrict the base of the upper integers set to another upper integers set. (Contributed by Mario Carneiro, 26-Dec-2013.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ (𝑀 ∈ ℤ → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑))
Theorem	rexanuz2 15310*	Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 26-Dec-2013.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓) ↔ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓))
Theorem	r19.29uz 15311*	A version of 19.29 1880 for upper integer quantifiers. (Contributed by Mario Carneiro, 10-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ ((∀𝑘 ∈ 𝑍 𝜑 ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓))
Theorem	r19.2uz 15312*	A version of r19.2z 4434 for upper integer quantifiers. (Contributed by Mario Carneiro, 15-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 → ∃𝑘 ∈ 𝑍 𝜑)
Theorem	rexuzre 15313*	Convert an upper real quantifier to an upper integer quantifier. (Contributed by Mario Carneiro, 7-May-2016.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ (𝑀 ∈ ℤ → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝑍 (𝑗 ≤ 𝑘 → 𝜑)))
Theorem	rexico 15314*	Restrict the base of an upper real quantifier to an upper real set. (Contributed by Mario Carneiro, 12-May-2016.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) → (∃𝑗 ∈ (𝐵[,)+∞)∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑) ↔ ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑)))
Theorem	cau3lem 15315*	Lemma for cau3 15316. (Contributed by Mario Carneiro, 15-Feb-2014.) (Revised by Mario Carneiro, 1-May-2014.)
⊢ 𝑍 ⊆ ℤ    &   ⊢ (𝜏 → 𝜓)    &   ⊢ ((𝐹‘𝑘) = (𝐹‘𝑗) → (𝜓 ↔ 𝜒))    &   ⊢ ((𝐹‘𝑘) = (𝐹‘𝑚) → (𝜓 ↔ 𝜃))    &   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜓) → (𝐺‘((𝐹‘𝑗)𝐷(𝐹‘𝑘))) = (𝐺‘((𝐹‘𝑘)𝐷(𝐹‘𝑗))))    &   ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜒) → (𝐺‘((𝐹‘𝑚)𝐷(𝐹‘𝑗))) = (𝐺‘((𝐹‘𝑗)𝐷(𝐹‘𝑚))))    &   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜃) ∧ (𝜒 ∧ 𝑥 ∈ ℝ)) → (((𝐺‘((𝐹‘𝑘)𝐷(𝐹‘𝑗))) < (𝑥 / 2) ∧ (𝐺‘((𝐹‘𝑗)𝐷(𝐹‘𝑚))) < (𝑥 / 2)) → (𝐺‘((𝐹‘𝑘)𝐷(𝐹‘𝑚))) < 𝑥))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜏 ∧ (𝐺‘((𝐹‘𝑘)𝐷(𝐹‘𝑗))) < 𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜏 ∧ ∀𝑚 ∈ (ℤ≥‘𝑘)(𝐺‘((𝐹‘𝑘)𝐷(𝐹‘𝑚))) < 𝑥)))
Theorem	cau3 15316*	Convert between three-quantifier and four-quantifier versions of the Cauchy criterion. (In particular, the four-quantifier version has no occurrence of 𝑗 in the assertion, so it can be used with rexanuz 15306 and friends.) (Contributed by Mario Carneiro, 15-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ ∀𝑚 ∈ (ℤ≥‘𝑘)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥))
Theorem	cau4 15317*	Change the base of a Cauchy criterion. (Contributed by Mario Carneiro, 18-Mar-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ 𝑊 = (ℤ≥‘𝑁)    ⇒   ⊢ (𝑁 ∈ 𝑍 → (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑊 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)))
Theorem	caubnd2 15318*	A Cauchy sequence of complex numbers is eventually bounded. (Contributed by Mario Carneiro, 14-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∃𝑦 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑦)
Theorem	caubnd 15319*	A Cauchy sequence of complex numbers is bounded. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 14-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ ((∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) < 𝑦)
Theorem	sqreulem 15320	Lemma for sqreu 15321: write a general complex square root in terms of the square root function over nonnegative reals. (Contributed by Mario Carneiro, 9-Jul-2013.)
⊢ 𝐵 = ((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴))))    ⇒   ⊢ ((𝐴 ∈ ℂ ∧ ((abs‘𝐴) + 𝐴) ≠ 0) → ((𝐵↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝐵) ∧ (i · 𝐵) ∉ ℝ+))
Theorem	sqreu 15321*	Existence and uniqueness for the square root function in general. (Contributed by Mario Carneiro, 9-Jul-2013.)
⊢ (𝐴 ∈ ℂ → ∃!𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+))
Theorem	sqrtcl 15322	Closure of the square root function over the complex numbers. (Contributed by Mario Carneiro, 10-Jul-2013.)
⊢ (𝐴 ∈ ℂ → (√‘𝐴) ∈ ℂ)
Theorem	sqrtthlem 15323	Lemma for sqrtth 15325. (Contributed by Mario Carneiro, 10-Jul-2013.)
⊢ (𝐴 ∈ ℂ → (((√‘𝐴)↑2) = 𝐴 ∧ 0 ≤ (ℜ‘(√‘𝐴)) ∧ (i · (√‘𝐴)) ∉ ℝ+))
Theorem	sqrtf 15324	Mapping domain and codomain of the square root function. (Contributed by Mario Carneiro, 13-Sep-2015.)
⊢ √:ℂ⟶ℂ
Theorem	sqrtth 15325	Square root theorem over the complex numbers. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 10-Jul-2013.)
⊢ (𝐴 ∈ ℂ → ((√‘𝐴)↑2) = 𝐴)
Theorem	sqrtrege0 15326	The square root function must make a choice between the two roots, which differ by a sign change. In the general complex case, the choice of "positive" and "negative" is not so clear. The convention we use is to take the root with positive real part, unless 𝐴 is a nonpositive real (in which case both roots have 0 real part); in this case we take the one in the positive imaginary direction. Another way to look at this is that we choose the root that is largest with respect to lexicographic order on the complex numbers (sorting by real part first, then by imaginary part as tie-breaker). (Contributed by Mario Carneiro, 10-Jul-2013.)
⊢ (𝐴 ∈ ℂ → 0 ≤ (ℜ‘(√‘𝐴)))
Theorem	eqsqrtor 15327	Solve an equation containing a square. (Contributed by Mario Carneiro, 23-Apr-2015.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑2) = 𝐵 ↔ (𝐴 = (√‘𝐵) ∨ 𝐴 = -(√‘𝐵))))
Theorem	eqsqrtd 15328	A deduction for showing that a number equals the square root of another. (Contributed by Mario Carneiro, 3-Apr-2015.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → (𝐴↑2) = 𝐵)    &   ⊢ (𝜑 → 0 ≤ (ℜ‘𝐴))    &   ⊢ (𝜑 → ¬ (i · 𝐴) ∈ ℝ+)    ⇒   ⊢ (𝜑 → 𝐴 = (√‘𝐵))
Theorem	eqsqrt2d 15329	A deduction for showing that a number equals the square root of another. (Contributed by Mario Carneiro, 3-Apr-2015.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → (𝐴↑2) = 𝐵)    &   ⊢ (𝜑 → 0 < (ℜ‘𝐴))    ⇒   ⊢ (𝜑 → 𝐴 = (√‘𝐵))
Theorem	amgm2 15330	Arithmetic-geometric mean inequality for 𝑛 = 2. (Contributed by Mario Carneiro, 2-Jul-2014.) (Proof shortened by AV, 9-Jul-2022.)
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (√‘(𝐴 · 𝐵)) ≤ ((𝐴 + 𝐵) / 2))
Theorem	sqrtthi 15331	Square root theorem. Theorem I.35 of [Apostol] p. 29. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (0 ≤ 𝐴 → ((√‘𝐴) · (√‘𝐴)) = 𝐴)
Theorem	sqrtcli 15332	The square root of a nonnegative real is a real. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (0 ≤ 𝐴 → (√‘𝐴) ∈ ℝ)
Theorem	sqrtgt0i 15333	The square root of a positive real is positive. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (0 < 𝐴 → 0 < (√‘𝐴))
Theorem	sqrtmsqi 15334	Square root of square. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (0 ≤ 𝐴 → (√‘(𝐴 · 𝐴)) = 𝐴)
Theorem	sqrtsqi 15335	Square root of square. (Contributed by NM, 11-Aug-1999.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (0 ≤ 𝐴 → (√‘(𝐴↑2)) = 𝐴)
Theorem	sqsqrti 15336	Square of square root. (Contributed by NM, 11-Aug-1999.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (0 ≤ 𝐴 → ((√‘𝐴)↑2) = 𝐴)
Theorem	sqrtge0i 15337	The square root of a nonnegative real is nonnegative. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (0 ≤ 𝐴 → 0 ≤ (√‘𝐴))
Theorem	absidi 15338	A nonnegative number is its own absolute value. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (0 ≤ 𝐴 → (abs‘𝐴) = 𝐴)
Theorem	absnidi 15339	A negative number is the negative of its own absolute value. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (𝐴 ≤ 0 → (abs‘𝐴) = -𝐴)
Theorem	leabsi 15340	A real number is less than or equal to its absolute value. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ 𝐴 ≤ (abs‘𝐴)
Theorem	absori 15341	The absolute value of a real number is either that number or its negative. (Contributed by NM, 30-Sep-1999.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ ((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴)
Theorem	absrei 15342	Absolute value of a real number. (Contributed by NM, 3-Aug-1999.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (abs‘𝐴) = (√‘(𝐴↑2))
Theorem	sqrtpclii 15343	The square root of a positive real is a real. (Contributed by Mario Carneiro, 6-Sep-2013.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 0 < 𝐴    ⇒   ⊢ (√‘𝐴) ∈ ℝ
Theorem	sqrtgt0ii 15344	The square root of a positive real is positive. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 0 < 𝐴    ⇒   ⊢ 0 < (√‘𝐴)
Theorem	sqrt11i 15345	The square root function is one-to-one. (Contributed by NM, 27-Jul-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → ((√‘𝐴) = (√‘𝐵) ↔ 𝐴 = 𝐵))
Theorem	sqrtmuli 15346	Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (√‘(𝐴 · 𝐵)) = ((√‘𝐴) · (√‘𝐵)))
Theorem	sqrtmulii 15347	Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 0 ≤ 𝐴    &   ⊢ 0 ≤ 𝐵    ⇒   ⊢ (√‘(𝐴 · 𝐵)) = ((√‘𝐴) · (√‘𝐵))
Theorem	sqrtmsq2i 15348	Relationship between square root and squares. (Contributed by NM, 31-Jul-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → ((√‘𝐴) = 𝐵 ↔ 𝐴 = (𝐵 · 𝐵)))
Theorem	sqrtlei 15349	Square root is monotonic. (Contributed by NM, 3-Aug-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴 ≤ 𝐵 ↔ (√‘𝐴) ≤ (√‘𝐵)))
Theorem	sqrtlti 15350	Square root is strictly monotonic. (Contributed by Roy F. Longton, 8-Aug-2005.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴 < 𝐵 ↔ (√‘𝐴) < (√‘𝐵)))
Theorem	abslti 15351	Absolute value and 'less than' relation. (Contributed by NM, 6-Apr-2005.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ ((abs‘𝐴) < 𝐵 ↔ (-𝐵 < 𝐴 ∧ 𝐴 < 𝐵))
Theorem	abslei 15352	Absolute value and 'less than or equal to' relation. (Contributed by NM, 6-Apr-2005.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ ((abs‘𝐴) ≤ 𝐵 ↔ (-𝐵 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵))
Theorem	cnsqrt00 15353	A square root of a complex number is zero iff its argument is 0. Version of sqrt00 15223 for complex numbers. (Contributed by AV, 26-Jan-2023.)
⊢ (𝐴 ∈ ℂ → ((√‘𝐴) = 0 ↔ 𝐴 = 0))
Theorem	absvalsqi 15354	Square of value of absolute value function. (Contributed by NM, 2-Oct-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴))
Theorem	absvalsq2i 15355	Square of value of absolute value function. (Contributed by NM, 2-Oct-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ ((abs‘𝐴)↑2) = (((ℜ‘𝐴)↑2) + ((ℑ‘𝐴)↑2))
Theorem	abscli 15356	Real closure of absolute value. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (abs‘𝐴) ∈ ℝ
Theorem	absge0i 15357	Absolute value is nonnegative. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ 0 ≤ (abs‘𝐴)
Theorem	absval2i 15358	Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (abs‘𝐴) = (√‘(((ℜ‘𝐴)↑2) + ((ℑ‘𝐴)↑2)))
Theorem	abs00i 15359	The absolute value of a number is zero iff the number is zero. Proposition 10-3.7(c) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ ((abs‘𝐴) = 0 ↔ 𝐴 = 0)
Theorem	absgt0i 15360	The absolute value of a nonzero number is positive. Remark in [Apostol] p. 363. (Contributed by NM, 1-Oct-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 ≠ 0 ↔ 0 < (abs‘𝐴))
Theorem	absnegi 15361	Absolute value of negative. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (abs‘-𝐴) = (abs‘𝐴)
Theorem	abscji 15362	The absolute value of a number and its conjugate are the same. Proposition 10-3.7(b) of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (abs‘(∗‘𝐴)) = (abs‘𝐴)
Theorem	releabsi 15363	The real part of a number is less than or equal to its absolute value. Proposition 10-3.7(d) of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (ℜ‘𝐴) ≤ (abs‘𝐴)
Theorem	abssubi 15364	Swapping order of subtraction doesn't change the absolute value. Example of [Apostol] p. 363. (Contributed by NM, 1-Oct-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (abs‘(𝐴 − 𝐵)) = (abs‘(𝐵 − 𝐴))
Theorem	absmuli 15365	Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133. (Contributed by NM, 1-Oct-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (abs‘(𝐴 · 𝐵)) = ((abs‘𝐴) · (abs‘𝐵))
Theorem	sqabsaddi 15366	Square of absolute value of sum. Proposition 10-3.7(g) of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ ((abs‘(𝐴 + 𝐵))↑2) = ((((abs‘𝐴)↑2) + ((abs‘𝐵)↑2)) + (2 · (ℜ‘(𝐴 · (∗‘𝐵)))))
Theorem	sqabssubi 15367	Square of absolute value of difference. (Contributed by Steve Rodriguez, 20-Jan-2007.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ ((abs‘(𝐴 − 𝐵))↑2) = ((((abs‘𝐴)↑2) + ((abs‘𝐵)↑2)) − (2 · (ℜ‘(𝐴 · (∗‘𝐵)))))
Theorem	absdivzi 15368	Absolute value distributes over division. (Contributed by NM, 26-Mar-2005.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐵 ≠ 0 → (abs‘(𝐴 / 𝐵)) = ((abs‘𝐴) / (abs‘𝐵)))
Theorem	abstrii 15369	Triangle inequality for absolute value. Proposition 10-3.7(h) of [Gleason] p. 133. This is Metamath 100 proof #91. (Contributed by NM, 2-Oct-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (abs‘(𝐴 + 𝐵)) ≤ ((abs‘𝐴) + (abs‘𝐵))
Theorem	abs3difi 15370	Absolute value of differences around common element. (Contributed by NM, 2-Oct-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    ⇒   ⊢ (abs‘(𝐴 − 𝐵)) ≤ ((abs‘(𝐴 − 𝐶)) + (abs‘(𝐶 − 𝐵)))
Theorem	abs3lemi 15371	Lemma involving absolute value of differences. (Contributed by NM, 2-Oct-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    &   ⊢ 𝐷 ∈ ℝ    ⇒   ⊢ (((abs‘(𝐴 − 𝐶)) < (𝐷 / 2) ∧ (abs‘(𝐶 − 𝐵)) < (𝐷 / 2)) → (abs‘(𝐴 − 𝐵)) < 𝐷)
Theorem	rpsqrtcld 15372	The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (√‘𝐴) ∈ ℝ+)
Theorem	sqrtgt0d 15373	The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 0 < (√‘𝐴))
Theorem	absnidd 15374	A negative number is the negative of its own absolute value. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 0)    ⇒   ⊢ (𝜑 → (abs‘𝐴) = -𝐴)
Theorem	leabsd 15375	A real number is less than or equal to its absolute value. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝐴 ≤ (abs‘𝐴))
Theorem	absord 15376	The absolute value of a real number is either that number or its negative. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴))
Theorem	absred 15377	Absolute value of a real number. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (abs‘𝐴) = (√‘(𝐴↑2)))
Theorem	resqrtcld 15378	The square root of a nonnegative real is a real. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    ⇒   ⊢ (𝜑 → (√‘𝐴) ∈ ℝ)
Theorem	sqrtmsqd 15379	Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    ⇒   ⊢ (𝜑 → (√‘(𝐴 · 𝐴)) = 𝐴)
Theorem	sqrtsqd 15380	Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    ⇒   ⊢ (𝜑 → (√‘(𝐴↑2)) = 𝐴)
Theorem	sqrtge0d 15381	The square root of a nonnegative real is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    ⇒   ⊢ (𝜑 → 0 ≤ (√‘𝐴))
Theorem	sqrtnegd 15382	The square root of a negative number. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    ⇒   ⊢ (𝜑 → (√‘-𝐴) = (i · (√‘𝐴)))
Theorem	absidd 15383	A nonnegative number is its own absolute value. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    ⇒   ⊢ (𝜑 → (abs‘𝐴) = 𝐴)
Theorem	sqrtdivd 15384	Square root distributes over division. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (√‘(𝐴 / 𝐵)) = ((√‘𝐴) / (√‘𝐵)))
Theorem	sqrtmuld 15385	Square root distributes over multiplication. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (√‘(𝐴 · 𝐵)) = ((√‘𝐴) · (√‘𝐵)))
Theorem	sqrtsq2d 15386	Relationship between square root and squares. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐵)    ⇒   ⊢ (𝜑 → ((√‘𝐴) = 𝐵 ↔ 𝐴 = (𝐵↑2)))
Theorem	sqrtled 15387	Square root is monotonic. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (√‘𝐴) ≤ (√‘𝐵)))
Theorem	sqrtltd 15388	Square root is strictly monotonic. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (√‘𝐴) < (√‘𝐵)))
Theorem	sqr11d 15389	The square root function is one-to-one. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐵)    &   ⊢ (𝜑 → (√‘𝐴) = (√‘𝐵))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	nn0absid 15390	A nonnegative integer is its own absolute value. (Contributed by AV, 22-Nov-2025.)
⊢ (𝑁 ∈ ℕ0 → (abs‘𝑁) = 𝑁)
Theorem	nn0absidi 15391	A nonnegative integer is its own absolute value (inference form). (Contributed by AV, 22-Nov-2025.)
⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ (abs‘𝑁) = 𝑁
Theorem	absltd 15392	Absolute value and 'less than' relation. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((abs‘𝐴) < 𝐵 ↔ (-𝐵 < 𝐴 ∧ 𝐴 < 𝐵)))
Theorem	absled 15393	Absolute value and 'less than or equal to' relation. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((abs‘𝐴) ≤ 𝐵 ↔ (-𝐵 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)))
Theorem	abssubge0d 15394	Absolute value of a nonnegative difference. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (abs‘(𝐵 − 𝐴)) = (𝐵 − 𝐴))
Theorem	abssuble0d 15395	Absolute value of a nonpositive difference. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) = (𝐵 − 𝐴))
Theorem	absdifltd 15396	The absolute value of a difference and 'less than' relation. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((abs‘(𝐴 − 𝐵)) < 𝐶 ↔ ((𝐵 − 𝐶) < 𝐴 ∧ 𝐴 < (𝐵 + 𝐶))))
Theorem	absdifled 15397	The absolute value of a difference and 'less than or equal to' relation. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((abs‘(𝐴 − 𝐵)) ≤ 𝐶 ↔ ((𝐵 − 𝐶) ≤ 𝐴 ∧ 𝐴 ≤ (𝐵 + 𝐶))))
Theorem	icodiamlt 15398	Two elements in a half-open interval have separation strictly less than the difference between the endpoints. (Contributed by Stefan O'Rear, 12-Sep-2014.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∧ 𝐷 ∈ (𝐴[,)𝐵))) → (abs‘(𝐶 − 𝐷)) < (𝐵 − 𝐴))
Theorem	abscld 15399	Real closure of absolute value. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ)
Theorem	sqrtcld 15400	Closure of the square root function over the complex numbers. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (√‘𝐴) ∈ ℂ)