P. 117 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 11601-11700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	sqabs 11601	The squares of two reals are equal iff their absolute values are equal. (Contributed by NM, 6-Mar-2009.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴↑2) = (𝐵↑2) ↔ (abs‘𝐴) = (abs‘𝐵)))
Theorem	absrele 11602	The absolute value of a complex number is greater than or equal to the absolute value of its real part. (Contributed by NM, 1-Apr-2005.)
⊢ (𝐴 ∈ ℂ → (abs‘(ℜ‘𝐴)) ≤ (abs‘𝐴))
Theorem	absimle 11603	The absolute value of a complex number is greater than or equal to the absolute value of its imaginary part. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)
⊢ (𝐴 ∈ ℂ → (abs‘(ℑ‘𝐴)) ≤ (abs‘𝐴))
Theorem	nn0abscl 11604	The absolute value of an integer is a nonnegative integer. (Contributed by NM, 27-Feb-2005.)
⊢ (𝐴 ∈ ℤ → (abs‘𝐴) ∈ ℕ0)
Theorem	zabscl 11605	The absolute value of an integer is an integer. (Contributed by Stefan O'Rear, 24-Sep-2014.)
⊢ (𝐴 ∈ ℤ → (abs‘𝐴) ∈ ℤ)
Theorem	ltabs 11606	A number which is less than its absolute value is negative. (Contributed by Jim Kingdon, 12-Aug-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < (abs‘𝐴)) → 𝐴 < 0)
Theorem	abslt 11607	Absolute value and 'less than' relation. (Contributed by NM, 6-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((abs‘𝐴) < 𝐵 ↔ (-𝐵 < 𝐴 ∧ 𝐴 < 𝐵)))
Theorem	absle 11608	Absolute value and 'less than or equal to' relation. (Contributed by NM, 6-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((abs‘𝐴) ≤ 𝐵 ↔ (-𝐵 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)))
Theorem	abssubap0 11609	If the absolute value of a complex number is less than a real, its difference from the real is apart from zero. (Contributed by Jim Kingdon, 12-Aug-2021.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ (abs‘𝐴) < 𝐵) → (𝐵 − 𝐴) # 0)
Theorem	abssubne0 11610	If the absolute value of a complex number is less than a real, its difference from the real is nonzero. See also abssubap0 11609 which is the same with not equal changed to apart. (Contributed by NM, 2-Nov-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ (abs‘𝐴) < 𝐵) → (𝐵 − 𝐴) ≠ 0)
Theorem	absdiflt 11611	The absolute value of a difference and 'less than' relation. (Contributed by Paul Chapman, 18-Sep-2007.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((abs‘(𝐴 − 𝐵)) < 𝐶 ↔ ((𝐵 − 𝐶) < 𝐴 ∧ 𝐴 < (𝐵 + 𝐶))))
Theorem	absdifle 11612	The absolute value of a difference and 'less than or equal to' relation. (Contributed by Paul Chapman, 18-Sep-2007.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((abs‘(𝐴 − 𝐵)) ≤ 𝐶 ↔ ((𝐵 − 𝐶) ≤ 𝐴 ∧ 𝐴 ≤ (𝐵 + 𝐶))))
Theorem	elicc4abs 11613	Membership in a symmetric closed real interval. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 ∈ ((𝐴 − 𝐵)[,](𝐴 + 𝐵)) ↔ (abs‘(𝐶 − 𝐴)) ≤ 𝐵))
Theorem	lenegsq 11614	Comparison to a nonnegative number based on comparison to squares. (Contributed by NM, 16-Jan-2006.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) → ((𝐴 ≤ 𝐵 ∧ -𝐴 ≤ 𝐵) ↔ (𝐴↑2) ≤ (𝐵↑2)))
Theorem	releabs 11615	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, 1-Apr-2005.)
⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ≤ (abs‘𝐴))
Theorem	recvalap 11616	Reciprocal expressed with a real denominator. (Contributed by Jim Kingdon, 13-Aug-2021.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (1 / 𝐴) = ((∗‘𝐴) / ((abs‘𝐴)↑2)))
Theorem	absidm 11617	The absolute value function is idempotent. (Contributed by NM, 20-Nov-2004.)
⊢ (𝐴 ∈ ℂ → (abs‘(abs‘𝐴)) = (abs‘𝐴))
Theorem	absgt0ap 11618	The absolute value of a number apart from zero is positive. (Contributed by Jim Kingdon, 13-Aug-2021.)
⊢ (𝐴 ∈ ℂ → (𝐴 # 0 ↔ 0 < (abs‘𝐴)))
Theorem	nnabscl 11619	The absolute value of a nonzero integer is a positive integer. (Contributed by Paul Chapman, 21-Mar-2011.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (abs‘𝑁) ∈ ℕ)
Theorem	abssub 11620	Swapping order of subtraction doesn't change the absolute value. (Contributed by NM, 1-Oct-1999.) (Proof shortened by Mario Carneiro, 29-May-2016.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (abs‘(𝐴 − 𝐵)) = (abs‘(𝐵 − 𝐴)))
Theorem	abssubge0 11621	Absolute value of a nonnegative difference. (Contributed by NM, 14-Feb-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (abs‘(𝐵 − 𝐴)) = (𝐵 − 𝐴))
Theorem	abssuble0 11622	Absolute value of a nonpositive difference. (Contributed by FL, 3-Jan-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (abs‘(𝐴 − 𝐵)) = (𝐵 − 𝐴))
Theorem	abstri 11623	Triangle inequality for absolute value. Proposition 10-3.7(h) of [Gleason] p. 133. (Contributed by NM, 7-Mar-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (abs‘(𝐴 + 𝐵)) ≤ ((abs‘𝐴) + (abs‘𝐵)))
Theorem	abs3dif 11624	Absolute value of differences around common element. (Contributed by FL, 9-Oct-2006.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (abs‘(𝐴 − 𝐵)) ≤ ((abs‘(𝐴 − 𝐶)) + (abs‘(𝐶 − 𝐵))))
Theorem	abs2dif 11625	Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘𝐴) − (abs‘𝐵)) ≤ (abs‘(𝐴 − 𝐵)))
Theorem	abs2dif2 11626	Difference of absolute values. (Contributed by Mario Carneiro, 14-Apr-2016.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (abs‘(𝐴 − 𝐵)) ≤ ((abs‘𝐴) + (abs‘𝐵)))
Theorem	abs2difabs 11627	Absolute value of difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (abs‘((abs‘𝐴) − (abs‘𝐵))) ≤ (abs‘(𝐴 − 𝐵)))
Theorem	recan 11628*	Cancellation law involving the real part of a complex number. (Contributed by NM, 12-May-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∀𝑥 ∈ ℂ (ℜ‘(𝑥 · 𝐴)) = (ℜ‘(𝑥 · 𝐵)) ↔ 𝐴 = 𝐵))
Theorem	absf 11629	Mapping domain and codomain of the absolute value function. (Contributed by NM, 30-Aug-2007.) (Revised by Mario Carneiro, 7-Nov-2013.)
⊢ abs:ℂ⟶ℝ
Theorem	abs3lem 11630	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	fzomaxdiflem 11631	Lemma for fzomaxdif 11632. (Contributed by Stefan O'Rear, 6-Sep-2015.)
⊢ (((𝐴 ∈ (𝐶..^𝐷) ∧ 𝐵 ∈ (𝐶..^𝐷)) ∧ 𝐴 ≤ 𝐵) → (abs‘(𝐵 − 𝐴)) ∈ (0..^(𝐷 − 𝐶)))
Theorem	fzomaxdif 11632	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	cau3lem 11633*	Lemma for cau3 11634. (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 11634*	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 11507 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 11635*	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 11636*	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	amgm2 11637	Arithmetic-geometric mean inequality for 𝑛 = 2. (Contributed by Mario Carneiro, 2-Jul-2014.)
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (√‘(𝐴 · 𝐵)) ≤ ((𝐴 + 𝐵) / 2))
Theorem	sqrtthi 11638	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 11639	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 11640	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 11641	Square root of square. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (0 ≤ 𝐴 → (√‘(𝐴 · 𝐴)) = 𝐴)
Theorem	sqrtsqi 11642	Square root of square. (Contributed by NM, 11-Aug-1999.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (0 ≤ 𝐴 → (√‘(𝐴↑2)) = 𝐴)
Theorem	sqsqrti 11643	Square of square root. (Contributed by NM, 11-Aug-1999.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (0 ≤ 𝐴 → ((√‘𝐴)↑2) = 𝐴)
Theorem	sqrtge0i 11644	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 11645	A nonnegative number is its own absolute value. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (0 ≤ 𝐴 → (abs‘𝐴) = 𝐴)
Theorem	absnidi 11646	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 11647	A real number is less than or equal to its absolute value. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ 𝐴 ≤ (abs‘𝐴)
Theorem	absrei 11648	Absolute value of a real number. (Contributed by NM, 3-Aug-1999.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (abs‘𝐴) = (√‘(𝐴↑2))
Theorem	sqrtpclii 11649	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 11650	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 11651	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 11652	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 11653	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 11654	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 11655	Square root is monotonic. (Contributed by NM, 3-Aug-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴 ≤ 𝐵 ↔ (√‘𝐴) ≤ (√‘𝐵)))
Theorem	sqrtlti 11656	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 11657	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 11658	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	absvalsqi 11659	Square of value of absolute value function. (Contributed by NM, 2-Oct-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴))
Theorem	absvalsq2i 11660	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 11661	Real closure of absolute value. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (abs‘𝐴) ∈ ℝ
Theorem	absge0i 11662	Absolute value is nonnegative. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ 0 ≤ (abs‘𝐴)
Theorem	absval2i 11663	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 11664	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	absgt0api 11665	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 11666	Absolute value of negative. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (abs‘-𝐴) = (abs‘𝐴)
Theorem	abscji 11667	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 11668	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 11669	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 11670	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 11671	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 11672	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	absdivapzi 11673	Absolute value distributes over division. (Contributed by Jim Kingdon, 13-Aug-2021.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐵 # 0 → (abs‘(𝐴 / 𝐵)) = ((abs‘𝐴) / (abs‘𝐵)))
Theorem	abstrii 11674	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 11675	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 11676	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 11677	The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (√‘𝐴) ∈ ℝ+)
Theorem	sqrtgt0d 11678	The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 0 < (√‘𝐴))
Theorem	absnidd 11679	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 11680	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	absred 11681	Absolute value of a real number. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (abs‘𝐴) = (√‘(𝐴↑2)))
Theorem	resqrtcld 11682	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 11683	Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    ⇒   ⊢ (𝜑 → (√‘(𝐴 · 𝐴)) = 𝐴)
Theorem	sqrtsqd 11684	Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    ⇒   ⊢ (𝜑 → (√‘(𝐴↑2)) = 𝐴)
Theorem	sqrtge0d 11685	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	absidd 11686	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 11687	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 11688	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 11689	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 11690	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 11691	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 11692	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	absltd 11693	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 11694	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 11695	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 11696	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 11697	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 11698	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 11699	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 11700	Real closure of absolute value. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ)