P. 98 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9701-9800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	abs3dif 9701	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 9702	Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘𝐴) − (abs‘𝐵)) ≤ (abs‘(𝐴 − 𝐵)))
Theorem	abs2dif2 9703	Difference of absolute values. (Contributed by Mario Carneiro, 14-Apr-2016.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (abs‘(𝐴 − 𝐵)) ≤ ((abs‘𝐴) + (abs‘𝐵)))
Theorem	abs2difabs 9704	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 9705*	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 9706	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 9707	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 9708	Lemma for fzomaxdif 9709. (Contributed by Stefan O'Rear, 6-Sep-2015.)
⊢ (((𝐴 ∈ (𝐶..^𝐷) ∧ 𝐵 ∈ (𝐶..^𝐷)) ∧ 𝐴 ≤ 𝐵) → (abs‘(𝐵 − 𝐴)) ∈ (0..^(𝐷 − 𝐶)))
Theorem	fzomaxdif 9709	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 9710*	Lemma for cau3 9711. (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 9711*	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 9587 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 9712*	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 9713*	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 9714	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 9715	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 9716	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 9717	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 9718	Square root of square. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (0 ≤ 𝐴 → (√‘(𝐴 · 𝐴)) = 𝐴)
Theorem	sqrtsqi 9719	Square root of square. (Contributed by NM, 11-Aug-1999.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (0 ≤ 𝐴 → (√‘(𝐴↑2)) = 𝐴)
Theorem	sqsqrti 9720	Square of square root. (Contributed by NM, 11-Aug-1999.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (0 ≤ 𝐴 → ((√‘𝐴)↑2) = 𝐴)
Theorem	sqrtge0i 9721	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 9722	A nonnegative number is its own absolute value. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (0 ≤ 𝐴 → (abs‘𝐴) = 𝐴)
Theorem	absnidi 9723	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 9724	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 9725	Absolute value of a real number. (Contributed by NM, 3-Aug-1999.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (abs‘𝐴) = (√‘(𝐴↑2))
Theorem	sqrtpclii 9726	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 9727	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 9728	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 9729	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 9730	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 9731	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 9732	Square root is monotonic. (Contributed by NM, 3-Aug-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴 ≤ 𝐵 ↔ (√‘𝐴) ≤ (√‘𝐵)))
Theorem	sqrtlti 9733	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 9734	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 9735	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 9736	Square of value of absolute value function. (Contributed by NM, 2-Oct-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴))
Theorem	absvalsq2i 9737	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 9738	Real closure of absolute value. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (abs‘𝐴) ∈ ℝ
Theorem	absge0i 9739	Absolute value is nonnegative. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ 0 ≤ (abs‘𝐴)
Theorem	absval2i 9740	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 9741	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 9742	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 9743	Absolute value of negative. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (abs‘-𝐴) = (abs‘𝐴)
Theorem	abscji 9744	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 9745	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 9746	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 9747	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 9748	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 9749	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 9750	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 9751	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 9752	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 9753	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 9754	The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (√‘𝐴) ∈ ℝ+)
Theorem	sqrtgt0d 9755	The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 0 < (√‘𝐴))
Theorem	absnidd 9756	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 9757	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 9758	Absolute value of a real number. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (abs‘𝐴) = (√‘(𝐴↑2)))
Theorem	resqrtcld 9759	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 9760	Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    ⇒   ⊢ (𝜑 → (√‘(𝐴 · 𝐴)) = 𝐴)
Theorem	sqrtsqd 9761	Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    ⇒   ⊢ (𝜑 → (√‘(𝐴↑2)) = 𝐴)
Theorem	sqrtge0d 9762	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 9763	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 9764	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 9765	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 9766	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 9767	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 9768	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 9769	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 9770	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 9771	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 9772	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 9773	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 9774	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 9775	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 9776	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 9777	Real closure of absolute value. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ)
Theorem	absvalsqd 9778	Square of value of absolute value function. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴)))
Theorem	absvalsq2d 9779	Square of value of absolute value function. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((abs‘𝐴)↑2) = (((ℜ‘𝐴)↑2) + ((ℑ‘𝐴)↑2)))
Theorem	absge0d 9780	Absolute value is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → 0 ≤ (abs‘𝐴))
Theorem	absval2d 9781	Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (abs‘𝐴) = (√‘(((ℜ‘𝐴)↑2) + ((ℑ‘𝐴)↑2))))
Theorem	abs00d 9782	The absolute value of a number is zero iff the number is zero. Proposition 10-3.7(c) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → (abs‘𝐴) = 0)    ⇒   ⊢ (𝜑 → 𝐴 = 0)
Theorem	absne0d 9783	The absolute value of a number is zero iff the number is zero. Proposition 10-3.7(c) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    ⇒   ⊢ (𝜑 → (abs‘𝐴) ≠ 0)
Theorem	absrpclapd 9784	The absolute value of a complex number apart from zero is a positive real. (Contributed by Jim Kingdon, 13-Aug-2021.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 # 0)    ⇒   ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ+)
Theorem	absnegd 9785	Absolute value of negative. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (abs‘-𝐴) = (abs‘𝐴))
Theorem	abscjd 9786	The absolute value of a number and its conjugate are the same. Proposition 10-3.7(b) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (abs‘(∗‘𝐴)) = (abs‘𝐴))
Theorem	releabsd 9787	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 Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (ℜ‘𝐴) ≤ (abs‘𝐴))
Theorem	absexpd 9788	Absolute value of positive integer exponentiation. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (abs‘(𝐴↑𝑁)) = ((abs‘𝐴)↑𝑁))
Theorem	abssubd 9789	Swapping order of subtraction doesn't change the absolute value. Example of [Apostol] p. 363. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) = (abs‘(𝐵 − 𝐴)))
Theorem	absmuld 9790	Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (abs‘(𝐴 · 𝐵)) = ((abs‘𝐴) · (abs‘𝐵)))
Theorem	absdivapd 9791	Absolute value distributes over division. (Contributed by Jim Kingdon, 13-Aug-2021.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 # 0)    ⇒   ⊢ (𝜑 → (abs‘(𝐴 / 𝐵)) = ((abs‘𝐴) / (abs‘𝐵)))
Theorem	abstrid 9792	Triangle inequality for absolute value. Proposition 10-3.7(h) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (abs‘(𝐴 + 𝐵)) ≤ ((abs‘𝐴) + (abs‘𝐵)))
Theorem	abs2difd 9793	Difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((abs‘𝐴) − (abs‘𝐵)) ≤ (abs‘(𝐴 − 𝐵)))
Theorem	abs2dif2d 9794	Difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) ≤ ((abs‘𝐴) + (abs‘𝐵)))
Theorem	abs2difabsd 9795	Absolute value of difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (abs‘((abs‘𝐴) − (abs‘𝐵))) ≤ (abs‘(𝐴 − 𝐵)))
Theorem	abs3difd 9796	Absolute value of differences around common element. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) ≤ ((abs‘(𝐴 − 𝐶)) + (abs‘(𝐶 − 𝐵))))
Theorem	abs3lemd 9797	Lemma involving absolute value of differences. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    &   ⊢ (𝜑 → (abs‘(𝐴 − 𝐶)) < (𝐷 / 2))    &   ⊢ (𝜑 → (abs‘(𝐶 − 𝐵)) < (𝐷 / 2))    ⇒   ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) < 𝐷)
Theorem	qdenre 9798*	The rational numbers are dense in ℝ: any real number can be approximated with arbitrary precision by a rational number. For order theoretic density, see qbtwnre 9111. (Contributed by BJ, 15-Oct-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℚ (abs‘(𝑥 − 𝐴)) < 𝐵)
3.8  Elementary limits and convergence
3.8.1  Limits
Syntax	cli 9799	Extend class notation with convergence relation for limits.
class ⇝
Definition	df-clim 9800*	Define the limit relation for complex number sequences. See clim 9802 for its relational expression. (Contributed by NM, 28-Aug-2005.)
⊢ ⇝ = {⟨𝑓, 𝑦⟩ ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥))}