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Theorem List for Intuitionistic Logic Explorer - 9701-9800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsqrtgt0ii 9701 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)
 
Theoremsqrt11i 9702 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))
 
Theoremsqrtmuli 9703 Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (√‘(𝐴 · 𝐵)) = ((√‘𝐴) · (√‘𝐵)))
 
Theoremsqrtmulii 9704 Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   0 ≤ 𝐴    &   0 ≤ 𝐵       (√‘(𝐴 · 𝐵)) = ((√‘𝐴) · (√‘𝐵))
 
Theoremsqrtmsq2i 9705 Relationship between square root and squares. (Contributed by NM, 31-Jul-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → ((√‘𝐴) = 𝐵𝐴 = (𝐵 · 𝐵)))
 
Theoremsqrtlei 9706 Square root is monotonic. (Contributed by NM, 3-Aug-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴𝐵 ↔ (√‘𝐴) ≤ (√‘𝐵)))
 
Theoremsqrtlti 9707 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)))
 
Theoremabslti 9708 Absolute value and 'less than' relation. (Contributed by NM, 6-Apr-2005.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((abs‘𝐴) < 𝐵 ↔ (-𝐵 < 𝐴𝐴 < 𝐵))
 
Theoremabslei 9709 Absolute value and 'less than or equal to' relation. (Contributed by NM, 6-Apr-2005.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((abs‘𝐴) ≤ 𝐵 ↔ (-𝐵𝐴𝐴𝐵))
 
Theoremabsvalsqi 9710 Square of value of absolute value function. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ ℂ       ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴))
 
Theoremabsvalsq2i 9711 Square of value of absolute value function. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ ℂ       ((abs‘𝐴)↑2) = (((ℜ‘𝐴)↑2) + ((ℑ‘𝐴)↑2))
 
Theoremabscli 9712 Real closure of absolute value. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℂ       (abs‘𝐴) ∈ ℝ
 
Theoremabsge0i 9713 Absolute value is nonnegative. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℂ       0 ≤ (abs‘𝐴)
 
Theoremabsval2i 9714 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)))
 
Theoremabs00i 9715 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)
 
Theoremabsgt0api 9716 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))
 
Theoremabsnegi 9717 Absolute value of negative. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℂ       (abs‘-𝐴) = (abs‘𝐴)
 
Theoremabscji 9718 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)
 
Theoremreleabsi 9719 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)
 
Theoremabssubi 9720 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‘(𝐴𝐵)) = (abs‘(𝐵𝐴))
 
Theoremabsmuli 9721 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))
 
Theoremsqabsaddi 9722 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)))))
 
Theoremsqabssubi 9723 Square of absolute value of difference. (Contributed by Steve Rodriguez, 20-Jan-2007.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       ((abs‘(𝐴𝐵))↑2) = ((((abs‘𝐴)↑2) + ((abs‘𝐵)↑2)) − (2 · (ℜ‘(𝐴 · (∗‘𝐵)))))
 
Theoremabsdivapzi 9724 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)))
 
Theoremabstrii 9725 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))
 
Theoremabs3difi 9726 Absolute value of differences around common element. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       (abs‘(𝐴𝐵)) ≤ ((abs‘(𝐴𝐶)) + (abs‘(𝐶𝐵)))
 
Theoremabs3lemi 9727 Lemma involving absolute value of differences. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℝ       (((abs‘(𝐴𝐶)) < (𝐷 / 2) ∧ (abs‘(𝐶𝐵)) < (𝐷 / 2)) → (abs‘(𝐴𝐵)) < 𝐷)
 
Theoremrpsqrtcld 9728 The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (√‘𝐴) ∈ ℝ+)
 
Theoremsqrtgt0d 9729 The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → 0 < (√‘𝐴))
 
Theoremabsnidd 9730 A negative number is the negative of its own absolute value. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 ≤ 0)       (𝜑 → (abs‘𝐴) = -𝐴)
 
Theoremleabsd 9731 A real number is less than or equal to its absolute value. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴 ≤ (abs‘𝐴))
 
Theoremabsred 9732 Absolute value of a real number. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (abs‘𝐴) = (√‘(𝐴↑2)))
 
Theoremresqrtcld 9733 The square root of a nonnegative real is a real. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)       (𝜑 → (√‘𝐴) ∈ ℝ)
 
Theoremsqrtmsqd 9734 Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)       (𝜑 → (√‘(𝐴 · 𝐴)) = 𝐴)
 
Theoremsqrtsqd 9735 Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)       (𝜑 → (√‘(𝐴↑2)) = 𝐴)
 
Theoremsqrtge0d 9736 The square root of a nonnegative real is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)       (𝜑 → 0 ≤ (√‘𝐴))
 
Theoremabsidd 9737 A nonnegative number is its own absolute value. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)       (𝜑 → (abs‘𝐴) = 𝐴)
 
Theoremsqrtdivd 9738 Square root distributes over division. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (√‘(𝐴 / 𝐵)) = ((√‘𝐴) / (√‘𝐵)))
 
Theoremsqrtmuld 9739 Square root distributes over multiplication. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐵)       (𝜑 → (√‘(𝐴 · 𝐵)) = ((√‘𝐴) · (√‘𝐵)))
 
Theoremsqrtsq2d 9740 Relationship between square root and squares. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐵)       (𝜑 → ((√‘𝐴) = 𝐵𝐴 = (𝐵↑2)))
 
Theoremsqrtled 9741 Square root is monotonic. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐵)       (𝜑 → (𝐴𝐵 ↔ (√‘𝐴) ≤ (√‘𝐵)))
 
Theoremsqrtltd 9742 Square root is strictly monotonic. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐵)       (𝜑 → (𝐴 < 𝐵 ↔ (√‘𝐴) < (√‘𝐵)))
 
Theoremsqr11d 9743 The square root function is one-to-one. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐵)    &   (𝜑 → (√‘𝐴) = (√‘𝐵))       (𝜑𝐴 = 𝐵)
 
Theoremabsltd 9744 Absolute value and 'less than' relation. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → ((abs‘𝐴) < 𝐵 ↔ (-𝐵 < 𝐴𝐴 < 𝐵)))
 
Theoremabsled 9745 Absolute value and 'less than or equal to' relation. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → ((abs‘𝐴) ≤ 𝐵 ↔ (-𝐵𝐴𝐴𝐵)))
 
Theoremabssubge0d 9746 Absolute value of a nonnegative difference. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)       (𝜑 → (abs‘(𝐵𝐴)) = (𝐵𝐴))
 
Theoremabssuble0d 9747 Absolute value of a nonpositive difference. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)       (𝜑 → (abs‘(𝐴𝐵)) = (𝐵𝐴))
 
Theoremabsdifltd 9748 The absolute value of a difference and 'less than' relation. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → ((abs‘(𝐴𝐵)) < 𝐶 ↔ ((𝐵𝐶) < 𝐴𝐴 < (𝐵 + 𝐶))))
 
Theoremabsdifled 9749 The absolute value of a difference and 'less than or equal to' relation. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → ((abs‘(𝐴𝐵)) ≤ 𝐶 ↔ ((𝐵𝐶) ≤ 𝐴𝐴 ≤ (𝐵 + 𝐶))))
 
Theoremicodiamlt 9750 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‘(𝐶𝐷)) < (𝐵𝐴))
 
Theoremabscld 9751 Real closure of absolute value. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (abs‘𝐴) ∈ ℝ)
 
Theoremabsvalsqd 9752 Square of value of absolute value function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴)))
 
Theoremabsvalsq2d 9753 Square of value of absolute value function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → ((abs‘𝐴)↑2) = (((ℜ‘𝐴)↑2) + ((ℑ‘𝐴)↑2)))
 
Theoremabsge0d 9754 Absolute value is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → 0 ≤ (abs‘𝐴))
 
Theoremabsval2d 9755 Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (abs‘𝐴) = (√‘(((ℜ‘𝐴)↑2) + ((ℑ‘𝐴)↑2))))
 
Theoremabs00d 9756 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.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑 → (abs‘𝐴) = 0)       (𝜑𝐴 = 0)
 
Theoremabsne0d 9757 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.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)       (𝜑 → (abs‘𝐴) ≠ 0)
 
Theoremabsrpclapd 9758 The absolute value of a complex number apart from zero is a positive real. (Contributed by Jim Kingdon, 13-Aug-2021.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)       (𝜑 → (abs‘𝐴) ∈ ℝ+)
 
Theoremabsnegd 9759 Absolute value of negative. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (abs‘-𝐴) = (abs‘𝐴))
 
Theoremabscjd 9760 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.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (abs‘(∗‘𝐴)) = (abs‘𝐴))
 
Theoremreleabsd 9761 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.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (ℜ‘𝐴) ≤ (abs‘𝐴))
 
Theoremabsexpd 9762 Absolute value of positive integer exponentiation. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (abs‘(𝐴𝑁)) = ((abs‘𝐴)↑𝑁))
 
Theoremabssubd 9763 Swapping order of subtraction doesn't change the absolute value. Example of [Apostol] p. 363. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (abs‘(𝐴𝐵)) = (abs‘(𝐵𝐴)))
 
Theoremabsmuld 9764 Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (abs‘(𝐴 · 𝐵)) = ((abs‘𝐴) · (abs‘𝐵)))
 
Theoremabsdivapd 9765 Absolute value distributes over division. (Contributed by Jim Kingdon, 13-Aug-2021.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → (abs‘(𝐴 / 𝐵)) = ((abs‘𝐴) / (abs‘𝐵)))
 
Theoremabstrid 9766 Triangle inequality for absolute value. Proposition 10-3.7(h) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (abs‘(𝐴 + 𝐵)) ≤ ((abs‘𝐴) + (abs‘𝐵)))
 
Theoremabs2difd 9767 Difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((abs‘𝐴) − (abs‘𝐵)) ≤ (abs‘(𝐴𝐵)))
 
Theoremabs2dif2d 9768 Difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (abs‘(𝐴𝐵)) ≤ ((abs‘𝐴) + (abs‘𝐵)))
 
Theoremabs2difabsd 9769 Absolute value of difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (abs‘((abs‘𝐴) − (abs‘𝐵))) ≤ (abs‘(𝐴𝐵)))
 
Theoremabs3difd 9770 Absolute value of differences around common element. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (abs‘(𝐴𝐵)) ≤ ((abs‘(𝐴𝐶)) + (abs‘(𝐶𝐵))))
 
Theoremabs3lemd 9771 Lemma involving absolute value of differences. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑 → (abs‘(𝐴𝐶)) < (𝐷 / 2))    &   (𝜑 → (abs‘(𝐶𝐵)) < (𝐷 / 2))       (𝜑 → (abs‘(𝐴𝐵)) < 𝐷)
 
Theoremqdenre 9772* The rational numbers are dense in : any real number can be approximated with arbitrary precision by a rational number. For order theoretic density, see qbtwnre 9109. (Contributed by BJ, 15-Oct-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℚ (abs‘(𝑥𝐴)) < 𝐵)
 
3.8  Elementary limits and convergence
 
3.8.1  Limits
 
Syntaxcli 9773 Extend class notation with convergence relation for limits.
class
 
Definitiondf-clim 9774* Define the limit relation for complex number sequences. See clim 9776 for its relational expression. (Contributed by NM, 28-Aug-2005.)
⇝ = {⟨𝑓, 𝑦⟩ ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝑓𝑘) ∈ ℂ ∧ (abs‘((𝑓𝑘) − 𝑦)) < 𝑥))}
 
Theoremclimrel 9775 The limit relation is a relation. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
Rel ⇝
 
Theoremclim 9776* Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴. This means that for any real 𝑥, no matter how small, there always exists an integer 𝑗 such that the absolute difference of any later complex number in the sequence and the limit is less than 𝑥. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝜑𝐹𝑉)    &   ((𝜑𝑘 ∈ ℤ) → (𝐹𝑘) = 𝐵)       (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥))))
 
Theoremclimcl 9777 Closure of the limit of a sequence of complex numbers. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐹𝐴𝐴 ∈ ℂ)
 
Theoremclim2 9778* Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴, with more general quantifier restrictions than clim 9776. (Contributed by NM, 6-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑉)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)       (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥))))
 
Theoremclim2c 9779* Express the predicate 𝐹 converges to 𝐴. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑉)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   (𝜑𝐴 ∈ ℂ)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℂ)       (𝜑 → (𝐹𝐴 ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘(𝐵𝐴)) < 𝑥))
 
Theoremclim0 9780* Express the predicate 𝐹 converges to 0. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑉)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)       (𝜑 → (𝐹 ⇝ 0 ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘𝐵) < 𝑥)))
 
Theoremclim0c 9781* Express the predicate 𝐹 converges to 0. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑉)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℂ)       (𝜑 → (𝐹 ⇝ 0 ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘𝐵) < 𝑥))
 
Theoremclimi 9782* Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐶 ∈ ℝ+)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   (𝜑𝐹𝐴)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝐶))
 
Theoremclimi2 9783* Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐶 ∈ ℝ+)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   (𝜑𝐹𝐴)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘(𝐵𝐴)) < 𝐶)
 
Theoremclimi0 9784* Convergence of a sequence of complex numbers to zero. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐶 ∈ ℝ+)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   (𝜑𝐹 ⇝ 0)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘𝐵) < 𝐶)
 
Theoremclimconst 9785* An (eventually) constant sequence converges to its value. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑉)    &   (𝜑𝐴 ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)       (𝜑𝐹𝐴)
 
Theoremclimconst2 9786 A constant sequence converges to its value. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
(ℤ𝑀) ⊆ 𝑍    &   𝑍 ∈ V       ((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℤ) → (𝑍 × {𝐴}) ⇝ 𝐴)
 
Theoremclimz 9787 The zero sequence converges to zero. (Contributed by NM, 2-Oct-1999.) (Revised by Mario Carneiro, 31-Jan-2014.)
(ℤ × {0}) ⇝ 0
 
Theoremclimuni 9788 An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 2-Oct-1999.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)
((𝐹𝐴𝐹𝐵) → 𝐴 = 𝐵)
 
Theoremfclim 9789 The limit relation is function-like, and with range the complex numbers. (Contributed by Mario Carneiro, 31-Jan-2014.)
⇝ :dom ⇝ ⟶ℂ
 
Theoremclimdm 9790 Two ways to express that a function has a limit. (The expression ( ⇝ ‘𝐹) is sometimes useful as a shorthand for "the unique limit of the function 𝐹"). (Contributed by Mario Carneiro, 18-Mar-2014.)
(𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ ( ⇝ ‘𝐹))
 
Theoremclimeu 9791* An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.)
(𝐹𝐴 → ∃!𝑥 𝐹𝑥)
 
Theoremclimreu 9792* An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.)
(𝐹𝐴 → ∃!𝑥 ∈ ℂ 𝐹𝑥)
 
Theoremclimmo 9793* An infinite sequence of complex numbers converges to at most one limit. (Contributed by Mario Carneiro, 13-Jul-2013.)
∃*𝑥 𝐹𝑥
 
Theoremclimeq 9794* Two functions that are eventually equal to one another have the same limit. (Contributed by Mario Carneiro, 5-Nov-2013.) (Revised by Mario Carneiro, 31-Jan-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐹𝑉)    &   (𝜑𝐺𝑊)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐺𝑘))       (𝜑 → (𝐹𝐴𝐺𝐴))
 
Theoremclimmpt 9795* Exhibit a function 𝐺 with the same convergence properties as the not-quite-function 𝐹. (Contributed by Mario Carneiro, 31-Jan-2014.)
𝑍 = (ℤ𝑀)    &   𝐺 = (𝑘𝑍 ↦ (𝐹𝑘))       ((𝑀 ∈ ℤ ∧ 𝐹𝑉) → (𝐹𝐴𝐺𝐴))
 
Theorem2clim 9796* If two sequences converge to each other, they converge to the same limit. (Contributed by NM, 24-Dec-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐺𝑉)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ ℂ)    &   (𝜑 → ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − (𝐺𝑘))) < 𝑥)    &   (𝜑𝐹𝐴)       (𝜑𝐺𝐴)
 
Theoremclimshftlemg 9797 A shifted function converges if the original function converges. (Contributed by Mario Carneiro, 5-Nov-2013.)
((𝑀 ∈ ℤ ∧ 𝐹𝑉) → (𝐹𝐴 → (𝐹 shift 𝑀) ⇝ 𝐴))
 
Theoremclimres 9798 A function restricted to upper integers converges iff the original function converges. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 31-Jan-2014.)
((𝑀 ∈ ℤ ∧ 𝐹𝑉) → ((𝐹 ↾ (ℤ𝑀)) ⇝ 𝐴𝐹𝐴))
 
Theoremclimshft 9799 A shifted function converges iff the original function converges. (Contributed by NM, 16-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
((𝑀 ∈ ℤ ∧ 𝐹𝑉) → ((𝐹 shift 𝑀) ⇝ 𝐴𝐹𝐴))
 
Theoremiserclim0 9800 The zero series converges to zero. (Contributed by Jim Kingdon, 19-Aug-2021.)
(𝑀 ∈ ℤ → seq𝑀( + , ((ℤ𝑀) × {0}), ℂ) ⇝ 0)
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