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Theorem List for Metamath Proof Explorer - 14101-14200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremabsori 14101 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)

Theoremabsrei 14102 Absolute value of a real number. (Contributed by NM, 3-Aug-1999.)
𝐴 ∈ ℝ       (abs‘𝐴) = (√‘(𝐴↑2))

Theoremsqrtpclii 14103 The square root of a positive real is a real. (Contributed by Mario Carneiro, 6-Sep-2013.)
𝐴 ∈ ℝ    &   0 < 𝐴       (√‘𝐴) ∈ ℝ

Theoremsqrtgt0ii 14104 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 14105 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 14106 Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (√‘(𝐴 · 𝐵)) = ((√‘𝐴) · (√‘𝐵)))

Theoremsqrtmulii 14107 Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   0 ≤ 𝐴    &   0 ≤ 𝐵       (√‘(𝐴 · 𝐵)) = ((√‘𝐴) · (√‘𝐵))

Theoremsqrtmsq2i 14108 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 14109 Square root is monotonic. (Contributed by NM, 3-Aug-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴𝐵 ↔ (√‘𝐴) ≤ (√‘𝐵)))

Theoremsqrtlti 14110 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 14111 Absolute value and 'less than' relation. (Contributed by NM, 6-Apr-2005.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((abs‘𝐴) < 𝐵 ↔ (-𝐵 < 𝐴𝐴 < 𝐵))

Theoremabslei 14112 Absolute value and 'less than or equal to' relation. (Contributed by NM, 6-Apr-2005.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((abs‘𝐴) ≤ 𝐵 ↔ (-𝐵𝐴𝐴𝐵))

Theoremabsvalsqi 14113 Square of value of absolute value function. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ ℂ       ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴))

Theoremabsvalsq2i 14114 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 14115 Real closure of absolute value. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℂ       (abs‘𝐴) ∈ ℝ

Theoremabsge0i 14116 Absolute value is nonnegative. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℂ       0 ≤ (abs‘𝐴)

Theoremabsval2i 14117 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 14118 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)

Theoremabsgt0i 14119 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 14120 Absolute value of negative. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℂ       (abs‘-𝐴) = (abs‘𝐴)

Theoremabscji 14121 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 14122 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 14123 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 14124 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 14125 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 14126 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)))))

Theoremabsdivzi 14127 Absolute value distributes over division. (Contributed by NM, 26-Mar-2005.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 ≠ 0 → (abs‘(𝐴 / 𝐵)) = ((abs‘𝐴) / (abs‘𝐵)))

Theoremabstrii 14128 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 14129 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 14130 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 14131 The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (√‘𝐴) ∈ ℝ+)

Theoremsqrtgt0d 14132 The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → 0 < (√‘𝐴))

Theoremabsnidd 14133 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 14134 A real number is less than or equal to its absolute value. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴 ≤ (abs‘𝐴))

Theoremabsord 14135 The absolute value of a real number is either that number or its negative. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → ((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴))

Theoremabsred 14136 Absolute value of a real number. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (abs‘𝐴) = (√‘(𝐴↑2)))

Theoremresqrtcld 14137 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 14138 Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)       (𝜑 → (√‘(𝐴 · 𝐴)) = 𝐴)

Theoremsqrtsqd 14139 Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)       (𝜑 → (√‘(𝐴↑2)) = 𝐴)

Theoremsqrtge0d 14140 The square root of a nonnegative real is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)       (𝜑 → 0 ≤ (√‘𝐴))

Theoremsqrtnegd 14141 The square root of a negative number. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)       (𝜑 → (√‘-𝐴) = (i · (√‘𝐴)))

Theoremabsidd 14142 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 14143 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 14144 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 14145 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 14146 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 14147 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 14148 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 14149 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 14150 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 14151 Absolute value of a nonnegative difference. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)       (𝜑 → (abs‘(𝐵𝐴)) = (𝐵𝐴))

Theoremabssuble0d 14152 Absolute value of a nonpositive difference. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)       (𝜑 → (abs‘(𝐴𝐵)) = (𝐵𝐴))

Theoremabsdifltd 14153 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 14154 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 14155 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 14156 Real closure of absolute value. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (abs‘𝐴) ∈ ℝ)

Theoremsqrtcld 14157 Closure of the square root function over the complex numbers. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (√‘𝐴) ∈ ℂ)

Theoremsqrtrege0d 14158 The real part of the square root function is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → 0 ≤ (ℜ‘(√‘𝐴)))

Theoremsqsqrtd 14159 Square root theorem. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → ((√‘𝐴)↑2) = 𝐴)

Theoremmsqsqrtd 14160 Square root theorem. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → ((√‘𝐴) · (√‘𝐴)) = 𝐴)

Theoremsqr00d 14161 A square root is zero iff its argument is 0. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑 → (√‘𝐴) = 0)       (𝜑𝐴 = 0)

Theoremabsvalsqd 14162 Square of value of absolute value function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴)))

Theoremabsvalsq2d 14163 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 14164 Absolute value is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → 0 ≤ (abs‘𝐴))

Theoremabsval2d 14165 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 14166 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 14167 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)

Theoremabsrpcld 14168 The absolute value of a nonzero number is a positive real. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)       (𝜑 → (abs‘𝐴) ∈ ℝ+)

Theoremabsnegd 14169 Absolute value of negative. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (abs‘-𝐴) = (abs‘𝐴))

Theoremabscjd 14170 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 14171 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 14172 Absolute value of positive integer exponentiation. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (abs‘(𝐴𝑁)) = ((abs‘𝐴)↑𝑁))

Theoremabssubd 14173 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 14174 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)))

Theoremabsdivd 14175 Absolute value distributes over division. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → (abs‘(𝐴 / 𝐵)) = ((abs‘𝐴) / (abs‘𝐵)))

Theoremabstrid 14176 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 14177 Difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((abs‘𝐴) − (abs‘𝐵)) ≤ (abs‘(𝐴𝐵)))

Theoremabs2dif2d 14178 Difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (abs‘(𝐴𝐵)) ≤ ((abs‘𝐴) + (abs‘𝐵)))

Theoremabs2difabsd 14179 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 14180 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 14181 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)

5.10  Elementary limits and convergence

5.10.1  Superior limit (lim sup)

Syntaxclsp 14182 Extend class notation to include the limsup function.
class lim sup

Definitiondf-limsup 14183* Define the superior limit of an infinite sequence of extended real numbers. Definition 12-4.1 of [Gleason] p. 175. See limsupval 14186 for its value. (Contributed by NM, 26-Oct-2005.) (Revised by AV, 11-Sep-2020.)
lim sup = (𝑥 ∈ V ↦ inf(ran (𝑘 ∈ ℝ ↦ sup(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ))

Theoremlimsupgord 14184 Ordering property of the superior limit function. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → sup(((𝐹 “ (𝐵[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ sup(((𝐹 “ (𝐴[,)+∞)) ∩ ℝ*), ℝ*, < ))

Theoremlimsupcl 14185 Closure of the superior limit. (Contributed by NM, 26-Oct-2005.) (Revised by AV, 12-Sep-2020.)
(𝐹𝑉 → (lim sup‘𝐹) ∈ ℝ*)

Theoremlimsupval 14186* The superior limit of an infinite sequence 𝐹 of extended real numbers, which is the infimum of the set of suprema of all upper infinite subsequences of 𝐹. Definition 12-4.1 of [Gleason] p. 175. (Contributed by NM, 26-Oct-2005.) (Revised by AV, 12-Sep-2014.)
𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))       (𝐹𝑉 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < ))

Theoremlimsupgf 14187* Closure of the superior limit function. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))       𝐺:ℝ⟶ℝ*

Theoremlimsupgval 14188* Value of the superior limit function. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))       (𝑀 ∈ ℝ → (𝐺𝑀) = sup(((𝐹 “ (𝑀[,)+∞)) ∩ ℝ*), ℝ*, < ))

Theoremlimsupgle 14189* The defining property of the superior limit function. (Contributed by Mario Carneiro, 5-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))       (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ*) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ*) → ((𝐺𝐶) ≤ 𝐴 ↔ ∀𝑗𝐵 (𝐶𝑗 → (𝐹𝑗) ≤ 𝐴)))

Theoremlimsuple 14190* The defining property of the superior limit. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)
𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))       ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ*𝐴 ∈ ℝ*) → (𝐴 ≤ (lim sup‘𝐹) ↔ ∀𝑗 ∈ ℝ 𝐴 ≤ (𝐺𝑗)))

Theoremlimsuplt 14191* The defining property of the superior limit. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)
𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))       ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ*𝐴 ∈ ℝ*) → ((lim sup‘𝐹) < 𝐴 ↔ ∃𝑗 ∈ ℝ (𝐺𝑗) < 𝐴))

Theoremlimsupval2 14192* The superior limit, relativized to an unbounded set. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)
𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))    &   (𝜑𝐹𝑉)    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑 → sup(𝐴, ℝ*, < ) = +∞)       (𝜑 → (lim sup‘𝐹) = inf((𝐺𝐴), ℝ*, < ))

Theoremlimsupgre 14193* If a sequence of real numbers has upper bounded limit supremum, then all the partial suprema are real. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)
𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))    &   𝑍 = (ℤ𝑀)       ((𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶ℝ ∧ (lim sup‘𝐹) < +∞) → 𝐺:ℝ⟶ℝ)

Theoremlimsupbnd1 14194* If a sequence is eventually at most 𝐴, then the limsup is also at most 𝐴. (The converse is only true if the less or equal is replaced by strictly less than; consider the sequence 1 / 𝑛 which is never less or equal to zero even though the limsup is.) (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)
(𝜑𝐵 ⊆ ℝ)    &   (𝜑𝐹:𝐵⟶ℝ*)    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝐴))       (𝜑 → (lim sup‘𝐹) ≤ 𝐴)

Theoremlimsupbnd2 14195* If a sequence is eventually greater than 𝐴, then the limsup is also greater than 𝐴. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)
(𝜑𝐵 ⊆ ℝ)    &   (𝜑𝐹:𝐵⟶ℝ*)    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑 → sup(𝐵, ℝ*, < ) = +∞)    &   (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗𝐴 ≤ (𝐹𝑗)))       (𝜑𝐴 ≤ (lim sup‘𝐹))

5.10.2  Limits

Syntaxcli 14196 Extend class notation with convergence relation for limits.
class

Syntaxcrli 14197 Extend class notation with real convergence relation for limits.
class 𝑟

Syntaxco1 14198 Extend class notation with the set of all eventually bounded functions.
class 𝑂(1)

Syntaxclo1 14199 Extend class notation with the set of all eventually upper bounded functions.
class ≤𝑂(1)

Definitiondf-clim 14200* Define the limit relation for complex number sequences. See clim 14206 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))}

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