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Theorem List for Intuitionistic Logic Explorer - 10901-11000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremabssub 10901 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‘(𝐴𝐵)) = (abs‘(𝐵𝐴)))
 
Theoremabssubge0 10902 Absolute value of a nonnegative difference. (Contributed by NM, 14-Feb-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (abs‘(𝐵𝐴)) = (𝐵𝐴))
 
Theoremabssuble0 10903 Absolute value of a nonpositive difference. (Contributed by FL, 3-Jan-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (abs‘(𝐴𝐵)) = (𝐵𝐴))
 
Theoremabstri 10904 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)))
 
Theoremabs3dif 10905 Absolute value of differences around common element. (Contributed by FL, 9-Oct-2006.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (abs‘(𝐴𝐵)) ≤ ((abs‘(𝐴𝐶)) + (abs‘(𝐶𝐵))))
 
Theoremabs2dif 10906 Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘𝐴) − (abs‘𝐵)) ≤ (abs‘(𝐴𝐵)))
 
Theoremabs2dif2 10907 Difference of absolute values. (Contributed by Mario Carneiro, 14-Apr-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (abs‘(𝐴𝐵)) ≤ ((abs‘𝐴) + (abs‘𝐵)))
 
Theoremabs2difabs 10908 Absolute value of difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (abs‘((abs‘𝐴) − (abs‘𝐵))) ≤ (abs‘(𝐴𝐵)))
 
Theoremrecan 10909* 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))
 
Theoremabsf 10910 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
 
Theoremabs3lem 10911 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))
 
Theoremfzomaxdiflem 10912 Lemma for fzomaxdif 10913. (Contributed by Stefan O'Rear, 6-Sep-2015.)
(((𝐴 ∈ (𝐶..^𝐷) ∧ 𝐵 ∈ (𝐶..^𝐷)) ∧ 𝐴𝐵) → (abs‘(𝐵𝐴)) ∈ (0..^(𝐷𝐶)))
 
Theoremfzomaxdif 10913 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‘(𝐴𝐵)) ∈ (0..^(𝐷𝐶)))
 
Theoremcau3lem 10914* Lemma for cau3 10915. (Contributed by Mario Carneiro, 15-Feb-2014.) (Revised by Mario Carneiro, 1-May-2014.)
𝑍 ⊆ ℤ    &   (𝜏𝜓)    &   ((𝐹𝑘) = (𝐹𝑗) → (𝜓𝜒))    &   ((𝐹𝑘) = (𝐹𝑚) → (𝜓𝜃))    &   ((𝜑𝜒𝜓) → (𝐺‘((𝐹𝑗)𝐷(𝐹𝑘))) = (𝐺‘((𝐹𝑘)𝐷(𝐹𝑗))))    &   ((𝜑𝜃𝜒) → (𝐺‘((𝐹𝑚)𝐷(𝐹𝑗))) = (𝐺‘((𝐹𝑗)𝐷(𝐹𝑚))))    &   ((𝜑 ∧ (𝜓𝜃) ∧ (𝜒𝑥 ∈ ℝ)) → (((𝐺‘((𝐹𝑘)𝐷(𝐹𝑗))) < (𝑥 / 2) ∧ (𝐺‘((𝐹𝑗)𝐷(𝐹𝑚))) < (𝑥 / 2)) → (𝐺‘((𝐹𝑘)𝐷(𝐹𝑚))) < 𝑥))       (𝜑 → (∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝜏 ∧ (𝐺‘((𝐹𝑘)𝐷(𝐹𝑗))) < 𝑥) ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝜏 ∧ ∀𝑚 ∈ (ℤ𝑘)(𝐺‘((𝐹𝑘)𝐷(𝐹𝑚))) < 𝑥)))
 
Theoremcau3 10915* 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 10788 and friends.) (Contributed by Mario Carneiro, 15-Feb-2014.)
𝑍 = (ℤ𝑀)       (∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥) ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ ∀𝑚 ∈ (ℤ𝑘)(abs‘((𝐹𝑘) − (𝐹𝑚))) < 𝑥))
 
Theoremcau4 10916* Change the base of a Cauchy criterion. (Contributed by Mario Carneiro, 18-Mar-2014.)
𝑍 = (ℤ𝑀)    &   𝑊 = (ℤ𝑁)       (𝑁𝑍 → (∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥) ↔ ∀𝑥 ∈ ℝ+𝑗𝑊𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥)))
 
Theoremcaubnd2 10917* A Cauchy sequence of complex numbers is eventually bounded. (Contributed by Mario Carneiro, 14-Feb-2014.)
𝑍 = (ℤ𝑀)       (∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥) → ∃𝑦 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘(𝐹𝑘)) < 𝑦)
 
Theoremamgm2 10918 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))
 
Theoremsqrtthi 10919 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)
 
Theoremsqrtcli 10920 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)
 
Theoremsqrtgt0i 10921 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))
 
Theoremsqrtmsqi 10922 Square root of square. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℝ       (0 ≤ 𝐴 → (√‘(𝐴 · 𝐴)) = 𝐴)
 
Theoremsqrtsqi 10923 Square root of square. (Contributed by NM, 11-Aug-1999.)
𝐴 ∈ ℝ       (0 ≤ 𝐴 → (√‘(𝐴↑2)) = 𝐴)
 
Theoremsqsqrti 10924 Square of square root. (Contributed by NM, 11-Aug-1999.)
𝐴 ∈ ℝ       (0 ≤ 𝐴 → ((√‘𝐴)↑2) = 𝐴)
 
Theoremsqrtge0i 10925 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))
 
Theoremabsidi 10926 A nonnegative number is its own absolute value. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℝ       (0 ≤ 𝐴 → (abs‘𝐴) = 𝐴)
 
Theoremabsnidi 10927 A negative number is the negative of its own absolute value. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℝ       (𝐴 ≤ 0 → (abs‘𝐴) = -𝐴)
 
Theoremleabsi 10928 A real number is less than or equal to its absolute value. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℝ       𝐴 ≤ (abs‘𝐴)
 
Theoremabsrei 10929 Absolute value of a real number. (Contributed by NM, 3-Aug-1999.)
𝐴 ∈ ℝ       (abs‘𝐴) = (√‘(𝐴↑2))
 
Theoremsqrtpclii 10930 The square root of a positive real is a real. (Contributed by Mario Carneiro, 6-Sep-2013.)
𝐴 ∈ ℝ    &   0 < 𝐴       (√‘𝐴) ∈ ℝ
 
Theoremsqrtgt0ii 10931 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 10932 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 10933 Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (√‘(𝐴 · 𝐵)) = ((√‘𝐴) · (√‘𝐵)))
 
Theoremsqrtmulii 10934 Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   0 ≤ 𝐴    &   0 ≤ 𝐵       (√‘(𝐴 · 𝐵)) = ((√‘𝐴) · (√‘𝐵))
 
Theoremsqrtmsq2i 10935 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 10936 Square root is monotonic. (Contributed by NM, 3-Aug-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴𝐵 ↔ (√‘𝐴) ≤ (√‘𝐵)))
 
Theoremsqrtlti 10937 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 10938 Absolute value and 'less than' relation. (Contributed by NM, 6-Apr-2005.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((abs‘𝐴) < 𝐵 ↔ (-𝐵 < 𝐴𝐴 < 𝐵))
 
Theoremabslei 10939 Absolute value and 'less than or equal to' relation. (Contributed by NM, 6-Apr-2005.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((abs‘𝐴) ≤ 𝐵 ↔ (-𝐵𝐴𝐴𝐵))
 
Theoremabsvalsqi 10940 Square of value of absolute value function. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ ℂ       ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴))
 
Theoremabsvalsq2i 10941 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 10942 Real closure of absolute value. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℂ       (abs‘𝐴) ∈ ℝ
 
Theoremabsge0i 10943 Absolute value is nonnegative. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℂ       0 ≤ (abs‘𝐴)
 
Theoremabsval2i 10944 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 10945 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 10946 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 10947 Absolute value of negative. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℂ       (abs‘-𝐴) = (abs‘𝐴)
 
Theoremabscji 10948 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 10949 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 10950 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 10951 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 10952 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 10953 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 10954 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 10955 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 10956 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 10957 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 10958 The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (√‘𝐴) ∈ ℝ+)
 
Theoremsqrtgt0d 10959 The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → 0 < (√‘𝐴))
 
Theoremabsnidd 10960 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 10961 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 10962 Absolute value of a real number. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (abs‘𝐴) = (√‘(𝐴↑2)))
 
Theoremresqrtcld 10963 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 10964 Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)       (𝜑 → (√‘(𝐴 · 𝐴)) = 𝐴)
 
Theoremsqrtsqd 10965 Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)       (𝜑 → (√‘(𝐴↑2)) = 𝐴)
 
Theoremsqrtge0d 10966 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 10967 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 10968 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 10969 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 10970 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 10971 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 10972 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 10973 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 10974 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 10975 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 10976 Absolute value of a nonnegative difference. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)       (𝜑 → (abs‘(𝐵𝐴)) = (𝐵𝐴))
 
Theoremabssuble0d 10977 Absolute value of a nonpositive difference. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)       (𝜑 → (abs‘(𝐴𝐵)) = (𝐵𝐴))
 
Theoremabsdifltd 10978 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 10979 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 10980 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 10981 Real closure of absolute value. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (abs‘𝐴) ∈ ℝ)
 
Theoremabsvalsqd 10982 Square of value of absolute value function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴)))
 
Theoremabsvalsq2d 10983 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 10984 Absolute value is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → 0 ≤ (abs‘𝐴))
 
Theoremabsval2d 10985 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 10986 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 10987 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 10988 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 10989 Absolute value of negative. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (abs‘-𝐴) = (abs‘𝐴))
 
Theoremabscjd 10990 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 10991 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 10992 Absolute value of positive integer exponentiation. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (abs‘(𝐴𝑁)) = ((abs‘𝐴)↑𝑁))
 
Theoremabssubd 10993 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 10994 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 10995 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 10996 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 10997 Difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((abs‘𝐴) − (abs‘𝐵)) ≤ (abs‘(𝐴𝐵)))
 
Theoremabs2dif2d 10998 Difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (abs‘(𝐴𝐵)) ≤ ((abs‘𝐴) + (abs‘𝐵)))
 
Theoremabs2difabsd 10999 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 11000 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))))
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