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Theorem List for Intuitionistic Logic Explorer - 10801-10900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremabsge0i 10801 Absolute value is nonnegative. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℂ       0 ≤ (abs‘𝐴)

Theoremabsval2i 10802 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 10803 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 10804 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 10805 Absolute value of negative. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℂ       (abs‘-𝐴) = (abs‘𝐴)

Theoremabscji 10806 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 10807 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 10808 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 10809 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 10810 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 10811 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 10812 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 10813 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 10814 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 10815 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 10816 The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (√‘𝐴) ∈ ℝ+)

Theoremsqrtgt0d 10817 The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → 0 < (√‘𝐴))

Theoremabsnidd 10818 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 10819 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 10820 Absolute value of a real number. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (abs‘𝐴) = (√‘(𝐴↑2)))

Theoremresqrtcld 10821 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 10822 Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)       (𝜑 → (√‘(𝐴 · 𝐴)) = 𝐴)

Theoremsqrtsqd 10823 Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)       (𝜑 → (√‘(𝐴↑2)) = 𝐴)

Theoremsqrtge0d 10824 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 10825 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 10826 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 10827 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 10828 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 10829 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 10830 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 10831 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 10832 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 10833 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 10834 Absolute value of a nonnegative difference. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)       (𝜑 → (abs‘(𝐵𝐴)) = (𝐵𝐴))

Theoremabssuble0d 10835 Absolute value of a nonpositive difference. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)       (𝜑 → (abs‘(𝐴𝐵)) = (𝐵𝐴))

Theoremabsdifltd 10836 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 10837 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 10838 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 10839 Real closure of absolute value. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (abs‘𝐴) ∈ ℝ)

Theoremabsvalsqd 10840 Square of value of absolute value function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴)))

Theoremabsvalsq2d 10841 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 10842 Absolute value is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → 0 ≤ (abs‘𝐴))

Theoremabsval2d 10843 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 10844 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 10845 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 10846 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 10847 Absolute value of negative. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (abs‘-𝐴) = (abs‘𝐴))

Theoremabscjd 10848 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 10849 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 10850 Absolute value of positive integer exponentiation. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (abs‘(𝐴𝑁)) = ((abs‘𝐴)↑𝑁))

Theoremabssubd 10851 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 10852 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 10853 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 10854 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 10855 Difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((abs‘𝐴) − (abs‘𝐵)) ≤ (abs‘(𝐴𝐵)))

Theoremabs2dif2d 10856 Difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (abs‘(𝐴𝐵)) ≤ ((abs‘𝐴) + (abs‘𝐵)))

Theoremabs2difabsd 10857 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 10858 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 10859 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 10860* 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 9921. (Contributed by BJ, 15-Oct-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℚ (abs‘(𝑥𝐴)) < 𝐵)

3.7.5  The maximum of two real numbers

Theoremmaxcom 10861 The maximum of two reals is commutative. Lemma 3.9 of [Geuvers], p. 10. (Contributed by Jim Kingdon, 21-Dec-2021.)
sup({𝐴, 𝐵}, ℝ, < ) = sup({𝐵, 𝐴}, ℝ, < )

Theoremmaxabsle 10862 An upper bound for {𝐴, 𝐵}. (Contributed by Jim Kingdon, 20-Dec-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ≤ (((𝐴 + 𝐵) + (abs‘(𝐴𝐵))) / 2))

Theoremmaxleim 10863 Value of maximum when we know which number is larger. (Contributed by Jim Kingdon, 21-Dec-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐵 → sup({𝐴, 𝐵}, ℝ, < ) = 𝐵))

Theoremmaxabslemab 10864 Lemma for maxabs 10867. A variation of maxleim 10863- that is, if we know which of two real numbers is larger, we know the maximum of the two. (Contributed by Jim Kingdon, 21-Dec-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑 → (((𝐴 + 𝐵) + (abs‘(𝐴𝐵))) / 2) = 𝐵)

Theoremmaxabslemlub 10865 Lemma for maxabs 10867. A least upper bound for {𝐴, 𝐵}. (Contributed by Jim Kingdon, 20-Dec-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐶 < (((𝐴 + 𝐵) + (abs‘(𝐴𝐵))) / 2))       (𝜑 → (𝐶 < 𝐴𝐶 < 𝐵))

Theoremmaxabslemval 10866* Lemma for maxabs 10867. Value of the supremum. (Contributed by Jim Kingdon, 22-Dec-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((((𝐴 + 𝐵) + (abs‘(𝐴𝐵))) / 2) ∈ ℝ ∧ ∀𝑥 ∈ {𝐴, 𝐵} ¬ (((𝐴 + 𝐵) + (abs‘(𝐴𝐵))) / 2) < 𝑥 ∧ ∀𝑥 ∈ ℝ (𝑥 < (((𝐴 + 𝐵) + (abs‘(𝐴𝐵))) / 2) → ∃𝑧 ∈ {𝐴, 𝐵}𝑥 < 𝑧)))

Theoremmaxabs 10867 Maximum of two real numbers in terms of absolute value. (Contributed by Jim Kingdon, 20-Dec-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → sup({𝐴, 𝐵}, ℝ, < ) = (((𝐴 + 𝐵) + (abs‘(𝐴𝐵))) / 2))

Theoremmaxcl 10868 The maximum of two real numbers is a real number. (Contributed by Jim Kingdon, 22-Dec-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → sup({𝐴, 𝐵}, ℝ, < ) ∈ ℝ)

Theoremmaxle1 10869 The maximum of two reals is no smaller than the first real. Lemma 3.10 of [Geuvers], p. 10. (Contributed by Jim Kingdon, 21-Dec-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ≤ sup({𝐴, 𝐵}, ℝ, < ))

Theoremmaxle2 10870 The maximum of two reals is no smaller than the second real. Lemma 3.10 of [Geuvers], p. 10. (Contributed by Jim Kingdon, 21-Dec-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ≤ sup({𝐴, 𝐵}, ℝ, < ))

Theoremmaxleast 10871 The maximum of two reals is a least upper bound. Lemma 3.11 of [Geuvers], p. 10. (Contributed by Jim Kingdon, 22-Dec-2021.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴𝐶𝐵𝐶)) → sup({𝐴, 𝐵}, ℝ, < ) ≤ 𝐶)

Theoremmaxleastb 10872 Two ways of saying the maximum of two numbers is less than or equal to a third. (Contributed by Jim Kingdon, 31-Jan-2022.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (sup({𝐴, 𝐵}, ℝ, < ) ≤ 𝐶 ↔ (𝐴𝐶𝐵𝐶)))

Theoremmaxleastlt 10873 The maximum as a least upper bound, in terms of less than. (Contributed by Jim Kingdon, 9-Feb-2022.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐶 < sup({𝐴, 𝐵}, ℝ, < ))) → (𝐶 < 𝐴𝐶 < 𝐵))

Theoremmaxleb 10874 Equivalence of and being equal to the maximum of two reals. Lemma 3.12 of [Geuvers], p. 10. (Contributed by Jim Kingdon, 21-Dec-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐵 ↔ sup({𝐴, 𝐵}, ℝ, < ) = 𝐵))

Theoremdfabsmax 10875 Absolute value of a real number in terms of maximum. Definition 3.13 of [Geuvers], p. 11. (Contributed by BJ and Jim Kingdon, 21-Dec-2021.)
(𝐴 ∈ ℝ → (abs‘𝐴) = sup({𝐴, -𝐴}, ℝ, < ))

Theoremmaxltsup 10876 Two ways of saying the maximum of two numbers is less than a third. (Contributed by Jim Kingdon, 10-Feb-2022.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (sup({𝐴, 𝐵}, ℝ, < ) < 𝐶 ↔ (𝐴 < 𝐶𝐵 < 𝐶)))

Theoremmax0addsup 10877 The sum of the positive and negative part functions is the absolute value function over the reals. (Contributed by Jim Kingdon, 30-Jan-2022.)
(𝐴 ∈ ℝ → (sup({𝐴, 0}, ℝ, < ) + sup({-𝐴, 0}, ℝ, < )) = (abs‘𝐴))

Theoremrexanre 10878* Combine two different upper real properties into one. (Contributed by Mario Carneiro, 8-May-2016.)
(𝐴 ⊆ ℝ → (∃𝑗 ∈ ℝ ∀𝑘𝐴 (𝑗𝑘 → (𝜑𝜓)) ↔ (∃𝑗 ∈ ℝ ∀𝑘𝐴 (𝑗𝑘𝜑) ∧ ∃𝑗 ∈ ℝ ∀𝑘𝐴 (𝑗𝑘𝜓))))

Theoremrexico 10879* Restrict the base of an upper real quantifier to an upper real set. (Contributed by Mario Carneiro, 12-May-2016.)
((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) → (∃𝑗 ∈ (𝐵[,)+∞)∀𝑘𝐴 (𝑗𝑘𝜑) ↔ ∃𝑗 ∈ ℝ ∀𝑘𝐴 (𝑗𝑘𝜑)))

Theoremmaxclpr 10880 The maximum of two real numbers is one of those numbers if and only if dichotomy (𝐴𝐵𝐵𝐴) holds. For example, this can be combined with zletric 8996 if one is dealing with integers, but real number dichotomy in general does not follow from our axioms. (Contributed by Jim Kingdon, 1-Feb-2022.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (sup({𝐴, 𝐵}, ℝ, < ) ∈ {𝐴, 𝐵} ↔ (𝐴𝐵𝐵𝐴)))

Theoremrpmaxcl 10881 The maximum of two positive real numbers is a positive real number. (Contributed by Jim Kingdon, 10-Nov-2023.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ+) → sup({𝐴, 𝐵}, ℝ, < ) ∈ ℝ+)

Theoremzmaxcl 10882 The maximum of two integers is an integer. (Contributed by Jim Kingdon, 27-Sep-2022.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → sup({𝐴, 𝐵}, ℝ, < ) ∈ ℤ)

Theorem2zsupmax 10883 Two ways to express the maximum of two integers. Because order of integers is decidable, we have more flexibility than for real numbers. (Contributed by Jim Kingdon, 22-Jan-2023.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → sup({𝐴, 𝐵}, ℝ, < ) = if(𝐴𝐵, 𝐵, 𝐴))

Theoremfimaxre2 10884* A nonempty finite set of real numbers has an upper bound. (Contributed by Jeff Madsen, 27-May-2011.) (Revised by Mario Carneiro, 13-Feb-2014.)
((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥)

Theoremnegfi 10885* The negation of a finite set of real numbers is finite. (Contributed by AV, 9-Aug-2020.)
((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin) → {𝑛 ∈ ℝ ∣ -𝑛𝐴} ∈ Fin)

3.7.6  The minimum of two real numbers

Theoremmincom 10886 The minimum of two reals is commutative. (Contributed by Jim Kingdon, 8-Feb-2021.)
inf({𝐴, 𝐵}, ℝ, < ) = inf({𝐵, 𝐴}, ℝ, < )

Theoremminmax 10887 Minimum expressed in terms of maximum. (Contributed by Jim Kingdon, 8-Feb-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → inf({𝐴, 𝐵}, ℝ, < ) = -sup({-𝐴, -𝐵}, ℝ, < ))

Theoremmincl 10888 The minumum of two real numbers is a real number. (Contributed by Jim Kingdon, 25-Apr-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → inf({𝐴, 𝐵}, ℝ, < ) ∈ ℝ)

Theoremmin1inf 10889 The minimum of two numbers is less than or equal to the first. (Contributed by Jim Kingdon, 8-Feb-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → inf({𝐴, 𝐵}, ℝ, < ) ≤ 𝐴)

Theoremmin2inf 10890 The minimum of two numbers is less than or equal to the second. (Contributed by Jim Kingdon, 9-Feb-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → inf({𝐴, 𝐵}, ℝ, < ) ≤ 𝐵)

Theoremlemininf 10891 Two ways of saying a number is less than or equal to the minimum of two others. (Contributed by NM, 3-Aug-2007.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ inf({𝐵, 𝐶}, ℝ, < ) ↔ (𝐴𝐵𝐴𝐶)))

Theoremltmininf 10892 Two ways of saying a number is less than the minimum of two others. (Contributed by Jim Kingdon, 10-Feb-2022.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < inf({𝐵, 𝐶}, ℝ, < ) ↔ (𝐴 < 𝐵𝐴 < 𝐶)))

Theoremminabs 10893 The minimum of two real numbers in terms of absolute value. (Contributed by Jim Kingdon, 15-May-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → inf({𝐴, 𝐵}, ℝ, < ) = (((𝐴 + 𝐵) − (abs‘(𝐴𝐵))) / 2))

Theoremminclpr 10894 The minimum of two real numbers is one of those numbers if and only if dichotomy (𝐴𝐵𝐵𝐴) holds. For example, this can be combined with zletric 8996 if one is dealing with integers, but real number dichotomy in general does not follow from our axioms. (Contributed by Jim Kingdon, 23-May-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (inf({𝐴, 𝐵}, ℝ, < ) ∈ {𝐴, 𝐵} ↔ (𝐴𝐵𝐵𝐴)))

Theoremrpmincl 10895 The minumum of two positive real numbers is a positive real number. (Contributed by Jim Kingdon, 25-Apr-2023.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ+) → inf({𝐴, 𝐵}, ℝ, < ) ∈ ℝ+)

Theorembdtrilem 10896 Lemma for bdtri 10897. (Contributed by Steven Nguyen and Jim Kingdon, 17-May-2023.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ+) → ((abs‘(𝐴𝐶)) + (abs‘(𝐵𝐶))) ≤ (𝐶 + (abs‘((𝐴 + 𝐵) − 𝐶))))

Theorembdtri 10897 Triangle inequality for bounded values. (Contributed by Jim Kingdon, 15-May-2023.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ+) → inf({(𝐴 + 𝐵), 𝐶}, ℝ, < ) ≤ (inf({𝐴, 𝐶}, ℝ, < ) + inf({𝐵, 𝐶}, ℝ, < )))

Theoremmul0inf 10898 Equality of a product with zero. A bit of a curiosity, in the sense that theorems like abs00ap 10720 and mulap0bd 8325 may better express the ideas behind it. (Contributed by Jim Kingdon, 31-Jul-2023.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵) = 0 ↔ inf({(abs‘𝐴), (abs‘𝐵)}, ℝ, < ) = 0))

3.7.7  The maximum of two extended reals

Theoremxrmaxleim 10899 Value of maximum when we know which extended real is larger. (Contributed by Jim Kingdon, 25-Apr-2023.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝐵 → sup({𝐴, 𝐵}, ℝ*, < ) = 𝐵))

Theoremxrmaxiflemcl 10900 Lemma for xrmaxif 10906. Closure. (Contributed by Jim Kingdon, 29-Apr-2023.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) ∈ ℝ*)

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