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Theorem List for Metamath Proof Explorer - 41901-42000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremxreqled 41901 Equality implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐴 = 𝐵)       (𝜑𝐴𝐵)

Theoremxrgepnfd 41902 An extended real greater than or equal to +∞ is +∞ (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑 → +∞ ≤ 𝐴)       (𝜑𝐴 = +∞)

Theoremxrge0nemnfd 41903 A nonnegative extended real is not minus infinity. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ (0[,]+∞))       (𝜑𝐴 ≠ -∞)

Theoremsupxrgere 41904* If a real number can be approximated from below by members of a set, then it is less than or equal to the supremum of the set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝜑    &   (𝜑𝐴 ⊆ ℝ*)    &   (𝜑𝐵 ∈ ℝ)    &   ((𝜑𝑥 ∈ ℝ+) → ∃𝑦𝐴 (𝐵𝑥) < 𝑦)       (𝜑𝐵 ≤ sup(𝐴, ℝ*, < ))

Theoremiuneqfzuzlem 41905* Lemma for iuneqfzuz 41906: here, inclusion is proven; aiuneqfzuz uses this lemma twice, to prove equality. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑍 = (ℤ𝑁)       (∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 𝑛𝑍 𝐴 𝑛𝑍 𝐵)

Theoremiuneqfzuz 41906* If two unions indexed by upper integers are equal if they agree on any partial indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑍 = (ℤ𝑁)       (∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 𝑛𝑍 𝐴 = 𝑛𝑍 𝐵)

Theoremxle2addd 41907 Adding both side of two inequalities. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐷 ∈ ℝ*)    &   (𝜑𝐴𝐶)    &   (𝜑𝐵𝐷)       (𝜑 → (𝐴 +𝑒 𝐵) ≤ (𝐶 +𝑒 𝐷))

Theoremsupxrgelem 41908* If an extended real number can be approximated from below by members of a set, then it is less than or equal to the supremum of the set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝜑    &   (𝜑𝐴 ⊆ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   ((𝜑𝑥 ∈ ℝ+) → ∃𝑦𝐴 𝐵 < (𝑦 +𝑒 𝑥))       (𝜑𝐵 ≤ sup(𝐴, ℝ*, < ))

Theoremsupxrge 41909* If an extended real number can be approximated from below by members of a set, then it is less than or equal to the supremum of the set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝜑    &   (𝜑𝐴 ⊆ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   ((𝜑𝑥 ∈ ℝ+) → ∃𝑦𝐴 𝐵 ≤ (𝑦 +𝑒 𝑥))       (𝜑𝐵 ≤ sup(𝐴, ℝ*, < ))

Theoremsuplesup 41910* If any element of 𝐴 can be approximated from below by members of 𝐵, then the supremum of 𝐴 is less than or equal to the supremum of 𝐵. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐵 ⊆ ℝ*)    &   (𝜑 → ∀𝑥𝐴𝑦 ∈ ℝ+𝑧𝐵 (𝑥𝑦) < 𝑧)       (𝜑 → sup(𝐴, ℝ*, < ) ≤ sup(𝐵, ℝ*, < ))

Theoreminfxrglb 41911* The infimum of a set of extended reals is less than an extended real if and only if the set contains a smaller number. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
((𝐴 ⊆ ℝ*𝐵 ∈ ℝ*) → (inf(𝐴, ℝ*, < ) < 𝐵 ↔ ∃𝑥𝐴 𝑥 < 𝐵))

Theoremxadd0ge2 41912 A number is less than or equal to itself plus a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ (0[,]+∞))       (𝜑𝐴 ≤ (𝐵 +𝑒 𝐴))

Theoremnepnfltpnf 41913 An extended real that is not +∞ is less than +∞. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴 ≠ +∞)    &   (𝜑𝐴 ∈ ℝ*)       (𝜑𝐴 < +∞)

Theoremltadd12dd 41914 Addition to both sides of 'less than'. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐴 < 𝐶)    &   (𝜑𝐵 < 𝐷)       (𝜑 → (𝐴 + 𝐵) < (𝐶 + 𝐷))

Theoremnemnftgtmnft 41915 An extended real that is not minus infinity, is larger than minus infinity. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
((𝐴 ∈ ℝ*𝐴 ≠ -∞) → -∞ < 𝐴)

Theoremxrgtso 41916 'Greater than' is a strict ordering on the extended reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
< Or ℝ*

Theoremrpex 41917 The positive reals form a set. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
+ ∈ V

Theoremxrge0ge0 41918 A nonnegative extended real is nonnegative. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝐴 ∈ (0[,]+∞) → 0 ≤ 𝐴)

Theoremxrssre 41919 A subset of extended reals that does not contain +∞ and -∞ is a subset of the reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴 ⊆ ℝ*)    &   (𝜑 → ¬ +∞ ∈ 𝐴)    &   (𝜑 → ¬ -∞ ∈ 𝐴)       (𝜑𝐴 ⊆ ℝ)

Theoremssuzfz 41920 A finite subset of the upper integers is a subset of a finite set of sequential integers. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐴𝑍)    &   (𝜑𝐴 ∈ Fin)       (𝜑𝐴 ⊆ (𝑀...sup(𝐴, ℝ, < )))

Theoremabsfun 41921 The absolute value is a function. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
Fun abs

Theoreminfrpge 41922* The infimum of a nonempty, bounded subset of extended reals can be approximated from above by an element of the set. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
𝑥𝜑    &   (𝜑𝐴 ⊆ ℝ*)    &   (𝜑𝐴 ≠ ∅)    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑥𝑦)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → ∃𝑧𝐴 𝑧 ≤ (inf(𝐴, ℝ*, < ) +𝑒 𝐵))

Theoremxrlexaddrp 41923* If an extended real number 𝐴 can be approximated from above, adding positive reals to 𝐵, then 𝐴 is less than or equal to 𝐵. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   ((𝜑𝑥 ∈ ℝ+) → 𝐴 ≤ (𝐵 +𝑒 𝑥))       (𝜑𝐴𝐵)

Theoremsupsubc 41924* The supremum function distributes over subtraction in a sense similar to that in supaddc 11595. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐴 ≠ ∅)    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥)    &   (𝜑𝐵 ∈ ℝ)    &   𝐶 = {𝑧 ∣ ∃𝑣𝐴 𝑧 = (𝑣𝐵)}       (𝜑 → (sup(𝐴, ℝ, < ) − 𝐵) = sup(𝐶, ℝ, < ))

Theoremxralrple2 41925* Show that 𝐴 is less than 𝐵 by showing that there is no positive bound on the difference. A variant on xralrple 12586. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝑥𝜑    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ (0[,)+∞))       (𝜑 → (𝐴𝐵 ↔ ∀𝑥 ∈ ℝ+ 𝐴 ≤ ((1 + 𝑥) · 𝐵)))

Theoremnnuzdisj 41926 The first 𝑁 elements of the set of nonnegative integers are distinct from any later members. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
((1...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅

Theoremltdivgt1 41927 Divsion by a number greater than 1. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (1 < 𝐵 ↔ (𝐴 / 𝐵) < 𝐴))

Theoremxrltned 41928 'Less than' implies not equal. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐴𝐵)

Theoremnnsplit 41929 Express the set of positive integers as the disjoint (see nnuzdisj 41926) union of the first 𝑁 values and the rest. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
(𝑁 ∈ ℕ → ℕ = ((1...𝑁) ∪ (ℤ‘(𝑁 + 1))))

Theoremdivdiv3d 41930 Division into a fraction. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) / 𝐶) = (𝐴 / (𝐶 · 𝐵)))

Theoremabslt2sqd 41931 Comparison of the square of two numbers. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (abs‘𝐴) < (abs‘𝐵))       (𝜑 → (𝐴↑2) < (𝐵↑2))

Theoremqenom 41932 The set of rational numbers is equinumerous to omega (the set of finite ordinal numbers). (Contributed by Glauco Siliprandi, 24-Dec-2020.)
ℚ ≈ ω

Theoremqct 41933 The set of rational numbers is countable. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
ℚ ≼ ω

Theoremxrltnled 41934 'Less than' in terms of 'less than or equal to'. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)       (𝜑 → (𝐴 < 𝐵 ↔ ¬ 𝐵𝐴))

Theoremlenlteq 41935 'less than or equal to' but not 'less than' implies 'equal' . (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑 → ¬ 𝐴 < 𝐵)       (𝜑𝐴 = 𝐵)

Theoremxrred 41936 An extended real that is neither minus infinity, nor plus infinity, is real. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐴 ≠ -∞)    &   (𝜑𝐴 ≠ +∞)       (𝜑𝐴 ∈ ℝ)

Theoremrr2sscn2 41937 The cartesian square of is a subset of the cartesian square of . (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(ℝ × ℝ) ⊆ (ℂ × ℂ)

Theoreminfxr 41938* The infimum of a set of extended reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑𝐴 ⊆ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑 → ∀𝑥𝐴 ¬ 𝑥 < 𝐵)    &   (𝜑 → ∀𝑥 ∈ ℝ (𝐵 < 𝑥 → ∃𝑦𝐴 𝑦 < 𝑥))       (𝜑 → inf(𝐴, ℝ*, < ) = 𝐵)

Theoreminfxrunb2 41939* The infimum of an unbounded-below set of extended reals is minus infinity. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝐴 ⊆ ℝ* → (∀𝑥 ∈ ℝ ∃𝑦𝐴 𝑦 < 𝑥 ↔ inf(𝐴, ℝ*, < ) = -∞))

Theoreminfxrbnd2 41940* The infimum of a bounded-below set of extended reals is greater than minus infinity. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝐴 ⊆ ℝ* → (∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑥𝑦 ↔ -∞ < inf(𝐴, ℝ*, < )))

Theoreminfleinflem1 41941 Lemma for infleinf 41943, case 𝐵 ≠ ∅ ∧ -∞ < inf(𝐵, ℝ*, < ). (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴 ⊆ ℝ*)    &   (𝜑𝐵 ⊆ ℝ*)    &   (𝜑𝑊 ∈ ℝ+)    &   (𝜑𝑋𝐵)    &   (𝜑𝑋 ≤ (inf(𝐵, ℝ*, < ) +𝑒 (𝑊 / 2)))    &   (𝜑𝑍𝐴)    &   (𝜑𝑍 ≤ (𝑋 +𝑒 (𝑊 / 2)))       (𝜑 → inf(𝐴, ℝ*, < ) ≤ (inf(𝐵, ℝ*, < ) +𝑒 𝑊))

Theoreminfleinflem2 41942 Lemma for infleinf 41943, when inf(𝐵, ℝ*, < ) = -∞. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴 ⊆ ℝ*)    &   (𝜑𝐵 ⊆ ℝ*)    &   (𝜑𝑅 ∈ ℝ)    &   (𝜑𝑋𝐵)    &   (𝜑𝑋 < (𝑅 − 2))    &   (𝜑𝑍𝐴)    &   (𝜑𝑍 ≤ (𝑋 +𝑒 1))       (𝜑𝑍 < 𝑅)

Theoreminfleinf 41943* If any element of 𝐵 can be approximated from above by members of 𝐴, then the infimum of 𝐴 is less than or equal to the infimum of 𝐵. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴 ⊆ ℝ*)    &   (𝜑𝐵 ⊆ ℝ*)    &   ((𝜑𝑥𝐵𝑦 ∈ ℝ+) → ∃𝑧𝐴 𝑧 ≤ (𝑥 +𝑒 𝑦))       (𝜑 → inf(𝐴, ℝ*, < ) ≤ inf(𝐵, ℝ*, < ))

Theoremxralrple4 41944* Show that 𝐴 is less than 𝐵 by showing that there is no positive bound on the difference. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → (𝐴𝐵 ↔ ∀𝑥 ∈ ℝ+ 𝐴 ≤ (𝐵 + (𝑥𝑁))))

Theoremxralrple3 41945* Show that 𝐴 is less than 𝐵 by showing that there is no positive bound on the difference. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐶)       (𝜑 → (𝐴𝐵 ↔ ∀𝑥 ∈ ℝ+ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥))))

Theoremeluzelzd 41946 A member of an upper set of integers is an integer. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑁 ∈ (ℤ𝑀))       (𝜑𝑁 ∈ ℤ)

Theoremsuplesup2 41947* If any element of 𝐴 is less than or equal to an element in 𝐵, then the supremum of 𝐴 is less than or equal to the supremum of 𝐵. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝐴 ⊆ ℝ*)    &   (𝜑𝐵 ⊆ ℝ*)    &   ((𝜑𝑥𝐴) → ∃𝑦𝐵 𝑥𝑦)       (𝜑 → sup(𝐴, ℝ*, < ) ≤ sup(𝐵, ℝ*, < ))

Theoremrecnnltrp 41948 𝑁 is a natural number large enough that its reciprocal is smaller than the given positive 𝐸. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑁 = ((⌊‘(1 / 𝐸)) + 1)       (𝐸 ∈ ℝ+ → (𝑁 ∈ ℕ ∧ (1 / 𝑁) < 𝐸))

Theoremfiminre2 41949* A nonempty finite set of real numbers is bounded below. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑥𝑦)

Theoremnnn0 41950 The set of positive integers is nonempty. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
ℕ ≠ ∅

Theoremfzct 41951 A finite set of sequential integer is countable. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝑁...𝑀) ≼ ω

Theoremrpgtrecnn 41952* Any positive real number is greater than the reciprocal of a positive integer. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝐴 ∈ ℝ+ → ∃𝑛 ∈ ℕ (1 / 𝑛) < 𝐴)

Theoremfzossuz 41953 A half-open integer interval is a subset of an upper set of integers. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝑀..^𝑁) ⊆ (ℤ𝑀)

Theoreminfrefilb 41954 The infimum of a finite set of reals is less than or equal to any of its elements. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
((𝐵 ⊆ ℝ ∧ 𝐵 ∈ Fin ∧ 𝐴𝐵) → inf(𝐵, ℝ, < ) ≤ 𝐴)

Theoreminfxrrefi 41955 The real and extended real infima match when the set is finite. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → inf(𝐴, ℝ*, < ) = inf(𝐴, ℝ, < ))

Theoremxrralrecnnle 41956* Show that 𝐴 is less than 𝐵 by showing that there is no positive bound on the difference. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑛𝜑    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴𝐵 ↔ ∀𝑛 ∈ ℕ 𝐴 ≤ (𝐵 + (1 / 𝑛))))

Theoremfzoct 41957 A finite set of sequential integer is countable. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝑁..^𝑀) ≼ ω

Theoremfrexr 41958 A function taking real values, is a function taking extended real values. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐹:𝐴⟶ℝ)       (𝜑𝐹:𝐴⟶ℝ*)

Theoremnnrecrp 41959 The reciprocal of a positive natural number is a positive real number. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝑁 ∈ ℕ → (1 / 𝑁) ∈ ℝ+)

Theoremqred 41960 A rational number is a real number. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℚ)       (𝜑𝐴 ∈ ℝ)

Theoremreclt0d 41961 The reciprocal of a negative number is negative. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 < 0)       (𝜑 → (1 / 𝐴) < 0)

Theoremlt0neg1dd 41962 If a number is negative, its negative is positive. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 < 0)       (𝜑 → 0 < -𝐴)

Theoremmnfled 41963 Minus infinity is less than or equal to any extended real. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ*)       (𝜑 → -∞ ≤ 𝐴)

Theoreminfxrcld 41964 The infimum of an arbitrary set of extended reals is an extended real. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ⊆ ℝ*)       (𝜑 → inf(𝐴, ℝ*, < ) ∈ ℝ*)

Theoremxrralrecnnge 41965* Show that 𝐴 is less than 𝐵 by showing that there is no positive bound on the difference. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑛𝜑    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ*)       (𝜑 → (𝐴𝐵 ↔ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵))

Theoremreclt0 41966 The reciprocal of a negative number is negative. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 ≠ 0)       (𝜑 → (𝐴 < 0 ↔ (1 / 𝐴) < 0))

Theoremltmulneg 41967 Multiplying by a negative number, swaps the order. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐶 < 0)       (𝜑 → (𝐴 < 𝐵 ↔ (𝐵 · 𝐶) < (𝐴 · 𝐶)))

Theoremallbutfi 41968* For all but finitely many. Some authors say "cofinitely many". Some authors say "ultimately". Compare with eliuniin 41671 and eliuniin2 41691 (here, the precondition can be dropped; see eliuniincex 41681). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑍 = (ℤ𝑀)    &   𝐴 = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐵       (𝑋𝐴 ↔ ∃𝑛𝑍𝑚 ∈ (ℤ𝑛)𝑋𝐵)

Theoremltdiv23neg 41969 Swap denominator with other side of 'less than', when both are negative. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐵 < 0)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐶 < 0)       (𝜑 → ((𝐴 / 𝐵) < 𝐶 ↔ (𝐴 / 𝐶) < 𝐵))

Theoremxreqnltd 41970 A consequence of trichotomy. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐴 = 𝐵)       (𝜑 → ¬ 𝐴 < 𝐵)

Theoremmnfnre2 41971 Minus infinity is not a real number. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
¬ -∞ ∈ ℝ

Theoremuzssre 41972 An upper set of integers is a subset of the Reals. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(ℤ𝑀) ⊆ ℝ

Theoremzssxr 41973 The integers are a subset of the extended reals. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
ℤ ⊆ ℝ*

Theoremfisupclrnmpt 41974* A nonempty finite indexed set contains its supremum. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑𝑅 Or 𝐴)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑𝐵 ≠ ∅)    &   ((𝜑𝑥𝐵) → 𝐶𝐴)       (𝜑 → sup(ran (𝑥𝐵𝐶), 𝐴, 𝑅) ∈ 𝐴)

Theoremsupxrunb3 41975* The supremum of an unbounded-above set of extended reals is plus infinity. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝐴 ⊆ ℝ* → (∀𝑥 ∈ ℝ ∃𝑦𝐴 𝑥𝑦 ↔ sup(𝐴, ℝ*, < ) = +∞))

Theoremelfzod 41976 Membership in a half-open integer interval. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐾 ∈ (ℤ𝑀))    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝐾 < 𝑁)       (𝜑𝐾 ∈ (𝑀..^𝑁))

Theoremfimaxre4 41977* A nonempty finite set of real numbers is bounded (image set version). (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)       (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)

Theoremren0 41978 The set of reals is nonempty. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
ℝ ≠ ∅

Theoremeluzelz2 41979 A member of an upper set of integers is an integer. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑍 = (ℤ𝑀)       (𝑁𝑍𝑁 ∈ ℤ)

Theoremresabs2d 41980 Absorption law for restriction. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐵𝐶)       (𝜑 → ((𝐴𝐵) ↾ 𝐶) = (𝐴𝐵))

Theoremuzid2 41981 Membership of the least member in an upper set of integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝑀 ∈ (ℤ𝑁) → 𝑀 ∈ (ℤ𝑀))

Theoremsupxrleubrnmpt 41982* The supremum of a nonempty bounded indexed set of extended reals is less than or equal to an upper bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)       (𝜑 → (sup(ran (𝑥𝐴𝐵), ℝ*, < ) ≤ 𝐶 ↔ ∀𝑥𝐴 𝐵𝐶))

Theoremuzssre2 41983 An upper set of integers is a subset of the Reals. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑍 = (ℤ𝑀)       𝑍 ⊆ ℝ

Theoremuzssd 41984 Subset relationship for two sets of upper integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝑁 ∈ (ℤ𝑀))       (𝜑 → (ℤ𝑁) ⊆ (ℤ𝑀))

Theoremeluzd 41985 Membership in an upper set of integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝑀𝑁)       (𝜑𝑁𝑍)

Theoreminfxrlbrnmpt2 41986* A member of a nonempty indexed set of reals is greater than or equal to the set's lower bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)    &   (𝜑𝐶𝐴)    &   (𝜑𝐷 ∈ ℝ*)    &   (𝑥 = 𝐶𝐵 = 𝐷)       (𝜑 → inf(ran (𝑥𝐴𝐵), ℝ*, < ) ≤ 𝐷)

Theoremxrre4 41987 An extended real is real iff it is not an infinty. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝐴 ∈ ℝ* → (𝐴 ∈ ℝ ↔ (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)))

Theoremuz0 41988 The upper integers function applied to a non-integer, is the empty set. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑀 ∈ ℤ → (ℤ𝑀) = ∅)

Theoremeluzelz2d 41989 A member of an upper set of integers is an integer. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑁𝑍)       (𝜑𝑁 ∈ ℤ)

Theoreminfleinf2 41990* If any element in 𝐵 is greater than or equal to an element in 𝐴, then the infimum of 𝐴 is less than or equal to the infimum of 𝐵. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑𝐴 ⊆ ℝ*)    &   (𝜑𝐵 ⊆ ℝ*)    &   ((𝜑𝑥𝐵) → ∃𝑦𝐴 𝑦𝑥)       (𝜑 → inf(𝐴, ℝ*, < ) ≤ inf(𝐵, ℝ*, < ))

Theoremunb2ltle 41991* "Unbounded below" expressed with < and with . (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝐴 ⊆ ℝ* → (∀𝑤 ∈ ℝ ∃𝑦𝐴 𝑦 < 𝑤 ↔ ∀𝑥 ∈ ℝ ∃𝑦𝐴 𝑦𝑥))

Theoremuzidd2 41992 Membership of the least member in an upper set of integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)       (𝜑𝑀𝑍)

Theoremuzssd2 41993 Subset relationship for two sets of upper integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑁𝑍)       (𝜑 → (ℤ𝑁) ⊆ 𝑍)

Theoremrexabslelem 41994* An indexed set of absolute values of real numbers is bounded if and only if the original values are bounded above and below. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)       (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 (abs‘𝐵) ≤ 𝑦 ↔ (∃𝑤 ∈ ℝ ∀𝑥𝐴 𝐵𝑤 ∧ ∃𝑧 ∈ ℝ ∀𝑥𝐴 𝑧𝐵)))

Theoremrexabsle 41995* An indexed set of absolute values of real numbers is bounded if and only if the original values are bounded above and below. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)       (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 (abs‘𝐵) ≤ 𝑦 ↔ (∃𝑤 ∈ ℝ ∀𝑥𝐴 𝐵𝑤 ∧ ∃𝑧 ∈ ℝ ∀𝑥𝐴 𝑧𝐵)))

Theoremallbutfiinf 41996* Given a "for all but finitely many" condition, the condition holds from 𝑁 on. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑍 = (ℤ𝑀)    &   𝐴 = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐵    &   (𝜑𝑋𝐴)    &   𝑁 = inf({𝑛𝑍 ∣ ∀𝑚 ∈ (ℤ𝑛)𝑋𝐵}, ℝ, < )       (𝜑 → (𝑁𝑍 ∧ ∀𝑚 ∈ (ℤ𝑁)𝑋𝐵))

Theoremsupxrrernmpt 41997* The real and extended real indexed suprema match when the indexed real supremum exists. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑𝐴 ≠ ∅)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)       (𝜑 → sup(ran (𝑥𝐴𝐵), ℝ*, < ) = sup(ran (𝑥𝐴𝐵), ℝ, < ))

Theoremsuprleubrnmpt 41998* The supremum of a nonempty bounded indexed set of reals is less than or equal to an upper bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑𝐴 ≠ ∅)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (sup(ran (𝑥𝐴𝐵), ℝ, < ) ≤ 𝐶 ↔ ∀𝑥𝐴 𝐵𝐶))

Theoreminfrnmptle 41999* An indexed infimum of extended reals is smaller than another indexed infimum of extended reals, when every indexed element is smaller than the corresponding one. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)    &   ((𝜑𝑥𝐴) → 𝐶 ∈ ℝ*)    &   ((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑 → inf(ran (𝑥𝐴𝐵), ℝ*, < ) ≤ inf(ran (𝑥𝐴𝐶), ℝ*, < ))

Theoreminfxrunb3 42000* The infimum of an unbounded-below set of extended reals is minus infinity. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝐴 ⊆ ℝ* → (∀𝑥 ∈ ℝ ∃𝑦𝐴 𝑦𝑥 ↔ inf(𝐴, ℝ*, < ) = -∞))

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