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Theorem List for Metamath Proof Explorer - 39401-39500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoreminfleinf 39401* If any element of 𝐵 can be approximated from above by members of 𝐴, then the infimum of 𝐴 is smaller 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 39402* 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 39403* 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 39404 A member of an upper set of integers is an integer. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑁 ∈ (ℤ𝑀))       (𝜑𝑁 ∈ ℤ)

Theoremsuplesup2 39405* If any element of 𝐴 is smaller or equal to an element in 𝐵, then the supremum of 𝐴 is smaller 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 39406 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 39407* 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 39408 The set of positive integers is nonempty. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
ℕ ≠ ∅

Theoremfzct 39409 A finite set of sequential integer is countable. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝑁...𝑀) ≼ ω

Theoremrpgtrecnn 39410* 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 39411 A half-open integer interval is a subset of an upper set of integers. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝑀..^𝑁) ⊆ (ℤ𝑀)

Theoremfzossz 39412 A half-open integer interval is a set of integers. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝑀..^𝑁) ⊆ ℤ

Theoreminfrefilb 39413 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 39414 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 39415* 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 39416 A finite set of sequential integer is countable. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝑁..^𝑀) ≼ ω

Theoremfrexr 39417 A function taking real values, is a function taking extended real values. Common case. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐹:𝐴⟶ℝ)       (𝜑𝐹:𝐴⟶ℝ*)

Theoremnnrecrp 39418 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 39419 A rational number is a real number. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℚ)       (𝜑𝐴 ∈ ℝ)

Theoremreclt0d 39420 The reciprocal of a negative number is negative. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 < 0)       (𝜑 → (1 / 𝐴) < 0)

Theoremlt0neg1dd 39421 If a number is negative, its negative is positive. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 < 0)       (𝜑 → 0 < -𝐴)

Theoremmnfled 39422 Minus infinity is less than or equal to any extended real. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ*)       (𝜑 → -∞ ≤ 𝐴)

Theoremxrleidd 39423 'Less than or equal to' is reflexive for extended reals. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ*)       (𝜑𝐴𝐴)

Theoremnegelrpd 39424 The negation of a negative number is in the positive real numbers. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 < 0)       (𝜑 → -𝐴 ∈ ℝ+)

Theoreminfxrcld 39425 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 39426* 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 39427 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 39428 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 39429* For all but finitely many. Some authors say "cofinitely many". Some authors say "ultimately". Compare with eliuniin 39099 and eliuniin2 39123 (here, the precondition can be dropped; see eliuniincex 39112). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑍 = (ℤ𝑀)    &   𝐴 = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐵       (𝑋𝐴 ↔ ∃𝑛𝑍𝑚 ∈ (ℤ𝑛)𝑋𝐵)

Theoremltdiv23neg 39430 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 39431 A consequence of trichotomy. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐴 = 𝐵)       (𝜑 → ¬ 𝐴 < 𝐵)

Theoremmnfnre2 39432 Minus infinity is not a real number. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
¬ -∞ ∈ ℝ

Theoremuzssre 39433 An upper set of integers is a subset of the Reals. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(ℤ𝑀) ⊆ ℝ

Theoremzssxr 39434 The integers are a subset of the extended reals. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
ℤ ⊆ ℝ*

Theoremfisupclrnmpt 39435* 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 39436* 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 39437 Membership in a half-open integer interval. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐾 ∈ (ℤ𝑀))    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝐾 < 𝑁)       (𝜑𝐾 ∈ (𝑀..^𝑁))

Theoremfimaxre4 39438* 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 39439 The set of reals is nonempty. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
ℝ ≠ ∅

Theoremeluzelz2 39440 A member of an upper set of integers is an integer. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑍 = (ℤ𝑀)       (𝑁𝑍𝑁 ∈ ℤ)

Theorempnfnre2 39441 Plus infinity is not a real number. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
¬ +∞ ∈ ℝ

Theoremresabs2d 39442 Absorption law for restriction. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐵𝐶)       (𝜑 → ((𝐴𝐵) ↾ 𝐶) = (𝐴𝐵))

Theoremuzid2 39443 Membership of the least member in an upper set of integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝑀 ∈ (ℤ𝑁) → 𝑀 ∈ (ℤ𝑀))

Theoremuzidd 39444 Membership of the least member in an upper set of integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝑀 ∈ ℤ)       (𝜑𝑀 ∈ (ℤ𝑀))

Theoremsupxrleubrnmpt 39445* 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 39446 An upper set of integers is a subset of the Reals. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑍 = (ℤ𝑀)       𝑍 ⊆ ℝ

Theoremuzssd 39447 Subset relationship for two sets of upper integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝑁 ∈ (ℤ𝑀))       (𝜑 → (ℤ𝑁) ⊆ (ℤ𝑀))

Theoremeluzd 39448 Membership in an upper set of integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝑀𝑁)       (𝜑𝑁𝑍)

Theoremelfzd 39449 Membership in a finite set of sequential integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝐾 ∈ ℤ)    &   (𝜑𝑀𝐾)    &   (𝜑𝐾𝑁)       (𝜑𝐾 ∈ (𝑀...𝑁))

Theoreminfxrlbrnmpt2 39450* 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 39451 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 39452 The upper integers function applied to a non integer, is the empty set. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑀 ∈ ℤ → (ℤ𝑀) = ∅)

Theoremeluzelz2d 39453 A member of an upper set of integers is an integer. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑁𝑍)       (𝜑𝑁 ∈ ℤ)

Theoreminfleinf2 39454* If any element in 𝐵 is larger or equal to an element in 𝐴, then the infimum of 𝐴 is smaller 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 39455* "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 39456 Membership of the least member in an upper set of integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)       (𝜑𝑀𝑍)

Theoremuzssd2 39457 Subset relationship for two sets of upper integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑁𝑍)       (𝜑 → (ℤ𝑁) ⊆ 𝑍)

Theoremrexabslelem 39458* 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 39459* 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 39460* 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 39461* 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 39462* 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 39463* 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 39464* 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*, < ) = -∞))

Theoremuzn0d 39465 The upper integers are all nonempty. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)       (𝜑𝑍 ≠ ∅)

Theoremuzssd3 39466 Subset relationship for two sets of upper integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑍 = (ℤ𝑀)       (𝑁𝑍 → (ℤ𝑁) ⊆ 𝑍)

Theoremrexabsle2 39467* 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 ↔ (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦 ∧ ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵)))

Theoreminfxrunb3rnmpt 39468* The infimum of an unbounded-below set of extended reals is minus infinity. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   𝑦𝜑    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)       (𝜑 → (∀𝑦 ∈ ℝ ∃𝑥𝐴 𝐵𝑦 ↔ inf(ran (𝑥𝐴𝐵), ℝ*, < ) = -∞))

Theoremsupxrre3rnmpt 39469* The indexed supremum of a nonempty set of reals, is real if and only if it is bounded-above . (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑𝐴 ≠ ∅)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)       (𝜑 → (sup(ran (𝑥𝐴𝐵), ℝ*, < ) ∈ ℝ ↔ ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦))

Theoremuzublem 39470* A set of reals, indexed by upper integers, is bound if and only if any upper part is bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑗𝜑    &   𝑗𝑋    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑌 ∈ ℝ)    &   𝑊 = sup(ran (𝑗 ∈ (𝑀...𝐾) ↦ 𝐵), ℝ, < )    &   𝑋 = if(𝑊𝑌, 𝑌, 𝑊)    &   (𝜑𝐾𝑍)    &   ((𝜑𝑗𝑍) → 𝐵 ∈ ℝ)    &   (𝜑 → ∀𝑗 ∈ (ℤ𝐾)𝐵𝑌)       (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝐵𝑥)

Theoremuzub 39471* A set of reals, indexed by upper integers, is bound if and only if any upper part is bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑗𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑗𝑍) → 𝐵 ∈ ℝ)       (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)𝐵𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝐵𝑥))

Theoremssrexr 39472 A subset of the reals is a subset of the extended reals (common case). (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐴 ⊆ ℝ)       (𝜑𝐴 ⊆ ℝ*)

Theoremsupxrmnf2 39473 Removing minus infinity from a set does not affect its supremum. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝐴 ⊆ ℝ* → sup((𝐴 ∖ {-∞}), ℝ*, < ) = sup(𝐴, ℝ*, < ))

Theoremsupxrcli 39474 The supremum of an arbitrary set of extended reals is an extended real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐴 ⊆ ℝ*       sup(𝐴, ℝ*, < ) ∈ ℝ*

Theoremuzid3 39475 Membership of the least member in an upper set of integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑍 = (ℤ𝑀)       (𝑁𝑍𝑁 ∈ (ℤ𝑁))

Theoreminfxrlesupxr 39476 The supremum of a nonempty set is larger than or equal to the infimum. The second condition is needed, see supxrltinfxr 39490. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐴 ⊆ ℝ*)    &   (𝜑𝐴 ≠ ∅)       (𝜑 → inf(𝐴, ℝ*, < ) ≤ sup(𝐴, ℝ*, < ))

Theoremxnegeqd 39477 Equality of two extended numbers with -𝑒 in front of them. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐴 = 𝐵)       (𝜑 → -𝑒𝐴 = -𝑒𝐵)

Theoremxnegrecl 39478 The extended real negative of a real number is real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝐴 ∈ ℝ → -𝑒𝐴 ∈ ℝ)

Theoremxnegnegi 39479 Extended real version of negneg 10316. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐴 ∈ ℝ*       -𝑒-𝑒𝐴 = 𝐴

Theoremxnegeqi 39480 Equality of two extended numbers with -𝑒 in front of them. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐴 = 𝐵       -𝑒𝐴 = -𝑒𝐵

Theoremnfxnegd 39481 Deduction version of nfxneg 39504. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑥𝐴)       (𝜑𝑥-𝑒𝐴)

Theoremxnegnegd 39482 Extended real version of negnegd 10368. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐴 ∈ ℝ*)       (𝜑 → -𝑒-𝑒𝐴 = 𝐴)

Theoremuzred 39483 An upper integer is a real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐴𝑍)       (𝜑𝐴 ∈ ℝ)

Theoremxnegcli 39484 Closure of extended real negative. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐴 ∈ ℝ*       -𝑒𝐴 ∈ ℝ*

Theoremsupminfrnmpt 39485* The indexed supremum of a bounded-above set of reals is the negation of the indexed infimum of that set's image under negation. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝜑    &   (𝜑𝐴 ≠ ∅)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)       (𝜑 → sup(ran (𝑥𝐴𝐵), ℝ, < ) = -inf(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))

Theoremceilged 39486 The ceiling of a real number is greater than or equal to that number. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴 ≤ (⌈‘𝐴))

Theoreminfxrpnf 39487 Adding plus infinity to a set does not affect its infimum. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝐴 ⊆ ℝ* → inf((𝐴 ∪ {+∞}), ℝ*, < ) = inf(𝐴, ℝ*, < ))

Theoreminfxrrnmptcl 39488* The infimum of an arbitrary indexed set of extended reals is an extended real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)       (𝜑 → inf(ran (𝑥𝐴𝐵), ℝ*, < ) ∈ ℝ*)

Theoremleneg2d 39489 Negative of one side of 'less than or equal to'. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴 ≤ -𝐵𝐵 ≤ -𝐴))

Theoremsupxrltinfxr 39490 The supremum of the empty set is strictly smaller than the infimum of the empty set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
sup(∅, ℝ*, < ) < inf(∅, ℝ*, < )

Theoremmax1d 39491 A number is less than or equal to the maximum of it and another. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑𝐴 ≤ if(𝐴𝐵, 𝐵, 𝐴))

Theoremceilcld 39492 Closure of the ceiling function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (⌈‘𝐴) ∈ ℤ)

Theoremsupxrleubrnmptf 39493 The supremum of a nonempty bounded indexed set of extended reals is less than or equal to an upper bound. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐶    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)       (𝜑 → (sup(ran (𝑥𝐴𝐵), ℝ*, < ) ≤ 𝐶 ↔ ∀𝑥𝐴 𝐵𝐶))

Theoremnleltd 39494 'Not less than or equal to' implies 'grater than'. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → ¬ 𝐵𝐴)       (𝜑𝐴 < 𝐵)

Theoremzxrd 39495 An integer is an extended real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐴 ∈ ℤ)       (𝜑𝐴 ∈ ℝ*)

Theoreminfxrgelbrnmpt 39496* The infimum of an indexed set of extended reals is greater than or equal to a lower bound. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)       (𝜑 → (𝐶 ≤ inf(ran (𝑥𝐴𝐵), ℝ*, < ) ↔ ∀𝑥𝐴 𝐶𝐵))

Theoremrphalfltd 39497 Half of a positive real is less than the original number. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (𝐴 / 2) < 𝐴)

Theoremuzssz2 39498 An upper set of integers is a subset of all integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑍 = (ℤ𝑀)       𝑍 ⊆ ℤ

Theorem1xr 39499 1 is an extended real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
1 ∈ ℝ*

Theoremleneg3d 39500 Negative of one side of 'less than or equal to'. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (-𝐴𝐵 ↔ -𝐵𝐴))

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