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Theorem List for Metamath Proof Explorer - 38501-38600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnemnftgtmnft 38501 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 38502 'Greater than' is a strict ordering on the extended reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
< Or ℝ*

Theoremrpex 38503 The positive reals form a set. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
+ ∈ V

Theoremxrge0ge0 38504 A nonnegative extended real is nonnegative. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝐴 ∈ (0[,]+∞) → 0 ≤ 𝐴)

Theoremxrssre 38505 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 38506 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 38507 The absolute value is a function. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
Fun abs

Theoreminfrpge 38508* The infimum of a non empty, 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 38509* If an extended real number 𝐴 can be approximated from above, adding positive reals to 𝐵, then 𝐴 is smaller or equal than 𝐵. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   ((𝜑𝑥 ∈ ℝ+) → 𝐴 ≤ (𝐵 +𝑒 𝑥))       (𝜑𝐴𝐵)

Theoremsupsubc 38510* The supremum function distributes over subtraction in a sense similar to that in supaddc 10867. (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 38511* Show that 𝐴 is less than 𝐵 by showing that there is no positive bound on the difference. A variant on xralrple 11910. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝑥𝜑    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ (0[,)+∞))       (𝜑 → (𝐴𝐵 ↔ ∀𝑥 ∈ ℝ+ 𝐴 ≤ ((1 + 𝑥) · 𝐵)))

Theoremnnuzdisj 38512 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 38513 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 38514 'Less than' implies not equal. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐴𝐵)

Theoremnnsplit 38515 Express the set of positive integers as the disjoint (see nnuzdisj 38512) 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 38516 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 38517 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 38518 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 38519 The set of rational numbers is countable. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
ℚ ≼ ω

Theoremxrltnled 38520 '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 38521 '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 38522 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 38523 ℝ^2 is a subset of CC^ 2. Common case. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(ℝ × ℝ) ⊆ (ℂ × ℂ)

Theoreminfxr 38524* 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 38525* 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 38526* 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 38527 Lemma for infleinf 38529, 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 38528 Lemma for infleinf 38529, 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 38529* 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 38530* 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 38531* 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 38532 A member of an upper set of integers is an integer. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑁 ∈ (ℤ𝑀))       (𝜑𝑁 ∈ ℤ)

Theoremsuplesup2 38533* 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 38534 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 38535* 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 38536 The set of positive integers is nonempty. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
ℕ ≠ ∅

Theoremfzct 38537 A finite set of sequential integer is countable. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝑁...𝑀) ≼ ω

Theoremrpgtrecnn 38538* 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 38539 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 38540 A half-open integer interval is a set of integers. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝑀..^𝑁) ⊆ ℤ

Theoreminfrefilb 38541 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 38542 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 38543* 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 38544 A finite set of sequential integer is countable. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝑁..^𝑀) ≼ ω

Theoremfrexr 38545 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 38546 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 38547 A rational number is a real number. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℚ)       (𝜑𝐴 ∈ ℝ)

Theoremreclt0d 38548 The reciprocal of a negative number is negative. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 < 0)       (𝜑 → (1 / 𝐴) < 0)

Theoremlt0neg1dd 38549 If a number is negative, its negative is positive. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 < 0)       (𝜑 → 0 < -𝐴)

Theoremmnfled 38550 Minus infinity is less than or equal to any extended real. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ*)       (𝜑 → -∞ ≤ 𝐴)

Theoremxrleidd 38551 'Less than or equal to' is reflexive for extended reals. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ*)       (𝜑𝐴𝐴)

Theoremnegelrpd 38552 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 38553 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 38554* 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 38555 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 38556 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 38557* For all but finitely many. Some authors say "cofinitely many". Some authors say "ultimately". Compare with eliuniin 38307 and eliuniin2 38335 (here, the precondition can be dropped; see eliuniincex 38323). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑍 = (ℤ𝑀)    &   𝐴 = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐵       (𝑋𝐴 ↔ ∃𝑛𝑍𝑚 ∈ (ℤ𝑛)𝑋𝐵)

Theoremltdiv23neg 38558 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))

21.31.4  Real intervals

Theoremgtnelioc 38559 A real number larger than the upper bound of a left open right closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐵 < 𝐶)       (𝜑 → ¬ 𝐶 ∈ (𝐴(,]𝐵))

Theoremioossioc 38560 An open interval is a subset of its right closure. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴(,)𝐵) ⊆ (𝐴(,]𝐵)

Theoremioondisj2 38561 A condition for two open intervals not to be disjoint. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵) ∧ (𝐶 ∈ ℝ*𝐷 ∈ ℝ*𝐶 < 𝐷)) ∧ (𝐴 < 𝐷𝐷𝐵)) → ((𝐴(,)𝐵) ∩ (𝐶(,)𝐷)) ≠ ∅)

Theoremioondisj1 38562 A condition for two open intervals not to be disjoint. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵) ∧ (𝐶 ∈ ℝ*𝐷 ∈ ℝ*𝐶 < 𝐷)) ∧ (𝐴𝐶𝐶 < 𝐵)) → ((𝐴(,)𝐵) ∩ (𝐶(,)𝐷)) ≠ ∅)

Theoremioosscn 38563 An open interval is a set of complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴(,)𝐵) ⊆ ℂ

Theoremioogtlb 38564 An element of a closed interval is greater than its lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝐶)

Theoremevthiccabs 38565* Extreme Value Theorem on y closed interval, for the absolute value of y continuous function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))       (𝜑 → (∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘(𝐹𝑦)) ≤ (abs‘(𝐹𝑥)) ∧ ∃𝑧 ∈ (𝐴[,]𝐵)∀𝑤 ∈ (𝐴[,]𝐵)(abs‘(𝐹𝑧)) ≤ (abs‘(𝐹𝑤))))

Theoremltnelicc 38566 A real number smaller than the lower bound of a closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐶 < 𝐴)       (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵))

Theoremeliood 38567 Membership in an open real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 < 𝐶)    &   (𝜑𝐶 < 𝐵)       (𝜑𝐶 ∈ (𝐴(,)𝐵))

Theoremiooabslt 38568 An upper bound for the distance from the center of an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ((𝐴𝐵)(,)(𝐴 + 𝐵)))       (𝜑 → (abs‘(𝐴𝐶)) < 𝐵)

Theoremgtnelicc 38569 A real number greater than the upper bound of a closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐵 < 𝐶)       (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵))

Theoremiooinlbub 38570 An open interval has empty intersection with its bounds. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴(,)𝐵) ∩ {𝐴, 𝐵}) = ∅

Theoremiocgtlb 38571 An element of a left open right closed interval is larger than its lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴(,]𝐵)) → 𝐴 < 𝐶)

Theoremiocleub 38572 An element of a left open right closed interval is smaller or equal to its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴(,]𝐵)) → 𝐶𝐵)

Theoremeliccd 38573 Membership in a closed real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴𝐶)    &   (𝜑𝐶𝐵)       (𝜑𝐶 ∈ (𝐴[,]𝐵))

Theoremiccssred 38574 A closed real interval is a set of reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)

Theoremeliccre 38575 A member of a closed interval of reals is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ∈ ℝ)

Theoremeliooshift 38576 Element of an open interval shifted by a displacement. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)       (𝜑 → (𝐴 ∈ (𝐵(,)𝐶) ↔ (𝐴 + 𝐷) ∈ ((𝐵 + 𝐷)(,)(𝐶 + 𝐷))))

Theoremeliocd 38577 Membership in a left open, right closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐴 < 𝐶)    &   (𝜑𝐶𝐵)       (𝜑𝐶 ∈ (𝐴(,]𝐵))

Theoremsnunioo2 38578 The closure of one end of an open real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵) → ((𝐴(,)𝐵) ∪ {𝐵}) = (𝐴(,]𝐵))

Theoremicoltub 38579 An element of a left closed right open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 < 𝐵)

Theoremtgiooss 38580 The restriction of the complex topology to a subset of reals, is a restriction of the standard topology on reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.) TODO (NM): Use rerest 22415 instead in fourierdlem48 39047, fourierdlem49 39048, fourierdlem62 39061 then delete this.
𝐽 = (TopOpen‘ℂfld)    &   𝐾 = (topGen‘ran (,))       (𝐴 ⊆ ℝ → (𝐽t 𝐴) = (𝐾t 𝐴))

Theoremeliocre 38581 A member of a left open, right closed interval of reals is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐵 ∈ ℝ ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐶 ∈ ℝ)

Theoremiooltub 38582 An element of an open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴(,)𝐵)) → 𝐶 < 𝐵)

Theoremioontr 38583 The interior of an interval in the standard topology on is the open interval itself. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (𝐴(,)𝐵)

Theoremeliccxr 38584 A member of a closed interval is a an extended real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ (𝐵[,]𝐶) → 𝐴 ∈ ℝ*)

Theoremsnunioo1 38585 The closure of one end of an open real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴}) = (𝐴[,)𝐵))

Theoremlbioc 38586 An left open right closed interval doesn't contain its left endpoint. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
¬ 𝐴 ∈ (𝐴(,]𝐵)

Theoremioomidp 38587 The midpoint is an element of the open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ((𝐴 + 𝐵) / 2) ∈ (𝐴(,)𝐵))

Theoremiccdifioo 38588 If the open inverval is removed from the closed interval, only the bounds are left. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) = {𝐴, 𝐵})

Theoremiccdifprioo 38589 An open interval is the closed interval without the bounds. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴[,]𝐵) ∖ {𝐴, 𝐵}) = (𝐴(,)𝐵))

Theoremioossioobi 38590 Biconditional form of ioossioo 12136. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐷 ∈ ℝ*)    &   (𝜑𝐶 < 𝐷)       (𝜑 → ((𝐶(,)𝐷) ⊆ (𝐴(,)𝐵) ↔ (𝐴𝐶𝐷𝐵)))

Theoremiccshift 38591* A closed interval shifted by a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑇 ∈ ℝ)       (𝜑 → ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)})

Theoremiccsuble 38592 An upper bound to the distance of two elements in a closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ (𝐴[,]𝐵))    &   (𝜑𝐷 ∈ (𝐴[,]𝐵))       (𝜑 → (𝐶𝐷) ≤ (𝐵𝐴))

Theoremiocopn 38593 A left open right closed interval is an open set of the standard topology restricted to an interval that contains the original interval and has the same upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ)    &   𝐾 = (topGen‘ran (,))    &   𝐽 = (𝐾t (𝐴(,]𝐵))    &   (𝜑𝐴𝐶)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐶(,]𝐵) ∈ 𝐽)

Theoremeliccelioc 38594 Membership in a closed interval and in a left open right closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ*)       (𝜑 → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐶𝐴)))

Theoremiooshift 38595* An open interval shifted by a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑇 ∈ ℝ)       (𝜑 → ((𝐴 + 𝑇)(,)(𝐵 + 𝑇)) = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)})

Theoremiccintsng 38596 Intersection of two adiacent closed intervals is a singleton. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐴𝐵𝐵𝐶)) → ((𝐴[,]𝐵) ∩ (𝐵[,]𝐶)) = {𝐵})

Theoremicoiccdif 38597 Left closed, right open interval gotten by a closed iterval taking away the upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴[,)𝐵) = ((𝐴[,]𝐵) ∖ {𝐵}))

Theoremicoopn 38598 A left closed right open interval is an open set of the standard topology restricted to an interval that contains the original interval and has the same lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   𝐾 = (topGen‘ran (,))    &   𝐽 = (𝐾t (𝐴[,)𝐵))    &   (𝜑𝐶𝐵)       (𝜑 → (𝐴[,)𝐶) ∈ 𝐽)

Theoremicoub 38599 A left-closed, right-open interval does not contain its upper bound. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝐴 ∈ ℝ* → ¬ 𝐵 ∈ (𝐴[,)𝐵))

Theoremeliccxrd 38600 Membership in a closed real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐴𝐶)    &   (𝜑𝐶𝐵)       (𝜑𝐶 ∈ (𝐴[,]𝐵))

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