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Theorem List for Metamath Proof Explorer - 41801-41900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfunimassd 41801* Sufficient condition for the image of a function being a subclass. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑 → Fun 𝐹)    &   ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)       (𝜑 → (𝐹𝐴) ⊆ 𝐵)

Theoremfimassd 41802 The image of a class is a subset of its codomain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐹:𝐴𝐵)       (𝜑 → (𝐹𝑋) ⊆ 𝐵)

Theoremfeqresmptf 41803* Express a restricted function as a mapping. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝐹    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐶𝐴)       (𝜑 → (𝐹𝐶) = (𝑥𝐶 ↦ (𝐹𝑥)))

Theoremelrnmpt1d 41804 Elementhood in an image set. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝐹 = (𝑥𝐴𝐵)    &   (𝜑𝑥𝐴)    &   (𝜑𝐵𝑉)       (𝜑𝐵 ∈ ran 𝐹)

Theoremdmresss 41805 The domain of a restriction is a subset of the original domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
dom (𝐴𝐵) ⊆ dom 𝐴

Theoremdmmptssf 41806 The domain of a mapping is a subset of its base class. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝐴    &   𝐹 = (𝑥𝐴𝐵)       dom 𝐹𝐴

Theoremdmmptdf2 41807 The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   𝑥𝐵    &   𝐴 = (𝑥𝐵𝐶)    &   ((𝜑𝑥𝐵) → 𝐶𝑉)       (𝜑 → dom 𝐴 = 𝐵)

Theoremdmuz 41808 Domain of the upper integers function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
dom ℤ = ℤ

Theoremfmptd2f 41809* Domain and codomain of the mapping operation; deduction form. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)

Theoremmpteq1df 41810 An equality theorem for the maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))

Theoremmptexf 41811 If the domain of a function given by maps-to notation is a set, the function is a set. Inference version of mptexg 6966. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝐴    &   𝐴 ∈ V       (𝑥𝐴𝐵) ∈ V

Theoremfvmpt4 41812* Value of a function given by the maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
((𝑥𝐴𝐵𝐶) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)

Theoremfmptf 41813* Functionality of the mapping operation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝐵    &   𝐹 = (𝑥𝐴𝐶)       (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)

Theoremresimass 41814 The image of a restriction is a subset of the original image. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
((𝐴𝐵) “ 𝐶) ⊆ (𝐴𝐶)

Theoremmptssid 41815 The mapping operation expressed with its actual domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝐴    &   𝐶 = {𝑥𝐴𝐵 ∈ V}       (𝑥𝐴𝐵) = (𝑥𝐶𝐵)

Theoremmptfnd 41816 The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.) (Revised by Thierry Arnoux, 10-May-2017.)
𝑥𝐴    &   𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑 → (𝑥𝐴𝐵) Fn 𝐴)

Theoremmpteq12da 41817 An equality inference for the maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑𝐴 = 𝐶)    &   ((𝜑𝑥𝐴) → 𝐵 = 𝐷)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))

Theoremrnmptlb 41818* Boundness below of the range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵)       (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)

Theoremrnmptbddlem 41819* Boundness of the range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)       (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)

Theoremrnmptbdd 41820* Boundness of the range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)       (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)

Theoremmptima2 41821* Image of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐶𝐴)       (𝜑 → ((𝑥𝐴𝐵) “ 𝐶) = ran (𝑥𝐶𝐵))

Theoremfunimaeq 41822* Membership relation for the values of a function whose image is a subclass. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑 → Fun 𝐹)    &   (𝜑 → Fun 𝐺)    &   (𝜑𝐴 ⊆ dom 𝐹)    &   (𝜑𝐴 ⊆ dom 𝐺)    &   ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐺𝑥))       (𝜑 → (𝐹𝐴) = (𝐺𝐴))

Theoremrnmptssf 41823* The range of an operation given by the maps-to notation as a subset. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝐶    &   𝐹 = (𝑥𝐴𝐵)       (∀𝑥𝐴 𝐵𝐶 → ran 𝐹𝐶)

Theoremrnmptbd2lem 41824* Boundness below of the range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧))

Theoremrnmptbd2 41825* Boundness below of the range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧))

Theoreminfnsuprnmpt 41826* The indexed infimum of real numbers is the negative of the indexed supremum of the negative values. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑𝐴 ≠ ∅)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵)       (𝜑 → inf(ran (𝑥𝐴𝐵), ℝ, < ) = -sup(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))

Theoremsuprclrnmpt 41827* Closure of the indexed supremum of a nonempty bounded set of reals. Range of a function in maps-to notation can be used, to express an indexed supremum. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑𝐴 ≠ ∅)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)       (𝜑 → sup(ran (𝑥𝐴𝐵), ℝ, < ) ∈ ℝ)

Theoremsuprubrnmpt2 41828* A member of a nonempty indexed set of reals is less than or equal to the set's upper bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)    &   (𝜑𝐶𝐴)    &   (𝜑𝐷 ∈ ℝ)    &   (𝑥 = 𝐶𝐵 = 𝐷)       (𝜑𝐷 ≤ sup(ran (𝑥𝐴𝐵), ℝ, < ))

Theoremsuprubrnmpt 41829* A member of a nonempty indexed set of reals is less than or equal to the set's upper bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)       ((𝜑𝑥𝐴) → 𝐵 ≤ sup(ran (𝑥𝐴𝐵), ℝ, < ))

Theoremrnmptssdf 41830* The range of an operation given by the maps-to notation as a subset. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   𝑥𝐶    &   𝐹 = (𝑥𝐴𝐵)    &   ((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑 → ran 𝐹𝐶)

Theoremrnmptbdlem 41831* Boundness above of the range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   𝑦𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))

Theoremrnmptbd 41832* Boundness above of the range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))

Theoremrnmptss2 41833* The range of an operation given by the maps-to notation as a subset. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑𝐴𝐵)    &   ((𝜑𝑥𝐴) → 𝐶𝑉)       (𝜑 → ran (𝑥𝐴𝐶) ⊆ ran (𝑥𝐵𝐶))

Theoremelmptima 41834* The image of a function in maps-to notation. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝐶𝑉 → (𝐶 ∈ ((𝑥𝐴𝐵) “ 𝐷) ↔ ∃𝑥 ∈ (𝐴𝐷)𝐶 = 𝐵))

Theoremralrnmpt3 41835* A restricted quantifier over an image set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝑦 = 𝐵 → (𝜓𝜒))       (𝜑 → (∀𝑦 ∈ ran (𝑥𝐴𝐵)𝜓 ↔ ∀𝑥𝐴 𝜒))

Theoremfvelima2 41836* Function value in an image. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
((𝐹 Fn 𝐴𝐵 ∈ (𝐹𝐶)) → ∃𝑥 ∈ (𝐴𝐶)(𝐹𝑥) = 𝐵)

Theoremfunresd 41837 A restriction of a function is a function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑 → Fun 𝐹)       (𝜑 → Fun (𝐹𝐴))

Theoremrnmptssbi 41838* The range of an operation given by the maps-to notation as a subset. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝜑    &   𝐹 = (𝑥𝐴𝐵)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑 → (ran 𝐹𝐶 ↔ ∀𝑥𝐴 𝐵𝐶))

Theoremfnfvelrnd 41839 A function's value belongs to its range. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐵𝐴)       (𝜑 → (𝐹𝐵) ∈ ran 𝐹)

Theoremimass2d 41840 Subset theorem for image. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐴𝐵)       (𝜑 → (𝐶𝐴) ⊆ (𝐶𝐵))

Theoremimassmpt 41841* Membership relation for the values of a function whose image is a subclass. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝜑    &   ((𝜑𝑥 ∈ (𝐴𝐶)) → 𝐵𝑉)    &   𝐹 = (𝑥𝐴𝐵)       (𝜑 → ((𝐹𝐶) ⊆ 𝐷 ↔ ∀𝑥 ∈ (𝐴𝐶)𝐵𝐷))

Theoremfpmd 41842 A total function is a partial function. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝐴)    &   (𝜑𝐹:𝐶𝐵)       (𝜑𝐹 ∈ (𝐵pm 𝐴))

Theoremfconst7 41843* An alternative way to express a constant function. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
𝑥𝜑    &   𝑥𝐹    &   (𝜑𝐹 Fn 𝐴)    &   (𝜑𝐵𝑉)    &   ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)       (𝜑𝐹 = (𝐴 × {𝐵}))

20.37.3  Ordering on real numbers - Real and complex numbers basic operations

Theoremsub2times 41844 Subtracting from a number, twice the number itself, gives negative the number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ℂ → (𝐴 − (2 · 𝐴)) = -𝐴)

Theoremabssubrp 41845 The distance of two distinct complex number is a strictly positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐴𝐵) → (abs‘(𝐴𝐵)) ∈ ℝ+)

Theoremelfzfzo 41846 Relationship between membership in a half-open finite set of sequential integers and membership in a finite set of sequential intergers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ (𝑀..^𝑁) ↔ (𝐴 ∈ (𝑀...𝑁) ∧ 𝐴 < 𝑁))

Theoremoddfl 41847 Odd number representation by using the floor function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐾 ∈ ℤ ∧ (𝐾 mod 2) ≠ 0) → 𝐾 = ((2 · (⌊‘(𝐾 / 2))) + 1))

Theoremabscosbd 41848 Bound for the absolute value of the cosine of a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ℝ → (abs‘(cos‘𝐴)) ≤ 1)

Theoremmul13d 41849 Commutative/associative law that swaps the first and the third factor in a triple product. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐶 · (𝐵 · 𝐴)))

Theoremnegpilt0 41850 Negative π is negative. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
-π < 0

Theoremdstregt0 41851* A complex number 𝐴 that is not real, has a distance from the reals that is strictly larger than 0. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ (ℂ ∖ ℝ))       (𝜑 → ∃𝑥 ∈ ℝ+𝑦 ∈ ℝ 𝑥 < (abs‘(𝐴𝑦)))

Theoremsubadd4b 41852 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)       (𝜑 → ((𝐴𝐵) + (𝐶𝐷)) = ((𝐴𝐷) + (𝐶𝐵)))

Theoremxrlttri5d 41853 Not equal and not larger implies smaller. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐴𝐵)    &   (𝜑 → ¬ 𝐵 < 𝐴)       (𝜑𝐴 < 𝐵)

Theoremneglt 41854 The negative of a positive number is less than the number itself. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ℝ+ → -𝐴 < 𝐴)

Theoremzltlesub 41855 If an integer 𝑁 is less than or equal to a real, and we subtract a quantity less than 1, then 𝑁 is less than or equal to the result. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑁 ∈ ℤ)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝑁𝐴)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐵 < 1)    &   (𝜑 → (𝐴𝐵) ∈ ℤ)       (𝜑𝑁 ≤ (𝐴𝐵))

Theoremdivlt0gt0d 41856 The ratio of a negative numerator and a positive denominator is negative. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐴 < 0)       (𝜑 → (𝐴 / 𝐵) < 0)

Theoremsubsub23d 41857 Swap subtrahend and result of subtraction. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴𝐵) = 𝐶 ↔ (𝐴𝐶) = 𝐵))

Theorem2timesgt 41858 Double of a positive real is larger than the real itself. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ℝ+𝐴 < (2 · 𝐴))

Theoremreopn 41859 The reals are open with respect to the standard topology. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
ℝ ∈ (topGen‘ran (,))

Theoremelfzop1le2 41860 A member in a half-open integer interval plus 1 is less than or equal to the upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐾 ∈ (𝑀..^𝑁) → (𝐾 + 1) ≤ 𝑁)

Theoremsub31 41861 Swap the first and third terms in a double subtraction. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 − (𝐵𝐶)) = (𝐶 − (𝐵𝐴)))

Theoremnnne1ge2 41862 A positive integer which is not 1 is greater than or equal to 2. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝑁 ∈ ℕ ∧ 𝑁 ≠ 1) → 2 ≤ 𝑁)

Theoremlefldiveq 41863 A closed enough, smaller real 𝐶 has the same floor of 𝐴 when both are divided by 𝐵. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐶 ∈ ((𝐴 − (𝐴 mod 𝐵))[,]𝐴))       (𝜑 → (⌊‘(𝐴 / 𝐵)) = (⌊‘(𝐶 / 𝐵)))

Theoremnegsubdi3d 41864 Distribution of negative over subtraction. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → -(𝐴𝐵) = (-𝐴 − -𝐵))

Theoremltdiv2dd 41865 Division of a positive number by both sides of 'less than'. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑𝐴 < 𝐵)       (𝜑 → (𝐶 / 𝐵) < (𝐶 / 𝐴))

Theoremabssinbd 41866 Bound for the absolute value of the sine of a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ℝ → (abs‘(sin‘𝐴)) ≤ 1)

Theoremhalffl 41867 Floor of (1 / 2). (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(⌊‘(1 / 2)) = 0

Theoremmonoords 41868* Ordering relation for a strictly monotonic sequence, increasing case. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘 ∈ (𝑀..^𝑁)) → (𝐹𝑘) < (𝐹‘(𝑘 + 1)))    &   (𝜑𝐼 ∈ (𝑀...𝑁))    &   (𝜑𝐽 ∈ (𝑀...𝑁))    &   (𝜑𝐼 < 𝐽)       (𝜑 → (𝐹𝐼) < (𝐹𝐽))

Theoremhashssle 41869 The size of a subset of a finite set is less than the size of the containing set. (Contributed by Glauco Siliprandi, 11-Dec-2019.) TODO (NM): usage (2 times) should be replaced by hashss 13766, and hashssle 41869 should be deleted afterwards.
((𝐴 ∈ Fin ∧ 𝐵𝐴) → (♯‘𝐵) ≤ (♯‘𝐴))

Theoremlttri5d 41870 Not equal and not larger implies smaller. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑 → ¬ 𝐵 < 𝐴)       (𝜑𝐴 < 𝐵)

Theoremfzisoeu 41871* A finite ordered set has a unique order isomorphism to a generic finite sequence of integers. This theorem generalizes fz1iso 13816 for the base index and also states the uniqueness condition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐻 ∈ Fin)    &   (𝜑 → < Or 𝐻)    &   (𝜑𝑀 ∈ ℤ)    &   𝑁 = ((♯‘𝐻) + (𝑀 − 1))       (𝜑 → ∃!𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻))

Theoremlt3addmuld 41872 If three real numbers are less than a fourth real number, the sum of the three real numbers is less than three times the third real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐴 < 𝐷)    &   (𝜑𝐵 < 𝐷)    &   (𝜑𝐶 < 𝐷)       (𝜑 → ((𝐴 + 𝐵) + 𝐶) < (3 · 𝐷))

Theoremabsnpncan2d 41873 Triangular inequality, combined with cancellation law for subtraction (applied twice). (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)       (𝜑 → (abs‘(𝐴𝐷)) ≤ (((abs‘(𝐴𝐵)) + (abs‘(𝐵𝐶))) + (abs‘(𝐶𝐷))))

Theoremfperiodmullem 41874* A function with period T is also periodic with period nonnegative multiple of T. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℂ)    &   (𝜑𝑇 ∈ ℝ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑋 ∈ ℝ)    &   ((𝜑𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))       (𝜑 → (𝐹‘(𝑋 + (𝑁 · 𝑇))) = (𝐹𝑋))

Theoremfperiodmul 41875* A function with period T is also periodic with period multiple of T. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℂ)    &   (𝜑𝑇 ∈ ℝ)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝑋 ∈ ℝ)    &   ((𝜑𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))       (𝜑 → (𝐹‘(𝑋 + (𝑁 · 𝑇))) = (𝐹𝑋))

Theoremupbdrech 41876* Choice of an upper bound for a nonempty bunded set (image set version). (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ≠ ∅)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)    &   𝐶 = sup({𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}, ℝ, < )       (𝜑 → (𝐶 ∈ ℝ ∧ ∀𝑥𝐴 𝐵𝐶))

Theoremlt4addmuld 41877 If four real numbers are less than a fifth real number, the sum of the four real numbers is less than four times the fifth real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐸 ∈ ℝ)    &   (𝜑𝐴 < 𝐸)    &   (𝜑𝐵 < 𝐸)    &   (𝜑𝐶 < 𝐸)    &   (𝜑𝐷 < 𝐸)       (𝜑 → (((𝐴 + 𝐵) + 𝐶) + 𝐷) < (4 · 𝐸))

Theoremabsnpncan3d 41878 Triangular inequality, combined with cancellation law for subtraction (applied three times). (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝐸 ∈ ℂ)       (𝜑 → (abs‘(𝐴𝐸)) ≤ ((((abs‘(𝐴𝐵)) + (abs‘(𝐵𝐶))) + (abs‘(𝐶𝐷))) + (abs‘(𝐷𝐸))))

Theoremupbdrech2 41879* Choice of an upper bound for a possibly empty bunded set (image set version). (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)    &   𝐶 = if(𝐴 = ∅, 0, sup({𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}, ℝ, < ))       (𝜑 → (𝐶 ∈ ℝ ∧ ∀𝑥𝐴 𝐵𝐶))

Theoremssfiunibd 41880* A finite union of bounded sets is bounded. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑧 𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑥𝐴) → ∃𝑦 ∈ ℝ ∀𝑧𝑥 𝐵𝑦)    &   (𝜑𝐶 𝐴)       (𝜑 → ∃𝑤 ∈ ℝ ∀𝑧𝐶 𝐵𝑤)

Theoremfzdifsuc2 41881 Remove a successor from the end of a finite set of sequential integers. Similar to fzdifsuc 12962, but with a weaker condition. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝑁 ∈ (ℤ‘(𝑀 − 1)) → (𝑀...𝑁) = ((𝑀...(𝑁 + 1)) ∖ {(𝑁 + 1)}))

Theoremfzsscn 41882 A finite sequence of integers is a set of complex numbers. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝑀...𝑁) ⊆ ℂ

Theoremdivcan8d 41883 A cancellation law for division. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝐵 ≠ 0)       (𝜑 → (𝐵 / (𝐴 · 𝐵)) = (1 / 𝐴))

Theoremdmmcand 41884 Cancellation law for division and multiplication. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) · (𝐵 · 𝐶)) = (𝐴 · 𝐶))

Theoremfzssre 41885 A finite sequence of integers is a set of real numbers. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝑀...𝑁) ⊆ ℝ

Theorembccld 41886 A binomial coefficient, in its extended domain, is a nonnegative integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐾 ∈ ℤ)       (𝜑 → (𝑁C𝐾) ∈ ℕ0)

Theoremleadd12dd 41887 Addition to both sides of 'less than or equal to'. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐴𝐶)    &   (𝜑𝐵𝐷)       (𝜑 → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷))

Theoremfzssnn0 41888 A finite set of sequential integers that is a subset of 0. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(0...𝑁) ⊆ ℕ0

Theoremxreqle 41889 Equality implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝐴 ∈ ℝ*𝐴 = 𝐵) → 𝐴𝐵)

Theoremxaddid2d 41890 0 is a left identity for extended real addition. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)       (𝜑 → (0 +𝑒 𝐴) = 𝐴)

Theoremxadd0ge 41891 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 ≤ (𝐴 +𝑒 𝐵))

Theoremelfzolem1 41892 A member in a half-open integer interval is less than or equal to the upper bound minus 1 . (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝐾 ∈ (𝑀..^𝑁) → 𝐾 ≤ (𝑁 − 1))

Theoremxrgtned 41893 'Greater than' implies not equal. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐵𝐴)

Theoremxrleneltd 41894 'Less than or equal to' and 'not equals' implies 'less than', for extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐴𝐵)    &   (𝜑𝐴𝐵)       (𝜑𝐴 < 𝐵)

Theoremxaddcomd 41895 The extended real addition operation is commutative. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)       (𝜑 → (𝐴 +𝑒 𝐵) = (𝐵 +𝑒 𝐴))

Theoremsupxrre3 41896* The supremum of a nonempty set of reals, is real if and only if it is bounded-above . (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (sup(𝐴, ℝ*, < ) ∈ ℝ ↔ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥))

Theoremuzfissfz 41897* For any finite subset of the upper integers, there is a finite set of sequential integers that includes it. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐴𝑍)    &   (𝜑𝐴 ∈ Fin)       (𝜑 → ∃𝑘𝑍 𝐴 ⊆ (𝑀...𝑘))

Theoremxleadd2d 41898 Addition of extended reals preserves the "less than or equal to" relation, in the right slot. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐶 +𝑒 𝐴) ≤ (𝐶 +𝑒 𝐵))

Theoremsuprltrp 41899* The supremum of a nonempty bounded set of reals can be approximated from below by elements of the set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐴 ≠ ∅)    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥)    &   (𝜑𝑋 ∈ ℝ+)       (𝜑 → ∃𝑧𝐴 (sup(𝐴, ℝ, < ) − 𝑋) < 𝑧)

Theoremxleadd1d 41900 Addition of extended reals preserves the "less than or equal to" relation, in the left slot. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐴 +𝑒 𝐶) ≤ (𝐵 +𝑒 𝐶))

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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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