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Theorem List for Metamath Proof Explorer - 38301-38400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremssmapsn 38301* A subset 𝐶 of a set exponentiation to a singleton, is its projection 𝐷 exponentiated to the singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑓𝐷    &   (𝜑𝐴𝑉)    &   (𝜑𝐶 ⊆ (𝐵𝑚 {𝐴}))    &   𝐷 = 𝑓𝐶 ran 𝑓       (𝜑𝐶 = (𝐷𝑚 {𝐴}))
 
Theoremiunmapsn 38302* The indexed union of set exponentiations to a singleton is equal to the set exponentiation of the indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵𝑊)    &   (𝜑𝐶𝑍)       (𝜑 𝑥𝐴 (𝐵𝑚 {𝐶}) = ( 𝑥𝐴 𝐵𝑚 {𝐶}))
 
Theoremabsfico 38303 Mapping domain and codomain of the absolute value function. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
abs:ℂ⟶(0[,)+∞)
 
Theoremicof 38304 The set of left-closed right-open intervals of extended reals maps to subsets of extended reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
[,):(ℝ* × ℝ*)⟶𝒫 ℝ*
 
Theoremrnmpt0 38305* The range of a function in map-to notation is empty if and only if its domain is empty. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   𝐹 = (𝑥𝐴𝐵)       (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅))
 
Theoremrnmptn0 38306* The range of a function in map-to notation is nonempty if the domain is nonempty. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   𝐹 = (𝑥𝐴𝐵)    &   (𝜑𝐴 ≠ ∅)       (𝜑 → ran 𝐹 ≠ ∅)
 
Theoremelpmrn 38307 The range of a partial function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝐹 ∈ (𝐴pm 𝐵) → ran 𝐹𝐴)
 
Theoremimaexi 38308 The image of a set is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐴𝑉       (𝐴𝐵) ∈ V
 
Theoremaxccdom 38309* Relax the constraint on ax-cc to dominance instead of equinumerosity. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑋 ≼ ω)    &   ((𝜑𝑧𝑋) → 𝑧 ≠ ∅)       (𝜑 → ∃𝑓(𝑓 Fn 𝑋 ∧ ∀𝑧𝑋 (𝑓𝑧) ∈ 𝑧))
 
Theoremdmmptdf 38310* The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   𝐴 = (𝑥𝐵𝐶)    &   ((𝜑𝑥𝐵) → 𝐶𝑉)       (𝜑 → dom 𝐴 = 𝐵)
 
Theoremelpmi2 38311 The domain of a partial function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝐹 ∈ (𝐴pm 𝐵) → dom 𝐹𝐵)
 
Theoremdmrelrnrel 38312* A relation preserving function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦)))    &   (𝜑𝐵𝐴)    &   (𝜑𝐶𝐴)    &   (𝜑𝐵𝑅𝐶)       (𝜑 → (𝐹𝐵)𝑆(𝐹𝐶))
 
Theoremfdmd 38313 The domain of a mapping. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐹:𝐴𝐵)       (𝜑 → dom 𝐹 = 𝐴)
 
Theoremfco3 38314 Functionality of a composition. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑 → Fun 𝐹)    &   (𝜑 → Fun 𝐺)       (𝜑 → (𝐹𝐺):(𝐺 “ dom 𝐹)⟶ran 𝐹)
 
Theoremdmexd 38315 The domain of a set is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴𝑉)       (𝜑 → dom 𝐴 ∈ V)
 
Theoremfvcod 38316 Value of a function composition. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑 → Fun 𝐺)    &   (𝜑𝐴 ∈ dom 𝐺)    &   𝐻 = (𝐹𝐺)       (𝜑 → (𝐻𝐴) = (𝐹‘(𝐺𝐴)))
 
Theoremfcod 38317 Composition of two mappings. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐹:𝐵𝐶)    &   (𝜑𝐺:𝐴𝐵)       (𝜑 → (𝐹𝐺):𝐴𝐶)
 
Theoremfreld 38318 A mapping is a relation. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐹:𝐴𝐵)       (𝜑 → Rel 𝐹)
 
Theoremfrnd 38319 The range of a mapping. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐹:𝐴𝐵)       (𝜑 → ran 𝐹𝐵)
 
Theoremelrnmpt2id 38320* Membership in the range of an operation class abstraction. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       ((𝑥𝐴𝑦𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝐶𝑉) → (𝑥𝐹𝑦) ∈ ran 𝐹)
 
Theoremfvmptelrn 38321* A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
(𝜑 → (𝑥𝐴𝐵):𝐴𝐶)       ((𝜑𝑥𝐴) → 𝐵𝐶)
 
Theoremaxccd 38322* An alternative version of the axiom of countable choice. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ≈ ω)    &   ((𝜑𝑥𝐴) → 𝑥 ≠ ∅)       (𝜑 → ∃𝑓𝑥𝐴 (𝑓𝑥) ∈ 𝑥)
 
Theoremaxccd2 38323* An alternative version of the axiom of countable choice. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ≼ ω)    &   ((𝜑𝑥𝐴) → 𝑥 ≠ ∅)       (𝜑 → ∃𝑓𝑥𝐴 (𝑓𝑥) ∈ 𝑥)
 
20.31.3  Ordering on real numbers - Real and complex numbers basic operations
 
Theoremsub2times 38324 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)
 
Theoremxrltled 38325 'Less than' implies 'less than or equal to', for extended reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐴𝐵)
 
Theoremabssubrp 38326 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 38327 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 38328 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 38329 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 38330 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 38331 Negative π is negative. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
-π < 0
 
Theoremdstregt0 38332* 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 38333 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 38334 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 38335 The negative of a positive number is less than the number itself. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ℝ+ → -𝐴 < 𝐴)
 
Theoremzltlesub 38336 If an integer 𝑁 is smaller or equal to a real, and we subtract a quantity smaller than 1, then 𝑁 is smaller 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 38337 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 38338 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 38339 Double of a positive real is larger than the real itself. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ℝ+𝐴 < (2 · 𝐴))
 
Theoremreopn 38340 The reals are open with respect to the standard topology. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
ℝ ∈ (topGen‘ran (,))
 
Theoremelfzop1le2 38341 A member in a half-open integer interval plus 1 is less or equal than the upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐾 ∈ (𝑀..^𝑁) → (𝐾 + 1) ≤ 𝑁)
 
Theoremsub31 38342 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 38343 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 38344 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 38345 Distribution of negative over subtraction. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → -(𝐴𝐵) = (-𝐴 − -𝐵))
 
Theoremltdiv2dd 38346 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))
 
Theoremabsnpncand 38347 Triangular inequality, combined with cancellation law for subtraction. (Contributed by Glauco Siliprandi, 11-Dec-2019.) TODO (NM): usage (2 times) should be replaced by abs3difd 13906, and absnpncand 38347 should be deleted afterwards.
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (abs‘(𝐴𝐶)) ≤ ((abs‘(𝐴𝐵)) + (abs‘(𝐵𝐶))))
 
Theoremabssinbd 38348 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 38349 Floor of (1 / 2). (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(⌊‘(1 / 2)) = 0
 
Theoremmonoords 38350* 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 38351 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 12923, and hashssle 38351 should be deleted afterwards.
((𝐴 ∈ Fin ∧ 𝐵𝐴) → (#‘𝐵) ≤ (#‘𝐴))
 
Theoremlttri5d 38352 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 38353* A finite ordered set has a unique order isomorphism to a generic finite sequence of integers. This theorem generalizes fz1iso 12968 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 38354 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 38355 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 38356* 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 38357* 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 38358* Choice of an upper bound for a non empty 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 38359 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 38360 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 38361* 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 38362* 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)
 
Theoremfz1ssfz0 38363 Subset relationship for finite sets of sequential integers. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(1...𝑁) ⊆ (0...𝑁)
 
Theoremsubdir2d 38364 Distribution of multiplication over subtraction. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (𝐶 · (𝐴𝐵)) = ((𝐶 · 𝐴) − (𝐶 · 𝐵)))
 
Theoremfzdifsuc2 38365 Remove a successor from the end of a finite set of sequential integers. Similar to fzdifsuc 12138, but with a weaker condition. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝑁 ∈ (ℤ‘(𝑀 − 1)) → (𝑀...𝑁) = ((𝑀...(𝑁 + 1)) ∖ {(𝑁 + 1)}))
 
Theoremfzsscn 38366 A finite sequence of integers is a set of complex numbers. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝑀...𝑁) ⊆ ℂ
 
Theoremdivcan8d 38367 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 38368 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 38369 A finite sequence of integers is a set of real numbers. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝑀...𝑁) ⊆ ℝ
 
Theoremelfzelzd 38370 A member of a finite set of sequential integer is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝐾 ∈ (𝑀...𝑁))       (𝜑𝐾 ∈ ℤ)
 
Theorembccld 38371 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 38372 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 38373 A finite set of sequential integers that is a subset of 0. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(0...𝑁) ⊆ ℕ0
 
Theoremxreqle 38374 Equality implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝐴 ∈ ℝ*𝐴 = 𝐵) → 𝐴𝐵)
 
Theoremxaddid2d 38375 0 is a left identity for extended real addition. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)       (𝜑 → (0 +𝑒 𝐴) = 𝐴)
 
Theoremxadd0ge 38376 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 38377 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 38378 'Greater than' implies not equal. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐵𝐴)
 
Theoremxrleneltd 38379 '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 38380 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 38381* 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 38382* 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 38383 Addition of extended reals preserves the "less than or equal" 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 38384* 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 38385 Addition of extended reals preserves the "less than or equal" 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))
 
Theoremxreqled 38386 Equality implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐴 = 𝐵)       (𝜑𝐴𝐵)
 
Theoremxrgepnfd 38387 An extended real greater or equal to +∞ is +∞ (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑 → +∞ ≤ 𝐴)       (𝜑𝐴 = +∞)
 
Theoremxrge0nemnfd 38388 A nonnegative extended real is not minus infinity. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ (0[,]+∞))       (𝜑𝐴 ≠ -∞)
 
Theoremsupxrgere 38389* If a real number can be approximated from below by members of a set, then it is smaller 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 38390* Lemma for iuneqfzuz 38391: 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 38391* 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 38392 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 38393* If an extended real number can be approximated from below by members of a set, then it is smaller 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 38394* If an extended real number can be approximated from below by members of a set, then it is smaller 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 38395* If any element of 𝐴 can be approximated from below by members of 𝐵, then the supremum of 𝐴 is smaller 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 38396* 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 38397 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 38398 An extended real that is not +∞ is less than +∞. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴 ≠ +∞)    &   (𝜑𝐴 ∈ ℝ*)       (𝜑𝐴 < +∞)
 
Theoremltadd12dd 38399 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 38400 An extended real that is not minus infinity, is larger than minus infinity. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
((𝐴 ∈ ℝ*𝐴 ≠ -∞) → -∞ < 𝐴)
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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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