Home Metamath Proof ExplorerTheorem List (p. 385 of 424) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-27159) Hilbert Space Explorer (27160-28684) Users' Mathboxes (28685-42360)

Theorem List for Metamath Proof Explorer - 38401-38500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremunirnmapsn 38401 Equality theorem for a subset of a set exponentiation, where the exponent is a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   𝐶 = {𝐴}    &   (𝜑𝑋 ⊆ (𝐵𝑚 𝐶))       (𝜑𝑋 = (ran 𝑋𝑚 𝐶))

Theoremiunmapss 38402* The indexed union of set exponentiations is a subset of the set exponentiation of the indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵𝑊)       (𝜑 𝑥𝐴 (𝐵𝑚 𝐶) ⊆ ( 𝑥𝐴 𝐵𝑚 𝐶))

Theoremssmapsn 38403* 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 38404* 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 38405 Mapping domain and codomain of the absolute value function. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
abs:ℂ⟶(0[,)+∞)

Theoremicof 38406 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 38407* 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 38408* 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 38409 The range of a partial function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝐹 ∈ (𝐴pm 𝐵) → ran 𝐹𝐴)

Theoremimaexi 38410 The image of a set is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐴𝑉       (𝐴𝐵) ∈ V

Theoremaxccdom 38411* 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 38412* 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 38413 The domain of a partial function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝐹 ∈ (𝐴pm 𝐵) → dom 𝐹𝐵)

Theoremdmrelrnrel 38414* 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 38415 The domain of a mapping. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐹:𝐴𝐵)       (𝜑 → dom 𝐹 = 𝐴)

Theoremfco3 38416 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 38417 The domain of a set is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴𝑉)       (𝜑 → dom 𝐴 ∈ V)

Theoremfvcod 38418 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 38419 Composition of two mappings. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐹:𝐵𝐶)    &   (𝜑𝐺:𝐴𝐵)       (𝜑 → (𝐹𝐺):𝐴𝐶)

Theoremfreld 38420 A mapping is a relation. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐹:𝐴𝐵)       (𝜑 → Rel 𝐹)

Theoremfrnd 38421 The range of a mapping. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐹:𝐴𝐵)       (𝜑 → ran 𝐹𝐵)

Theoremelrnmpt2id 38422* 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 38423* A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
(𝜑 → (𝑥𝐴𝐵):𝐴𝐶)       ((𝜑𝑥𝐴) → 𝐵𝐶)

Theoremaxccd 38424* 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 38425* An alternative version of the axiom of countable choice. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ≼ ω)    &   ((𝜑𝑥𝐴) → 𝑥 ≠ ∅)       (𝜑 → ∃𝑓𝑥𝐴 (𝑓𝑥) ∈ 𝑥)

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

Theoremsub2times 38426 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 38427 '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 38428 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 38429 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 38430 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 38431 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 38432 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 38433 Negative π is negative. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
-π < 0

Theoremdstregt0 38434* 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 38435 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 38436 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 38437 The negative of a positive number is less than the number itself. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ℝ+ → -𝐴 < 𝐴)

Theoremzltlesub 38438 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 38439 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 38440 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 38441 Double of a positive real is larger than the real itself. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ℝ+𝐴 < (2 · 𝐴))

Theoremreopn 38442 The reals are open with respect to the standard topology. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
ℝ ∈ (topGen‘ran (,))

Theoremelfzop1le2 38443 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 38444 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 38445 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 38446 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 38447 Distribution of negative over subtraction. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → -(𝐴𝐵) = (-𝐴 − -𝐵))

Theoremltdiv2dd 38448 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 38449 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 14047, and absnpncand 38449 should be deleted afterwards.
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (abs‘(𝐴𝐶)) ≤ ((abs‘(𝐴𝐵)) + (abs‘(𝐵𝐶))))

Theoremabssinbd 38450 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 38451 Floor of (1 / 2). (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(⌊‘(1 / 2)) = 0

Theoremmonoords 38452* 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 38453 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 13058, and hashssle 38453 should be deleted afterwards.
((𝐴 ∈ Fin ∧ 𝐵𝐴) → (#‘𝐵) ≤ (#‘𝐴))

Theoremlttri5d 38454 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 38455* A finite ordered set has a unique order isomorphism to a generic finite sequence of integers. This theorem generalizes fz1iso 13103 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 38456 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 38457 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 38458* 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 38459* 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 38460* 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 38461 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 38462 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 38463* 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 38464* 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 38465 Subset relationship for finite sets of sequential integers. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(1...𝑁) ⊆ (0...𝑁)

Theoremfzdifsuc2 38466 Remove a successor from the end of a finite set of sequential integers. Similar to fzdifsuc 12270, but with a weaker condition. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝑁 ∈ (ℤ‘(𝑀 − 1)) → (𝑀...𝑁) = ((𝑀...(𝑁 + 1)) ∖ {(𝑁 + 1)}))

Theoremfzsscn 38467 A finite sequence of integers is a set of complex numbers. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝑀...𝑁) ⊆ ℂ

Theoremdivcan8d 38468 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 38469 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 38470 A finite sequence of integers is a set of real numbers. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝑀...𝑁) ⊆ ℝ

Theoremelfzelzd 38471 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 38472 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 38473 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 38474 A finite set of sequential integers that is a subset of 0. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(0...𝑁) ⊆ ℕ0

Theoremxreqle 38475 Equality implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝐴 ∈ ℝ*𝐴 = 𝐵) → 𝐴𝐵)

Theoremxaddid2d 38476 0 is a left identity for extended real addition. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)       (𝜑 → (0 +𝑒 𝐴) = 𝐴)

Theoremxadd0ge 38477 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 38478 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 38479 'Greater than' implies not equal. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐵𝐴)

Theoremxrleneltd 38480 '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 38481 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 38482* 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 38483* 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 38484 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 38485* 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 38486 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 38487 Equality implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐴 = 𝐵)       (𝜑𝐴𝐵)

Theoremxrgepnfd 38488 An extended real greater or equal to +∞ is +∞ (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑 → +∞ ≤ 𝐴)       (𝜑𝐴 = +∞)

Theoremxrge0nemnfd 38489 A nonnegative extended real is not minus infinity. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ (0[,]+∞))       (𝜑𝐴 ≠ -∞)

Theoremsupxrgere 38490* 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 38491* Lemma for iuneqfzuz 38492: 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 38492* 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 38493 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 38494* 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 38495* 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 38496* 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 38497* 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 38498 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 38499 An extended real that is not +∞ is less than +∞. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴 ≠ +∞)    &   (𝜑𝐴 ∈ ℝ*)       (𝜑𝐴 < +∞)

Theoremltadd12dd 38500 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))

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42360
 Copyright terms: Public domain < Previous  Next >