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Theorem List for Metamath Proof Explorer - 41401-41500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrnmptbd2 41401* 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 41402* 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 41403* 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 41404* 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 41405* 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 41406* 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 41407* 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 41408* 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 41409* 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 41410* 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 41411* 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 41412* Function value in an image. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
((𝐹 Fn 𝐴𝐵 ∈ (𝐹𝐶)) → ∃𝑥 ∈ (𝐴𝐶)(𝐹𝑥) = 𝐵)
 
Theoremfunresd 41413 A restriction of a function is a function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑 → Fun 𝐹)       (𝜑 → Fun (𝐹𝐴))
 
Theoremrnmptssbi 41414* 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 41415 A function's value belongs to its range. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐵𝐴)       (𝜑 → (𝐹𝐵) ∈ ran 𝐹)
 
Theoremimass2d 41416 Subset theorem for image. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐴𝐵)       (𝜑 → (𝐶𝐴) ⊆ (𝐶𝐵))
 
Theoremimassmpt 41417* 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 41418 A total function is a partial function. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝐴)    &   (𝜑𝐹:𝐶𝐵)       (𝜑𝐹 ∈ (𝐵pm 𝐴))
 
Theoremfconst7 41419* 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}))
 
20.36.3  Ordering on real numbers - Real and complex numbers basic operations
 
Theoremsub2times 41420 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 41421 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 41422 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 41423 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 41424 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 41425 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 41426 Negative π is negative. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
-π < 0
 
Theoremdstregt0 41427* 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 41428 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 41429 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 41430 The negative of a positive number is less than the number itself. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ℝ+ → -𝐴 < 𝐴)
 
Theoremzltlesub 41431 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 41432 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 41433 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 41434 Double of a positive real is larger than the real itself. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ℝ+𝐴 < (2 · 𝐴))
 
Theoremreopn 41435 The reals are open with respect to the standard topology. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
ℝ ∈ (topGen‘ran (,))
 
Theoremelfzop1le2 41436 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 41437 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 41438 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 41439 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 41440 Distribution of negative over subtraction. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → -(𝐴𝐵) = (-𝐴 − -𝐵))
 
Theoremltdiv2dd 41441 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 41442 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 41443 Floor of (1 / 2). (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(⌊‘(1 / 2)) = 0
 
Theoremmonoords 41444* 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 41445 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 13760, and hashssle 41445 should be deleted afterwards.
((𝐴 ∈ Fin ∧ 𝐵𝐴) → (♯‘𝐵) ≤ (♯‘𝐴))
 
Theoremlttri5d 41446 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 41447* A finite ordered set has a unique order isomorphism to a generic finite sequence of integers. This theorem generalizes fz1iso 13810 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 41448 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 41449 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 41450* 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 41451* 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 41452* 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 41453 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 41454 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 41455* 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 41456* 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 41457 Remove a successor from the end of a finite set of sequential integers. Similar to fzdifsuc 12957, but with a weaker condition. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝑁 ∈ (ℤ‘(𝑀 − 1)) → (𝑀...𝑁) = ((𝑀...(𝑁 + 1)) ∖ {(𝑁 + 1)}))
 
Theoremfzsscn 41458 A finite sequence of integers is a set of complex numbers. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝑀...𝑁) ⊆ ℂ
 
Theoremdivcan8d 41459 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 41460 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 41461 A finite sequence of integers is a set of real numbers. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝑀...𝑁) ⊆ ℝ
 
Theoremelfzelzd 41462 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 41463 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 41464 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 41465 A finite set of sequential integers that is a subset of 0. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(0...𝑁) ⊆ ℕ0
 
Theoremxreqle 41466 Equality implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝐴 ∈ ℝ*𝐴 = 𝐵) → 𝐴𝐵)
 
Theoremxaddid2d 41467 0 is a left identity for extended real addition. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)       (𝜑 → (0 +𝑒 𝐴) = 𝐴)
 
Theoremxadd0ge 41468 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 41469 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 41470 'Greater than' implies not equal. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐵𝐴)
 
Theoremxrleneltd 41471 '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 41472 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 41473* 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 41474* 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 41475 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 41476* 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 41477 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))
 
Theoremxreqled 41478 Equality implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐴 = 𝐵)       (𝜑𝐴𝐵)
 
Theoremxrgepnfd 41479 An extended real greater than or equal to +∞ is +∞ (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑 → +∞ ≤ 𝐴)       (𝜑𝐴 = +∞)
 
Theoremxrge0nemnfd 41480 A nonnegative extended real is not minus infinity. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ (0[,]+∞))       (𝜑𝐴 ≠ -∞)
 
Theoremsupxrgere 41481* If a real number can be approximated from below by members of a set, then it is less than 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 41482* Lemma for iuneqfzuz 41483: 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 41483* 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 41484 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 41485* If an extended real number can be approximated from below by members of a set, then it is less than 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 41486* If an extended real number can be approximated from below by members of a set, then it is less than 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 41487* If any element of 𝐴 can be approximated from below by members of 𝐵, then the supremum of 𝐴 is less than 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 41488* 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 41489 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 41490 An extended real that is not +∞ is less than +∞. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴 ≠ +∞)    &   (𝜑𝐴 ∈ ℝ*)       (𝜑𝐴 < +∞)
 
Theoremltadd12dd 41491 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 41492 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 41493 'Greater than' is a strict ordering on the extended reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
< Or ℝ*
 
Theoremrpex 41494 The positive reals form a set. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
+ ∈ V
 
Theoremxrge0ge0 41495 A nonnegative extended real is nonnegative. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝐴 ∈ (0[,]+∞) → 0 ≤ 𝐴)
 
Theoremxrssre 41496 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 41497 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 41498 The absolute value is a function. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
Fun abs
 
Theoreminfrpge 41499* The infimum of a nonempty, 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 41500* If an extended real number 𝐴 can be approximated from above, adding positive reals to 𝐵, then 𝐴 is less than or equal to 𝐵. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   ((𝜑𝑥 ∈ ℝ+) → 𝐴 ≤ (𝐵 +𝑒 𝑥))       (𝜑𝐴𝐵)
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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 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44804
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