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Theorem List for Metamath Proof Explorer - 39301-39400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremelfzfzo 39301 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 39302 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 39303 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 39304 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 39305 Negative π is negative. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
-π < 0

Theoremdstregt0 39306* 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 39307 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 39308 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 39309 The negative of a positive number is less than the number itself. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ℝ+ → -𝐴 < 𝐴)

Theoremzltlesub 39310 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 39311 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 39312 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 39313 Double of a positive real is larger than the real itself. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ℝ+𝐴 < (2 · 𝐴))

Theoremreopn 39314 The reals are open with respect to the standard topology. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
ℝ ∈ (topGen‘ran (,))

Theoremelfzop1le2 39315 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 39316 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 39317 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 39318 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 39319 Distribution of negative over subtraction. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → -(𝐴𝐵) = (-𝐴 − -𝐵))

Theoremltdiv2dd 39320 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 39321 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 14180, and absnpncand 39321 should be deleted afterwards.
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (abs‘(𝐴𝐶)) ≤ ((abs‘(𝐴𝐵)) + (abs‘(𝐵𝐶))))

Theoremabssinbd 39322 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 39323 Floor of (1 / 2). (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(⌊‘(1 / 2)) = 0

Theoremmonoords 39324* 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 39325 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 13180, and hashssle 39325 should be deleted afterwards.
((𝐴 ∈ Fin ∧ 𝐵𝐴) → (#‘𝐵) ≤ (#‘𝐴))

Theoremlttri5d 39326 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 39327* A finite ordered set has a unique order isomorphism to a generic finite sequence of integers. This theorem generalizes fz1iso 13229 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 39328 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 39329 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 39330* 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 39331* 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 39332* 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 39333 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 39334 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 39335* 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 39336* 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 39337 Subset relationship for finite sets of sequential integers. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(1...𝑁) ⊆ (0...𝑁)

Theoremfzdifsuc2 39338 Remove a successor from the end of a finite set of sequential integers. Similar to fzdifsuc 12385, but with a weaker condition. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝑁 ∈ (ℤ‘(𝑀 − 1)) → (𝑀...𝑁) = ((𝑀...(𝑁 + 1)) ∖ {(𝑁 + 1)}))

Theoremfzsscn 39339 A finite sequence of integers is a set of complex numbers. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝑀...𝑁) ⊆ ℂ

Theoremdivcan8d 39340 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 39341 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 39342 A finite sequence of integers is a set of real numbers. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝑀...𝑁) ⊆ ℝ

Theoremelfzelzd 39343 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 39344 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 39345 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 39346 A finite set of sequential integers that is a subset of 0. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(0...𝑁) ⊆ ℕ0

Theoremxreqle 39347 Equality implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝐴 ∈ ℝ*𝐴 = 𝐵) → 𝐴𝐵)

Theoremxaddid2d 39348 0 is a left identity for extended real addition. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)       (𝜑 → (0 +𝑒 𝐴) = 𝐴)

Theoremxadd0ge 39349 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 39350 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 39351 'Greater than' implies not equal. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐵𝐴)

Theoremxrleneltd 39352 '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 39353 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 39354* 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 39355* 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 39356 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 39357* 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 39358 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 39359 Equality implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐴 = 𝐵)       (𝜑𝐴𝐵)

Theoremxrgepnfd 39360 An extended real greater or equal to +∞ is +∞ (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑 → +∞ ≤ 𝐴)       (𝜑𝐴 = +∞)

Theoremxrge0nemnfd 39361 A nonnegative extended real is not minus infinity. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ (0[,]+∞))       (𝜑𝐴 ≠ -∞)

Theoremsupxrgere 39362* 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 39363* Lemma for iuneqfzuz 39364: 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 39364* 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 39365 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 39366* 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 39367* 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 39368* 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 39369* 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 39370 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 39371 An extended real that is not +∞ is less than +∞. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴 ≠ +∞)    &   (𝜑𝐴 ∈ ℝ*)       (𝜑𝐴 < +∞)

Theoremltadd12dd 39372 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 39373 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 39374 'Greater than' is a strict ordering on the extended reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
< Or ℝ*

Theoremrpex 39375 The positive reals form a set. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
+ ∈ V

Theoremxrge0ge0 39376 A nonnegative extended real is nonnegative. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝐴 ∈ (0[,]+∞) → 0 ≤ 𝐴)

Theoremxrssre 39377 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 39378 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 39379 The absolute value is a function. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
Fun abs

Theoreminfrpge 39380* The infimum of a non empty, bounded subset of extended reals can be approximated from above by an element of the set. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
𝑥𝜑    &   (𝜑𝐴 ⊆ ℝ*)    &   (𝜑𝐴 ≠ ∅)    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑥𝑦)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → ∃𝑧𝐴 𝑧 ≤ (inf(𝐴, ℝ*, < ) +𝑒 𝐵))

Theoremxrlexaddrp 39381* If an extended real number 𝐴 can be approximated from above, adding positive reals to 𝐵, then 𝐴 is smaller or equal than 𝐵. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   ((𝜑𝑥 ∈ ℝ+) → 𝐴 ≤ (𝐵 +𝑒 𝑥))       (𝜑𝐴𝐵)

Theoremsupsubc 39382* The supremum function distributes over subtraction in a sense similar to that in supaddc 10975. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐴 ≠ ∅)    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥)    &   (𝜑𝐵 ∈ ℝ)    &   𝐶 = {𝑧 ∣ ∃𝑣𝐴 𝑧 = (𝑣𝐵)}       (𝜑 → (sup(𝐴, ℝ, < ) − 𝐵) = sup(𝐶, ℝ, < ))

Theoremxralrple2 39383* Show that 𝐴 is less than 𝐵 by showing that there is no positive bound on the difference. A variant on xralrple 12021. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝑥𝜑    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ (0[,)+∞))       (𝜑 → (𝐴𝐵 ↔ ∀𝑥 ∈ ℝ+ 𝐴 ≤ ((1 + 𝑥) · 𝐵)))

Theoremnnuzdisj 39384 The first 𝑁 elements of the set of nonnegative integers are distinct from any later members. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
((1...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅

Theoremltdivgt1 39385 Divsion by a number greater than 1. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (1 < 𝐵 ↔ (𝐴 / 𝐵) < 𝐴))

Theoremxrltned 39386 'Less than' implies not equal. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐴𝐵)

Theoremnnsplit 39387 Express the set of positive integers as the disjoint (see nnuzdisj 39384) union of the first 𝑁 values and the rest. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
(𝑁 ∈ ℕ → ℕ = ((1...𝑁) ∪ (ℤ‘(𝑁 + 1))))

Theoremdivdiv3d 39388 Division into a fraction. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) / 𝐶) = (𝐴 / (𝐶 · 𝐵)))

Theoremabslt2sqd 39389 Comparison of the square of two numbers. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (abs‘𝐴) < (abs‘𝐵))       (𝜑 → (𝐴↑2) < (𝐵↑2))

Theoremqenom 39390 The set of rational numbers is equinumerous to omega (the set of finite ordinal numbers). (Contributed by Glauco Siliprandi, 24-Dec-2020.)
ℚ ≈ ω

Theoremqct 39391 The set of rational numbers is countable. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
ℚ ≼ ω

Theoremxrltnled 39392 'Less than' in terms of 'less than or equal to'. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)       (𝜑 → (𝐴 < 𝐵 ↔ ¬ 𝐵𝐴))

Theoremlenlteq 39393 'less than or equal to' but not 'less than' implies 'equal' . (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑 → ¬ 𝐴 < 𝐵)       (𝜑𝐴 = 𝐵)

Theoremxrred 39394 An extended real that is neither minus infinity, nor plus infinity, is real. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐴 ≠ -∞)    &   (𝜑𝐴 ≠ +∞)       (𝜑𝐴 ∈ ℝ)

Theoremrr2sscn2 39395 ℝ^2 is a subset of CC^ 2. Common case. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(ℝ × ℝ) ⊆ (ℂ × ℂ)

Theoreminfxr 39396* The infimum of a set of extended reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑𝐴 ⊆ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑 → ∀𝑥𝐴 ¬ 𝑥 < 𝐵)    &   (𝜑 → ∀𝑥 ∈ ℝ (𝐵 < 𝑥 → ∃𝑦𝐴 𝑦 < 𝑥))       (𝜑 → inf(𝐴, ℝ*, < ) = 𝐵)

Theoreminfxrunb2 39397* The infimum of an unbounded-below set of extended reals is minus infinity. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝐴 ⊆ ℝ* → (∀𝑥 ∈ ℝ ∃𝑦𝐴 𝑦 < 𝑥 ↔ inf(𝐴, ℝ*, < ) = -∞))

Theoreminfxrbnd2 39398* The infimum of a bounded-below set of extended reals is greater than minus infinity. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝐴 ⊆ ℝ* → (∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑥𝑦 ↔ -∞ < inf(𝐴, ℝ*, < )))

Theoreminfleinflem1 39399 Lemma for infleinf 39401, case 𝐵 ≠ ∅ ∧ -∞ < inf(𝐵, ℝ*, < ). (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴 ⊆ ℝ*)    &   (𝜑𝐵 ⊆ ℝ*)    &   (𝜑𝑊 ∈ ℝ+)    &   (𝜑𝑋𝐵)    &   (𝜑𝑋 ≤ (inf(𝐵, ℝ*, < ) +𝑒 (𝑊 / 2)))    &   (𝜑𝑍𝐴)    &   (𝜑𝑍 ≤ (𝑋 +𝑒 (𝑊 / 2)))       (𝜑 → inf(𝐴, ℝ*, < ) ≤ (inf(𝐵, ℝ*, < ) +𝑒 𝑊))

Theoreminfleinflem2 39400 Lemma for infleinf 39401, when inf(𝐵, ℝ*, < ) = -∞. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴 ⊆ ℝ*)    &   (𝜑𝐵 ⊆ ℝ*)    &   (𝜑𝑅 ∈ ℝ)    &   (𝜑𝑋𝐵)    &   (𝜑𝑋 < (𝑅 − 2))    &   (𝜑𝑍𝐴)    &   (𝜑𝑍 ≤ (𝑋 +𝑒 1))       (𝜑𝑍 < 𝑅)

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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 42301-42316
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