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Theorem List for Metamath Proof Explorer - 43301-43400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsupsubc 43301* The supremum function distributes over subtraction in a sense similar to that in supaddc 12056. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
(πœ‘ β†’ 𝐴 βŠ† ℝ)    &   (πœ‘ β†’ 𝐴 β‰  βˆ…)    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ 𝐴 𝑦 ≀ π‘₯)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   πΆ = {𝑧 ∣ βˆƒπ‘£ ∈ 𝐴 𝑧 = (𝑣 βˆ’ 𝐡)}    β‡’   (πœ‘ β†’ (sup(𝐴, ℝ, < ) βˆ’ 𝐡) = sup(𝐢, ℝ, < ))
 
Theoremxralrple2 43302* Show that 𝐴 is less than 𝐡 by showing that there is no positive bound on the difference. A variant on xralrple 13053. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
β„²π‘₯πœ‘    &   (πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ (0[,)+∞))    β‡’   (πœ‘ β†’ (𝐴 ≀ 𝐡 ↔ βˆ€π‘₯ ∈ ℝ+ 𝐴 ≀ ((1 + π‘₯) Β· 𝐡)))
 
Theoremnnuzdisj 43303 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) ∩ (β„€β‰₯β€˜(𝑁 + 1))) = βˆ…
 
Theoremltdivgt1 43304 Divsion by a number greater than 1. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
(πœ‘ β†’ 𝐴 ∈ ℝ+)    &   (πœ‘ β†’ 𝐡 ∈ ℝ+)    β‡’   (πœ‘ β†’ (1 < 𝐡 ↔ (𝐴 / 𝐡) < 𝐴))
 
Theoremxrltned 43305 'Less than' implies not equal. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐴 < 𝐡)    β‡’   (πœ‘ β†’ 𝐴 β‰  𝐡)
 
Theoremnnsplit 43306 Express the set of positive integers as the disjoint (see nnuzdisj 43303) union of the first 𝑁 values and the rest. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
(𝑁 ∈ β„• β†’ β„• = ((1...𝑁) βˆͺ (β„€β‰₯β€˜(𝑁 + 1))))
 
Theoremdivdiv3d 43307 Division into a fraction. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ 𝐢 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 β‰  0)    &   (πœ‘ β†’ 𝐢 β‰  0)    β‡’   (πœ‘ β†’ ((𝐴 / 𝐡) / 𝐢) = (𝐴 / (𝐢 Β· 𝐡)))
 
Theoremabslt2sqd 43308 Comparison of the square of two numbers. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ (absβ€˜π΄) < (absβ€˜π΅))    β‡’   (πœ‘ β†’ (𝐴↑2) < (𝐡↑2))
 
Theoremqenom 43309 The set of rational numbers is equinumerous to omega (the set of finite ordinal numbers). (Contributed by Glauco Siliprandi, 24-Dec-2020.)
β„š β‰ˆ Ο‰
 
Theoremqct 43310 The set of rational numbers is countable. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
β„š β‰Ό Ο‰
 
Theoremxrltnled 43311 'Less than' in terms of 'less than or equal to'. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    β‡’   (πœ‘ β†’ (𝐴 < 𝐡 ↔ Β¬ 𝐡 ≀ 𝐴))
 
Theoremlenlteq 43312 'less than or equal to' but not 'less than' implies 'equal' . (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 ≀ 𝐡)    &   (πœ‘ β†’ Β¬ 𝐴 < 𝐡)    β‡’   (πœ‘ β†’ 𝐴 = 𝐡)
 
Theoremxrred 43313 An extended real that is neither minus infinity, nor plus infinity, is real. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐴 β‰  -∞)    &   (πœ‘ β†’ 𝐴 β‰  +∞)    β‡’   (πœ‘ β†’ 𝐴 ∈ ℝ)
 
Theoremrr2sscn2 43314 The cartesian square of ℝ is a subset of the cartesian square of β„‚. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(ℝ Γ— ℝ) βŠ† (β„‚ Γ— β„‚)
 
Theoreminfxr 43315* The infimum of a set of extended reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
β„²π‘₯πœ‘    &   β„²π‘¦πœ‘    &   (πœ‘ β†’ 𝐴 βŠ† ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 Β¬ π‘₯ < 𝐡)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ ℝ (𝐡 < π‘₯ β†’ βˆƒπ‘¦ ∈ 𝐴 𝑦 < π‘₯))    β‡’   (πœ‘ β†’ inf(𝐴, ℝ*, < ) = 𝐡)
 
Theoreminfxrunb2 43316* The infimum of an unbounded-below set of extended reals is minus infinity. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝐴 βŠ† ℝ* β†’ (βˆ€π‘₯ ∈ ℝ βˆƒπ‘¦ ∈ 𝐴 𝑦 < π‘₯ ↔ inf(𝐴, ℝ*, < ) = -∞))
 
Theoreminfxrbnd2 43317* The infimum of a bounded-below set of extended reals is greater than minus infinity. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝐴 βŠ† ℝ* β†’ (βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ 𝐴 π‘₯ ≀ 𝑦 ↔ -∞ < inf(𝐴, ℝ*, < )))
 
Theoreminfleinflem1 43318 Lemma for infleinf 43320, case 𝐡 β‰  βˆ… ∧ -∞ < inf(𝐡, ℝ*, < ). (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(πœ‘ β†’ 𝐴 βŠ† ℝ*)    &   (πœ‘ β†’ 𝐡 βŠ† ℝ*)    &   (πœ‘ β†’ π‘Š ∈ ℝ+)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ 𝑋 ≀ (inf(𝐡, ℝ*, < ) +𝑒 (π‘Š / 2)))    &   (πœ‘ β†’ 𝑍 ∈ 𝐴)    &   (πœ‘ β†’ 𝑍 ≀ (𝑋 +𝑒 (π‘Š / 2)))    β‡’   (πœ‘ β†’ inf(𝐴, ℝ*, < ) ≀ (inf(𝐡, ℝ*, < ) +𝑒 π‘Š))
 
Theoreminfleinflem2 43319 Lemma for infleinf 43320, when inf(𝐡, ℝ*, < ) = -∞. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(πœ‘ β†’ 𝐴 βŠ† ℝ*)    &   (πœ‘ β†’ 𝐡 βŠ† ℝ*)    &   (πœ‘ β†’ 𝑅 ∈ ℝ)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ 𝑋 < (𝑅 βˆ’ 2))    &   (πœ‘ β†’ 𝑍 ∈ 𝐴)    &   (πœ‘ β†’ 𝑍 ≀ (𝑋 +𝑒 1))    β‡’   (πœ‘ β†’ 𝑍 < 𝑅)
 
Theoreminfleinf 43320* If any element of 𝐡 can be approximated from above by members of 𝐴, then the infimum of 𝐴 is less than or equal to the infimum of 𝐡. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(πœ‘ β†’ 𝐴 βŠ† ℝ*)    &   (πœ‘ β†’ 𝐡 βŠ† ℝ*)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ ℝ+) β†’ βˆƒπ‘§ ∈ 𝐴 𝑧 ≀ (π‘₯ +𝑒 𝑦))    β‡’   (πœ‘ β†’ inf(𝐴, ℝ*, < ) ≀ inf(𝐡, ℝ*, < ))
 
Theoremxralrple4 43321* Show that 𝐴 is less than 𝐡 by showing that there is no positive bound on the difference. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    β‡’   (πœ‘ β†’ (𝐴 ≀ 𝐡 ↔ βˆ€π‘₯ ∈ ℝ+ 𝐴 ≀ (𝐡 + (π‘₯↑𝑁))))
 
Theoremxralrple3 43322* Show that 𝐴 is less than 𝐡 by showing that there is no positive bound on the difference. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    &   (πœ‘ β†’ 0 ≀ 𝐢)    β‡’   (πœ‘ β†’ (𝐴 ≀ 𝐡 ↔ βˆ€π‘₯ ∈ ℝ+ 𝐴 ≀ (𝐡 + (𝐢 Β· π‘₯))))
 
Theoremeluzelzd 43323 A member of an upper set of integers is an integer. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    β‡’   (πœ‘ β†’ 𝑁 ∈ β„€)
 
Theoremsuplesup2 43324* If any element of 𝐴 is less than or equal to an element in 𝐡, then the supremum of 𝐴 is less than or equal to the supremum of 𝐡. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(πœ‘ β†’ 𝐴 βŠ† ℝ*)    &   (πœ‘ β†’ 𝐡 βŠ† ℝ*)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ βˆƒπ‘¦ ∈ 𝐡 π‘₯ ≀ 𝑦)    β‡’   (πœ‘ β†’ sup(𝐴, ℝ*, < ) ≀ sup(𝐡, ℝ*, < ))
 
Theoremrecnnltrp 43325 𝑁 is a natural number large enough that its reciprocal is smaller than the given positive 𝐸. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑁 = ((βŒŠβ€˜(1 / 𝐸)) + 1)    β‡’   (𝐸 ∈ ℝ+ β†’ (𝑁 ∈ β„• ∧ (1 / 𝑁) < 𝐸))
 
Theoremnnn0 43326 The set of positive integers is nonempty. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
β„• β‰  βˆ…
 
Theoremfzct 43327 A finite set of sequential integer is countable. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝑁...𝑀) β‰Ό Ο‰
 
Theoremrpgtrecnn 43328* Any positive real number is greater than the reciprocal of a positive integer. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝐴 ∈ ℝ+ β†’ βˆƒπ‘› ∈ β„• (1 / 𝑛) < 𝐴)
 
Theoremfzossuz 43329 A half-open integer interval is a subset of an upper set of integers. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝑀..^𝑁) βŠ† (β„€β‰₯β€˜π‘€)
 
Theoreminfxrrefi 43330 The real and extended real infima match when the set is finite. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
((𝐴 βŠ† ℝ ∧ 𝐴 ∈ Fin ∧ 𝐴 β‰  βˆ…) β†’ inf(𝐴, ℝ*, < ) = inf(𝐴, ℝ, < ))
 
Theoremxrralrecnnle 43331* Show that 𝐴 is less than 𝐡 by showing that there is no positive bound on the difference. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
β„²π‘›πœ‘    &   (πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ (𝐴 ≀ 𝐡 ↔ βˆ€π‘› ∈ β„• 𝐴 ≀ (𝐡 + (1 / 𝑛))))
 
Theoremfzoct 43332 A finite set of sequential integer is countable. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝑁..^𝑀) β‰Ό Ο‰
 
Theoremfrexr 43333 A function taking real values, is a function taking extended real values. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝐹:π΄βŸΆβ„)    β‡’   (πœ‘ β†’ 𝐹:π΄βŸΆβ„*)
 
Theoremnnrecrp 43334 The reciprocal of a positive natural number is a positive real number. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝑁 ∈ β„• β†’ (1 / 𝑁) ∈ ℝ+)
 
Theoremreclt0d 43335 The reciprocal of a negative number is negative. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 0)    β‡’   (πœ‘ β†’ (1 / 𝐴) < 0)
 
Theoremlt0neg1dd 43336 If a number is negative, its negative is positive. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 0)    β‡’   (πœ‘ β†’ 0 < -𝐴)
 
Theoremmnfled 43337 Minus infinity is less than or equal to any extended real. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    β‡’   (πœ‘ β†’ -∞ ≀ 𝐴)
 
Theoreminfxrcld 43338 The infimum of an arbitrary set of extended reals is an extended real. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝐴 βŠ† ℝ*)    β‡’   (πœ‘ β†’ inf(𝐴, ℝ*, < ) ∈ ℝ*)
 
Theoremxrralrecnnge 43339* Show that 𝐴 is less than 𝐡 by showing that there is no positive bound on the difference. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
β„²π‘›πœ‘    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    β‡’   (πœ‘ β†’ (𝐴 ≀ 𝐡 ↔ βˆ€π‘› ∈ β„• (𝐴 βˆ’ (1 / 𝑛)) ≀ 𝐡))
 
Theoremreclt0 43340 The reciprocal of a negative number is negative. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 β‰  0)    β‡’   (πœ‘ β†’ (𝐴 < 0 ↔ (1 / 𝐴) < 0))
 
Theoremltmulneg 43341 Multiplying by a negative number, swaps the order. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    &   (πœ‘ β†’ 𝐢 < 0)    β‡’   (πœ‘ β†’ (𝐴 < 𝐡 ↔ (𝐡 Β· 𝐢) < (𝐴 Β· 𝐢)))
 
Theoremallbutfi 43342* For all but finitely many. Some authors say "cofinitely many". Some authors say "ultimately". Compare with eliuniin 43043 and eliuniin2 43064 (here, the precondition can be dropped; see eliuniincex 43053). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   π΄ = βˆͺ 𝑛 ∈ 𝑍 ∩ π‘š ∈ (β„€β‰₯β€˜π‘›)𝐡    β‡’   (𝑋 ∈ 𝐴 ↔ βˆƒπ‘› ∈ 𝑍 βˆ€π‘š ∈ (β„€β‰₯β€˜π‘›)𝑋 ∈ 𝐡)
 
Theoremltdiv23neg 43343 Swap denominator with other side of 'less than', when both are negative. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 < 0)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    &   (πœ‘ β†’ 𝐢 < 0)    β‡’   (πœ‘ β†’ ((𝐴 / 𝐡) < 𝐢 ↔ (𝐴 / 𝐢) < 𝐡))
 
Theoremxreqnltd 43344 A consequence of trichotomy. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐴 = 𝐡)    β‡’   (πœ‘ β†’ Β¬ 𝐴 < 𝐡)
 
Theoremmnfnre2 43345 Minus infinity is not a real number. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Β¬ -∞ ∈ ℝ
 
Theoremzssxr 43346 The integers are a subset of the extended reals. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
β„€ βŠ† ℝ*
 
Theoremfisupclrnmpt 43347* A nonempty finite indexed set contains its supremum. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
β„²π‘₯πœ‘    &   (πœ‘ β†’ 𝑅 Or 𝐴)    &   (πœ‘ β†’ 𝐡 ∈ Fin)    &   (πœ‘ β†’ 𝐡 β‰  βˆ…)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ 𝐢 ∈ 𝐴)    β‡’   (πœ‘ β†’ sup(ran (π‘₯ ∈ 𝐡 ↦ 𝐢), 𝐴, 𝑅) ∈ 𝐴)
 
Theoremsupxrunb3 43348* The supremum of an unbounded-above set of extended reals is plus infinity. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝐴 βŠ† ℝ* β†’ (βˆ€π‘₯ ∈ ℝ βˆƒπ‘¦ ∈ 𝐴 π‘₯ ≀ 𝑦 ↔ sup(𝐴, ℝ*, < ) = +∞))
 
Theoremelfzod 43349 Membership in a half-open integer interval. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(πœ‘ β†’ 𝐾 ∈ (β„€β‰₯β€˜π‘€))    &   (πœ‘ β†’ 𝑁 ∈ β„€)    &   (πœ‘ β†’ 𝐾 < 𝑁)    β‡’   (πœ‘ β†’ 𝐾 ∈ (𝑀..^𝑁))
 
Theoremfimaxre4 43350* A nonempty finite set of real numbers is bounded (image set version). (Contributed by Glauco Siliprandi, 23-Oct-2021.)
β„²π‘₯πœ‘    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ βˆƒπ‘¦ ∈ ℝ βˆ€π‘₯ ∈ 𝐴 𝐡 ≀ 𝑦)
 
Theoremren0 43351 The set of reals is nonempty. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
ℝ β‰  βˆ…
 
Theoremeluzelz2 43352 A member of an upper set of integers is an integer. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑍 = (β„€β‰₯β€˜π‘€)    β‡’   (𝑁 ∈ 𝑍 β†’ 𝑁 ∈ β„€)
 
Theoremresabs2d 43353 Absorption law for restriction. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(πœ‘ β†’ 𝐡 βŠ† 𝐢)    β‡’   (πœ‘ β†’ ((𝐴 β†Ύ 𝐡) β†Ύ 𝐢) = (𝐴 β†Ύ 𝐡))
 
Theoremuzid2 43354 Membership of the least member in an upper set of integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝑀 ∈ (β„€β‰₯β€˜π‘) β†’ 𝑀 ∈ (β„€β‰₯β€˜π‘€))
 
Theoremsupxrleubrnmpt 43355* The supremum of a nonempty bounded indexed set of extended reals is less than or equal to an upper bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
β„²π‘₯πœ‘    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ ℝ*)    β‡’   (πœ‘ β†’ (sup(ran (π‘₯ ∈ 𝐴 ↦ 𝐡), ℝ*, < ) ≀ 𝐢 ↔ βˆ€π‘₯ ∈ 𝐴 𝐡 ≀ 𝐢))
 
Theoremuzssre2 43356 An upper set of integers is a subset of the Reals. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑍 = (β„€β‰₯β€˜π‘€)    β‡’   π‘ βŠ† ℝ
 
Theoremuzssd 43357 Subset relationship for two sets of upper integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    β‡’   (πœ‘ β†’ (β„€β‰₯β€˜π‘) βŠ† (β„€β‰₯β€˜π‘€))
 
Theoremeluzd 43358 Membership in an upper set of integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„€)    &   (πœ‘ β†’ 𝑀 ≀ 𝑁)    β‡’   (πœ‘ β†’ 𝑁 ∈ 𝑍)
 
Theoreminfxrlbrnmpt2 43359* A member of a nonempty indexed set of reals is greater than or equal to the set's lower bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
β„²π‘₯πœ‘    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ 𝐴)    &   (πœ‘ β†’ 𝐷 ∈ ℝ*)    &   (π‘₯ = 𝐢 β†’ 𝐡 = 𝐷)    β‡’   (πœ‘ β†’ inf(ran (π‘₯ ∈ 𝐴 ↦ 𝐡), ℝ*, < ) ≀ 𝐷)
 
Theoremxrre4 43360 An extended real is real iff it is not an infinty. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝐴 ∈ ℝ* β†’ (𝐴 ∈ ℝ ↔ (𝐴 β‰  -∞ ∧ 𝐴 β‰  +∞)))
 
Theoremuz0 43361 The upper integers function applied to a non-integer, is the empty set. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(Β¬ 𝑀 ∈ β„€ β†’ (β„€β‰₯β€˜π‘€) = βˆ…)
 
Theoremeluzelz2d 43362 A member of an upper set of integers is an integer. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑁 ∈ 𝑍)    β‡’   (πœ‘ β†’ 𝑁 ∈ β„€)
 
Theoreminfleinf2 43363* If any element in 𝐡 is greater than or equal to an element in 𝐴, then the infimum of 𝐴 is less than or equal to the infimum of 𝐡. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
β„²π‘₯πœ‘    &   β„²π‘¦πœ‘    &   (πœ‘ β†’ 𝐴 βŠ† ℝ*)    &   (πœ‘ β†’ 𝐡 βŠ† ℝ*)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ βˆƒπ‘¦ ∈ 𝐴 𝑦 ≀ π‘₯)    β‡’   (πœ‘ β†’ inf(𝐴, ℝ*, < ) ≀ inf(𝐡, ℝ*, < ))
 
Theoremunb2ltle 43364* "Unbounded below" expressed with < and with ≀. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝐴 βŠ† ℝ* β†’ (βˆ€π‘€ ∈ ℝ βˆƒπ‘¦ ∈ 𝐴 𝑦 < 𝑀 ↔ βˆ€π‘₯ ∈ ℝ βˆƒπ‘¦ ∈ 𝐴 𝑦 ≀ π‘₯))
 
Theoremuzidd2 43365 Membership of the least member in an upper set of integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   π‘ = (β„€β‰₯β€˜π‘€)    β‡’   (πœ‘ β†’ 𝑀 ∈ 𝑍)
 
Theoremuzssd2 43366 Subset relationship for two sets of upper integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑁 ∈ 𝑍)    β‡’   (πœ‘ β†’ (β„€β‰₯β€˜π‘) βŠ† 𝑍)
 
Theoremrexabslelem 43367* An indexed set of absolute values of real numbers is bounded if and only if the original values are bounded above and below. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
β„²π‘₯πœ‘    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ (βˆƒπ‘¦ ∈ ℝ βˆ€π‘₯ ∈ 𝐴 (absβ€˜π΅) ≀ 𝑦 ↔ (βˆƒπ‘€ ∈ ℝ βˆ€π‘₯ ∈ 𝐴 𝐡 ≀ 𝑀 ∧ βˆƒπ‘§ ∈ ℝ βˆ€π‘₯ ∈ 𝐴 𝑧 ≀ 𝐡)))
 
Theoremrexabsle 43368* An indexed set of absolute values of real numbers is bounded if and only if the original values are bounded above and below. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
β„²π‘₯πœ‘    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ (βˆƒπ‘¦ ∈ ℝ βˆ€π‘₯ ∈ 𝐴 (absβ€˜π΅) ≀ 𝑦 ↔ (βˆƒπ‘€ ∈ ℝ βˆ€π‘₯ ∈ 𝐴 𝐡 ≀ 𝑀 ∧ βˆƒπ‘§ ∈ ℝ βˆ€π‘₯ ∈ 𝐴 𝑧 ≀ 𝐡)))
 
Theoremallbutfiinf 43369* Given a "for all but finitely many" condition, the condition holds from 𝑁 on. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   π΄ = βˆͺ 𝑛 ∈ 𝑍 ∩ π‘š ∈ (β„€β‰₯β€˜π‘›)𝐡    &   (πœ‘ β†’ 𝑋 ∈ 𝐴)    &   π‘ = inf({𝑛 ∈ 𝑍 ∣ βˆ€π‘š ∈ (β„€β‰₯β€˜π‘›)𝑋 ∈ 𝐡}, ℝ, < )    β‡’   (πœ‘ β†’ (𝑁 ∈ 𝑍 ∧ βˆ€π‘š ∈ (β„€β‰₯β€˜π‘)𝑋 ∈ 𝐡))
 
Theoremsupxrrernmpt 43370* The real and extended real indexed suprema match when the indexed real supremum exists. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
β„²π‘₯πœ‘    &   (πœ‘ β†’ 𝐴 β‰  βˆ…)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ βˆƒπ‘¦ ∈ ℝ βˆ€π‘₯ ∈ 𝐴 𝐡 ≀ 𝑦)    β‡’   (πœ‘ β†’ sup(ran (π‘₯ ∈ 𝐴 ↦ 𝐡), ℝ*, < ) = sup(ran (π‘₯ ∈ 𝐴 ↦ 𝐡), ℝ, < ))
 
Theoremsuprleubrnmpt 43371* The supremum of a nonempty bounded indexed set of reals is less than or equal to an upper bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
β„²π‘₯πœ‘    &   (πœ‘ β†’ 𝐴 β‰  βˆ…)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ βˆƒπ‘¦ ∈ ℝ βˆ€π‘₯ ∈ 𝐴 𝐡 ≀ 𝑦)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    β‡’   (πœ‘ β†’ (sup(ran (π‘₯ ∈ 𝐴 ↦ 𝐡), ℝ, < ) ≀ 𝐢 ↔ βˆ€π‘₯ ∈ 𝐴 𝐡 ≀ 𝐢))
 
Theoreminfrnmptle 43372* An indexed infimum of extended reals is smaller than another indexed infimum of extended reals, when every indexed element is smaller than the corresponding one. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
β„²π‘₯πœ‘    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ*)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐢 ∈ ℝ*)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ≀ 𝐢)    β‡’   (πœ‘ β†’ inf(ran (π‘₯ ∈ 𝐴 ↦ 𝐡), ℝ*, < ) ≀ inf(ran (π‘₯ ∈ 𝐴 ↦ 𝐢), ℝ*, < ))
 
Theoreminfxrunb3 43373* The infimum of an unbounded-below set of extended reals is minus infinity. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝐴 βŠ† ℝ* β†’ (βˆ€π‘₯ ∈ ℝ βˆƒπ‘¦ ∈ 𝐴 𝑦 ≀ π‘₯ ↔ inf(𝐴, ℝ*, < ) = -∞))
 
Theoremuzn0d 43374 The upper integers are all nonempty. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   π‘ = (β„€β‰₯β€˜π‘€)    β‡’   (πœ‘ β†’ 𝑍 β‰  βˆ…)
 
Theoremuzssd3 43375 Subset relationship for two sets of upper integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑍 = (β„€β‰₯β€˜π‘€)    β‡’   (𝑁 ∈ 𝑍 β†’ (β„€β‰₯β€˜π‘) βŠ† 𝑍)
 
Theoremrexabsle2 43376* An indexed set of absolute values of real numbers is bounded if and only if the original values are bounded above and below. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
β„²π‘₯πœ‘    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ (βˆƒπ‘¦ ∈ ℝ βˆ€π‘₯ ∈ 𝐴 (absβ€˜π΅) ≀ 𝑦 ↔ (βˆƒπ‘¦ ∈ ℝ βˆ€π‘₯ ∈ 𝐴 𝐡 ≀ 𝑦 ∧ βˆƒπ‘¦ ∈ ℝ βˆ€π‘₯ ∈ 𝐴 𝑦 ≀ 𝐡)))
 
Theoreminfxrunb3rnmpt 43377* The infimum of an unbounded-below set of extended reals is minus infinity. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
β„²π‘₯πœ‘    &   β„²π‘¦πœ‘    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ*)    β‡’   (πœ‘ β†’ (βˆ€π‘¦ ∈ ℝ βˆƒπ‘₯ ∈ 𝐴 𝐡 ≀ 𝑦 ↔ inf(ran (π‘₯ ∈ 𝐴 ↦ 𝐡), ℝ*, < ) = -∞))
 
Theoremsupxrre3rnmpt 43378* The indexed supremum of a nonempty set of reals, is real if and only if it is bounded-above . (Contributed by Glauco Siliprandi, 23-Oct-2021.)
β„²π‘₯πœ‘    &   (πœ‘ β†’ 𝐴 β‰  βˆ…)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ (sup(ran (π‘₯ ∈ 𝐴 ↦ 𝐡), ℝ*, < ) ∈ ℝ ↔ βˆƒπ‘¦ ∈ ℝ βˆ€π‘₯ ∈ 𝐴 𝐡 ≀ 𝑦))
 
Theoremuzublem 43379* A set of reals, indexed by upper integers, is bound if and only if any upper part is bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
β„²π‘—πœ‘    &   β„²π‘—𝑋    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   π‘ = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ π‘Œ ∈ ℝ)    &   π‘Š = sup(ran (𝑗 ∈ (𝑀...𝐾) ↦ 𝐡), ℝ, < )    &   π‘‹ = if(π‘Š ≀ π‘Œ, π‘Œ, π‘Š)    &   (πœ‘ β†’ 𝐾 ∈ 𝑍)    &   ((πœ‘ ∧ 𝑗 ∈ 𝑍) β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ βˆ€π‘— ∈ (β„€β‰₯β€˜πΎ)𝐡 ≀ π‘Œ)    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘— ∈ 𝑍 𝐡 ≀ π‘₯)
 
Theoremuzub 43380* A set of reals, indexed by upper integers, is bound if and only if any upper part is bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
β„²π‘—πœ‘    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   π‘ = (β„€β‰₯β€˜π‘€)    &   ((πœ‘ ∧ 𝑗 ∈ 𝑍) β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ (βˆƒπ‘₯ ∈ ℝ βˆƒπ‘˜ ∈ 𝑍 βˆ€π‘— ∈ (β„€β‰₯β€˜π‘˜)𝐡 ≀ π‘₯ ↔ βˆƒπ‘₯ ∈ ℝ βˆ€π‘— ∈ 𝑍 𝐡 ≀ π‘₯))
 
Theoremssrexr 43381 A subset of the reals is a subset of the extended reals. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(πœ‘ β†’ 𝐴 βŠ† ℝ)    β‡’   (πœ‘ β†’ 𝐴 βŠ† ℝ*)
 
Theoremsupxrmnf2 43382 Removing minus infinity from a set does not affect its supremum. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝐴 βŠ† ℝ* β†’ sup((𝐴 βˆ– {-∞}), ℝ*, < ) = sup(𝐴, ℝ*, < ))
 
Theoremsupxrcli 43383 The supremum of an arbitrary set of extended reals is an extended real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐴 βŠ† ℝ*    β‡’   sup(𝐴, ℝ*, < ) ∈ ℝ*
 
Theoremuzid3 43384 Membership of the least member in an upper set of integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑍 = (β„€β‰₯β€˜π‘€)    β‡’   (𝑁 ∈ 𝑍 β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘))
 
Theoreminfxrlesupxr 43385 The supremum of a nonempty set is greater than or equal to the infimum. The second condition is needed, see supxrltinfxr 43398. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(πœ‘ β†’ 𝐴 βŠ† ℝ*)    &   (πœ‘ β†’ 𝐴 β‰  βˆ…)    β‡’   (πœ‘ β†’ inf(𝐴, ℝ*, < ) ≀ sup(𝐴, ℝ*, < ))
 
Theoremxnegeqd 43386 Equality of two extended numbers with -𝑒 in front of them. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(πœ‘ β†’ 𝐴 = 𝐡)    β‡’   (πœ‘ β†’ -𝑒𝐴 = -𝑒𝐡)
 
Theoremxnegrecl 43387 The extended real negative of a real number is real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝐴 ∈ ℝ β†’ -𝑒𝐴 ∈ ℝ)
 
Theoremxnegnegi 43388 Extended real version of negneg 11385. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐴 ∈ ℝ*    β‡’   -𝑒-𝑒𝐴 = 𝐴
 
Theoremxnegeqi 43389 Equality of two extended numbers with -𝑒 in front of them. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐴 = 𝐡    β‡’   -𝑒𝐴 = -𝑒𝐡
 
Theoremnfxnegd 43390 Deduction version of nfxneg 43410. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(πœ‘ β†’ β„²π‘₯𝐴)    β‡’   (πœ‘ β†’ β„²π‘₯-𝑒𝐴)
 
Theoremxnegnegd 43391 Extended real version of negnegd 11437. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    β‡’   (πœ‘ β†’ -𝑒-𝑒𝐴 = 𝐴)
 
Theoremuzred 43392 An upper integer is a real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝐴 ∈ 𝑍)    β‡’   (πœ‘ β†’ 𝐴 ∈ ℝ)
 
Theoremxnegcli 43393 Closure of extended real negative. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐴 ∈ ℝ*    β‡’   -𝑒𝐴 ∈ ℝ*
 
Theoremsupminfrnmpt 43394* The indexed supremum of a bounded-above set of reals is the negation of the indexed infimum of that set's image under negation. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
β„²π‘₯πœ‘    &   (πœ‘ β†’ 𝐴 β‰  βˆ…)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ βˆƒπ‘¦ ∈ ℝ βˆ€π‘₯ ∈ 𝐴 𝐡 ≀ 𝑦)    β‡’   (πœ‘ β†’ sup(ran (π‘₯ ∈ 𝐴 ↦ 𝐡), ℝ, < ) = -inf(ran (π‘₯ ∈ 𝐴 ↦ -𝐡), ℝ, < ))
 
Theoreminfxrpnf 43395 Adding plus infinity to a set does not affect its infimum. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝐴 βŠ† ℝ* β†’ inf((𝐴 βˆͺ {+∞}), ℝ*, < ) = inf(𝐴, ℝ*, < ))
 
Theoreminfxrrnmptcl 43396* The infimum of an arbitrary indexed set of extended reals is an extended real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
β„²π‘₯πœ‘    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ*)    β‡’   (πœ‘ β†’ inf(ran (π‘₯ ∈ 𝐴 ↦ 𝐡), ℝ*, < ) ∈ ℝ*)
 
Theoremleneg2d 43397 Negative of one side of 'less than or equal to'. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ (𝐴 ≀ -𝐡 ↔ 𝐡 ≀ -𝐴))
 
Theoremsupxrltinfxr 43398 The supremum of the empty set is strictly smaller than the infimum of the empty set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
sup(βˆ…, ℝ*, < ) < inf(βˆ…, ℝ*, < )
 
Theoremmax1d 43399 A number is less than or equal to the maximum of it and another. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ 𝐴 ≀ if(𝐴 ≀ 𝐡, 𝐡, 𝐴))
 
Theoremsupxrleubrnmptf 43400 The supremum of a nonempty bounded indexed set of extended reals is less than or equal to an upper bound. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
β„²π‘₯πœ‘    &   β„²π‘₯𝐴    &   β„²π‘₯𝐢    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ ℝ*)    β‡’   (πœ‘ β†’ (sup(ran (π‘₯ ∈ 𝐴 ↦ 𝐡), ℝ*, < ) ≀ 𝐢 ↔ βˆ€π‘₯ ∈ 𝐴 𝐡 ≀ 𝐢))
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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 42301-42400 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 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-46948
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