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Theorem List for Metamath Proof Explorer - 42401-42500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremiccpartxr 42401 If there is a partition, then all intermediate points and bounds are extended real numbers. (Contributed by AV, 11-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))    &   (𝜑𝐼 ∈ (0...𝑀))       (𝜑 → (𝑃𝐼) ∈ ℝ*)

Theoremiccpartgtprec 42402 If there is a partition, then all intermediate points and the upper bound are strictly greater than the preceeding intermediate points or lower bound. (Contributed by AV, 11-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))    &   (𝜑𝐼 ∈ (1...𝑀))       (𝜑 → (𝑃‘(𝐼 − 1)) < (𝑃𝐼))

Theoremiccpartipre 42403 If there is a partition, then all intermediate points are real numbers. (Contributed by AV, 11-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))    &   (𝜑𝐼 ∈ (1..^𝑀))       (𝜑 → (𝑃𝐼) ∈ ℝ)

Theoremiccpartiltu 42404* If there is a partition, then all intermediate points are strictly less than the upper bound. (Contributed by AV, 12-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))       (𝜑 → ∀𝑖 ∈ (1..^𝑀)(𝑃𝑖) < (𝑃𝑀))

Theoremiccpartigtl 42405* If there is a partition, then all intermediate points are strictly greater than the lower bound. (Contributed by AV, 12-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))       (𝜑 → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃𝑖))

Theoremiccpartlt 42406 If there is a partition, then the lower bound is strictly less than the upper bound. Corresponds to fourierdlem11 41276 in GS's mathbox. (Contributed by AV, 12-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))       (𝜑 → (𝑃‘0) < (𝑃𝑀))

Theoremiccpartltu 42407* If there is a partition, then all intermediate points and the lower bound are strictly less than the upper bound. (Contributed by AV, 14-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))       (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃𝑀))

Theoremiccpartgtl 42408* If there is a partition, then all intermediate points and the upper bound are strictly greater than the lower bound. (Contributed by AV, 14-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))       (𝜑 → ∀𝑖 ∈ (1...𝑀)(𝑃‘0) < (𝑃𝑖))

Theoremiccpartgt 42409* If there is a partition, then all intermediate points and the bounds are strictly ordered. (Contributed by AV, 18-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))       (𝜑 → ∀𝑖 ∈ (0...𝑀)∀𝑗 ∈ (0...𝑀)(𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))

Theoremiccpartleu 42410* If there is a partition, then all intermediate points and the lower and the upper bound are less than or equal to the upper bound. (Contributed by AV, 14-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))       (𝜑 → ∀𝑖 ∈ (0...𝑀)(𝑃𝑖) ≤ (𝑃𝑀))

Theoremiccpartgel 42411* If there is a partition, then all intermediate points and the upper and the lower bound are greater than or equal to the lower bound. (Contributed by AV, 14-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))       (𝜑 → ∀𝑖 ∈ (0...𝑀)(𝑃‘0) ≤ (𝑃𝑖))

Theoremiccpartrn 42412 If there is a partition, then all intermediate points and bounds are contained in a closed interval of extended reals. (Contributed by AV, 14-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))       (𝜑 → ran 𝑃 ⊆ ((𝑃‘0)[,](𝑃𝑀)))

Theoremiccpartf 42413 The range of the partition is between its starting point and its ending point. Corresponds to fourierdlem15 41280 in GS's mathbox. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 14-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))       (𝜑𝑃:(0...𝑀)⟶((𝑃‘0)[,](𝑃𝑀)))

Theoremiccpartel 42414 If there is a partition, then all intermediate points and bounds are contained in a closed interval of extended reals. (Contributed by AV, 14-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))       ((𝜑𝐼 ∈ (0...𝑀)) → (𝑃𝐼) ∈ ((𝑃‘0)[,](𝑃𝑀)))

Theoremiccelpart 42415* An element of any partitioned half-open interval of extended reals is an element of a part of this partition. (Contributed by AV, 18-Jul-2020.)
(𝑀 ∈ ℕ → ∀𝑝 ∈ (RePart‘𝑀)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))

Theoremiccpartiun 42416* A half-open interval of extended reals is the union of the parts of its partition. (Contributed by AV, 18-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))       (𝜑 → ((𝑃‘0)[,)(𝑃𝑀)) = 𝑖 ∈ (0..^𝑀)((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))))

Theoremicceuelpartlem 42417 Lemma for icceuelpart 42418. (Contributed by AV, 19-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))       (𝜑 → ((𝐼 ∈ (0..^𝑀) ∧ 𝐽 ∈ (0..^𝑀)) → (𝐼 < 𝐽 → (𝑃‘(𝐼 + 1)) ≤ (𝑃𝐽))))

Theoremicceuelpart 42418* An element of a partitioned half-open interval of extended reals is an element of exactly one part of the partition. (Contributed by AV, 19-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))       ((𝜑𝑋 ∈ ((𝑃‘0)[,)(𝑃𝑀))) → ∃!𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))))

Theoremiccpartdisj 42419* The segments of a partitioned half-open interval of extended reals are a disjoint collection. (Contributed by AV, 19-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))       (𝜑Disj 𝑖 ∈ (0..^𝑀)((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))))

Theoremiccpartnel 42420 A point of a partition is not an element of any open interval determined by the partition. Corresponds to fourierdlem12 41277 in GS's mathbox. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 8-Jul-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑃 ∈ (RePart‘𝑀))    &   (𝜑𝑋 ∈ ran 𝑃)       ((𝜑𝐼 ∈ (0..^𝑀)) → ¬ 𝑋 ∈ ((𝑃𝐼)(,)(𝑃‘(𝐼 + 1))))

20.36.8  Shifting functions with an integer range domain

Theoremfargshiftfv 42421* If a class is a function, then the values of the "shifted function" correspond to the function values of the class. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
𝐺 = (𝑥 ∈ (0..^(♯‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))       ((𝑁 ∈ ℕ0𝐹:(1...𝑁)⟶dom 𝐸) → (𝑋 ∈ (0..^𝑁) → (𝐺𝑋) = (𝐹‘(𝑋 + 1))))

Theoremfargshiftf 42422* If a class is a function, then also its "shifted function" is a function. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
𝐺 = (𝑥 ∈ (0..^(♯‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))       ((𝑁 ∈ ℕ0𝐹:(1...𝑁)⟶dom 𝐸) → 𝐺:(0..^(♯‘𝐹))⟶dom 𝐸)

Theoremfargshiftf1 42423* If a function is 1-1, then also the shifted function is 1-1. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
𝐺 = (𝑥 ∈ (0..^(♯‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))       ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–1-1→dom 𝐸) → 𝐺:(0..^(♯‘𝐹))–1-1→dom 𝐸)

Theoremfargshiftfo 42424* If a function is onto, then also the shifted function is onto. (Contributed by Alexander van der Vekens, 24-Nov-2017.)
𝐺 = (𝑥 ∈ (0..^(♯‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))       ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → 𝐺:(0..^(♯‘𝐹))–onto→dom 𝐸)

Theoremfargshiftfva 42425* The values of a shifted function correspond to the value of the original function. (Contributed by Alexander van der Vekens, 24-Nov-2017.)
𝐺 = (𝑥 ∈ (0..^(♯‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))       ((𝑁 ∈ ℕ0𝐹:(1...𝑁)⟶dom 𝐸) → (∀𝑘 ∈ (1...𝑁)(𝐸‘(𝐹𝑘)) = 𝑘 / 𝑥𝑃 → ∀𝑙 ∈ (0..^𝑁)(𝐸‘(𝐺𝑙)) = (𝑙 + 1) / 𝑥𝑃))

20.36.9  Words over a set (extension)

20.36.9.1  Last symbol of a word - extension

Theoremlswn0 42426 The last symbol of a not empty word exists. The empty set must be excluded as symbol, because otherwise, it cannot be distinguished between valid cases ( is the last symbol) and invalid cases ( means that no last symbol exists. This is because of the special definition of a function in set.mm. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
((𝑊 ∈ Word 𝑉 ∧ ∅ ∉ 𝑉 ∧ (♯‘𝑊) ≠ 0) → (lastS‘𝑊) ≠ ∅)

20.36.10  Unordered pairs

20.36.10.1  Set of unordered pairs

Theoremsprid 42427 Two identical representations of the class of all unordered pairs. (Contributed by AV, 21-Nov-2021.)
{𝑝 ∣ ∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏}} = {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}}

Theoremelsprel 42428* An unordered pair is an element of all unordered pairs. At least one of the two elements of the unordered pair must be a set. Otherwise, the unordered pair would be the empty set, see prprc 4534, which is not an element of all unordered pairs, see spr0nelg 42429. (Contributed by AV, 21-Nov-2021.)
((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} ∈ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}})

Theoremspr0nelg 42429* The empty set is not an element of all unordered pairs. (Contributed by AV, 21-Nov-2021.)
∅ ∉ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}}

Syntaxcspr 42430 Extend class notation with set of pairs.
class Pairs

Definitiondf-spr 42431* Define the function which maps a set 𝑣 to the set of pairs consisting of elements of the set 𝑣. (Contributed by AV, 21-Nov-2021.)
Pairs = (𝑣 ∈ V ↦ {𝑝 ∣ ∃𝑎𝑣𝑏𝑣 𝑝 = {𝑎, 𝑏}})

Theoremsprval 42432* The set of all unordered pairs over a given set 𝑉. (Contributed by AV, 21-Nov-2021.)
(𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})

Theoremsprvalpw 42433* The set of all unordered pairs over a given set 𝑉, expressed by a restricted class abstraction. (Contributed by AV, 21-Nov-2021.)
(𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})

Theoremsprssspr 42434* The set of all unordered pairs over a given set 𝑉 is a subset of the set of all unordered pairs. (Contributed by AV, 21-Nov-2021.)
(Pairs‘𝑉) ⊆ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}}

Theoremspr0el 42435 The empty set is not an unordered pair over any set 𝑉. (Contributed by AV, 21-Nov-2021.)
∅ ∉ (Pairs‘𝑉)

Theoremsprvalpwn0 42436* The set of all unordered pairs over a given set 𝑉, expressed by a restricted class abstraction. (Contributed by AV, 21-Nov-2021.)
(𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})

Theoremsprel 42437* An element of the set of all unordered pairs over a given set 𝑉 is a pair of elements of the set 𝑉. (Contributed by AV, 22-Nov-2021.)
(𝑋 ∈ (Pairs‘𝑉) → ∃𝑎𝑉𝑏𝑉 𝑋 = {𝑎, 𝑏})

Theoremprssspr 42438* An element of a subset of the set of all unordered pairs over a given set 𝑉, is a pair of elements of the set 𝑉. (Contributed by AV, 22-Nov-2021.)
((𝑃 ⊆ (Pairs‘𝑉) ∧ 𝑋𝑃) → ∃𝑎𝑉𝑏𝑉 𝑋 = {𝑎, 𝑏})

Theoremprelspr 42439 An unordered pair of elements of a fixed set 𝑉 belongs to the set of all unordered pairs over the set 𝑉. (Contributed by AV, 21-Nov-2021.)
((𝑉𝑊 ∧ (𝑋𝑉𝑌𝑉)) → {𝑋, 𝑌} ∈ (Pairs‘𝑉))

Theoremprsprel 42440 The elements of a pair from the set of all unordered pairs over a given set 𝑉 are elements of the set 𝑉. (Contributed by AV, 22-Nov-2021.)
(({𝑋, 𝑌} ∈ (Pairs‘𝑉) ∧ (𝑋𝑈𝑌𝑊)) → (𝑋𝑉𝑌𝑉))

Theoremprsssprel 42441 The elements of a pair from a subset of the set of all unordered pairs over a given set 𝑉 are elements of the set 𝑉. (Contributed by AV, 21-Nov-2021.)
((𝑃 ⊆ (Pairs‘𝑉) ∧ {𝑋, 𝑌} ∈ 𝑃 ∧ (𝑋𝑈𝑌𝑊)) → (𝑋𝑉𝑌𝑉))

Theoremsprvalpwle2 42442* The set of all unordered pairs over a given set 𝑉, expressed by a restricted class abstraction. (Contributed by AV, 24-Nov-2021.)
(𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})

Theoremsprsymrelfvlem 42443* Lemma for sprsymrelf 42448 and sprsymrelfv 42447. (Contributed by AV, 19-Nov-2021.)
(𝑃 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))

Theoremsprsymrelf1lem 42444* Lemma for sprsymrelf1 42449. (Contributed by AV, 22-Nov-2021.)
((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}} → 𝑎𝑏))

Theoremsprsymrelfolem1 42445* Lemma 1 for sprsymrelfo 42450. (Contributed by AV, 22-Nov-2021.)
𝑄 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)}       𝑄 ∈ 𝒫 (Pairs‘𝑉)

Theoremsprsymrelfolem2 42446* Lemma 2 for sprsymrelfo 42450. (Contributed by AV, 23-Nov-2021.)
𝑄 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)}       ((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) → (𝑥𝑅𝑦 ↔ ∃𝑐𝑄 𝑐 = {𝑥, 𝑦}))

Theoremsprsymrelfv 42447* The value of the function 𝐹 which maps a subset of the set of pairs over a fixed set 𝑉 to the relation relating two elements of the set 𝑉 iff they are in a pair of the subset. (Contributed by AV, 19-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}    &   𝐹 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})       (𝑋𝑃 → (𝐹𝑋) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑋 𝑐 = {𝑥, 𝑦}})

Theoremsprsymrelf 42448* The mapping 𝐹 is a function from the subsets of the set of pairs over a fixed set 𝑉 into the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 19-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}    &   𝐹 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})       𝐹:𝑃𝑅

Theoremsprsymrelf1 42449* The mapping 𝐹 is a one-to-one function from the subsets of the set of pairs over a fixed set 𝑉 into the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 19-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}    &   𝐹 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})       𝐹:𝑃1-1𝑅

Theoremsprsymrelfo 42450* The mapping 𝐹 is a function from the subsets of the set of pairs over a fixed set 𝑉 onto the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 23-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}    &   𝐹 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})       (𝑉𝑊𝐹:𝑃onto𝑅)

Theoremsprsymrelf1o 42451* The mapping 𝐹 is a bijection between the subsets of the set of pairs over a fixed set 𝑉 into the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 23-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}    &   𝐹 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})       (𝑉𝑊𝐹:𝑃1-1-onto𝑅)

Theoremsprbisymrel 42452* There is a bijection between the subsets of the set of pairs over a fixed set 𝑉 and the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 23-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}       (𝑉𝑊 → ∃𝑓 𝑓:𝑃1-1-onto𝑅)

Theoremsprsymrelen 42453* The class 𝑃 of subsets of the set of pairs over a fixed set 𝑉 and the class 𝑅 of symmetric relations on the fixed set 𝑉 are equinumerous. (Contributed by AV, 27-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}       (𝑉𝑊𝑃𝑅)

20.36.10.2  Proper (unordered) pairs

Proper (unordered) pairs are unordered pairs with exactly 2 elements. The set of proper pairs with elements of a class 𝑉 is defined by {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}.

For example, {1, 2} is a proper pair, because 1 ≠ 2 ( see 1ne2 11595). Examples for not proper unordered pairs are {1, 1} = {1} (see preqsn 4626), {1, V} = {1} (see prprc2 4533) or {V, V} = ∅ (see prprc 4534).

Theoremprpair 42454* Characterization of a proper pair: A class is a proper pair iff it consists of exactly two different sets. (Contributed by AV, 11-Mar-2023.)
𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}       (𝑋𝑃 ↔ ∃𝑎𝑉𝑏𝑉 (𝑋 = {𝑎, 𝑏} ∧ 𝑎𝑏))

Theoremprproropf1olem0 42455 Lemma 0 for prproropf1o 42460. Remark: 𝑂, the set of ordered ordered pairs, i.e., ordered pairs in which the first component is less than the second component, can alternatively be written as 𝑂 = {𝑥 ∈ (𝑉 × 𝑉) ∣ (1st𝑥)𝑅(2nd𝑥)} or even as 𝑂 = {𝑥 ∈ (𝑉 × 𝑉) ∣ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝑅}, by which the relationship between ordered and unordered pair is immediately visible. (Contributed by AV, 18-Mar-2023.)
𝑂 = (𝑅 ∩ (𝑉 × 𝑉))       (𝑊𝑂 ↔ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊)))

Theoremprproropf1olem1 42456* Lemma 1 for prproropf1o 42460. (Contributed by AV, 12-Mar-2023.)
𝑂 = (𝑅 ∩ (𝑉 × 𝑉))    &   𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}       ((𝑅 Or 𝑉𝑊𝑂) → {(1st𝑊), (2nd𝑊)} ∈ 𝑃)

Theoremprproropf1olem2 42457* Lemma 2 for prproropf1o 42460. (Contributed by AV, 13-Mar-2023.)
𝑂 = (𝑅 ∩ (𝑉 × 𝑉))    &   𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}       ((𝑅 Or 𝑉𝑋𝑃) → ⟨inf(𝑋, 𝑉, 𝑅), sup(𝑋, 𝑉, 𝑅)⟩ ∈ 𝑂)

Theoremprproropf1olem3 42458* Lemma 3 for prproropf1o 42460. (Contributed by AV, 13-Mar-2023.)
𝑂 = (𝑅 ∩ (𝑉 × 𝑉))    &   𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}    &   𝐹 = (𝑝𝑃 ↦ ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩)       ((𝑅 Or 𝑉𝑊𝑂) → (𝐹‘{(1st𝑊), (2nd𝑊)}) = ⟨(1st𝑊), (2nd𝑊)⟩)

Theoremprproropf1olem4 42459* Lemma 4 for prproropf1o 42460. (Contributed by AV, 14-Mar-2023.)
𝑂 = (𝑅 ∩ (𝑉 × 𝑉))    &   𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}    &   𝐹 = (𝑝𝑃 ↦ ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩)       ((𝑅 Or 𝑉𝑊𝑃𝑍𝑃) → ((𝐹𝑍) = (𝐹𝑊) → 𝑍 = 𝑊))

Theoremprproropf1o 42460* There is a bijection between the set of proper pairs and the set of ordered ordered pairs, i.e., ordered pairs in which the first component is less than the second component. (Contributed by AV, 15-Mar-2023.)
𝑂 = (𝑅 ∩ (𝑉 × 𝑉))    &   𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}    &   𝐹 = (𝑝𝑃 ↦ ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩)       (𝑅 Or 𝑉𝐹:𝑃1-1-onto𝑂)

Theoremprproropen 42461* The set of proper pairs and the set of ordered ordered pairs, i.e., ordered pairs in which the first component is less than the second component, are equinumerous. (Contributed by AV, 15-Mar-2023.)
𝑂 = (𝑅 ∩ (𝑉 × 𝑉))    &   𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}       ((𝑉𝑊𝑅 Or 𝑉) → 𝑂𝑃)

Theoremprproropreud 42462* There is exactly one ordered ordered pair fulfilling a wff iff there is exactly one proper pair fulfilling an equivalent wff. (Contributed by AV, 20-Mar-2023.)
𝑂 = (𝑅 ∩ (𝑉 × 𝑉))    &   𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}    &   (𝜑𝑅 Or 𝑉)    &   (𝑥 = ⟨inf(𝑦, 𝑉, 𝑅), sup(𝑦, 𝑉, 𝑅)⟩ → (𝜓𝜒))    &   (𝑥 = 𝑧 → (𝜓𝜃))       (𝜑 → (∃!𝑥𝑂 𝜓 ↔ ∃!𝑦𝑃 𝜒))

Theorempairreueq 42463* Two equivalent representations of the existence of a unique proper pair. (Contributed by AV, 1-Mar-2023.)
𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}       (∃!𝑝𝑃 𝜑 ↔ ∃!𝑝 ∈ 𝒫 𝑉((♯‘𝑝) = 2 ∧ 𝜑))

Theorempaireqne 42464* Two sets are not equal iff there is exactly one proper pair whose elements are either one of these sets. (Contributed by AV, 27-Jan-2023.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}       (𝜑 → (∃!𝑝𝑃𝑥𝑝 (𝑥 = 𝐴𝑥 = 𝐵) ↔ 𝐴𝐵))

20.36.10.3  Set of proper unordered pairs

Syntaxcprpr 42465 Extend class notation with set of proper unordered pairs.
class Pairsproper

Definitiondf-prpr 42466* Define the function which maps a set 𝑣 to the set of proper unordered pairs consisting of exactly two (different) elements of the set 𝑣. (Contributed by AV, 29-Apr-2023.)
Pairsproper = (𝑣 ∈ V ↦ {𝑝 ∣ ∃𝑎𝑣𝑏𝑣 (𝑎𝑏𝑝 = {𝑎, 𝑏})})

Theoremprprval 42467* The set of all proper unordered pairs over a given set 𝑉. (Contributed by AV, 29-Apr-2023.)
(𝑉𝑊 → (Pairsproper𝑉) = {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})})

Theoremprprvalpw 42468* The set of all proper unordered pairs over a given set 𝑉, expressed by a restricted class abstraction. (Contributed by AV, 29-Apr-2023.)
(𝑉𝑊 → (Pairsproper𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})})

Theoremprprelb 42469 An element of the set of all proper unordered pairs over a given set 𝑉 is a subset of 𝑉 of size two. (Contributed by AV, 29-Apr-2023.)
(𝑉𝑊 → (𝑃 ∈ (Pairsproper𝑉) ↔ (𝑃 ∈ 𝒫 𝑉 ∧ (♯‘𝑃) = 2)))

Theoremprprelprb 42470* A set is an element of the set of all proper unordered pairs over a given set 𝑋 iff it is a pair of different elements of the set 𝑋. (Contributed by AV, 7-May-2023.)
(𝑃 ∈ (Pairsproper𝑋) ↔ (𝑋 ∈ V ∧ ∃𝑎𝑋𝑏𝑋 (𝑃 = {𝑎, 𝑏} ∧ 𝑎𝑏)))

Theoremprprspr2 42471* The set of all proper unordered pairs over a given set 𝑉 is the set of all unordered pairs over that set of size two. (Contributed by AV, 29-Apr-2023.)
(Pairsproper𝑉) = {𝑝 ∈ (Pairs‘𝑉) ∣ (♯‘𝑝) = 2}

Theoremprprsprreu 42472* There is a unique proper unordered pair over a given set 𝑉 fulfilling a wff iff there is a unique unordered pair over 𝑉 of size two fulfilling this wff. (Contributed by AV, 30-Apr-2023.)
(𝑉𝑊 → (∃!𝑝 ∈ (Pairsproper𝑉)𝜑 ↔ ∃!𝑝 ∈ (Pairs‘𝑉)((♯‘𝑝) = 2 ∧ 𝜑)))

Theoremprprreueq 42473* There is a unique proper unordered pair over a given set 𝑉 fulfilling a wff iff there is a unique subset of 𝑉 of size two fulfilling this wff. (Contributed by AV, 29-Apr-2023.)
(𝑉𝑊 → (∃!𝑝 ∈ (Pairsproper𝑉)𝜑 ↔ ∃!𝑝 ∈ 𝒫 𝑉((♯‘𝑝) = 2 ∧ 𝜑)))

20.36.11  Number theory (extension)

20.36.11.1  Fermat numbers

At first, the (sequence of) Fermat numbers FermatNo (the 𝑛-th Fermat number is denoted as (FermatNo‘𝑛)) is defined, see df-fmtno 42475, and basic theorems are provided. Afterwards, it is shown that the first five Fermat numbers are prime, the (first) five Fermat primes, see fmtnofz04prm 42524, but that the fifth Fermat number (counting starts at 0!) is not prime, see fmtno5nprm 42530. The fourth Fermat number (i.e., the fifth Fermat prime) (FermatNo‘4) = 65537 is currently the biggest number proven to be prime in set.mm, see 65537prm 42523 (previously, it was 4001, see 4001prm 16261).

Another important result of this section is Goldbach's theorem goldbachth 42494, showing that two different Fermut numbers are coprime. By this, it can be proven that there is an infinite number of primes, see prminf2 42535.

Finally, it is shown that every prime of the form ((2↑𝑘) + 1) must be a Fermat number (i.e., a Fermat prime), see 2pwp1prmfmtno 42539.

Syntaxcfmtno 42474 Extend class notation with the Fermat numbers.
class FermatNo

Definitiondf-fmtno 42475 Define the function that enumerates the Fermat numbers, see definition in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
FermatNo = (𝑛 ∈ ℕ0 ↦ ((2↑(2↑𝑛)) + 1))

Theoremfmtno 42476 The 𝑁 th Fermat number. (Contributed by AV, 13-Jun-2021.)
(𝑁 ∈ ℕ0 → (FermatNo‘𝑁) = ((2↑(2↑𝑁)) + 1))

Theoremfmtnoge3 42477 Each Fermat number is greater than or equal to 3. (Contributed by AV, 4-Aug-2021.)
(𝑁 ∈ ℕ0 → (FermatNo‘𝑁) ∈ (ℤ‘3))

Theoremfmtnonn 42478 Each Fermat number is a positive integer. (Contributed by AV, 26-Jul-2021.) (Proof shortened by AV, 4-Aug-2021.)
(𝑁 ∈ ℕ0 → (FermatNo‘𝑁) ∈ ℕ)

Theoremfmtnom1nn 42479 A Fermat number minus one is a power of a power of two. (Contributed by AV, 29-Jul-2021.)
(𝑁 ∈ ℕ0 → ((FermatNo‘𝑁) − 1) = (2↑(2↑𝑁)))

Theoremfmtnoodd 42480 Each Fermat number is odd. (Contributed by AV, 26-Jul-2021.)
(𝑁 ∈ ℕ0 → ¬ 2 ∥ (FermatNo‘𝑁))

Theoremfmtnorn 42481* A Fermat number is a function value of the enumeration of the Fermat numbers. (Contributed by AV, 3-Aug-2021.)
(𝐹 ∈ ran FermatNo ↔ ∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹)

Theoremfmtnof1 42482 The enumeration of the Fermat numbers is a one-one function into the positive integers. (Contributed by AV, 3-Aug-2021.)
FermatNo:ℕ01-1→ℕ

Theoremfmtnoinf 42483 The set of Fermat numbers is infinite. (Contributed by AV, 3-Aug-2021.)
ran FermatNo ∉ Fin

Theoremfmtnorec1 42484 The first recurrence relation for Fermat numbers, see Wikipedia "Fermat number", https://en.wikipedia.org/wiki/Fermat_number#Basic_properties, 22-Jul-2021. (Contributed by AV, 22-Jul-2021.)
(𝑁 ∈ ℕ0 → (FermatNo‘(𝑁 + 1)) = ((((FermatNo‘𝑁) − 1)↑2) + 1))

Theoremsqrtpwpw2p 42485 The floor of the square root of 2 to the power of 2 to the power of a positive integer plus a bounded nonnegative integer. (Contributed by AV, 28-Jul-2021.)
((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ0𝑀 < ((2↑((2↑(𝑁 − 1)) + 1)) + 1)) → (⌊‘(√‘((2↑(2↑𝑁)) + 𝑀))) = (2↑(2↑(𝑁 − 1))))

Theoremfmtnosqrt 42486 The floor of the square root of a Fermat number. (Contributed by AV, 28-Jul-2021.)
(𝑁 ∈ ℕ → (⌊‘(√‘(FermatNo‘𝑁))) = (2↑(2↑(𝑁 − 1))))

Theoremfmtno0 42487 The 0 th Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
(FermatNo‘0) = 3

Theoremfmtno1 42488 The 1 st Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
(FermatNo‘1) = 5

Theoremfmtnorec2lem 42489* Lemma for fmtnorec2 42490 (induction step). (Contributed by AV, 29-Jul-2021.)
(𝑦 ∈ ℕ0 → ((FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) → (FermatNo‘((𝑦 + 1) + 1)) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + 2)))

Theoremfmtnorec2 42490* The second recurrence relation for Fermat numbers, see ProofWiki "Product of Sequence of Fermat Numbers plus 2", 29-Jul-2021, https://proofwiki.org/wiki/Product_of_Sequence_of_Fermat_Numbers_plus_2 or Wikipedia "Fermat number", 29-Jul-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties. (Contributed by AV, 29-Jul-2021.)
(𝑁 ∈ ℕ0 → (FermatNo‘(𝑁 + 1)) = (∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛) + 2))

Theoremfmtnodvds 42491 Any Fermat number divides a greater Fermat number minus 2. Corrolary of fmtnorec2 42490, see ProofWiki "Product of Sequence of Fermat Numbers plus 2/Corollary", 31-Jul-2021. (Contributed by AV, 1-Aug-2021.)
((𝑁 ∈ ℕ0𝑀 ∈ ℕ) → (FermatNo‘𝑁) ∥ ((FermatNo‘(𝑁 + 𝑀)) − 2))

Theoremgoldbachthlem1 42492 Lemma 1 for goldbachth 42494. (Contributed by AV, 1-Aug-2021.)
((𝑁 ∈ ℕ0𝑀 ∈ ℕ0𝑀 < 𝑁) → (FermatNo‘𝑀) ∥ ((FermatNo‘𝑁) − 2))

Theoremgoldbachthlem2 42493 Lemma 2 for goldbachth 42494. (Contributed by AV, 1-Aug-2021.)
((𝑁 ∈ ℕ0𝑀 ∈ ℕ0𝑀 < 𝑁) → ((FermatNo‘𝑁) gcd (FermatNo‘𝑀)) = 1)

Theoremgoldbachth 42494 Goldbach's theorem: Two different Fermat numbers are coprime. See ProofWiki "Goldbach's theorem", 31-Jul-2021, https://proofwiki.org/wiki/Goldbach%27s_Theorem or Wikipedia "Fermat number", 31-Jul-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties. (Contributed by AV, 1-Aug-2021.)
((𝑁 ∈ ℕ0𝑀 ∈ ℕ0𝑁𝑀) → ((FermatNo‘𝑁) gcd (FermatNo‘𝑀)) = 1)

Theoremfmtnorec3 42495* The third recurrence relation for Fermat numbers, see Wikipedia "Fermat number", 31-Jul-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties. (Contributed by AV, 2-Aug-2021.)
(𝑁 ∈ (ℤ‘2) → (FermatNo‘𝑁) = ((FermatNo‘(𝑁 − 1)) + ((2↑(2↑(𝑁 − 1))) · ∏𝑛 ∈ (0...(𝑁 − 2))(FermatNo‘𝑛))))

Theoremfmtnorec4 42496 The fourth recurrence relation for Fermat numbers, see Wikipedia "Fermat number", 31-Jul-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties. (Contributed by AV, 31-Jul-2021.)
(𝑁 ∈ (ℤ‘2) → (FermatNo‘𝑁) = (((FermatNo‘(𝑁 − 1))↑2) − (2 · (((FermatNo‘(𝑁 − 2)) − 1)↑2))))

Theoremfmtno2 42497 The 2 nd Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
(FermatNo‘2) = 17

Theoremfmtno3 42498 The 3 rd Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
(FermatNo‘3) = 257

Theoremfmtno4 42499 The 4 th Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
(FermatNo‘4) = 65537

Theoremfmtno5lem1 42500 Lemma 1 for fmtno5 42504. (Contributed by AV, 22-Jul-2021.)
(65536 · 6) = 393216

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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 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