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Theorem List for Metamath Proof Explorer - 44901-45000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnaryrcl 44901 Reverse closure for n-ary (endo)functions. (Contributed by AV, 14-May-2024.)
𝐼 = (0..^𝑁)       (𝐹 ∈ (𝑋 NaryF 𝑁) → (𝑋 ∈ V ∧ 𝑁 ∈ ℕ0))
 
Theoremnaryfvalelfv 44902 The value of an n-ary (endo)function on a set 𝑋 is an element of 𝑋. (Contributed by AV, 14-May-2024.)
𝐼 = (0..^𝑁)       ((𝐹 ∈ (𝑋 NaryF 𝑁) ∧ 𝐴:𝐼𝑋) → (𝐹𝐴) ∈ 𝑋)
 
Theorem0aryfvalel 44903* A nullary (endo)function on a set 𝑋 is a singleton of an ordered pair with the empty set as first component. A nullary function represents a constant: (𝐹‘∅) = 𝐶 with 𝐶𝑋, see also 0aryfvalelfv 44904. Instead of (𝐹‘∅), nullary functions are usually written as 𝐹() in literature. (Contributed by AV, 15-May-2024.)
(𝑋𝑉 → (𝐹 ∈ (𝑋 NaryF 0) ↔ ∃𝑥𝑋 𝐹 = {⟨∅, 𝑥⟩}))
 
Theorem0aryfvalelfv 44904* The value of a nullary (endo)function on a set 𝑋. (Contributed by AV, 19-May-2024.)
(𝐹 ∈ (𝑋 NaryF 0) → ∃𝑥𝑋 (𝐹‘∅) = 𝑥)
 
Theorem1aryfvalel 44905 A unary (endo)function on a set 𝑋. (Contributed by AV, 15-May-2024.)
(𝑋𝑉 → (𝐹 ∈ (𝑋 NaryF 1) ↔ 𝐹:(𝑋m {0})⟶𝑋))
 
Theoremfv1arycl 44906 Closure of a unary (endo)function. (Contributed by AV, 18-May-2024.)
((𝐺 ∈ (𝑋 NaryF 1) ∧ 𝐴𝑋) → (𝐺‘{⟨0, 𝐴⟩}) ∈ 𝑋)
 
Theoremnarympt1 44907* A unary (endo)function in maps-to notation. (Contributed by AV, 16-May-2024.)
𝐹 = (𝑥 ∈ (𝑋m {0}) ↦ (𝐴‘(𝑥‘0)))       ((𝑋𝑉𝐴:𝑋𝑋) → 𝐹 ∈ (𝑋 NaryF 1))
 
Theoremnarympt1fv 44908* The value of a unary (endo)function in maps-to notation. (Contributed by AV, 16-May-2024.)
𝐹 = (𝑥 ∈ (𝑋m {0}) ↦ (𝐴‘(𝑥‘0)))       ((𝑋𝑉𝐵𝑋) → (𝐹‘{⟨0, 𝐵⟩}) = (𝐴𝐵))
 
Theoremnarymaptfv 44909* The value of the mapping of unary (endo)functions. (Contributed by AV, 18-May-2024.)
𝐻 = ( ∈ (𝑋 NaryF 1) ↦ (𝑥𝑋 ↦ (‘{⟨0, 𝑥⟩})))       (𝐹 ∈ (𝑋 NaryF 1) → (𝐻𝐹) = (𝑥𝑋 ↦ (𝐹‘{⟨0, 𝑥⟩})))
 
Theoremnarymaptf 44910* The mapping of unary (endo)functions is a function into the set of endofunctions. (Contributed by AV, 18-May-2024.)
𝐻 = ( ∈ (𝑋 NaryF 1) ↦ (𝑥𝑋 ↦ (‘{⟨0, 𝑥⟩})))       (𝑋𝑉𝐻:(𝑋 NaryF 1)⟶(𝑋m 𝑋))
 
Theoremnarymaptf1 44911* The mapping of unary (endo)functions is a one-to-one function into the set of endofunctions. (Contributed by AV, 19-May-2024.)
𝐻 = ( ∈ (𝑋 NaryF 1) ↦ (𝑥𝑋 ↦ (‘{⟨0, 𝑥⟩})))       (𝑋𝑉𝐻:(𝑋 NaryF 1)–1-1→(𝑋m 𝑋))
 
Theoremnarymaptfo 44912* The mapping of unary (endo)functions is a function onto the set of endofunctions. (Contributed by AV, 18-May-2024.)
𝐻 = ( ∈ (𝑋 NaryF 1) ↦ (𝑥𝑋 ↦ (‘{⟨0, 𝑥⟩})))       (𝑋𝑉𝐻:(𝑋 NaryF 1)–onto→(𝑋m 𝑋))
 
Theoremnarymaptf1o 44913* The mapping of unary (endo)functions is a one-to-one function onto the set of endofunctions (Contributed by AV, 19-May-2024.)
𝐻 = ( ∈ (𝑋 NaryF 1) ↦ (𝑥𝑋 ↦ (‘{⟨0, 𝑥⟩})))       (𝑋𝑉𝐻:(𝑋 NaryF 1)–1-1-onto→(𝑋m 𝑋))
 
Theoremnaryenef 44914 The set of unary (endo)functions and the set of endofunctions are equinumerous. (Contributed by AV, 19-May-2024.)
(𝑋 NaryF 1) ≈ (𝑋m 𝑋)
 
Theoremnaryenefmnd 44915 The set of unary (endo)functions and the base set of the monoid of endofunctions are equinumerous. (Contributed by AV, 19-May-2024.)
(𝑋 NaryF 1) ≈ (Base‘(EndoFMnd‘𝑋))
 
20.41.22.14  The Ackermann function

According to Wikipedia ("Ackermann function", 8-May-2024, https://en.wikipedia.org/wiki/Ackermann_function): "In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive. ... One common version is the two-argument Ackermann-Péter function developed by Rózsa Péter and Raphael Robinson. Its value grows very rapidly; for example, A(4,2) results in 2^65536-3 [see ackval42 44955)], an integer of 19,729 decimal digits."

In the following, the Ackermann function is defined as iterated 1-ary function (also mentioned in Wikipedia), see df-ack 44919, based on a definition IterComp of "the n-th iterate of (a class/function) f", see df-itco 44918. As an illustration, we have ((IterComp‘𝐹)‘3) = (𝐹 ∘ (𝐹𝐹))) (see itcoval3 44924).

The following recursive definition of the Ackermann function follows immediately from the definition df-ack 44919: ((Ack‘(𝑀 + 1))‘𝑁) = (((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))‘1)).

That the definition df-ack 44919 is equivalent to Péter's definition is proven by the following three theorems:

ackval0val 44945: ((Ack‘0)‘𝑀) = (𝑀 + 1) ackvalsuc0val 44946: ((Ack‘(𝑀 + 1))‘0) = ((Ack‘𝑀)‘1) ackvalsucsucval 44947: ((Ack‘(𝑀 + 1))‘(𝑁 + 1)) = ((Ack‘𝑀)‘((Ack‘(𝑀 + 1))‘𝑁)).

The initial values of the Ackermann function are calculated in the following four theorems:

ackval0012 44948: 𝐴(0, 0) = 1, 𝐴(0, 1) = 2, 𝐴(0, 2) = 3 ackval1012 44949: 𝐴(1, 0) = 2, 𝐴(1, 1) = 3, 𝐴(1, 3) = 4 ackval2012 44950: 𝐴(2, 0) = 3, 𝐴(2, 1) = 5, 𝐴(2, 3) = 7 ackval3012 44951: 𝐴(3, 0) = 5, 𝐴(3, 1) = 13, 𝐴(3, 3) = 29

 
Syntaxcitco 44916 Extend the definition of a class to include iterated functions.
class IterComp
 
Syntaxcack 44917 Extend the definition of a class to include the Ackermann function operator.
class Ack
 
Definitiondf-itco 44918* Define a function (recursively) that returns the n-th iterate of a class (usually a function) with regard to composition. (Contributed by Thierry Arnoux, 28-Apr-2024.) (Revised by AV, 2-May-2024.)
IterComp = (𝑓 ∈ V ↦ seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝑓𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝑓), 𝑓))))
 
Definitiondf-ack 44919* Define the Ackermann function (recursively). (Contributed by Thierry Arnoux, 28-Apr-2024.) (Revised by AV, 2-May-2024.)
Ack = seq0((𝑓 ∈ V, 𝑗 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ (((IterComp‘𝑓)‘(𝑛 + 1))‘1))), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)), 𝑖)))
 
Theoremitcoval 44920* The value of the function that returns the n-th iterate of a class (usually a function) with regard to composition. (Contributed by AV, 2-May-2024.)
(𝐹𝑉 → (IterComp‘𝐹) = seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹))))
 
Theoremitcoval0 44921 A function iterated zero times (defined as identity function). (Contributed by AV, 2-May-2024.)
(𝐹𝑉 → ((IterComp‘𝐹)‘0) = ( I ↾ dom 𝐹))
 
Theoremitcoval1 44922 A function iterated once. (Contributed by AV, 2-May-2024.)
((Rel 𝐹𝐹𝑉) → ((IterComp‘𝐹)‘1) = 𝐹)
 
Theoremitcoval2 44923 A function iterated twice. (Contributed by AV, 2-May-2024.)
((Rel 𝐹𝐹𝑉) → ((IterComp‘𝐹)‘2) = (𝐹𝐹))
 
Theoremitcoval3 44924 A function iterated three times. (Contributed by AV, 2-May-2024.)
((Rel 𝐹𝐹𝑉) → ((IterComp‘𝐹)‘3) = (𝐹 ∘ (𝐹𝐹)))
 
Theoremitcoval0mpt 44925* A mapping iterated zero times (defined as identity function). (Contributed by AV, 4-May-2024.)
𝐹 = (𝑛𝐴𝐵)       ((𝐴𝑉 ∧ ∀𝑛𝐴 𝐵𝑊) → ((IterComp‘𝐹)‘0) = (𝑛𝐴𝑛))
 
Theoremitcovalsuc 44926* The value of the function that returns the n-th iterate of a function with regard to composition at a successor. (Contributed by AV, 4-May-2024.)
((𝐹𝑉𝑌 ∈ ℕ0 ∧ ((IterComp‘𝐹)‘𝑌) = 𝐺) → ((IterComp‘𝐹)‘(𝑌 + 1)) = (𝐺(𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔))𝐹))
 
Theoremitcovalsucov 44927 The value of the function that returns the n-th iterate of a function with regard to composition at a successor. (Contributed by AV, 4-May-2024.)
((𝐹𝑉𝑌 ∈ ℕ0 ∧ ((IterComp‘𝐹)‘𝑌) = 𝐺) → ((IterComp‘𝐹)‘(𝑌 + 1)) = (𝐹𝐺))
 
Theoremitcovalendof 44928 The n-th iterate of an endofunction is an endofunction. (Contributed by AV, 7-May-2024.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐴)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → ((IterComp‘𝐹)‘𝑁):𝐴𝐴)
 
Theoremitcovalpclem1 44929* Lemma 1 for itcovalpc 44931: induction basis. (Contributed by AV, 4-May-2024.)
𝐹 = (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶))       (𝐶 ∈ ℕ0 → ((IterComp‘𝐹)‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 0))))
 
Theoremitcovalpclem2 44930* Lemma 2 for itcovalpc 44931: induction step. (Contributed by AV, 4-May-2024.)
𝐹 = (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶))       ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → (((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦))) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1))))))
 
Theoremitcovalpc 44931* The value of the function that returns the n-th iterate of the "plus a constant" function with regard to composition. (Contributed by AV, 4-May-2024.)
𝐹 = (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶))       ((𝐼 ∈ ℕ0𝐶 ∈ ℕ0) → ((IterComp‘𝐹)‘𝐼) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝐼))))
 
Theoremitcovalt2lem2lem1 44932 Lemma 1 for itcovalt2lem2 44935. (Contributed by AV, 6-May-2024.)
(((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → (((𝑁 + 𝐶) · 𝑌) − 𝐶) ∈ ℕ0)
 
Theoremitcovalt2lem2lem2 44933 Lemma 2 for itcovalt2lem2 44935. (Contributed by AV, 7-May-2024.)
(((𝑌 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → ((2 · (((𝑁 + 𝐶) · (2↑𝑌)) − 𝐶)) + 𝐶) = (((𝑁 + 𝐶) · (2↑(𝑌 + 1))) − 𝐶))
 
Theoremitcovalt2lem1 44934* Lemma 1 for itcovalt2 44936: induction basis. (Contributed by AV, 5-May-2024.)
𝐹 = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 𝐶))       (𝐶 ∈ ℕ0 → ((IterComp‘𝐹)‘0) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑0)) − 𝐶)))
 
Theoremitcovalt2lem2 44935* Lemma 2 for itcovalt2 44936: induction step. (Contributed by AV, 7-May-2024.)
𝐹 = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 𝐶))       ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → (((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶)) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑(𝑦 + 1))) − 𝐶))))
 
Theoremitcovalt2 44936* The value of the function that returns the n-th iterate of the "times 2 plus a constant" function with regard to composition. (Contributed by AV, 7-May-2024.)
𝐹 = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 𝐶))       ((𝐼 ∈ ℕ0𝐶 ∈ ℕ0) → ((IterComp‘𝐹)‘𝐼) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝐼)) − 𝐶)))
 
Theoremackvalsuc1mpt 44937* The Ackermann function at a successor of the first argument as a mapping of the second argument. (Contributed by Thierry Arnoux, 28-Apr-2024.) (Revised by AV, 4-May-2024.)
(𝑀 ∈ ℕ0 → (Ack‘(𝑀 + 1)) = (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘𝑀))‘(𝑛 + 1))‘1)))
 
Theoremackvalsuc1 44938 The Ackermann function at a successor of the first argument and an arbitrary second argument. (Contributed by Thierry Arnoux, 28-Apr-2024.) (Revised by AV, 4-May-2024.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → ((Ack‘(𝑀 + 1))‘𝑁) = (((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))‘1))
 
Theoremackval0 44939 The Ackermann function at 0. (Contributed by AV, 2-May-2024.)
(Ack‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 1))
 
Theoremackval1 44940 The Ackermann function at 1. (Contributed by AV, 4-May-2024.)
(Ack‘1) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2))
 
Theoremackval2 44941 The Ackermann function at 2. (Contributed by AV, 4-May-2024.)
(Ack‘2) = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 3))
 
Theoremackval3 44942 The Ackermann function at 3. (Contributed by AV, 7-May-2024.)
(Ack‘3) = (𝑛 ∈ ℕ0 ↦ ((2↑(𝑛 + 3)) − 3))
 
Theoremackendofnn0 44943 The Ackermann function at any nonnegative integer is an endofunction on the nonnegative integers. (Contributed by AV, 8-May-2024.)
(𝑀 ∈ ℕ0 → (Ack‘𝑀):ℕ0⟶ℕ0)
 
Theoremackfnnn0 44944 The Ackermann function at any nonnegative integer is a function on the nonnegative integers. (Contributed by AV, 4-May-2024.) (Proof shortened by AV, 8-May-2024.)
(𝑀 ∈ ℕ0 → (Ack‘𝑀) Fn ℕ0)
 
Theoremackval0val 44945 The Ackermann function at 0 (for the first argument). This is the first equation of Péter's definition of the Ackermann function. (Contributed by AV, 4-May-2024.)
(𝑀 ∈ ℕ0 → ((Ack‘0)‘𝑀) = (𝑀 + 1))
 
Theoremackvalsuc0val 44946 The Ackermann function at a successor (of the first argument). This is the second equation of Péter's definition of the Ackermann function. (Contributed by AV, 4-May-2024.)
(𝑀 ∈ ℕ0 → ((Ack‘(𝑀 + 1))‘0) = ((Ack‘𝑀)‘1))
 
Theoremackvalsucsucval 44947 The Ackermann function at the successors. This is the third equation of Péter's definition of the Ackermann function. (Contributed by AV, 8-May-2024.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → ((Ack‘(𝑀 + 1))‘(𝑁 + 1)) = ((Ack‘𝑀)‘((Ack‘(𝑀 + 1))‘𝑁)))
 
Theoremackval0012 44948 The Ackermann function at (0,0), (0,1), (0,2). (Contributed by AV, 2-May-2024.)
⟨((Ack‘0)‘0), ((Ack‘0)‘1), ((Ack‘0)‘2)⟩ = ⟨1, 2, 3⟩
 
Theoremackval1012 44949 The Ackermann function at (1,0), (1,1), (1,2). (Contributed by AV, 4-May-2024.)
⟨((Ack‘1)‘0), ((Ack‘1)‘1), ((Ack‘1)‘2)⟩ = ⟨2, 3, 4⟩
 
Theoremackval2012 44950 The Ackermann function at (2,0), (2,1), (2,2). (Contributed by AV, 4-May-2024.)
⟨((Ack‘2)‘0), ((Ack‘2)‘1), ((Ack‘2)‘2)⟩ = ⟨3, 5, 7⟩
 
Theoremackval3012 44951 The Ackermann function at (3,0), (3,1), (3,2). (Contributed by AV, 7-May-2024.)
⟨((Ack‘3)‘0), ((Ack‘3)‘1), ((Ack‘3)‘2)⟩ = ⟨5, 13, 29⟩
 
Theoremackval40 44952 The Ackermann function at (4,0). (Contributed by AV, 9-May-2024.)
((Ack‘4)‘0) = 13
 
Theoremackval41a 44953 The Ackermann function at (4,1). (Contributed by AV, 9-May-2024.)
((Ack‘4)‘1) = ((2↑16) − 3)
 
Theoremackval41 44954 The Ackermann function at (4,1). (Contributed by AV, 9-May-2024.)
((Ack‘4)‘1) = 65533
 
Theoremackval42 44955 The Ackermann function at (4,2). (Contributed by AV, 9-May-2024.)
((Ack‘4)‘2) = ((2↑65536) − 3)
 
Theoremackval42a 44956 The Ackermann function at (4,2), expressed with powers of 2. (Contributed by AV, 9-May-2024.)
((Ack‘4)‘2) = ((2↑(2↑(2↑(2↑2)))) − 3)
 
Theoremackval50 44957 The Ackermann function at (5,0). (Contributed by AV, 9-May-2024.)
((Ack‘5)‘0) = 65533
 
20.41.23  Elementary geometry (extension)
 
20.41.23.1  Auxiliary theorems
 
Theoremfv1prop 44958 The function value of unordered pair of ordered pairs with first components 1 and 2 at 1. (Contributed by AV, 4-Feb-2023.)
(𝐴𝑉 → ({⟨1, 𝐴⟩, ⟨2, 𝐵⟩}‘1) = 𝐴)
 
Theoremfv2prop 44959 The function value of unordered pair of ordered pairs with first components 1 and 2 at 1. (Contributed by AV, 4-Feb-2023.)
(𝐵𝑉 → ({⟨1, 𝐴⟩, ⟨2, 𝐵⟩}‘2) = 𝐵)
 
Theoremsubmuladdmuld 44960 Transformation of a sum of a product of a difference and a product with the subtrahend of the difference. (Contributed by AV, 2-Feb-2023.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)       (𝜑 → (((𝐴𝐵) · 𝐶) + (𝐵 · 𝐷)) = ((𝐴 · 𝐶) + (𝐵 · (𝐷𝐶))))
 
Theoremaffinecomb1 44961* Combination of two real affine combinations, one class variable resolved. (Contributed by AV, 22-Jan-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵𝐶)    &   (𝜑𝐸 ∈ ℝ)    &   (𝜑𝐹 ∈ ℝ)    &   (𝜑𝐺 ∈ ℝ)    &   𝑆 = ((𝐺𝐹) / (𝐶𝐵))       (𝜑 → (∃𝑡 ∈ ℝ (𝐴 = (((1 − 𝑡) · 𝐵) + (𝑡 · 𝐶)) ∧ 𝐸 = (((1 − 𝑡) · 𝐹) + (𝑡 · 𝐺))) ↔ 𝐸 = ((𝑆 · (𝐴𝐵)) + 𝐹)))
 
Theoremaffinecomb2 44962* Combination of two real affine combinations, presented without fraction. (Contributed by AV, 22-Jan-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵𝐶)    &   (𝜑𝐸 ∈ ℝ)    &   (𝜑𝐹 ∈ ℝ)    &   (𝜑𝐺 ∈ ℝ)       (𝜑 → (∃𝑡 ∈ ℝ (𝐴 = (((1 − 𝑡) · 𝐵) + (𝑡 · 𝐶)) ∧ 𝐸 = (((1 − 𝑡) · 𝐹) + (𝑡 · 𝐺))) ↔ ((𝐶𝐵) · 𝐸) = (((𝐺𝐹) · 𝐴) + ((𝐹 · 𝐶) − (𝐵 · 𝐺)))))
 
Theoremaffineid 44963 Identity of an affine combination. (Contributed by AV, 2-Feb-2023.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝑇 ∈ ℂ)       (𝜑 → (((1 − 𝑇) · 𝐴) + (𝑇 · 𝐴)) = 𝐴)
 
Theorem1subrec1sub 44964 Subtract the reciprocal of 1 minus a number from 1 results in the number divided by the number minus 1. (Contributed by AV, 15-Feb-2023.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 1) → (1 − (1 / (1 − 𝐴))) = (𝐴 / (𝐴 − 1)))
 
Theoremresum2sqcl 44965 The sum of two squares of real numbers is a real number. (Contributed by AV, 7-Feb-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))       ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝑄 ∈ ℝ)
 
Theoremresum2sqgt0 44966 The sum of the square of a nonzero real number and the square of another real number is greater than zero. (Contributed by AV, 7-Feb-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))       (((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℝ) → 0 < 𝑄)
 
Theoremresum2sqrp 44967 The sum of the square of a nonzero real number and the square of another real number is a positive real number. (Contributed by AV, 2-May-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))       (((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℝ) → 𝑄 ∈ ℝ+)
 
Theoremresum2sqorgt0 44968 The sum of the square of two real numbers is greater than zero if at least one of the real numbers is nonzero. (Contributed by AV, 26-Feb-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))       ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐴 ≠ 0 ∨ 𝐵 ≠ 0)) → 0 < 𝑄)
 
Theoremreorelicc 44969 Membership in and outside of a closed real interval. (Contributed by AV, 15-Feb-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 < 𝐴𝐶 ∈ (𝐴[,]𝐵) ∨ 𝐵 < 𝐶))
 
20.41.23.2  Real euclidean space of dimension 2
 
Theoremrrx2pxel 44970 The x-coordinate of a point in a real Euclidean space of dimension 2 is a real number. (Contributed by AV, 2-Feb-2023.)
𝐼 = {1, 2}    &   𝑃 = (ℝ ↑m 𝐼)       (𝑋𝑃 → (𝑋‘1) ∈ ℝ)
 
Theoremrrx2pyel 44971 The y-coordinate of a point in a real Euclidean space of dimension 2 is a real number. (Contributed by AV, 2-Feb-2023.)
𝐼 = {1, 2}    &   𝑃 = (ℝ ↑m 𝐼)       (𝑋𝑃 → (𝑋‘2) ∈ ℝ)
 
Theoremprelrrx2 44972 An unordered pair of ordered pairs with first components 1 and 2 and real numbers as second components is a point in a real Euclidean space of dimension 2. (Contributed by AV, 4-Feb-2023.)
𝐼 = {1, 2}    &   𝑃 = (ℝ ↑m 𝐼)       ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → {⟨1, 𝐴⟩, ⟨2, 𝐵⟩} ∈ 𝑃)
 
Theoremprelrrx2b 44973 An unordered pair of ordered pairs with first components 1 and 2 and real numbers as second components is a point in a real Euclidean space of dimension 2, determined by its coordinates. (Contributed by AV, 7-May-2023.)
𝐼 = {1, 2}    &   𝑃 = (ℝ ↑m 𝐼)       (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ)) → ((𝑍𝑃 ∧ (((𝑍‘1) = 𝐴 ∧ (𝑍‘2) = 𝐵) ∨ ((𝑍‘1) = 𝑋 ∧ (𝑍‘2) = 𝑌))) ↔ 𝑍 ∈ {{⟨1, 𝐴⟩, ⟨2, 𝐵⟩}, {⟨1, 𝑋⟩, ⟨2, 𝑌⟩}}))
 
Theoremrrx2pnecoorneor 44974 If two different points 𝑋 and 𝑌 in a real Euclidean space of dimension 2 are different, then they are different at least at one coordinate. (Contributed by AV, 26-Feb-2023.)
𝐼 = {1, 2}    &   𝑃 = (ℝ ↑m 𝐼)       ((𝑋𝑃𝑌𝑃𝑋𝑌) → ((𝑋‘1) ≠ (𝑌‘1) ∨ (𝑋‘2) ≠ (𝑌‘2)))
 
Theoremrrx2pnedifcoorneor 44975 If two different points 𝑋 and 𝑌 in a real Euclidean space of dimension 2 are different, then at least one difference of two corresponding coordinates is not 0. (Contributed by AV, 26-Feb-2023.)
𝐼 = {1, 2}    &   𝑃 = (ℝ ↑m 𝐼)    &   𝐴 = ((𝑌‘1) − (𝑋‘1))    &   𝐵 = ((𝑌‘2) − (𝑋‘2))       ((𝑋𝑃𝑌𝑃𝑋𝑌) → (𝐴 ≠ 0 ∨ 𝐵 ≠ 0))
 
Theoremrrx2pnedifcoorneorr 44976 If two different points 𝑋 and 𝑌 in a real Euclidean space of dimension 2 are different, then at least one difference of two corresponding coordinates is not 0. (Contributed by AV, 26-Feb-2023.)
𝐼 = {1, 2}    &   𝑃 = (ℝ ↑m 𝐼)    &   𝐴 = ((𝑌‘1) − (𝑋‘1))    &   𝐵 = ((𝑋‘2) − (𝑌‘2))       ((𝑋𝑃𝑌𝑃𝑋𝑌) → (𝐴 ≠ 0 ∨ 𝐵 ≠ 0))
 
Theoremrrx2xpref1o 44977* There is a bijection between the set of ordered pairs of real numbers (the cartesian product of the real numbers) and the set of points in the two dimensional Euclidean plane (represented as mappings from {1, 2} to the real numbers). (Contributed by AV, 12-Mar-2023.)
𝑅 = (ℝ ↑m {1, 2})    &   𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})       𝐹:(ℝ × ℝ)–1-1-onto𝑅
 
Theoremrrx2xpreen 44978 The set of points in the two dimensional Euclidean plane and the set of ordered pairs of real numbers (the cartesian product of the real numbers) are equinumerous. (Contributed by AV, 12-Mar-2023.)
𝑅 = (ℝ ↑m {1, 2})       𝑅 ≈ (ℝ × ℝ)
 
Theoremrrx2plord 44979* The lexicographical ordering for points in the two dimensional Euclidean plane: a point is less than another point iff its first coordinate is less than the first coordinate of the other point, or the first coordinates of both points are equal and the second coordinate of the first point is less than the second coordinate of the other point: 𝑎, 𝑏⟩ ≤ ⟨𝑥, 𝑦 iff (𝑎 < 𝑥 ∨ (𝑎 = 𝑥𝑏𝑦)). (Contributed by AV, 12-Mar-2023.)
𝑂 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑅𝑦𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))}       ((𝑋𝑅𝑌𝑅) → (𝑋𝑂𝑌 ↔ ((𝑋‘1) < (𝑌‘1) ∨ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) < (𝑌‘2)))))
 
Theoremrrx2plord1 44980* The lexicographical ordering for points in the two dimensional Euclidean plane: a point is less than another point if its first coordinate is less than the first coordinate of the other point. (Contributed by AV, 12-Mar-2023.)
𝑂 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑅𝑦𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))}       ((𝑋𝑅𝑌𝑅 ∧ (𝑋‘1) < (𝑌‘1)) → 𝑋𝑂𝑌)
 
Theoremrrx2plord2 44981* The lexicographical ordering for points in the two dimensional Euclidean plane: if the first coordinates of two points are equal, a point is less than another point iff the second coordinate of the point is less than the second coordinate of the other point. (Contributed by AV, 12-Mar-2023.)
𝑂 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑅𝑦𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))}    &   𝑅 = (ℝ ↑m {1, 2})       ((𝑋𝑅𝑌𝑅 ∧ (𝑋‘1) = (𝑌‘1)) → (𝑋𝑂𝑌 ↔ (𝑋‘2) < (𝑌‘2)))
 
Theoremrrx2plordisom 44982* The set of points in the two dimensional Euclidean plane with the lexicographical ordering is isomorphic to the cartesian product of the real numbers with the lexicographical ordering implied by the ordering of the real numbers. (Contributed by AV, 12-Mar-2023.)
𝑂 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑅𝑦𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))}    &   𝑅 = (ℝ ↑m {1, 2})    &   𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st𝑥) < (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) < (2nd𝑦))))}       𝐹 Isom 𝑇, 𝑂 ((ℝ × ℝ), 𝑅)
 
Theoremrrx2plordso 44983* The lexicographical ordering for points in the two dimensional Euclidean plane is a strict total ordering. (Contributed by AV, 12-Mar-2023.)
𝑂 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑅𝑦𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))}    &   𝑅 = (ℝ ↑m {1, 2})       𝑂 Or 𝑅
 
Theoremehl2eudisval0 44984 The Euclidean distance of a point to the origin in a real Euclidean space of dimension 2. (Contributed by AV, 26-Feb-2023.)
𝐸 = (𝔼hil‘2)    &   𝑋 = (ℝ ↑m {1, 2})    &   𝐷 = (dist‘𝐸)    &    0 = ({1, 2} × {0})       (𝐹𝑋 → (𝐹𝐷 0 ) = (√‘(((𝐹‘1)↑2) + ((𝐹‘2)↑2))))
 
Theoremehl2eudis0lt 44985 An upper bound of the Euclidean distance of a point to the origin in a real Euclidean space of dimension 2. (Contributed by AV, 9-May-2023.)
𝐸 = (𝔼hil‘2)    &   𝑋 = (ℝ ↑m {1, 2})    &   𝐷 = (dist‘𝐸)    &    0 = ({1, 2} × {0})       ((𝐹𝑋𝑅 ∈ ℝ+) → ((𝐹𝐷 0 ) < 𝑅 ↔ (((𝐹‘1)↑2) + ((𝐹‘2)↑2)) < (𝑅↑2)))
 
20.41.23.3  Spheres and lines in real Euclidean spaces
 
Syntaxcline 44986 Declare the syntax for lines in generalized real Euclidean spaces.
class LineM
 
Syntaxcsph 44987 Declare the syntax for spheres in generalized real Euclidean spaces.
class Sphere
 
Definitiondf-line 44988* Definition of lines passing through two different points in a left module (or any extended structure having a base set, an addition, and a scalar multiplication). (Contributed by AV, 14-Jan-2023.)
LineM = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤), 𝑦 ∈ ((Base‘𝑤) ∖ {𝑥}) ↦ {𝑝 ∈ (Base‘𝑤) ∣ ∃𝑡 ∈ (Base‘(Scalar‘𝑤))𝑝 = ((((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡)( ·𝑠𝑤)𝑥)(+g𝑤)(𝑡( ·𝑠𝑤)𝑦))}))
 
Definitiondf-sph 44989* Definition of spheres for given centers and radii in a metric space (or more generally, in a distance space, see distspace 22914, or even in any extended structure having a base set and a distance function into the real numbers. (Contributed by AV, 14-Jan-2023.)
Sphere = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤), 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟}))
 
Theoremlines 44990* The lines passing through two different points in a left module (or any extended structure having a base set, an addition, and a scalar multiplication). (Contributed by AV, 14-Jan-2023.)
𝐵 = (Base‘𝑊)    &   𝐿 = (LineM𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑆)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)    &    = (-g𝑆)    &    1 = (1r𝑆)       (𝑊𝑉𝐿 = (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑥) + (𝑡 · 𝑦))}))
 
Theoremline 44991* The line passing through the two different points 𝑋 and 𝑌 in a left module (or any extended structure having a base set, an addition, and a scalar multiplication). (Contributed by AV, 14-Jan-2023.)
𝐵 = (Base‘𝑊)    &   𝐿 = (LineM𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑆)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)    &    = (-g𝑆)    &    1 = (1r𝑆)       ((𝑊𝑉 ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) → (𝑋𝐿𝑌) = {𝑝𝐵 ∣ ∃𝑡𝐾 𝑝 = ((( 1 𝑡) · 𝑋) + (𝑡 · 𝑌))})
 
Theoremrrxlines 44992* Definition of lines passing through two different points in a generalized real Euclidean space of finite dimension. (Contributed by AV, 14-Jan-2023.)
𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝐿 = (LineM𝐸)    &    · = ( ·𝑠𝐸)    &    + = (+g𝐸)       (𝐼 ∈ Fin → 𝐿 = (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑝𝑃 ∣ ∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))}))
 
Theoremrrxline 44993* The line passing through the two different points 𝑋 and 𝑌 in a generalized real Euclidean space of finite dimension. (Contributed by AV, 14-Jan-2023.)
𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝐿 = (LineM𝐸)    &    · = ( ·𝑠𝐸)    &    + = (+g𝐸)       ((𝐼 ∈ Fin ∧ (𝑋𝑃𝑌𝑃𝑋𝑌)) → (𝑋𝐿𝑌) = {𝑝𝑃 ∣ ∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡) · 𝑋) + (𝑡 · 𝑌))})
 
Theoremrrxlinesc 44994* Definition of lines passing through two different points in a generalized real Euclidean space of finite dimension, expressed by their coordinates. (Contributed by AV, 13-Feb-2023.)
𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝐿 = (LineM𝐸)       (𝐼 ∈ Fin → 𝐿 = (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑝𝑃 ∣ ∃𝑡 ∈ ℝ ∀𝑖𝐼 (𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))}))
 
Theoremrrxlinec 44995* The line passing through the two different points 𝑋 and 𝑌 in a generalized real Euclidean space of finite dimension, expressed by its coordinates. Remark: This proof is shorter and requires less distinct variables than the proof using rrxlinesc 44994. (Contributed by AV, 13-Feb-2023.)
𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝐿 = (LineM𝐸)       ((𝐼 ∈ Fin ∧ (𝑋𝑃𝑌𝑃𝑋𝑌)) → (𝑋𝐿𝑌) = {𝑝𝑃 ∣ ∃𝑡 ∈ ℝ ∀𝑖𝐼 (𝑝𝑖) = (((1 − 𝑡) · (𝑋𝑖)) + (𝑡 · (𝑌𝑖)))})
 
Theoremeenglngeehlnmlem1 44996* Lemma 1 for eenglngeehlnm 44998. (Contributed by AV, 15-Feb-2023.)
(((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) → ((∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖)))) → ∃𝑡 ∈ ℝ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))))
 
Theoremeenglngeehlnmlem2 44997* Lemma 2 for eenglngeehlnm 44998. (Contributed by AV, 15-Feb-2023.)
(((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) → (∃𝑡 ∈ ℝ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))) → (∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))))))
 
Theoremeenglngeehlnm 44998 The line definition in the Tarski structure for the Euclidean geometry (see elntg 26769) corresponds to the definition of lines passing through two different points in a left module (see rrxlines 44992). (Contributed by AV, 16-Feb-2023.)
(𝑁 ∈ ℕ → (LineG‘(EEG‘𝑁)) = (LineM‘(𝔼hil𝑁)))
 
Theoremrrx2line 44999* The line passing through the two different points 𝑋 and 𝑌 in a real Euclidean space of dimension 2. (Contributed by AV, 22-Jan-2023.) (Proof shortened by AV, 13-Feb-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝐿 = (LineM𝐸)       ((𝑋𝑃𝑌𝑃𝑋𝑌) → (𝑋𝐿𝑌) = {𝑝𝑃 ∣ ∃𝑡 ∈ ℝ ((𝑝‘1) = (((1 − 𝑡) · (𝑋‘1)) + (𝑡 · (𝑌‘1))) ∧ (𝑝‘2) = (((1 − 𝑡) · (𝑋‘2)) + (𝑡 · (𝑌‘2))))})
 
Theoremrrx2vlinest 45000* The vertical line passing through the two different points 𝑋 and 𝑌 in a real Euclidean space of dimension 2 in "standard form". (Contributed by AV, 2-Feb-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝐿 = (LineM𝐸)       ((𝑋𝑃𝑌𝑃 ∧ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) ≠ (𝑌‘2))) → (𝑋𝐿𝑌) = {𝑝𝑃 ∣ (𝑝‘1) = (𝑋‘1)})
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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 45101-45178
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