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Theorem List for Metamath Proof Explorer - 39101-39200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrenpncan3 39101 Cancellation law for real subtraction. Compare npncan3 10913. (Contributed by Steven Nguyen, 28-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 𝐵) + (𝐶 𝐴)) = (𝐶 𝐵))
 
Theoremrepnpcan 39102 Cancellation law for addition and real subtraction. Compare pnpcan 10914. (Contributed by Steven Nguyen, 19-May-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) − (𝐴 + 𝐶)) = (𝐵 𝐶))
 
Theoremreppncan 39103 Cancellation law for mixed addition and real subtraction. Compare ppncan 10917. (Contributed by SN, 3-Sep-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐶) + (𝐵 𝐶)) = (𝐴 + 𝐵))
 
Theoremresubidaddid1lem 39104 Lemma for resubidaddid1 39105. A special case of npncan 10896. (Contributed by Steven Nguyen, 8-Jan-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑 → (𝐴 𝐵) = (𝐵 𝐶))       (𝜑 → ((𝐴 𝐵) + (𝐵 𝐶)) = (𝐴 𝐶))
 
Theoremresubidaddid1 39105 Any real number subtracted from itself forms a left additive identity. (Contributed by Steven Nguyen, 8-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 𝐴) + 𝐵) = 𝐵)
 
Theoremresubdi 39106 Distribution of multiplication over real subtraction. (Contributed by Steven Nguyen, 3-Jun-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 · (𝐵 𝐶)) = ((𝐴 · 𝐵) − (𝐴 · 𝐶)))
 
Theoremre1m1e0m0 39107 Equality of two left-additive identities. See resubidaddid1 39105. Uses ax-i2m1 10594. (Contributed by SN, 25-Dec-2023.)
(1 − 1) = (0 − 0)
 
Theoremsn-00idlem1 39108 Lemma for sn-00id 39111. (Contributed by SN, 25-Dec-2023.)
(𝐴 ∈ ℝ → (𝐴 · (0 − 0)) = (𝐴 𝐴))
 
Theoremsn-00idlem2 39109 Lemma for sn-00id 39111. (Contributed by SN, 25-Dec-2023.)
((0 − 0) ≠ 0 → (0 − 0) = 1)
 
Theoremsn-00idlem3 39110 Lemma for sn-00id 39111. (Contributed by SN, 25-Dec-2023.)
((0 − 0) = 1 → (0 + 0) = 0)
 
Theoremsn-00id 39111 00id 10804 proven without ax-mulcom 10590 but using ax-1ne0 10595. (Though note that the current version of 00id 10804 can be changed to avoid ax-icn 10585, ax-addcl 10586, ax-mulcl 10588, ax-i2m1 10594, ax-cnre 10599. Most of this is by using 0cnALT3 39033 instead of 0cn 10622). (Contributed by SN, 25-Dec-2023.) (Proof modification is discouraged.)
(0 + 0) = 0
 
Theoremre0m0e0 39112 Real number version of 0m0e0 11746 proven without ax-mulcom 10590. (Contributed by SN, 23-Jan-2024.)
(0 − 0) = 0
 
Theoremreaddid2 39113 Real number version of addid2 10812. (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ ℝ → (0 + 𝐴) = 𝐴)
 
Theoremsn-addid2 39114 addid2 10812 without ax-mulcom 10590. (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)
 
Theoremremul02 39115 Real number version of mul02 10807 proven without ax-mulcom 10590. (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ ℝ → (0 · 𝐴) = 0)
 
Theoremsn-0ne2 39116 0ne2 11833 without ax-mulcom 10590. (Contributed by SN, 23-Jan-2024.)
0 ≠ 2
 
Theoremremul01 39117 Real number version of mul01 10808 proven without ax-mulcom 10590. (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ ℝ → (𝐴 · 0) = 0)
 
Theoremresubid 39118 Subtraction of a real number from itself (compare subid 10894). (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ ℝ → (𝐴 𝐴) = 0)
 
Theoremreaddid1 39119 Real number version of addid1 10809, without ax-mulcom 10590. (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ ℝ → (𝐴 + 0) = 𝐴)
 
Theoremresubid1 39120 Real number version of subid1 10895, without ax-mulcom 10590. (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ ℝ → (𝐴 0) = 𝐴)
 
Theoremrenegneg 39121 A real number is equal to the negative of its negative. Compare negneg 10925. (Contributed by SN, 13-Feb-2024.)
(𝐴 ∈ ℝ → (0 − (0 − 𝐴)) = 𝐴)
 
Theoremreaddcan2 39122 Commuted version of readdcan 10803 without ax-mulcom 10590. (Contributed by SN, 21-Feb-2024.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵))
 
Theoremsn-ltaddpos 39123 ltaddpos 11119 without ax-mulcom 10590. (Contributed by SN, 13-Feb-2024.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴𝐵 < (𝐵 + 𝐴)))
 
Theoremrelt0neg1 39124 Comparison of a real and its negative to zero. Compare lt0neg1 11135. (Contributed by SN, 13-Feb-2024.)
(𝐴 ∈ ℝ → (𝐴 < 0 ↔ 0 < (0 − 𝐴)))
 
Theoremrelt0neg2 39125 Comparison of a real and its negative to zero. Compare lt0neg2 11136. (Contributed by SN, 13-Feb-2024.)
(𝐴 ∈ ℝ → (0 < 𝐴 ↔ (0 − 𝐴) < 0))
 
Theoremsn-0lt1 39126 0lt1 11151 without ax-mulcom 10590. (Contributed by SN, 13-Feb-2024.)
0 < 1
 
Theoremsn-ltp1 39127 ltp1 11469 without ax-mulcom 10590. (Contributed by SN, 13-Feb-2024.)
(𝐴 ∈ ℝ → 𝐴 < (𝐴 + 1))
 
Theoremremulinvcom 39128 A left multiplicative inverse is a right multiplicative inverse. Proven without ax-mulcom 10590. (Contributed by SN, 5-Feb-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (𝐴 · 𝐵) = 1)       (𝜑 → (𝐵 · 𝐴) = 1)
 
Theoremremulid2 39129 Commuted version of ax-1rid 10596 and real number version of mulid2 10629 without ax-mulcom 10590. (Contributed by SN, 5-Feb-2024.)
(𝐴 ∈ ℝ → (1 · 𝐴) = 𝐴)
 
Theoremremulcand 39130 Commuted version of remulcan2d 39036 without ax-mulcom 10590. (Contributed by SN, 21-Feb-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐶 · 𝐴) = (𝐶 · 𝐵) ↔ 𝐴 = 𝐵))
 
20.25.5  Projective spaces

Looking at a corner in 3D space, one can see three right angles. It is impossible to draw three lines in 2D space such that any two of these lines are perpendicular, but a good enough representation is made by casting lines from the 2D surface. Points along the same cast line are collapsed into one point on the 2D surface.

In many cases, the 2D surface is smaller than whatever needs to be represented. If the lines cast were perpendicular to the 2D surface, then only areas as small as the 2D surface could be represented. To fix this, the lines need to get further apart as they go farther from the 2D surface. On the other side of the 2D surface the lines will get closer together and intersect at a point. (Because it's defined that way).

From this perspective, two parallel lines in 3D space will be represented by two lines that seem to intersect at a point "at infinity". Considering all maximal classes of parallel lines on a 2D plane in 3D space, these classes will all appear to intersect at different points at infinity, forming a line at infinity. Therefore the real projective plane can be thought of as the real affine plane together with the line at infinity.

The projective plane takes care of some exceptions that may be found in the affine plane. For example, consider the curve that is the zeroes of 𝑦 = 𝑥↑2. Any line connecting the point (0, 1) to the x-axis intersects with the curve twice, except for the vertical line between (0, 1) and (0, 0). In the projective plane, the curve becomes an ellipse and there is no exception.

While it may not seem like it, points at infinity and points corresponding to the affine plane are the same type of point. Consider a line going through the origin in 3D (affine) space. Either it intersects the plane 𝑧 = 1 once, or it is entirely within the plane 𝑧 = 0. If it is entirely within the plane 𝑧 = 0, then it corresponds to the point at infinity intersecting all lines on the plane 𝑧 = 1 with the same slope. Else it corresponds to the point in the 2D plane 𝑧 = 1 that it intersects. So there is a bijection between 3D lines through the origin and points on the real projective plane.

The concept of projective spaces generalizes the projective plane to any dimension.

 
Syntaxcprjsp 39131 Extend class notation with the projective space function.
class ℙ𝕣𝕠𝕛
 
Definitiondf-prjsp 39132* Define the projective space function. In the bijection between 3D lines through the origin and points in the projective plane (see section comment), this is equivalent to making any two 3D points (excluding the origin) equivalent iff one is a multiple of another. This definition does not quite give all the properties needed, since the scalars of a left vector space can be "less dense" than the vectors (for example, equivocating rational multiples of real numbers). (Contributed by BJ and Steven Nguyen, 29-Apr-2023.)
ℙ𝕣𝕠𝕛 = (𝑣 ∈ LVec ↦ ((Base‘𝑣) ∖ {(0g𝑣)}) / 𝑏(𝑏 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑏𝑦𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠𝑣)𝑦))}))
 
Theoremprjspval 39133* Value of the projective space function, which is also known as the projectivization of 𝑉. (Contributed by Steven Nguyen, 29-Apr-2023.)
𝐵 = ((Base‘𝑉) ∖ {(0g𝑉)})    &    · = ( ·𝑠𝑉)    &   𝑆 = (Scalar‘𝑉)    &   𝐾 = (Base‘𝑆)       (𝑉 ∈ LVec → (ℙ𝕣𝕠𝕛‘𝑉) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))}))
 
Theoremprjsprel 39134* Utility theorem regarding the relation used in ℙ𝕣𝕠𝕛. (Contributed by Steven Nguyen, 29-Apr-2023.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))}       (𝑋 𝑌 ↔ ((𝑋𝐵𝑌𝐵) ∧ ∃𝑚𝐾 𝑋 = (𝑚 · 𝑌)))
 
Theoremprjspertr 39135* The relation in ℙ𝕣𝕠𝕛 is transitive. (Contributed by Steven Nguyen, 1-May-2023.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))}    &   𝐵 = ((Base‘𝑉) ∖ {(0g𝑉)})    &   𝑆 = (Scalar‘𝑉)    &    · = ( ·𝑠𝑉)    &   𝐾 = (Base‘𝑆)       ((𝑉 ∈ LMod ∧ (𝑋 𝑌𝑌 𝑍)) → 𝑋 𝑍)
 
Theoremprjsperref 39136* The relation in ℙ𝕣𝕠𝕛 is reflexive. (Contributed by Steven Nguyen, 30-Apr-2023.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))}    &   𝐵 = ((Base‘𝑉) ∖ {(0g𝑉)})    &   𝑆 = (Scalar‘𝑉)    &    · = ( ·𝑠𝑉)    &   𝐾 = (Base‘𝑆)       (𝑉 ∈ LMod → (𝑋𝐵𝑋 𝑋))
 
Theoremprjspersym 39137* The relation in ℙ𝕣𝕠𝕛 is symmetric. (Contributed by Steven Nguyen, 1-May-2023.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))}    &   𝐵 = ((Base‘𝑉) ∖ {(0g𝑉)})    &   𝑆 = (Scalar‘𝑉)    &    · = ( ·𝑠𝑉)    &   𝐾 = (Base‘𝑆)       ((𝑉 ∈ LVec ∧ 𝑋 𝑌) → 𝑌 𝑋)
 
Theoremprjsper 39138* The relation in ℙ𝕣𝕠𝕛 is an equivalence relation. (Contributed by Steven Nguyen, 1-May-2023.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))}    &   𝐵 = ((Base‘𝑉) ∖ {(0g𝑉)})    &   𝑆 = (Scalar‘𝑉)    &    · = ( ·𝑠𝑉)    &   𝐾 = (Base‘𝑆)       (𝑉 ∈ LVec → Er 𝐵)
 
Theoremprjspreln0 39139* Two nonzero vectors are equivalent by a nonzero scalar. (Contributed by Steven Nguyen, 31-May-2023.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))}    &   𝐵 = ((Base‘𝑉) ∖ {(0g𝑉)})    &   𝑆 = (Scalar‘𝑉)    &    · = ( ·𝑠𝑉)    &   𝐾 = (Base‘𝑆)    &    0 = (0g𝑆)       (𝑉 ∈ LVec → (𝑋 𝑌 ↔ ((𝑋𝐵𝑌𝐵) ∧ ∃𝑚 ∈ (𝐾 ∖ { 0 })𝑋 = (𝑚 · 𝑌))))
 
Theoremprjspvs 39140* A nonzero multiple of a vector is equivalent to the vector. (Contributed by Steven Nguyen, 6-Jun-2023.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))}    &   𝐵 = ((Base‘𝑉) ∖ {(0g𝑉)})    &   𝑆 = (Scalar‘𝑉)    &    · = ( ·𝑠𝑉)    &   𝐾 = (Base‘𝑆)    &    0 = (0g𝑆)       ((𝑉 ∈ LVec ∧ 𝑋𝐵𝑁 ∈ (𝐾 ∖ { 0 })) → (𝑁 · 𝑋) 𝑋)
 
Theoremprjsprellsp 39141* Two vectors are equivalent iff their spans are equal. (Contributed by Steven Nguyen, 31-May-2023.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))}    &   𝐵 = ((Base‘𝑉) ∖ {(0g𝑉)})    &   𝑆 = (Scalar‘𝑉)    &    · = ( ·𝑠𝑉)    &   𝐾 = (Base‘𝑆)    &   𝑁 = (LSpan‘𝑉)       ((𝑉 ∈ LVec ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 𝑌 ↔ (𝑁‘{𝑋}) = (𝑁‘{𝑌})))
 
Theoremprjspeclsp 39142* The vectors equivalent to a vector 𝑋 are the nonzero vectors in the span of 𝑋. (Contributed by Steven Nguyen, 6-Jun-2023.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))}    &   𝐵 = ((Base‘𝑉) ∖ {(0g𝑉)})    &   𝑆 = (Scalar‘𝑉)    &    · = ( ·𝑠𝑉)    &   𝐾 = (Base‘𝑆)    &   𝑁 = (LSpan‘𝑉)       ((𝑉 ∈ LVec ∧ 𝑋𝐵) → [𝑋] = ((𝑁‘{𝑋}) ∖ {(0g𝑉)}))
 
Theoremprjspval2 39143* Alternate definition of projective space. (Contributed by Steven Nguyen, 7-Jun-2023.)
0 = (0g𝑉)    &   𝐵 = ((Base‘𝑉) ∖ { 0 })    &   𝑁 = (LSpan‘𝑉)       (𝑉 ∈ LVec → (ℙ𝕣𝕠𝕛‘𝑉) = 𝑧𝐵 {((𝑁‘{𝑧}) ∖ { 0 })})
 
Syntaxcprjspn 39144 Extend class notation with the n-dimensional projective space function.
class ℙ𝕣𝕠𝕛n
 
Definitiondf-prjspn 39145* Define the n-dimensional projective space function. A projective space of dimension 1 is a projective line, and a projective space of dimension 2 is a projective plane. Compare df-ehl 23918. This space is considered n-dimensional because the vector space (𝑘 freeLMod (0...𝑛)) is (n+1)-dimensional and the ℙ𝕣𝕠𝕛 function returns equivalence classes with respect to a linear (1-dimensional) relation. (Contributed by BJ and Steven Nguyen, 29-Apr-2023.)
ℙ𝕣𝕠𝕛n = (𝑛 ∈ ℕ0, 𝑘 ∈ DivRing ↦ (ℙ𝕣𝕠𝕛‘(𝑘 freeLMod (0...𝑛))))
 
Theoremprjspnval 39146 Value of the n-dimensional projective space function. (Contributed by Steven Nguyen, 1-May-2023.)
((𝑁 ∈ ℕ0𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛n𝐾) = (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))))
 
Theoremprjspnval2 39147* Value of the n-dimensional projective space function, expanded. (Contributed by Steven Nguyen, 15-Jul-2023.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝑆 𝑥 = (𝑙 · 𝑦))}    &   𝑊 = (𝐾 freeLMod (0...𝑁))    &   𝐵 = ((Base‘𝑊) ∖ {(0g𝑊)})    &   𝑆 = (Base‘𝐾)    &    · = ( ·𝑠𝑊)       ((𝑁 ∈ ℕ0𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛n𝐾) = (𝐵 / ))
 
Theorem0prjspnlem 39148 Lemma for 0prjspn 39150. The given unit vector is a nonzero vector. (Contributed by Steven Nguyen, 16-Jul-2023.)
𝐵 = ((Base‘𝑊) ∖ {(0g𝑊)})    &   𝑊 = (𝐾 freeLMod (0...0))    &    1 = ((𝐾 unitVec (0...0))‘0)       (𝐾 ∈ DivRing → 1𝐵)
 
Theorem0prjspnrel 39149* In the zero-dimensional projective space, all vectors are equivalent to the unit vector. (Contributed by Steven Nguyen, 7-Jun-2023.)
= {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝑆 𝑥 = (𝑙 · 𝑦))}    &   𝐵 = ((Base‘𝑊) ∖ {(0g𝑊)})    &    · = ( ·𝑠𝑊)    &   𝑆 = (Base‘𝐾)    &   𝑊 = (𝐾 freeLMod (0...0))    &    1 = ((𝐾 unitVec (0...0))‘0)       ((𝐾 ∈ DivRing ∧ 𝑋𝐵) → 𝑋 1 )
 
Theorem0prjspn 39150 A zero-dimensional projective space has only 1 point. (Contributed by Steven Nguyen, 9-Jun-2023.)
𝑊 = (𝐾 freeLMod (0...0))    &   𝐵 = ((Base‘𝑊) ∖ {(0g𝑊)})       (𝐾 ∈ DivRing → (0ℙ𝕣𝕠𝕛n𝐾) = {𝐵})
 
20.25.6  Equivalent formulations of Fermat's Last Theorem
 
Theoremdffltz 39151* Fermat's Last Theorem (FLT) for nonzero integers is equivalent to the original scope of natural numbers. The backwards direction takes (𝑎𝑛) + (𝑏𝑛) = (𝑐𝑛), and adds the negative of any negative term to both sides, thus creating the corresponding equation with only positive integers. There are six combinations of negativity, so the proof is particularly long. (Contributed by Steven Nguyen, 27-Feb-2023.)
(∀𝑛 ∈ (ℤ‘3)∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ∀𝑧 ∈ ℕ ((𝑥𝑛) + (𝑦𝑛)) ≠ (𝑧𝑛) ↔ ∀𝑛 ∈ (ℤ‘3)∀𝑎 ∈ (ℤ ∖ {0})∀𝑏 ∈ (ℤ ∖ {0})∀𝑐 ∈ (ℤ ∖ {0})((𝑎𝑛) + (𝑏𝑛)) ≠ (𝑐𝑛))
 
Theoremfltne 39152 If a counterexample to FLT exists, its addends are not equal. (Contributed by Steven Nguyen, 1-Jun-2023.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐶 ∈ ℕ)    &   (𝜑𝑁 ∈ (ℤ‘3))    &   (𝜑 → ((𝐴𝑁) + (𝐵𝑁)) = (𝐶𝑁))       (𝜑𝐴𝐵)
 
Theoremfltltc 39153 (𝐶𝑁) is the largest term and therefore 𝐵 < 𝐶. (Contributed by Steven Nguyen, 22-Aug-2023.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐶 ∈ ℕ)    &   (𝜑𝑁 ∈ (ℤ‘3))    &   (𝜑 → ((𝐴𝑁) + (𝐵𝑁)) = (𝐶𝑁))       (𝜑𝐵 < 𝐶)
 
Theoremfltnltalem 39154 Lemma for fltnlta 39155. A lower bound for 𝐴 based on pwdif 15213. (Contributed by Steven Nguyen, 22-Aug-2023.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐶 ∈ ℕ)    &   (𝜑𝑁 ∈ (ℤ‘3))    &   (𝜑 → ((𝐴𝑁) + (𝐵𝑁)) = (𝐶𝑁))       (𝜑 → ((𝐶𝐵) · ((𝐶↑(𝑁 − 1)) + ((𝑁 − 1) · (𝐵↑(𝑁 − 1))))) < (𝐴𝑁))
 
Theoremfltnlta 39155 𝑁 is less than 𝐴. See https://www.youtu.be/EymVXkPWxyc for an outline. (Contributed by Steven Nguyen, 24-Aug-2023.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐶 ∈ ℕ)    &   (𝜑𝑁 ∈ (ℤ‘3))    &   (𝜑 → ((𝐴𝑁) + (𝐵𝑁)) = (𝐶𝑁))    &   (𝜑𝐴 < 𝐵)       (𝜑𝑁 < 𝐴)
 
20.26  Mathbox for Igor Ieskov
 
Theorembinom2d 39156 Deduction form of binom2. (Contributed by Igor Ieskov, 14-Dec-2023.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵)↑2) = (((𝐴↑2) + (2 · (𝐴 · 𝐵))) + (𝐵↑2)))
 
Theoremcu3addd 39157 Cube of sum of three numbers. (Contributed by Igor Ieskov, 14-Dec-2023.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (((𝐴 + 𝐵) + 𝐶)↑3) = (((((𝐴↑3) + (3 · ((𝐴↑2) · 𝐵))) + ((3 · (𝐴 · (𝐵↑2))) + (𝐵↑3))) + (((3 · ((𝐴↑2) · 𝐶)) + (((3 · 2) · (𝐴 · 𝐵)) · 𝐶)) + (3 · ((𝐵↑2) · 𝐶)))) + (((3 · (𝐴 · (𝐶↑2))) + (3 · (𝐵 · (𝐶↑2)))) + (𝐶↑3))))
 
Theoremsqnegd 39158 The square of the negative of a number. (Contributed by Igor Ieskov, 21-Jan-2024.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (-𝐴↑2) = (𝐴↑2))
 
Theoremnegexpidd 39159 The sum of a real number to the power of N and the negative of the number to the power of N equals zero if N is a nonnegative odd integer. (Contributed by Igor Ieskov, 21-Jan-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → ¬ 2 ∥ 𝑁)       (𝜑 → ((𝐴𝑁) + (-𝐴𝑁)) = 0)
 
Theoremrexlimdv3d 39160* An extended version of rexlimdvv 3293 to include three set variables. (Contributed by Igor Ieskov, 21-Jan-2024.)
(𝜑 → ((𝑥𝐴𝑦𝐵𝑧𝐶) → (𝜓𝜒)))       (𝜑 → (∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜓𝜒))
 
Theorem3cubeslem1 39161 Lemma for 3cubes 39167. (Contributed by Igor Ieskov, 22-Jan-2024.)
(𝜑𝐴 ∈ ℚ)       (𝜑 → 0 < (((𝐴 + 1)↑2) − 𝐴))
 
Theorem3cubeslem2 39162 Lemma for 3cubes 39167. Used to show that the denominators in 3cubeslem4 39166 are nonzero. (Contributed by Igor Ieskov, 22-Jan-2024.)
(𝜑𝐴 ∈ ℚ)       (𝜑 → ¬ ((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) + 3) = 0)
 
Theorem3cubeslem3l 39163 Lemma for 3cubes 39167. (Contributed by Igor Ieskov, 22-Jan-2024.)
(𝜑𝐴 ∈ ℚ)       (𝜑 → (𝐴 · (((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) + 3)↑3)) = (((𝐴↑7) · (3↑9)) + (((𝐴↑6) · (3↑9)) + (((𝐴↑5) · ((3↑8) + (3↑8))) + (((𝐴↑4) · (((3↑7) · 2) + (3↑6))) + (((𝐴↑3) · ((3↑6) + (3↑6))) + (((𝐴↑2) · (3↑5)) + (𝐴 · (3↑3)))))))))
 
Theorem3cubeslem3r 39164 Lemma for 3cubes 39167. (Contributed by Igor Ieskov, 22-Jan-2024.)
(𝜑𝐴 ∈ ℚ)       (𝜑 → ((((((3↑3) · (𝐴↑3)) − 1)↑3) + (((-((3↑3) · (𝐴↑3)) + ((3↑2) · 𝐴)) + 1)↑3)) + ((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴))↑3)) = (((𝐴↑7) · (3↑9)) + (((𝐴↑6) · (3↑9)) + (((𝐴↑5) · ((3↑8) + (3↑8))) + (((𝐴↑4) · (((3↑7) · 2) + (3↑6))) + (((𝐴↑3) · ((3↑6) + (3↑6))) + (((𝐴↑2) · (3↑5)) + (𝐴 · (3↑3)))))))))
 
Theorem3cubeslem3 39165 Lemma for 3cubes 39167. (Contributed by Igor Ieskov, 22-Jan-2024.)
(𝜑𝐴 ∈ ℚ)       (𝜑 → (𝐴 · (((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) + 3)↑3)) = ((((((3↑3) · (𝐴↑3)) − 1)↑3) + (((-((3↑3) · (𝐴↑3)) + ((3↑2) · 𝐴)) + 1)↑3)) + ((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴))↑3)))
 
Theorem3cubeslem4 39166 Lemma for 3cubes 39167. This is Ryley's explicit formula for decomposing a rational 𝐴 into a sum of three rational cubes. (Contributed by Igor Ieskov, 22-Jan-2024.)
(𝜑𝐴 ∈ ℚ)       (𝜑𝐴 = (((((((3↑3) · (𝐴↑3)) − 1) / ((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) + 3))↑3) + ((((-((3↑3) · (𝐴↑3)) + ((3↑2) · 𝐴)) + 1) / ((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) + 3))↑3)) + (((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) / ((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) + 3))↑3)))
 
Theorem3cubes 39167* Every rational number is a sum of three rational cubes. (S. Ryley, The Ladies' Diary 122 (1825), 35) (Contributed by Igor Ieskov, 22-Jan-2024.)
(𝐴 ∈ ℚ ↔ ∃𝑎 ∈ ℚ ∃𝑏 ∈ ℚ ∃𝑐 ∈ ℚ 𝐴 = (((𝑎↑3) + (𝑏↑3)) + (𝑐↑3)))
 
20.27  Mathbox for OpenAI
 
TheoremrntrclfvOAI 39168 The range of the transitive closure is equal to the range of the relation. (Contributed by OpenAI, 7-Jul-2020.)
(𝑅𝑉 → ran (t+‘𝑅) = ran 𝑅)
 
20.28  Mathbox for Stefan O'Rear
 
20.28.1  Additional elementary logic and set theory
 
Theoremmoxfr 39169* Transfer at-most-one between related expressions. (Contributed by Stefan O'Rear, 12-Feb-2015.)
𝐴 ∈ V    &   ∃!𝑦 𝑥 = 𝐴    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
 
20.28.2  Additional theory of functions
 
Theoremimaiinfv 39170* Indexed intersection of an image. (Contributed by Stefan O'Rear, 22-Feb-2015.)
((𝐹 Fn 𝐴𝐵𝐴) → 𝑥𝐵 (𝐹𝑥) = (𝐹𝐵))
 
20.28.3  Additional topology
 
Theoremelrfi 39171* Elementhood in a set of relative finite intersections. (Contributed by Stefan O'Rear, 22-Feb-2015.)
((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) ↔ ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣)))
 
Theoremelrfirn 39172* Elementhood in a set of relative finite intersections of an indexed family of sets. (Contributed by Stefan O'Rear, 22-Feb-2015.)
((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ ran 𝐹)) ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 𝑦𝑣 (𝐹𝑦))))
 
Theoremelrfirn2 39173* Elementhood in a set of relative finite intersections of an indexed family of sets (implicit). (Contributed by Stefan O'Rear, 22-Feb-2015.)
((𝐵𝑉 ∧ ∀𝑦𝐼 𝐶𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ ran (𝑦𝐼𝐶))) ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 𝑦𝑣 𝐶)))
 
Theoremcmpfiiin 39174* In a compact topology, a system of closed sets with nonempty finite intersections has a nonempty intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.)
𝑋 = 𝐽    &   (𝜑𝐽 ∈ Comp)    &   ((𝜑𝑘𝐼) → 𝑆 ∈ (Clsd‘𝐽))    &   ((𝜑 ∧ (𝑙𝐼𝑙 ∈ Fin)) → (𝑋 𝑘𝑙 𝑆) ≠ ∅)       (𝜑 → (𝑋 𝑘𝐼 𝑆) ≠ ∅)
 
20.28.4  Characterization of closure operators. Kuratowski closure axioms
 
Theoremismrcd1 39175* Any function from the subsets of a set to itself, which is extensive (satisfies mrcssid 16878), isotone (satisfies mrcss 16877), and idempotent (satisfies mrcidm 16880) has a collection of fixed points which is a Moore collection, and itself is the closure operator for that collection. This can be taken as an alternate definition for the closure operators. This is the first half, ismrcd2 39176 is the second. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝜑𝐵𝑉)    &   (𝜑𝐹:𝒫 𝐵⟶𝒫 𝐵)    &   ((𝜑𝑥𝐵) → 𝑥 ⊆ (𝐹𝑥))    &   ((𝜑𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥))    &   ((𝜑𝑥𝐵) → (𝐹‘(𝐹𝑥)) = (𝐹𝑥))       (𝜑 → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵))
 
Theoremismrcd2 39176* Second half of ismrcd1 39175. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝜑𝐵𝑉)    &   (𝜑𝐹:𝒫 𝐵⟶𝒫 𝐵)    &   ((𝜑𝑥𝐵) → 𝑥 ⊆ (𝐹𝑥))    &   ((𝜑𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥))    &   ((𝜑𝑥𝐵) → (𝐹‘(𝐹𝑥)) = (𝐹𝑥))       (𝜑𝐹 = (mrCls‘dom (𝐹 ∩ I )))
 
Theoremistopclsd 39177* A closure function which satisfies sscls 21594, clsidm 21605, cls0 21618, and clsun 33574 defines a (unique) topology which it is the closure function on. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝜑𝐵𝑉)    &   (𝜑𝐹:𝒫 𝐵⟶𝒫 𝐵)    &   ((𝜑𝑥𝐵) → 𝑥 ⊆ (𝐹𝑥))    &   ((𝜑𝑥𝐵) → (𝐹‘(𝐹𝑥)) = (𝐹𝑥))    &   (𝜑 → (𝐹‘∅) = ∅)    &   ((𝜑𝑥𝐵𝑦𝐵) → (𝐹‘(𝑥𝑦)) = ((𝐹𝑥) ∪ (𝐹𝑦)))    &   𝐽 = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐹‘(𝐵𝑧)) = (𝐵𝑧)}       (𝜑 → (𝐽 ∈ (TopOn‘𝐵) ∧ (cls‘𝐽) = 𝐹))
 
Theoremismrc 39178* A function is a Moore closure operator iff it satisfies mrcssid 16878, mrcss 16877, and mrcidm 16880. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐹 ∈ (mrCls “ (Moore‘𝐵)) ↔ (𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))))
 
20.28.5  Algebraic closure systems
 
Syntaxcnacs 39179 Class of Noetherian closure systems.
class NoeACS
 
Definitiondf-nacs 39180* Define a closure system of Noetherian type (not standard terminology) as an algebraic system where all closed sets are finitely generated. (Contributed by Stefan O'Rear, 4-Apr-2015.)
NoeACS = (𝑥 ∈ V ↦ {𝑐 ∈ (ACS‘𝑥) ∣ ∀𝑠𝑐𝑔 ∈ (𝒫 𝑥 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)})
 
Theoremisnacs 39181* Expand definition of Noetherian-type closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝐹 = (mrCls‘𝐶)       (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔)))
 
Theoremnacsfg 39182* In a Noetherian-type closure system, all closed sets are finitely generated. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝐹 = (mrCls‘𝐶)       ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑆𝐶) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹𝑔))
 
Theoremisnacs2 39183 Express Noetherian-type closure system with fewer quantifiers. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝐹 = (mrCls‘𝐶)       (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ (𝐹 “ (𝒫 𝑋 ∩ Fin)) = 𝐶))
 
Theoremmrefg2 39184* Slight variation on finite generation for closure systems. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝐹 = (mrCls‘𝐶)       (𝐶 ∈ (Moore‘𝑋) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 = (𝐹𝑔)))
 
Theoremmrefg3 39185* Slight variation on finite generation for closure systems. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝐹 = (mrCls‘𝐶)       ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 ⊆ (𝐹𝑔)))
 
Theoremnacsacs 39186 A closure system of Noetherian type is algebraic. (Contributed by Stefan O'Rear, 4-Apr-2015.)
(𝐶 ∈ (NoeACS‘𝑋) → 𝐶 ∈ (ACS‘𝑋))
 
Theoremisnacs3 39187* A choice-free order equivalent to the Noetherian condition on a closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
(𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝑠)))
 
Theoremincssnn0 39188* Transitivity induction of subsets, lemma for nacsfix 39189. (Contributed by Stefan O'Rear, 4-Apr-2015.)
((∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1)) ∧ 𝐴 ∈ ℕ0𝐵 ∈ (ℤ𝐴)) → (𝐹𝐴) ⊆ (𝐹𝐵))
 
Theoremnacsfix 39189* An increasing sequence of closed sets in a Noetherian-type closure system eventually fixates. (Contributed by Stefan O'Rear, 4-Apr-2015.)
((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ∃𝑦 ∈ ℕ0𝑧 ∈ (ℤ𝑦)(𝐹𝑧) = (𝐹𝑦))
 
20.28.6  Miscellanea 1. Map utilities
 
Theoremconstmap 39190 A constant (represented without dummy variables) is an element of a function set.

Note: In the following development, we will be quite often quantifying over functions and points in N-dimensional space (which are equivalent to functions from an "index set"). Many of the following theorems exist to transfer standard facts about functions to elements of function sets. (Contributed by Stefan O'Rear, 30-Aug-2014.) (Revised by Stefan O'Rear, 5-May-2015.)

𝐴 ∈ V    &   𝐶 ∈ V       (𝐵𝐶 → (𝐴 × {𝐵}) ∈ (𝐶m 𝐴))
 
Theoremmapco2g 39191 Renaming indices in a tuple, with sethood as antecedents. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Mario Carneiro, 5-May-2015.)
((𝐸 ∈ V ∧ 𝐴 ∈ (𝐵m 𝐶) ∧ 𝐷:𝐸𝐶) → (𝐴𝐷) ∈ (𝐵m 𝐸))
 
Theoremmapco2 39192 Post-composition (renaming indices) of a mapping viewed as a point. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)
𝐸 ∈ V       ((𝐴 ∈ (𝐵m 𝐶) ∧ 𝐷:𝐸𝐶) → (𝐴𝐷) ∈ (𝐵m 𝐸))
 
Theoremmapfzcons 39193 Extending a one-based mapping by adding a tuple at the end results in another mapping. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)
𝑀 = (𝑁 + 1)       ((𝑁 ∈ ℕ0𝐴 ∈ (𝐵m (1...𝑁)) ∧ 𝐶𝐵) → (𝐴 ∪ {⟨𝑀, 𝐶⟩}) ∈ (𝐵m (1...𝑀)))
 
Theoremmapfzcons1 39194 Recover prefix mapping from an extended mapping. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)
𝑀 = (𝑁 + 1)       (𝐴 ∈ (𝐵m (1...𝑁)) → ((𝐴 ∪ {⟨𝑀, 𝐶⟩}) ↾ (1...𝑁)) = 𝐴)
 
Theoremmapfzcons1cl 39195 A nonempty mapping has a prefix. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)
𝑀 = (𝑁 + 1)       (𝐴 ∈ (𝐵m (1...𝑀)) → (𝐴 ↾ (1...𝑁)) ∈ (𝐵m (1...𝑁)))
 
Theoremmapfzcons2 39196 Recover added element from an extended mapping. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)
𝑀 = (𝑁 + 1)       ((𝐴 ∈ (𝐵m (1...𝑁)) ∧ 𝐶𝐵) → ((𝐴 ∪ {⟨𝑀, 𝐶⟩})‘𝑀) = 𝐶)
 
20.28.7  Miscellanea for polynomials
 
Theoremmptfcl 39197* Interpret range of a maps-to notation as a constraint on the definition. (Contributed by Stefan O'Rear, 10-Oct-2014.)
((𝑡𝐴𝐵):𝐴𝐶 → (𝑡𝐴𝐵𝐶))
 
20.28.8  Multivariate polynomials over the integers
 
Syntaxcmzpcl 39198 Extend class notation to include pre-polynomial rings.
class mzPolyCld
 
Syntaxcmzp 39199 Extend class notation to include polynomial rings.
class mzPoly
 
Definitiondf-mzpcl 39200* Define the polynomially closed function rings over an arbitrary index set 𝑣. The set (mzPolyCld‘𝑣) contains all sets of functions from (ℤ ↑m 𝑣) to which include all constants and projections and are closed under addition and multiplication. This is a "temporary" set used to define the polynomial function ring itself (mzPoly‘𝑣); see df-mzp 39201. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPolyCld = (𝑣 ∈ V ↦ {𝑝 ∈ 𝒫 (ℤ ↑m (ℤ ↑m 𝑣)) ∣ ((∀𝑖 ∈ ℤ ((ℤ ↑m 𝑣) × {𝑖}) ∈ 𝑝 ∧ ∀𝑗𝑣 (𝑥 ∈ (ℤ ↑m 𝑣) ↦ (𝑥𝑗)) ∈ 𝑝) ∧ ∀𝑓𝑝𝑔𝑝 ((𝑓f + 𝑔) ∈ 𝑝 ∧ (𝑓f · 𝑔) ∈ 𝑝))})
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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 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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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