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Theorem List for Metamath Proof Explorer - 45601-45700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremitcovalt2lem2lem2 45601 Lemma 2 for itcovalt2lem2 45603. (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 45602* Lemma 1 for itcovalt2 45604: 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 45603* Lemma 2 for itcovalt2 45604: 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 45604* 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 45605* 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 45606 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 45607 The Ackermann function at 0. (Contributed by AV, 2-May-2024.)
(Ack‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 1))
 
Theoremackval1 45608 The Ackermann function at 1. (Contributed by AV, 4-May-2024.)
(Ack‘1) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2))
 
Theoremackval2 45609 The Ackermann function at 2. (Contributed by AV, 4-May-2024.)
(Ack‘2) = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 3))
 
Theoremackval3 45610 The Ackermann function at 3. (Contributed by AV, 7-May-2024.)
(Ack‘3) = (𝑛 ∈ ℕ0 ↦ ((2↑(𝑛 + 3)) − 3))
 
Theoremackendofnn0 45611 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 45612 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 45613 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 45614 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 45615 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 45616 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 45617 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 45618 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 45619 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 45620 The Ackermann function at (4,0). (Contributed by AV, 9-May-2024.)
((Ack‘4)‘0) = 13
 
Theoremackval41a 45621 The Ackermann function at (4,1). (Contributed by AV, 9-May-2024.)
((Ack‘4)‘1) = ((2↑16) − 3)
 
Theoremackval41 45622 The Ackermann function at (4,1). (Contributed by AV, 9-May-2024.)
((Ack‘4)‘1) = 65533
 
Theoremackval42 45623 The Ackermann function at (4,2). (Contributed by AV, 9-May-2024.)
((Ack‘4)‘2) = ((2↑65536) − 3)
 
Theoremackval42a 45624 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 45625 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 45626 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 45627 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 45628 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 45629* 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 45630* 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 45631 Identity of an affine combination. (Contributed by AV, 2-Feb-2023.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝑇 ∈ ℂ)       (𝜑 → (((1 − 𝑇) · 𝐴) + (𝑇 · 𝐴)) = 𝐴)
 
Theorem1subrec1sub 45632 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 45633 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 45634 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 45635 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 45636 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 45637 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 45638 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 45639 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 45640 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 45641 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 45642 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 45643 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 45644 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 45645* 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 45646 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 45647* 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 45648* 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 45649* 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 45650* 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 45651* 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 45652 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 45653 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 45654 Declare the syntax for lines in generalized real Euclidean spaces.
class LineM
 
Syntaxcsph 45655 Declare the syntax for spheres in generalized real Euclidean spaces.
class Sphere
 
Definitiondf-line 45656* 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 45657* Definition of spheres for given centers and radii in a metric space (or more generally, in a distance space, see distspace 23081, 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 45658* 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 45659* 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 45660* 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 45661* 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 45662* 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 45663* 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 45662. (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 45664* Lemma 1 for eenglngeehlnm 45666. (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 45665* Lemma 2 for eenglngeehlnm 45666. (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 45666 The line definition in the Tarski structure for the Euclidean geometry (see elntg 26942) corresponds to the definition of lines passing through two different points in a left module (see rrxlines 45660). (Contributed by AV, 16-Feb-2023.)
(𝑁 ∈ ℕ → (LineG‘(EEG‘𝑁)) = (LineM‘(𝔼hil𝑁)))
 
Theoremrrx2line 45667* 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 45668* 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)})
 
Theoremrrx2linest 45669* The 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)    &   𝐴 = ((𝑌‘1) − (𝑋‘1))    &   𝐵 = ((𝑌‘2) − (𝑋‘2))    &   𝐶 = (((𝑋‘2) · (𝑌‘1)) − ((𝑋‘1) · (𝑌‘2)))       ((𝑋𝑃𝑌𝑃𝑋𝑌) → (𝑋𝐿𝑌) = {𝑝𝑃 ∣ (𝐴 · (𝑝‘2)) = ((𝐵 · (𝑝‘1)) + 𝐶)})
 
Theoremrrx2linesl 45670* The line passing through the two different points 𝑋 and 𝑌 in a real Euclidean space of dimension 2, expressed by the slope 𝑆 between the two points ("point-slope form"), sometimes also written as ((𝑝‘2) − (𝑋‘2)) = (𝑆 · ((𝑝‘1) − (𝑋‘1))). (Contributed by AV, 22-Jan-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝐿 = (LineM𝐸)    &   𝑆 = (((𝑌‘2) − (𝑋‘2)) / ((𝑌‘1) − (𝑋‘1)))       ((𝑋𝑃𝑌𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) → (𝑋𝐿𝑌) = {𝑝𝑃 ∣ (𝑝‘2) = ((𝑆 · ((𝑝‘1) − (𝑋‘1))) + (𝑋‘2))})
 
Theoremrrx2linest2 45671* The line passing through the two different points 𝑋 and 𝑌 in a real Euclidean space of dimension 2 in another "standard form" (usually with (𝑝‘1) = 𝑥 and (𝑝‘2) = 𝑦). (Contributed by AV, 23-Feb-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝐿 = (LineM𝐸)    &   𝐴 = ((𝑋‘2) − (𝑌‘2))    &   𝐵 = ((𝑌‘1) − (𝑋‘1))    &   𝐶 = (((𝑋‘2) · (𝑌‘1)) − ((𝑋‘1) · (𝑌‘2)))       ((𝑋𝑃𝑌𝑃𝑋𝑌) → (𝑋𝐿𝑌) = {𝑝𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶})
 
Theoremelrrx2linest2 45672 The line passing through the two different points 𝑋 and 𝑌 in a real Euclidean space of dimension 2 in another "standard form" (usually with (𝑝‘1) = 𝑥 and (𝑝‘2) = 𝑦). (Contributed by AV, 23-Feb-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝐿 = (LineM𝐸)    &   𝐴 = ((𝑋‘2) − (𝑌‘2))    &   𝐵 = ((𝑌‘1) − (𝑋‘1))    &   𝐶 = (((𝑋‘2) · (𝑌‘1)) − ((𝑋‘1) · (𝑌‘2)))       ((𝑋𝑃𝑌𝑃𝑋𝑌) → (𝐺 ∈ (𝑋𝐿𝑌) ↔ (𝐺𝑃 ∧ ((𝐴 · (𝐺‘1)) + (𝐵 · (𝐺‘2))) = 𝐶)))
 
Theoremspheres 45673* The spheres for given centers and radii in a metric space (or any extensible structure having a base set and a distance function). (Contributed by AV, 22-Jan-2023.)
𝐵 = (Base‘𝑊)    &   𝑆 = (Sphere‘𝑊)    &   𝐷 = (dist‘𝑊)       (𝑊𝑉𝑆 = (𝑥𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}))
 
Theoremsphere 45674* A sphere with center 𝑋 and radius 𝑅 in a metric space (or any extensible structure having a base set and a distance function). (Contributed by AV, 22-Jan-2023.)
𝐵 = (Base‘𝑊)    &   𝑆 = (Sphere‘𝑊)    &   𝐷 = (dist‘𝑊)       ((𝑊𝑉𝑋𝐵𝑅 ∈ (0[,]+∞)) → (𝑋𝑆𝑅) = {𝑝𝐵 ∣ (𝑝𝐷𝑋) = 𝑅})
 
Theoremrrxsphere 45675* The sphere with center 𝑀 and radius 𝑅 in a generalized real Euclidean space of finite dimension. Remark: this theorem holds also for the degenerate case 𝑅 < 0 (negative radius): in this case, (𝑀𝑆𝑅) is empty. (Contributed by AV, 5-Feb-2023.)
𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝐷 = (dist‘𝐸)    &   𝑆 = (Sphere‘𝐸)       ((𝐼 ∈ Fin ∧ 𝑀𝑃𝑅 ∈ ℝ) → (𝑀𝑆𝑅) = {𝑝𝑃 ∣ (𝑝𝐷𝑀) = 𝑅})
 
Theorem2sphere 45676* The sphere with center 𝑀 and radius 𝑅 in a two dimensional Euclidean space is a circle. (Contributed by AV, 5-Feb-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝑆 = (Sphere‘𝐸)    &   𝐶 = {𝑝𝑃 ∣ ((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2)) = (𝑅↑2)}       ((𝑀𝑃𝑅 ∈ (0[,)+∞)) → (𝑀𝑆𝑅) = 𝐶)
 
Theorem2sphere0 45677* The sphere around the origin 0 (see rrx0 24161) with radius 𝑅 in a two dimensional Euclidean space is a circle. (Contributed by AV, 5-Feb-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝑆 = (Sphere‘𝐸)    &    0 = (𝐼 × {0})    &   𝐶 = {𝑝𝑃 ∣ (((𝑝‘1)↑2) + ((𝑝‘2)↑2)) = (𝑅↑2)}       (𝑅 ∈ (0[,)+∞) → ( 0 𝑆𝑅) = 𝐶)
 
Theoremline2ylem 45678* Lemma for line2y 45682. This proof is based on counterexamples for the following cases: 1. 𝐶 ≠ 0: p = (0,0) (LHS of bicondional is false, RHS is true); 2. 𝐶 = 0 ∧ 𝐵 ≠ 0: p = (1,-A/B) (LHS of bicondional is true, RHS is false); 3. 𝐴 = 𝐵 = 𝐶 = 0: p = (1,1) (LHS of bicondional is true, RHS is false). (Contributed by AV, 4-Feb-2023.)
𝐼 = {1, 2}    &   𝑃 = (ℝ ↑m 𝐼)       ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (∀𝑝𝑃 (((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶 ↔ (𝑝‘1) = 0) → (𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0)))
 
Theoremline2 45679* Example for a line 𝐺 passing through two different points in "standard form". (Contributed by AV, 3-Feb-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝐿 = (LineM𝐸)    &   𝐺 = {𝑝𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶}    &   𝑋 = {⟨1, 0⟩, ⟨2, (𝐶 / 𝐵)⟩}    &   𝑌 = {⟨1, 1⟩, ⟨2, ((𝐶𝐴) / 𝐵)⟩}       ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) ∧ 𝐶 ∈ ℝ) → 𝐺 = (𝑋𝐿𝑌))
 
Theoremline2xlem 45680* Lemma for line2x 45681. This proof is based on counterexamples for the following cases: 1. 𝑀 ≠ (𝐶 / 𝐵): p = (0,C/B) (LHS of bicondional is true, RHS is false); 2. 𝐴 ≠ 0 ∧ 𝑀 = (𝐶 / 𝐵): p = (1,C/B) (LHS of bicondional is false, RHS is true). (Contributed by AV, 4-Feb-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝐿 = (LineM𝐸)    &   𝐺 = {𝑝𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶}    &   𝑋 = {⟨1, 0⟩, ⟨2, 𝑀⟩}    &   𝑌 = {⟨1, 1⟩, ⟨2, 𝑀⟩}       (((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) ∧ 𝐶 ∈ ℝ) ∧ 𝑀 ∈ ℝ) → (∀𝑝𝑃 (((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶 ↔ (𝑝‘2) = 𝑀) → (𝐴 = 0 ∧ 𝑀 = (𝐶 / 𝐵))))
 
Theoremline2x 45681* Example for a horizontal line 𝐺 passing through two different points in "standard form". (Contributed by AV, 3-Feb-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝐿 = (LineM𝐸)    &   𝐺 = {𝑝𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶}    &   𝑋 = {⟨1, 0⟩, ⟨2, 𝑀⟩}    &   𝑌 = {⟨1, 1⟩, ⟨2, 𝑀⟩}       (((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) ∧ 𝐶 ∈ ℝ) ∧ 𝑀 ∈ ℝ) → (𝐺 = (𝑋𝐿𝑌) ↔ (𝐴 = 0 ∧ 𝑀 = (𝐶 / 𝐵))))
 
Theoremline2y 45682* Example for a vertical line 𝐺 passing through two different points in "standard form". (Contributed by AV, 3-Feb-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝐿 = (LineM𝐸)    &   𝐺 = {𝑝𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶}    &   𝑋 = {⟨1, 0⟩, ⟨2, 𝑀⟩}    &   𝑌 = {⟨1, 0⟩, ⟨2, 𝑁⟩}       (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀𝑁)) → (𝐺 = (𝑋𝐿𝑌) ↔ (𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0)))
 
Theoremitsclc0lem1 45683 Lemma for theorems about intersections of lines and circles in a real Euclidean space of dimension 2 . (Contributed by AV, 2-May-2023.)
(((𝑆 ∈ ℝ ∧ 𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ (𝑉 ∈ ℝ ∧ 0 ≤ 𝑉) ∧ (𝑊 ∈ ℝ ∧ 𝑊 ≠ 0)) → (((𝑆 · 𝑈) + (𝑇 · (√‘𝑉))) / 𝑊) ∈ ℝ)
 
Theoremitsclc0lem2 45684 Lemma for theorems about intersections of lines and circles in a real Euclidean space of dimension 2 . (Contributed by AV, 3-May-2023.)
(((𝑆 ∈ ℝ ∧ 𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ (𝑉 ∈ ℝ ∧ 0 ≤ 𝑉) ∧ (𝑊 ∈ ℝ ∧ 𝑊 ≠ 0)) → (((𝑆 · 𝑈) − (𝑇 · (√‘𝑉))) / 𝑊) ∈ ℝ)
 
Theoremitsclc0lem3 45685 Lemma for theorems about intersections of lines and circles in a real Euclidean space of dimension 2 . (Contributed by AV, 2-May-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))       (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝑅 ∈ ℝ) → 𝐷 ∈ ℝ)
 
Theoremitscnhlc0yqe 45686 Lemma for itsclc0 45698. Quadratic equation for the y-coordinate of the intersection points of a nonhorizontal line and a circle. (Contributed by AV, 6-Feb-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝑇 = -(2 · (𝐵 · 𝐶))    &   𝑈 = ((𝐶↑2) − ((𝐴↑2) · (𝑅↑2)))       ((((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝑅 ∈ ℝ+ ∧ (𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ)) → ((((𝑋↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑋) + (𝐵 · 𝑌)) = 𝐶) → ((𝑄 · (𝑌↑2)) + ((𝑇 · 𝑌) + 𝑈)) = 0))
 
Theoremitschlc0yqe 45687 Lemma for itsclc0 45698. Quadratic equation for the y-coordinate of the intersection points of a horizontal line and a circle. (Contributed by AV, 25-Feb-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝑇 = -(2 · (𝐵 · 𝐶))    &   𝑈 = ((𝐶↑2) − ((𝐴↑2) · (𝑅↑2)))       ((((𝐴 ∈ ℝ ∧ 𝐴 = 0) ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝑅 ∈ ℝ+ ∧ (𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ)) → ((((𝑋↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑋) + (𝐵 · 𝑌)) = 𝐶) → ((𝑄 · (𝑌↑2)) + ((𝑇 · 𝑌) + 𝑈)) = 0))
 
Theoremitsclc0yqe 45688 Lemma for itsclc0 45698. Quadratic equation for the y-coordinate of the intersection points of an arbitrary line and a circle. This theorem holds even for degenerate lines (𝐴 = 𝐵 = 0). (Contributed by AV, 25-Feb-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝑇 = -(2 · (𝐵 · 𝐶))    &   𝑈 = ((𝐶↑2) − ((𝐴↑2) · (𝑅↑2)))       (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝑅 ∈ ℝ+ ∧ (𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ)) → ((((𝑋↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑋) + (𝐵 · 𝑌)) = 𝐶) → ((𝑄 · (𝑌↑2)) + ((𝑇 · 𝑌) + 𝑈)) = 0))
 
Theoremitsclc0yqsollem1 45689 Lemma 1 for itsclc0yqsol 45691. (Contributed by AV, 6-Feb-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝑇 = -(2 · (𝐵 · 𝐶))    &   𝑈 = ((𝐶↑2) − ((𝐴↑2) · (𝑅↑2)))    &   𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))       (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ 𝑅 ∈ ℂ) → ((𝑇↑2) − (4 · (𝑄 · 𝑈))) = ((4 · (𝐴↑2)) · 𝐷))
 
Theoremitsclc0yqsollem2 45690 Lemma 2 for itsclc0yqsol 45691. (Contributed by AV, 6-Feb-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝑇 = -(2 · (𝐵 · 𝐶))    &   𝑈 = ((𝐶↑2) − ((𝐴↑2) · (𝑅↑2)))    &   𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))       (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝑅 ∈ ℝ ∧ 0 ≤ 𝐷) → (√‘((𝑇↑2) − (4 · (𝑄 · 𝑈)))) = ((2 · (abs‘𝐴)) · (√‘𝐷)))
 
Theoremitsclc0yqsol 45691 Lemma for itsclc0 45698. Solutions of the quadratic equations for the y-coordinate of the intersection points of a (nondegenerate) line and a circle. (Contributed by AV, 7-Feb-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))       ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 ≠ 0 ∨ 𝐵 ≠ 0)) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷) ∧ (𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ)) → ((((𝑋↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑋) + (𝐵 · 𝑌)) = 𝐶) → (𝑌 = (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄) ∨ 𝑌 = (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄))))
 
Theoremitscnhlc0xyqsol 45692 Lemma for itsclc0 45698. Solutions of the quadratic equations for the coordinates of the intersection points of a nonhorizontal line and a circle. (Contributed by AV, 8-Feb-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))       ((((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷) ∧ (𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ)) → ((((𝑋↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑋) + (𝐵 · 𝑌)) = 𝐶) → ((𝑋 = (((𝐴 · 𝐶) + (𝐵 · (√‘𝐷))) / 𝑄) ∧ 𝑌 = (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄)) ∨ (𝑋 = (((𝐴 · 𝐶) − (𝐵 · (√‘𝐷))) / 𝑄) ∧ 𝑌 = (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄)))))
 
Theoremitschlc0xyqsol1 45693 Lemma for itsclc0 45698. Solutions of the quadratic equations for the coordinates of the intersection points of a horizontal line and a circle. (Contributed by AV, 25-Feb-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))       ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 = 0 ∧ 𝐵 ≠ 0)) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷) ∧ (𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ)) → ((((𝑋↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑋) + (𝐵 · 𝑌)) = 𝐶) → (𝑌 = (𝐶 / 𝐵) ∧ (𝑋 = -((√‘𝐷) / 𝐵) ∨ 𝑋 = ((√‘𝐷) / 𝐵)))))
 
Theoremitschlc0xyqsol 45694 Lemma for itsclc0 45698. Solutions of the quadratic equations for the coordinates of the intersection points of a horizontal line and a circle. (Contributed by AV, 8-Feb-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))       ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 = 0 ∧ 𝐵 ≠ 0)) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷) ∧ (𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ)) → ((((𝑋↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑋) + (𝐵 · 𝑌)) = 𝐶) → ((𝑋 = (((𝐴 · 𝐶) + (𝐵 · (√‘𝐷))) / 𝑄) ∧ 𝑌 = (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄)) ∨ (𝑋 = (((𝐴 · 𝐶) − (𝐵 · (√‘𝐷))) / 𝑄) ∧ 𝑌 = (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄)))))
 
Theoremitsclc0xyqsol 45695 Lemma for itsclc0 45698. Solutions of the quadratic equations for the coordinates of the intersection points of a (nondegenerate) line and a circle. (Contributed by AV, 25-Feb-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))       ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 ≠ 0 ∨ 𝐵 ≠ 0)) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷) ∧ (𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ)) → ((((𝑋↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑋) + (𝐵 · 𝑌)) = 𝐶) → ((𝑋 = (((𝐴 · 𝐶) + (𝐵 · (√‘𝐷))) / 𝑄) ∧ 𝑌 = (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄)) ∨ (𝑋 = (((𝐴 · 𝐶) − (𝐵 · (√‘𝐷))) / 𝑄) ∧ 𝑌 = (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄)))))
 
Theoremitsclc0xyqsolr 45696 Lemma for itsclc0 45698. Solutions of the quadratic equations for the coordinates of the intersection points of a (nondegenerate) line and a circle. (Contributed by AV, 2-May-2023.) (Revised by AV, 14-May-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))       (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 ≠ 0 ∨ 𝐵 ≠ 0) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷)) → (((𝑋 = (((𝐴 · 𝐶) + (𝐵 · (√‘𝐷))) / 𝑄) ∧ 𝑌 = (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄)) ∨ (𝑋 = (((𝐴 · 𝐶) − (𝐵 · (√‘𝐷))) / 𝑄) ∧ 𝑌 = (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄))) → (((𝑋↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑋) + (𝐵 · 𝑌)) = 𝐶)))
 
Theoremitsclc0xyqsolb 45697 Lemma for itsclc0 45698. Solutions of the quadratic equations for the coordinates of the intersection points of a (nondegenerate) line and a circle. (Contributed by AV, 2-May-2023.) (Revised by AV, 14-May-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))       ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 ≠ 0 ∨ 𝐵 ≠ 0)) ∧ ((𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷) ∧ (𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ))) → ((((𝑋↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑋) + (𝐵 · 𝑌)) = 𝐶) ↔ ((𝑋 = (((𝐴 · 𝐶) + (𝐵 · (√‘𝐷))) / 𝑄) ∧ 𝑌 = (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄)) ∨ (𝑋 = (((𝐴 · 𝐶) − (𝐵 · (√‘𝐷))) / 𝑄) ∧ 𝑌 = (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄)))))
 
Theoremitsclc0 45698* The intersection points of a line 𝐿 and a circle around the origin. (Contributed by AV, 25-Feb-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝑆 = (Sphere‘𝐸)    &    0 = (𝐼 × {0})    &   𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))    &   𝐿 = {𝑝𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶}       (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 ≠ 0 ∨ 𝐵 ≠ 0) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷)) → ((𝑋 ∈ ( 0 𝑆𝑅) ∧ 𝑋𝐿) → (((𝑋‘1) = (((𝐴 · 𝐶) + (𝐵 · (√‘𝐷))) / 𝑄) ∧ (𝑋‘2) = (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄)) ∨ ((𝑋‘1) = (((𝐴 · 𝐶) − (𝐵 · (√‘𝐷))) / 𝑄) ∧ (𝑋‘2) = (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄)))))
 
Theoremitsclc0b 45699* The intersection points of a (nondegenerate) line through two points and a circle around the origin. (Contributed by AV, 2-May-2023.) (Revised by AV, 14-May-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝑆 = (Sphere‘𝐸)    &    0 = (𝐼 × {0})    &   𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))    &   𝐿 = {𝑝𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶}       (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 ≠ 0 ∨ 𝐵 ≠ 0) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷)) → ((𝑋 ∈ ( 0 𝑆𝑅) ∧ 𝑋𝐿) ↔ (𝑋𝑃 ∧ (((𝑋‘1) = (((𝐴 · 𝐶) + (𝐵 · (√‘𝐷))) / 𝑄) ∧ (𝑋‘2) = (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄)) ∨ ((𝑋‘1) = (((𝐴 · 𝐶) − (𝐵 · (√‘𝐷))) / 𝑄) ∧ (𝑋‘2) = (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄))))))
 
Theoremitsclinecirc0 45700 The intersection points of a line through two different points 𝑌 and 𝑍 and a circle around the origin, using the definition of a line in a two dimensional Euclidean space. (Contributed by AV, 25-Feb-2023.) (Proof shortened by AV, 16-May-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝑆 = (Sphere‘𝐸)    &    0 = (𝐼 × {0})    &   𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))    &   𝐿 = (LineM𝐸)    &   𝐴 = ((𝑌‘2) − (𝑍‘2))    &   𝐵 = ((𝑍‘1) − (𝑌‘1))    &   𝐶 = (((𝑌‘2) · (𝑍‘1)) − ((𝑌‘1) · (𝑍‘2)))       (((𝑌𝑃𝑍𝑃𝑌𝑍) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷)) → ((𝑋 ∈ ( 0 𝑆𝑅) ∧ 𝑋 ∈ (𝑌𝐿𝑍)) → (((𝑋‘1) = (((𝐴 · 𝐶) + (𝐵 · (√‘𝐷))) / 𝑄) ∧ (𝑋‘2) = (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄)) ∨ ((𝑋‘1) = (((𝐴 · 𝐶) − (𝐵 · (√‘𝐷))) / 𝑄) ∧ (𝑋‘2) = (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄)))))
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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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454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46009
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