P. 491 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 49001-49100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ackvalsucsucval 49001	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 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((Ack‘(𝑀 + 1))‘(𝑁 + 1)) = ((Ack‘𝑀)‘((Ack‘(𝑀 + 1))‘𝑁)))
Theorem	ackval0012 49002	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⟩
Theorem	ackval1012 49003	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⟩
Theorem	ackval2012 49004	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⟩
Theorem	ackval3012 49005	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⟩
Theorem	ackval40 49006	The Ackermann function at (4,0). (Contributed by AV, 9-May-2024.)
⊢ ((Ack‘4)‘0) = ;13
Theorem	ackval41a 49007	The Ackermann function at (4,1). (Contributed by AV, 9-May-2024.)
⊢ ((Ack‘4)‘1) = ((2↑;16) − 3)
Theorem	ackval41 49008	The Ackermann function at (4,1). (Contributed by AV, 9-May-2024.)
⊢ ((Ack‘4)‘1) = ;;;;65533
Theorem	ackval42 49009	The Ackermann function at (4,2). (Contributed by AV, 9-May-2024.)
⊢ ((Ack‘4)‘2) = ((2↑;;;;65536) − 3)
Theorem	ackval42a 49010	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)
Theorem	ackval50 49011	The Ackermann function at (5,0). (Contributed by AV, 9-May-2024.)
⊢ ((Ack‘5)‘0) = ;;;;65533
21.48.22  Elementary geometry (extension)
21.48.22.1  Auxiliary theorems
Theorem	fv1prop 49012	The function value of unordered pair of ordered pairs with first components 1 and 2 at 1. (Contributed by AV, 4-Feb-2023.)
⊢ (𝐴 ∈ 𝑉 → ({⟨1, 𝐴⟩, ⟨2, 𝐵⟩}‘1) = 𝐴)
Theorem	fv2prop 49013	The function value of unordered pair of ordered pairs with first components 1 and 2 at 1. (Contributed by AV, 4-Feb-2023.)
⊢ (𝐵 ∈ 𝑉 → ({⟨1, 𝐴⟩, ⟨2, 𝐵⟩}‘2) = 𝐵)
Theorem	submuladdmuld 49014	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.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐷 ∈ ℂ)    ⇒   ⊢ (𝜑 → (((𝐴 − 𝐵) · 𝐶) + (𝐵 · 𝐷)) = ((𝐴 · 𝐶) + (𝐵 · (𝐷 − 𝐶))))
Theorem	affinecomb1 49015*	Combination of two real affine combinations, one class variable resolved. (Contributed by AV, 22-Jan-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    &   ⊢ (𝜑 → 𝐸 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ ℝ)    &   ⊢ (𝜑 → 𝐺 ∈ ℝ)    &   ⊢ 𝑆 = ((𝐺 − 𝐹) / (𝐶 − 𝐵))    ⇒   ⊢ (𝜑 → (∃𝑡 ∈ ℝ (𝐴 = (((1 − 𝑡) · 𝐵) + (𝑡 · 𝐶)) ∧ 𝐸 = (((1 − 𝑡) · 𝐹) + (𝑡 · 𝐺))) ↔ 𝐸 = ((𝑆 · (𝐴 − 𝐵)) + 𝐹)))
Theorem	affinecomb2 49016*	Combination of two real affine combinations, presented without fraction. (Contributed by AV, 22-Jan-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    &   ⊢ (𝜑 → 𝐸 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ ℝ)    &   ⊢ (𝜑 → 𝐺 ∈ ℝ)    ⇒   ⊢ (𝜑 → (∃𝑡 ∈ ℝ (𝐴 = (((1 − 𝑡) · 𝐵) + (𝑡 · 𝐶)) ∧ 𝐸 = (((1 − 𝑡) · 𝐹) + (𝑡 · 𝐺))) ↔ ((𝐶 − 𝐵) · 𝐸) = (((𝐺 − 𝐹) · 𝐴) + ((𝐹 · 𝐶) − (𝐵 · 𝐺)))))
Theorem	affineid 49017	Identity of an affine combination. (Contributed by AV, 2-Feb-2023.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝑇 ∈ ℂ)    ⇒   ⊢ (𝜑 → (((1 − 𝑇) · 𝐴) + (𝑇 · 𝐴)) = 𝐴)
Theorem	1subrec1sub 49018	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)))
Theorem	resum2sqcl 49019	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)
Theorem	resum2sqgt0 49020	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)
Theorem	resum2sqrp 49021	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+)
Theorem	resum2sqorgt0 49022	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)
Theorem	reorelicc 49023	Membership in and outside of a closed real interval. (Contributed by AV, 15-Feb-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 < 𝐴 ∨ 𝐶 ∈ (𝐴[,]𝐵) ∨ 𝐵 < 𝐶))
21.48.22.2  Real euclidean space of dimension 2
Theorem	rrx2pxel 49024	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)    ⇒   ⊢ (𝑋 ∈ 𝑃 → (𝑋‘1) ∈ ℝ)
Theorem	rrx2pyel 49025	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)    ⇒   ⊢ (𝑋 ∈ 𝑃 → (𝑋‘2) ∈ ℝ)
Theorem	prelrrx2 49026	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)
Theorem	prelrrx2b 49027	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)) → ((𝑍 ∈ 𝑃 ∧ (((𝑍‘1) = 𝐴 ∧ (𝑍‘2) = 𝐵) ∨ ((𝑍‘1) = 𝑋 ∧ (𝑍‘2) = 𝑌))) ↔ 𝑍 ∈ {{⟨1, 𝐴⟩, ⟨2, 𝐵⟩}, {⟨1, 𝑋⟩, ⟨2, 𝑌⟩}}))
Theorem	rrx2pnecoorneor 49028	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)    ⇒   ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → ((𝑋‘1) ≠ (𝑌‘1) ∨ (𝑋‘2) ≠ (𝑌‘2)))
Theorem	rrx2pnedifcoorneor 49029	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))    ⇒   ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (𝐴 ≠ 0 ∨ 𝐵 ≠ 0))
Theorem	rrx2pnedifcoorneorr 49030	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))    ⇒   ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (𝐴 ≠ 0 ∨ 𝐵 ≠ 0))
Theorem	rrx2xpref1o 49031*	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-onto→𝑅
Theorem	rrx2xpreen 49032	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)
Theorem	rrx2plord 49033*	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 = 𝑥 ∧ 𝑏 ≤ 𝑦)). (Contributed by AV, 12-Mar-2023.)
⊢ 𝑂 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))}    ⇒   ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (𝑋𝑂𝑌 ↔ ((𝑋‘1) < (𝑌‘1) ∨ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) < (𝑌‘2)))))
Theorem	rrx2plord1 49034*	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⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))}    ⇒   ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ∧ (𝑋‘1) < (𝑌‘1)) → 𝑋𝑂𝑌)
Theorem	rrx2plord2 49035*	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⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))}    &   ⊢ 𝑅 = (ℝ ↑m {1, 2})    ⇒   ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ∧ (𝑋‘1) = (𝑌‘1)) → (𝑋𝑂𝑌 ↔ (𝑋‘2) < (𝑌‘2)))
Theorem	rrx2plordisom 49036*	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⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) ∧ ((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)) ∧ ((1st ‘𝑥) < (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) < (2nd ‘𝑦))))}    ⇒   ⊢ 𝐹 Isom 𝑇, 𝑂 ((ℝ × ℝ), 𝑅)
Theorem	rrx2plordso 49037*	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⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))}    &   ⊢ 𝑅 = (ℝ ↑m {1, 2})    ⇒   ⊢ 𝑂 Or 𝑅
Theorem	ehl2eudisval0 49038	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})    ⇒   ⊢ (𝐹 ∈ 𝑋 → (𝐹𝐷 0 ) = (√‘(((𝐹‘1)↑2) + ((𝐹‘2)↑2))))
Theorem	ehl2eudis0lt 49039	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})    ⇒   ⊢ ((𝐹 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+) → ((𝐹𝐷 0 ) < 𝑅 ↔ (((𝐹‘1)↑2) + ((𝐹‘2)↑2)) < (𝑅↑2)))
21.48.22.3  Spheres and lines in real Euclidean spaces
Syntax	cline 49040	Declare the syntax for lines in generalized real Euclidean spaces.
class LineM
Syntax	csph 49041	Declare the syntax for spheres in generalized real Euclidean spaces.
class Sphere
Definition	df-line 49042*	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)( ·𝑠 ‘𝑤)𝑥)(+g‘𝑤)(𝑡( ·𝑠 ‘𝑤)𝑦))}))
Definition	df-sph 49043*	Definition of spheres for given centers and radii in a metric space (or more generally, in a distance space, see distspace 24264, 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}))
Theorem	lines 49044*	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 = (LineM‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ − = (-g‘𝑆)    &   ⊢ 1 = (1r‘𝑆)    ⇒   ⊢ (𝑊 ∈ 𝑉 → 𝐿 = (𝑥 ∈ 𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))}))
Theorem	line 49045*	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 = (LineM‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ − = (-g‘𝑆)    &   ⊢ 1 = (1r‘𝑆)    ⇒   ⊢ ((𝑊 ∈ 𝑉 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ≠ 𝑌)) → (𝑋𝐿𝑌) = {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑋) + (𝑡 · 𝑌))})
Theorem	rrxlines 49046*	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 = (LineM‘𝐸)    &   ⊢ · = ( ·𝑠 ‘𝐸)    &   ⊢ + = (+g‘𝐸)    ⇒   ⊢ (𝐼 ∈ Fin → 𝐿 = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))}))
Theorem	rrxline 49047*	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 = (LineM‘𝐸)    &   ⊢ · = ( ·𝑠 ‘𝐸)    &   ⊢ + = (+g‘𝐸)    ⇒   ⊢ ((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) → (𝑋𝐿𝑌) = {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡) · 𝑋) + (𝑡 · 𝑌))})
Theorem	rrxlinesc 49048*	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 = (LineM‘𝐸)    ⇒   ⊢ (𝐼 ∈ Fin → 𝐿 = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ ∀𝑖 ∈ 𝐼 (𝑝‘𝑖) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖)))}))
Theorem	rrxlinec 49049*	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 49048. (Contributed by AV, 13-Feb-2023.)
⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝐿 = (LineM‘𝐸)    ⇒   ⊢ ((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) → (𝑋𝐿𝑌) = {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ ∀𝑖 ∈ 𝐼 (𝑝‘𝑖) = (((1 − 𝑡) · (𝑋‘𝑖)) + (𝑡 · (𝑌‘𝑖)))})
Theorem	eenglngeehlnmlem1 49050*	Lemma 1 for eenglngeehlnm 49052. (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)(𝑝‘𝑖) = (((1 − 𝑘) · (𝑥‘𝑖)) + (𝑘 · (𝑦‘𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖)))) → ∃𝑡 ∈ ℝ ∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖)))))
Theorem	eenglngeehlnmlem2 49051*	Lemma 2 for eenglngeehlnm 49052. (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)(𝑝‘𝑖) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖))) → (∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − 𝑘) · (𝑥‘𝑖)) + (𝑘 · (𝑦‘𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))))))
Theorem	eenglngeehlnm 49052	The line definition in the Tarski structure for the Euclidean geometry (see elntg 29061) corresponds to the definition of lines passing through two different points in a left module (see rrxlines 49046). (Contributed by AV, 16-Feb-2023.)
⊢ (𝑁 ∈ ℕ → (LineG‘(EEG‘𝑁)) = (LineM‘(𝔼hil‘𝑁)))
Theorem	rrx2line 49053*	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 = (LineM‘𝐸)    ⇒   ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (𝑋𝐿𝑌) = {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ ((𝑝‘1) = (((1 − 𝑡) · (𝑋‘1)) + (𝑡 · (𝑌‘1))) ∧ (𝑝‘2) = (((1 − 𝑡) · (𝑋‘2)) + (𝑡 · (𝑌‘2))))})
Theorem	rrx2vlinest 49054*	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 = (LineM‘𝐸)    ⇒   ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) ≠ (𝑌‘2))) → (𝑋𝐿𝑌) = {𝑝 ∈ 𝑃 ∣ (𝑝‘1) = (𝑋‘1)})
Theorem	rrx2linest 49055*	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 = (LineM‘𝐸)    &   ⊢ 𝐴 = ((𝑌‘1) − (𝑋‘1))    &   ⊢ 𝐵 = ((𝑌‘2) − (𝑋‘2))    &   ⊢ 𝐶 = (((𝑋‘2) · (𝑌‘1)) − ((𝑋‘1) · (𝑌‘2)))    ⇒   ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (𝑋𝐿𝑌) = {𝑝 ∈ 𝑃 ∣ (𝐴 · (𝑝‘2)) = ((𝐵 · (𝑝‘1)) + 𝐶)})
Theorem	rrx2linesl 49056*	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 = (LineM‘𝐸)    &   ⊢ 𝑆 = (((𝑌‘2) − (𝑋‘2)) / ((𝑌‘1) − (𝑋‘1)))    ⇒   ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) → (𝑋𝐿𝑌) = {𝑝 ∈ 𝑃 ∣ (𝑝‘2) = ((𝑆 · ((𝑝‘1) − (𝑋‘1))) + (𝑋‘2))})
Theorem	rrx2linest2 49057*	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 = (LineM‘𝐸)    &   ⊢ 𝐴 = ((𝑋‘2) − (𝑌‘2))    &   ⊢ 𝐵 = ((𝑌‘1) − (𝑋‘1))    &   ⊢ 𝐶 = (((𝑋‘2) · (𝑌‘1)) − ((𝑋‘1) · (𝑌‘2)))    ⇒   ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (𝑋𝐿𝑌) = {𝑝 ∈ 𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶})
Theorem	elrrx2linest2 49058	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 = (LineM‘𝐸)    &   ⊢ 𝐴 = ((𝑋‘2) − (𝑌‘2))    &   ⊢ 𝐵 = ((𝑌‘1) − (𝑋‘1))    &   ⊢ 𝐶 = (((𝑋‘2) · (𝑌‘1)) − ((𝑋‘1) · (𝑌‘2)))    ⇒   ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (𝐺 ∈ (𝑋𝐿𝑌) ↔ (𝐺 ∈ 𝑃 ∧ ((𝐴 · (𝐺‘1)) + (𝐵 · (𝐺‘2))) = 𝐶)))
Theorem	spheres 49059*	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)    ⇒   ⊢ (𝑊 ∈ 𝑉 → 𝑆 = (𝑥 ∈ 𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}))
Theorem	sphere 49060*	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)    ⇒   ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑅 ∈ (0[,]+∞)) → (𝑋𝑆𝑅) = {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑋) = 𝑅})
Theorem	rrxsphere 49061*	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 ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) → (𝑀𝑆𝑅) = {𝑝 ∈ 𝑃 ∣ (𝑝𝐷𝑀) = 𝑅})
Theorem	2sphere 49062*	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 = {𝑝 ∈ 𝑃 ∣ ((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2)) = (𝑅↑2)}    ⇒   ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) → (𝑀𝑆𝑅) = 𝐶)
Theorem	2sphere0 49063*	The sphere around the origin 0 (see rrx0 25357) 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 = {𝑝 ∈ 𝑃 ∣ (((𝑝‘1)↑2) + ((𝑝‘2)↑2)) = (𝑅↑2)}    ⇒   ⊢ (𝑅 ∈ (0[,)+∞) → ( 0 𝑆𝑅) = 𝐶)
Theorem	line2ylem 49064*	Lemma for line2y 49068. This proof is based on counterexamples for the following cases: 1. 𝐶 ≠ 0: p = (0,0) (LHS of biconditional is false, RHS is true); 2. 𝐶 = 0 ∧ 𝐵 ≠ 0: p = (1,-A/B) (LHS of biconditional is true, RHS is false); 3. 𝐴 = 𝐵 = 𝐶 = 0: p = (1,1) (LHS of biconditional is true, RHS is false). (Contributed by AV, 4-Feb-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    ⇒   ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (∀𝑝 ∈ 𝑃 (((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶 ↔ (𝑝‘1) = 0) → (𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0)))
Theorem	line2 49065*	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 = (LineM‘𝐸)    &   ⊢ 𝐺 = {𝑝 ∈ 𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶}    &   ⊢ 𝑋 = {⟨1, 0⟩, ⟨2, (𝐶 / 𝐵)⟩}    &   ⊢ 𝑌 = {⟨1, 1⟩, ⟨2, ((𝐶 − 𝐴) / 𝐵)⟩}    ⇒   ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) ∧ 𝐶 ∈ ℝ) → 𝐺 = (𝑋𝐿𝑌))
Theorem	line2xlem 49066*	Lemma for line2x 49067. This proof is based on counterexamples for the following cases: 1. 𝑀 ≠ (𝐶 / 𝐵): p = (0,C/B) (LHS of biconditional is true, RHS is false); 2. 𝐴 ≠ 0 ∧ 𝑀 = (𝐶 / 𝐵): p = (1,C/B) (LHS of biconditional is false, RHS is true). (Contributed by AV, 4-Feb-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝐿 = (LineM‘𝐸)    &   ⊢ 𝐺 = {𝑝 ∈ 𝑃 ∣ ((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) → (∀𝑝 ∈ 𝑃 (((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶 ↔ (𝑝‘2) = 𝑀) → (𝐴 = 0 ∧ 𝑀 = (𝐶 / 𝐵))))
Theorem	line2x 49067*	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 = (LineM‘𝐸)    &   ⊢ 𝐺 = {𝑝 ∈ 𝑃 ∣ ((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))))
Theorem	line2y 49068*	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 = (LineM‘𝐸)    &   ⊢ 𝐺 = {𝑝 ∈ 𝑃 ∣ ((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 ∧ 𝑀 ≠ 𝑁)) → (𝐺 = (𝑋𝐿𝑌) ↔ (𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0)))
Theorem	itsclc0lem1 49069	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)
Theorem	itsclc0lem2 49070	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)
Theorem	itsclc0lem3 49071	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)
Theorem	itscnhlc0yqe 49072	Lemma for itsclc0 49084. 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))
Theorem	itschlc0yqe 49073	Lemma for itsclc0 49084. 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))
Theorem	itsclc0yqe 49074	Lemma for itsclc0 49084. 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))
Theorem	itsclc0yqsollem1 49075	Lemma 1 for itsclc0yqsol 49077. (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))
Theorem	itsclc0yqsollem2 49076	Lemma 2 for itsclc0yqsol 49077. (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)))
Theorem	itsclc0yqsol 49077	Lemma for itsclc0 49084. 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))))
Theorem	itscnhlc0xyqsol 49078	Lemma for itsclc0 49084. 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)))))
Theorem	itschlc0xyqsol1 49079	Lemma for itsclc0 49084. 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)))))
Theorem	itschlc0xyqsol 49080	Lemma for itsclc0 49084. 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)))))
Theorem	itsclc0xyqsol 49081	Lemma for itsclc0 49084. 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)))))
Theorem	itsclc0xyqsolr 49082	Lemma for itsclc0 49084. 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)))
Theorem	itsclc0xyqsolb 49083	Lemma for itsclc0 49084. 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)))))
Theorem	itsclc0 49084*	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 = {𝑝 ∈ 𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶}    ⇒   ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 ≠ 0 ∨ 𝐵 ≠ 0) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷)) → ((𝑋 ∈ ( 0 𝑆𝑅) ∧ 𝑋 ∈ 𝐿) → (((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)))))
Theorem	itsclc0b 49085*	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 = {𝑝 ∈ 𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶}    ⇒   ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 ≠ 0 ∨ 𝐵 ≠ 0) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷)) → ((𝑋 ∈ ( 0 𝑆𝑅) ∧ 𝑋 ∈ 𝐿) ↔ (𝑋 ∈ 𝑃 ∧ (((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))))))
Theorem	itsclinecirc0 49086	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 = (LineM‘𝐸)    &   ⊢ 𝐴 = ((𝑌‘2) − (𝑍‘2))    &   ⊢ 𝐵 = ((𝑍‘1) − (𝑌‘1))    &   ⊢ 𝐶 = (((𝑌‘2) · (𝑍‘1)) − ((𝑌‘1) · (𝑍‘2)))    ⇒   ⊢ (((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) ∧ (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)))))
Theorem	itsclinecirc0b 49087	The intersection points of a line through two different points and a circle around the origin, using the definition of a line in a two dimensional Euclidean space. (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 = (LineM‘𝐸)    &   ⊢ 𝐴 = ((𝑋‘2) − (𝑌‘2))    &   ⊢ 𝐵 = ((𝑌‘1) − (𝑋‘1))    &   ⊢ 𝐶 = (((𝑋‘2) · (𝑌‘1)) − ((𝑋‘1) · (𝑌‘2)))    ⇒   ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷)) → ((𝑍 ∈ ( 0 𝑆𝑅) ∧ 𝑍 ∈ (𝑋𝐿𝑌)) ↔ (𝑍 ∈ 𝑃 ∧ (((𝑍‘1) = (((𝐴 · 𝐶) + (𝐵 · (√‘𝐷))) / 𝑄) ∧ (𝑍‘2) = (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄)) ∨ ((𝑍‘1) = (((𝐴 · 𝐶) − (𝐵 · (√‘𝐷))) / 𝑄) ∧ (𝑍‘2) = (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄))))))
Theorem	itsclinecirc0in 49088	The intersection points of a line through two different points and a circle around the origin, using the definition of a line in a two dimensional Euclidean space, expressed as intersection. (Contributed by AV, 7-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 = (LineM‘𝐸)    &   ⊢ 𝐴 = ((𝑋‘2) − (𝑌‘2))    &   ⊢ 𝐵 = ((𝑌‘1) − (𝑋‘1))    &   ⊢ 𝐶 = (((𝑋‘2) · (𝑌‘1)) − ((𝑋‘1) · (𝑌‘2)))    ⇒   ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷)) → (( 0 𝑆𝑅) ∩ (𝑋𝐿𝑌)) = {{⟨1, (((𝐴 · 𝐶) + (𝐵 · (√‘𝐷))) / 𝑄)⟩, ⟨2, (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄)⟩}, {⟨1, (((𝐴 · 𝐶) − (𝐵 · (√‘𝐷))) / 𝑄)⟩, ⟨2, (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄)⟩}})
Theorem	itsclquadb 49089*	Quadratic equation for the y-coordinate of the intersection points of a line and a circle. (Contributed by AV, 22-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+ ∧ 𝑌 ∈ ℝ) → (∃𝑥 ∈ ℝ (((𝑥↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑥) + (𝐵 · 𝑌)) = 𝐶) ↔ ((𝑄 · (𝑌↑2)) + ((𝑇 · 𝑌) + 𝑈)) = 0))
Theorem	itsclquadeu 49090*	Quadratic equation for the y-coordinate of the intersection points of a line and a circle. (Contributed by AV, 23-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+ ∧ 𝑌 ∈ ℝ) → (∃!𝑥 ∈ ℝ (((𝑥↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑥) + (𝐵 · 𝑌)) = 𝐶) ↔ ((𝑄 · (𝑌↑2)) + ((𝑇 · 𝑌) + 𝑈)) = 0))
Theorem	2itscplem1 49091	Lemma 1 for 2itscp 49094. (Contributed by AV, 4-Mar-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ∈ ℝ)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    &   ⊢ 𝐷 = (𝑋 − 𝐴)    &   ⊢ 𝐸 = (𝐵 − 𝑌)    ⇒   ⊢ (𝜑 → ((((𝐸↑2) · (𝐵↑2)) + ((𝐷↑2) · (𝐴↑2))) − (2 · ((𝐷 · 𝐴) · (𝐸 · 𝐵)))) = (((𝐷 · 𝐴) − (𝐸 · 𝐵))↑2))
Theorem	2itscplem2 49092	Lemma 2 for 2itscp 49094. (Contributed by AV, 4-Mar-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ∈ ℝ)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    &   ⊢ 𝐷 = (𝑋 − 𝐴)    &   ⊢ 𝐸 = (𝐵 − 𝑌)    &   ⊢ 𝐶 = ((𝐷 · 𝐵) + (𝐸 · 𝐴))    ⇒   ⊢ (𝜑 → (𝐶↑2) = ((((𝐷↑2) · (𝐵↑2)) + (2 · ((𝐷 · 𝐴) · (𝐸 · 𝐵)))) + ((𝐸↑2) · (𝐴↑2))))
Theorem	2itscplem3 49093	Lemma D for 2itscp 49094. (Contributed by AV, 4-Mar-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ∈ ℝ)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    &   ⊢ 𝐷 = (𝑋 − 𝐴)    &   ⊢ 𝐸 = (𝐵 − 𝑌)    &   ⊢ 𝐶 = ((𝐷 · 𝐵) + (𝐸 · 𝐴))    &   ⊢ (𝜑 → 𝑅 ∈ ℝ)    &   ⊢ 𝑄 = ((𝐸↑2) + (𝐷↑2))    &   ⊢ 𝑆 = (((𝑅↑2) · 𝑄) − (𝐶↑2))    ⇒   ⊢ (𝜑 → 𝑆 = ((((𝐸↑2) · ((𝑅↑2) − (𝐴↑2))) + ((𝐷↑2) · ((𝑅↑2) − (𝐵↑2)))) − (2 · ((𝐷 · 𝐴) · (𝐸 · 𝐵)))))
Theorem	2itscp 49094	A condition for a quadratic equation with real coefficients (for the intersection points of a line with a circle) to have (exactly) two different real solutions. (Contributed by AV, 5-Mar-2023.) (Revised by AV, 16-May-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ∈ ℝ)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    &   ⊢ 𝐷 = (𝑋 − 𝐴)    &   ⊢ 𝐸 = (𝐵 − 𝑌)    &   ⊢ 𝐶 = ((𝐷 · 𝐵) + (𝐸 · 𝐴))    &   ⊢ (𝜑 → 𝑅 ∈ ℝ)    &   ⊢ (𝜑 → ((𝐴↑2) + (𝐵↑2)) < (𝑅↑2))    &   ⊢ (𝜑 → (𝐵 ≠ 𝑌 ∨ 𝐴 ≠ 𝑋))    &   ⊢ 𝑄 = ((𝐸↑2) + (𝐷↑2))    &   ⊢ 𝑆 = (((𝑅↑2) · 𝑄) − (𝐶↑2))    ⇒   ⊢ (𝜑 → 0 < 𝑆)
Theorem	itscnhlinecirc02plem1 49095	Lemma 1 for itscnhlinecirc02p 49098. (Contributed by AV, 6-Mar-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ∈ ℝ)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    &   ⊢ 𝐷 = (𝑋 − 𝐴)    &   ⊢ 𝐸 = (𝐵 − 𝑌)    &   ⊢ 𝐶 = ((𝐷 · 𝐵) + (𝐸 · 𝐴))    &   ⊢ (𝜑 → 𝑅 ∈ ℝ)    &   ⊢ (𝜑 → ((𝐴↑2) + (𝐵↑2)) < (𝑅↑2))    &   ⊢ (𝜑 → 𝐵 ≠ 𝑌)    ⇒   ⊢ (𝜑 → 0 < ((-(2 · (𝐷 · 𝐶))↑2) − (4 · (((𝐸↑2) + (𝐷↑2)) · ((𝐶↑2) − ((𝐸↑2) · (𝑅↑2)))))))
Theorem	itscnhlinecirc02plem2 49096	Lemma 2 for itscnhlinecirc02p 49098. (Contributed by AV, 10-Mar-2023.)
⊢ 𝐷 = (𝑋 − 𝐴)    &   ⊢ 𝐸 = (𝐵 − 𝑌)    &   ⊢ 𝐶 = ((𝐵 · 𝑋) − (𝐴 · 𝑌))    ⇒   ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) ∧ 𝐵 ≠ 𝑌) ∧ (𝑅 ∈ ℝ ∧ ((𝐴↑2) + (𝐵↑2)) < (𝑅↑2))) → 0 < ((-(2 · (𝐷 · 𝐶))↑2) − (4 · (((𝐸↑2) + (𝐷↑2)) · ((𝐶↑2) − ((𝐸↑2) · (𝑅↑2)))))))
Theorem	itscnhlinecirc02plem3 49097	Lemma 3 for itscnhlinecirc02p 49098. (Contributed by AV, 10-Mar-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝑆 = (Sphere‘𝐸)    &   ⊢ 0 = (𝐼 × {0})    &   ⊢ 𝐿 = (LineM‘𝐸)    &   ⊢ 𝐷 = (dist‘𝐸)    ⇒   ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘2) ≠ (𝑌‘2)) ∧ (𝑅 ∈ ℝ+ ∧ (𝑋𝐷 0 ) < 𝑅)) → 0 < ((-(2 · (((𝑌‘1) − (𝑋‘1)) · (((𝑋‘2) · (𝑌‘1)) − ((𝑋‘1) · (𝑌‘2)))))↑2) − (4 · (((((𝑋‘2) − (𝑌‘2))↑2) + (((𝑌‘1) − (𝑋‘1))↑2)) · (((((𝑋‘2) · (𝑌‘1)) − ((𝑋‘1) · (𝑌‘2)))↑2) − ((((𝑋‘2) − (𝑌‘2))↑2) · (𝑅↑2)))))))
Theorem	itscnhlinecirc02p 49098*	Intersection of a nonhorizontal line with a circle: A nonhorizontal line passing through a point within a circle around the origin intersects the circle at exactly two different points. (Contributed by AV, 28-Jan-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝑆 = (Sphere‘𝐸)    &   ⊢ 0 = (𝐼 × {0})    &   ⊢ 𝐿 = (LineM‘𝐸)    &   ⊢ 𝐷 = (dist‘𝐸)    &   ⊢ 𝑍 = {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}    ⇒   ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘2) ≠ (𝑌‘2)) ∧ (𝑅 ∈ ℝ+ ∧ (𝑋𝐷 0 ) < 𝑅)) → ∃!𝑠 ∈ 𝒫 ℝ((♯‘𝑠) = 2 ∧ ∀𝑦 ∈ 𝑠 ∃!𝑥 ∈ ℝ (𝑍 ∈ ( 0 𝑆𝑅) ∧ 𝑍 ∈ (𝑋𝐿𝑌))))
Theorem	inlinecirc02plem 49099*	Lemma for inlinecirc02p 49100. (Contributed by AV, 7-May-2023.) (Revised by AV, 15-May-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝑆 = (Sphere‘𝐸)    &   ⊢ 0 = (𝐼 × {0})    &   ⊢ 𝐿 = (LineM‘𝐸)    &   ⊢ 𝑄 = ((𝐴↑2) + (𝐵↑2))    &   ⊢ 𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))    &   ⊢ 𝐴 = ((𝑋‘2) − (𝑌‘2))    &   ⊢ 𝐵 = ((𝑌‘1) − (𝑋‘1))    &   ⊢ 𝐶 = (((𝑋‘2) · (𝑌‘1)) − ((𝑋‘1) · (𝑌‘2)))    ⇒   ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) ∧ (𝑅 ∈ ℝ+ ∧ 0 < 𝐷)) → ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ((( 0 𝑆𝑅) ∩ (𝑋𝐿𝑌)) = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏))
Theorem	inlinecirc02p 49100	Intersection of a line with a circle: A line passing through a point within a circle around the origin intersects the circle at exactly two different points. (Contributed by AV, 9-May-2023.) (Revised by AV, 16-May-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝑆 = (Sphere‘𝐸)    &   ⊢ 0 = (𝐼 × {0})    &   ⊢ 𝐿 = (LineM‘𝐸)    &   ⊢ 𝐷 = (dist‘𝐸)    ⇒   ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) ∧ (𝑅 ∈ ℝ+ ∧ (𝑋𝐷 0 ) < 𝑅)) → (( 0 𝑆𝑅) ∩ (𝑋𝐿𝑌)) ∈ (Pairsproper‘𝑃))