P. 487 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 48601-48700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ackval0 48601	The Ackermann function at 0. (Contributed by AV, 2-May-2024.)
⊢ (Ack‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 1))
Theorem	ackval1 48602	The Ackermann function at 1. (Contributed by AV, 4-May-2024.)
⊢ (Ack‘1) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2))
Theorem	ackval2 48603	The Ackermann function at 2. (Contributed by AV, 4-May-2024.)
⊢ (Ack‘2) = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 3))
Theorem	ackval3 48604	The Ackermann function at 3. (Contributed by AV, 7-May-2024.)
⊢ (Ack‘3) = (𝑛 ∈ ℕ0 ↦ ((2↑(𝑛 + 3)) − 3))
Theorem	ackendofnn0 48605	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)
Theorem	ackfnnn0 48606	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)
Theorem	ackval0val 48607	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))
Theorem	ackvalsuc0val 48608	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))
Theorem	ackvalsucsucval 48609	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 48610	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 48611	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 48612	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 48613	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 48614	The Ackermann function at (4,0). (Contributed by AV, 9-May-2024.)
⊢ ((Ack‘4)‘0) = ;13
Theorem	ackval41a 48615	The Ackermann function at (4,1). (Contributed by AV, 9-May-2024.)
⊢ ((Ack‘4)‘1) = ((2↑;16) − 3)
Theorem	ackval41 48616	The Ackermann function at (4,1). (Contributed by AV, 9-May-2024.)
⊢ ((Ack‘4)‘1) = ;;;;65533
Theorem	ackval42 48617	The Ackermann function at (4,2). (Contributed by AV, 9-May-2024.)
⊢ ((Ack‘4)‘2) = ((2↑;;;;65536) − 3)
Theorem	ackval42a 48618	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 48619	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 48620	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 48621	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 48622	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 48623*	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 48624*	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 48625	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 48626	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 48627	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 48628	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 48629	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 48630	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 48631	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 48632	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 48633	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 48634	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 48635	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 48636	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 48637	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 48638	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 48639*	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 48640	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 48641*	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 48642*	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 48643*	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 48644*	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 48645*	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 48646	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 48647	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 48648	Declare the syntax for lines in generalized real Euclidean spaces.
class LineM
Syntax	csph 48649	Declare the syntax for spheres in generalized real Euclidean spaces.
class Sphere
Definition	df-line 48650*	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 48651*	Definition of spheres for given centers and radii in a metric space (or more generally, in a distance space, see distspace 24326, 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 48652*	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 48653*	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 48654*	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 48655*	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 48656*	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 48657*	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 48656. (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 48658*	Lemma 1 for eenglngeehlnm 48660. (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 48659*	Lemma 2 for eenglngeehlnm 48660. (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 48660	The line definition in the Tarski structure for the Euclidean geometry (see elntg 28999) corresponds to the definition of lines passing through two different points in a left module (see rrxlines 48654). (Contributed by AV, 16-Feb-2023.)
⊢ (𝑁 ∈ ℕ → (LineG‘(EEG‘𝑁)) = (LineM‘(𝔼hil‘𝑁)))
Theorem	rrx2line 48661*	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 48662*	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 48663*	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 48664*	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 48665*	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 48666	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 48667*	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 48668*	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 48669*	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 48670*	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 48671*	The sphere around the origin 0 (see rrx0 25431) 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 48672*	Lemma for line2y 48676. 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 48673*	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 48674*	Lemma for line2x 48675. 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 48675*	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 48676*	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 48677	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 48678	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 48679	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 48680	Lemma for itsclc0 48692. 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 48681	Lemma for itsclc0 48692. 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 48682	Lemma for itsclc0 48692. 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 48683	Lemma 1 for itsclc0yqsol 48685. (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 48684	Lemma 2 for itsclc0yqsol 48685. (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 48685	Lemma for itsclc0 48692. 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 48686	Lemma for itsclc0 48692. 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 48687	Lemma for itsclc0 48692. 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 48688	Lemma for itsclc0 48692. 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 48689	Lemma for itsclc0 48692. 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 48690	Lemma for itsclc0 48692. 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 48691	Lemma for itsclc0 48692. 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 48692*	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 48693*	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 48694	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 48695	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 48696	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 48697*	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 48698*	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 48699	Lemma 1 for 2itscp 48702. (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 48700	Lemma 2 for 2itscp 48702. (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))))