Home | Metamath
Proof Explorer Theorem List (p. 456 of 458) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | Metamath Proof Explorer
(1-28805) |
Hilbert Space Explorer
(28806-30328) |
Users' Mathboxes
(30329-45797) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ackval41a 45501 | The Ackermann function at (4,1). (Contributed by AV, 9-May-2024.) |
⊢ ((Ack‘4)‘1) = ((2↑;16) − 3) | ||
Theorem | ackval41 45502 | The Ackermann function at (4,1). (Contributed by AV, 9-May-2024.) |
⊢ ((Ack‘4)‘1) = ;;;;65533 | ||
Theorem | ackval42 45503 | The Ackermann function at (4,2). (Contributed by AV, 9-May-2024.) |
⊢ ((Ack‘4)‘2) = ((2↑;;;;65536) − 3) | ||
Theorem | ackval42a 45504 | 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 45505 | The Ackermann function at (5,0). (Contributed by AV, 9-May-2024.) |
⊢ ((Ack‘5)‘0) = ;;;;65533 | ||
Theorem | fv1prop 45506 | 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 45507 | 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 45508 | 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 45509* | 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 45510* | 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 45511 | 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 45512 | 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 45513 | 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 45514 | 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 45515 | 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 45516 | 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 45517 | 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)) | ||
Theorem | rrx2pxel 45518 | 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 45519 | 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 45520 | 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 45521 | 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 45522 | 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 45523 | 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 45524 | 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 45525* | 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 45526 | 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 45527* | 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 45528* | 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 45529* | 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 45530* | 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 45531* | 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 45532 | 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 45533 | 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))) | ||
Syntax | cline 45534 | Declare the syntax for lines in generalized real Euclidean spaces. |
class LineM | ||
Syntax | csph 45535 | Declare the syntax for spheres in generalized real Euclidean spaces. |
class Sphere | ||
Definition | df-line 45536* | 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 45537* | Definition of spheres for given centers and radii in a metric space (or more generally, in a distance space, see distspace 23023, 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 45538* | 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 45539* | 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 45540* | 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 45541* | 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 45542* | 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 45543* | 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 45542. (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 45544* | Lemma 1 for eenglngeehlnm 45546. (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 45545* | Lemma 2 for eenglngeehlnm 45546. (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 45546 | The line definition in the Tarski structure for the Euclidean geometry (see elntg 26882) corresponds to the definition of lines passing through two different points in a left module (see rrxlines 45540). (Contributed by AV, 16-Feb-2023.) |
⊢ (𝑁 ∈ ℕ → (LineG‘(EEG‘𝑁)) = (LineM‘(𝔼hil‘𝑁))) | ||
Theorem | rrx2line 45547* | 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 45548* | 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 45549* | 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 45550* | 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 45551* | 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 45552 | 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 45553* | 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 45554* | 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 45555* | 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 45556* | 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 45557* | The sphere around the origin 0 (see rrx0 24102) 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 45558* | Lemma for line2y 45562. 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) → (∀𝑝 ∈ 𝑃 (((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶 ↔ (𝑝‘1) = 0) → (𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0))) | ||
Theorem | line2 45559* | 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 45560* | Lemma for line2x 45561. 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 = (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 45561* | 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 45562* | 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 45563 | 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 45564 | 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 45565 | 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 45566 | Lemma for itsclc0 45578. 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 45567 | Lemma for itsclc0 45578. 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 45568 | Lemma for itsclc0 45578. 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 45569 | Lemma 1 for itsclc0yqsol 45571. (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 45570 | Lemma 2 for itsclc0yqsol 45571. (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 45571 | Lemma for itsclc0 45578. 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 45572 | Lemma for itsclc0 45578. 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 45573 | Lemma for itsclc0 45578. 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 45574 | Lemma for itsclc0 45578. 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 45575 | Lemma for itsclc0 45578. 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 45576 | Lemma for itsclc0 45578. 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 45577 | Lemma for itsclc0 45578. 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 45578* | 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 45579* | 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 45580 | 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 45581 | 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 45582 | 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 45583* | 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 45584* | 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 45585 | Lemma 1 for 2itscp 45588. (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 45586 | Lemma 2 for 2itscp 45588. (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 45587 | Lemma D for 2itscp 45588. (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 45588 | 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 45589 | Lemma 1 for itscnhlinecirc02p 45592. (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 45590 | Lemma 2 for itscnhlinecirc02p 45592. (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 45591 | Lemma 3 for itscnhlinecirc02p 45592. (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 45592* | 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 45593* | Lemma for inlinecirc02p 45594. (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 45594 | 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)) | ||
Theorem | inlinecirc02preu 45595* | 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, expressed with restricted uniqueness (and without the definition of proper pairs). (Contributed 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)) → ∃!𝑝 ∈ 𝒫 𝑃((♯‘𝑝) = 2 ∧ 𝑝 = (( 0 𝑆𝑅) ∩ (𝑋𝐿𝑌)))) | ||
Theorem | pm4.71da 45596 | Deduction converting a biconditional to a biconditional with conjunction. Variant of pm4.71d 565. (Contributed by Zhi Wang, 30-Aug-2024.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜒))) | ||
Theorem | logic1 45597 | Distribution of implication over biconditional with replacement (deduction form). (Contributed by Zhi Wang, 30-Aug-2024.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → (𝜓 → (𝜃 ↔ 𝜏))) ⇒ ⊢ (𝜑 → ((𝜓 → 𝜃) ↔ (𝜒 → 𝜏))) | ||
Theorem | logic1a 45598 | Variant of logic1 45597. (Contributed by Zhi Wang, 30-Aug-2024.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ ((𝜑 ∧ 𝜓) → (𝜃 ↔ 𝜏)) ⇒ ⊢ (𝜑 → ((𝜓 → 𝜃) ↔ (𝜒 → 𝜏))) | ||
Theorem | logic2 45599 | Variant of logic1 45597. (Contributed by Zhi Wang, 30-Aug-2024.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏))) ⇒ ⊢ (𝜑 → ((𝜓 → 𝜃) ↔ (𝜒 → 𝜏))) | ||
Theorem | pm5.32dav 45600 | Distribution of implication over biconditional (deduction form). Variant of pm5.32da 582. (Contributed by Zhi Wang, 30-Aug-2024.) |
⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) ⇒ ⊢ (𝜑 → ((𝜒 ∧ 𝜓) ↔ (𝜃 ∧ 𝜓))) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |