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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ehl2eudisval0 46801 | 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 46802 | 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 46803 | Declare the syntax for lines in generalized real Euclidean spaces. |
class LineM | ||
Syntax | csph 46804 | Declare the syntax for spheres in generalized real Euclidean spaces. |
class Sphere | ||
Definition | df-line 46805* | 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 46806* | Definition of spheres for given centers and radii in a metric space (or more generally, in a distance space, see distspace 23669, 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 46807* | 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 46808* | 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 46809* | 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 46810* | 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 46811* | 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 46812* | 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 46811. (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 46813* | Lemma 1 for eenglngeehlnm 46815. (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 46814* | Lemma 2 for eenglngeehlnm 46815. (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 46815 | The line definition in the Tarski structure for the Euclidean geometry (see elntg 27933) corresponds to the definition of lines passing through two different points in a left module (see rrxlines 46809). (Contributed by AV, 16-Feb-2023.) |
⊢ (𝑁 ∈ ℕ → (LineG‘(EEG‘𝑁)) = (LineM‘(𝔼hil‘𝑁))) | ||
Theorem | rrx2line 46816* | 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 46817* | 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 46818* | 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 46819* | 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 46820* | 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 46821 | 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 46822* | 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 46823* | 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 46824* | 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 46825* | 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 46826* | The sphere around the origin 0 (see rrx0 24761) 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 46827* | Lemma for line2y 46831. 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 46828* | 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 46829* | Lemma for line2x 46830. 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 46830* | 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 46831* | 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 46832 | 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 46833 | 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 46834 | 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 46835 | Lemma for itsclc0 46847. 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 46836 | Lemma for itsclc0 46847. 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 46837 | Lemma for itsclc0 46847. 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 46838 | Lemma 1 for itsclc0yqsol 46840. (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 46839 | Lemma 2 for itsclc0yqsol 46840. (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 46840 | Lemma for itsclc0 46847. 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 46841 | Lemma for itsclc0 46847. 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 46842 | Lemma for itsclc0 46847. 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 46843 | Lemma for itsclc0 46847. 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 46844 | Lemma for itsclc0 46847. 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 46845 | Lemma for itsclc0 46847. 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 46846 | Lemma for itsclc0 46847. 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 46847* | 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 46848* | 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 46849 | 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 46850 | 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 46851 | 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 46852* | 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 46853* | 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 46854 | Lemma 1 for 2itscp 46857. (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 46855 | Lemma 2 for 2itscp 46857. (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 46856 | Lemma D for 2itscp 46857. (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 46857 | 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 46858 | Lemma 1 for itscnhlinecirc02p 46861. (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 46859 | Lemma 2 for itscnhlinecirc02p 46861. (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 46860 | Lemma 3 for itscnhlinecirc02p 46861. (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 46861* | 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 46862* | Lemma for inlinecirc02p 46863. (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 46863 | 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 46864* | 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 46865 | Deduction converting a biconditional to a biconditional with conjunction. Variant of pm4.71d 562. (Contributed by Zhi Wang, 30-Aug-2024.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜒))) | ||
Theorem | logic1 46866 | 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 46867 | Variant of logic1 46866. (Contributed by Zhi Wang, 30-Aug-2024.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ ((𝜑 ∧ 𝜓) → (𝜃 ↔ 𝜏)) ⇒ ⊢ (𝜑 → ((𝜓 → 𝜃) ↔ (𝜒 → 𝜏))) | ||
Theorem | logic2 46868 | Variant of logic1 46866. (Contributed by Zhi Wang, 30-Aug-2024.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏))) ⇒ ⊢ (𝜑 → ((𝜓 → 𝜃) ↔ (𝜒 → 𝜏))) | ||
Theorem | pm5.32dav 46869 | Distribution of implication over biconditional (deduction form). Variant of pm5.32da 579. (Contributed by Zhi Wang, 30-Aug-2024.) |
⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) ⇒ ⊢ (𝜑 → ((𝜒 ∧ 𝜓) ↔ (𝜃 ∧ 𝜓))) | ||
Theorem | pm5.32dra 46870 | Reverse distribution of implication over biconditional (deduction form). (Contributed by Zhi Wang, 6-Sep-2024.) |
⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜃))) ⇒ ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) | ||
Theorem | exp12bd 46871 | The import-export theorem (impexp 451) for biconditionals (deduction form). (Contributed by Zhi Wang, 3-Sep-2024.) |
⊢ (𝜑 → (((𝜓 ∧ 𝜒) → 𝜃) ↔ ((𝜏 ∧ 𝜂) → 𝜁))) ⇒ ⊢ (𝜑 → ((𝜓 → (𝜒 → 𝜃)) ↔ (𝜏 → (𝜂 → 𝜁)))) | ||
Theorem | mpbiran3d 46872 | Equivalence with a conjunction one of whose conjuncts is a consequence of the other. Deduction form. (Contributed by Zhi Wang, 24-Sep-2024.) |
⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃))) & ⊢ ((𝜑 ∧ 𝜒) → 𝜃) ⇒ ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | ||
Theorem | mpbiran4d 46873 | Equivalence with a conjunction one of whose conjuncts is a consequence of the other. Deduction form. (Contributed by Zhi Wang, 27-Sep-2024.) |
⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃))) & ⊢ ((𝜑 ∧ 𝜃) → 𝜒) ⇒ ⊢ (𝜑 → (𝜓 ↔ 𝜃)) | ||
Theorem | dtrucor3 46874* | An example of how ax-5 1913 without a distinct variable condition causes paradox in models of at least two objects. The hypothesis "dtrucor3.1" is provable from dtru 5393 in the ZF set theory. axc16nf 2254 and euae 2659 demonstrate that the violation of dtru 5393 leads to a model with only one object assuming its existence (ax-6 1971). The conclusion is also provable in the empty model ( see emptyal 1911). See also nf5 2278 and nf5i 2142 for the relation between unconditional ax-5 1913 and being not free. (Contributed by Zhi Wang, 23-Sep-2024.) |
⊢ ¬ ∀𝑥 𝑥 = 𝑦 & ⊢ (𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑦) ⇒ ⊢ ∀𝑥 𝑥 = 𝑦 | ||
Theorem | ralbidb 46875* | Formula-building rule for restricted universal quantifier and additional condition (deduction form). See ralbidc 46876 for a more generalized form. (Contributed by Zhi Wang, 6-Sep-2024.) |
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ∧ 𝜓))) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜒 ↔ 𝜃)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥 ∈ 𝐵 (𝜓 → 𝜃))) | ||
Theorem | ralbidc 46876* | Formula-building rule for restricted universal quantifier and additional condition (deduction form). A variant of ralbidb 46875. (Contributed by Zhi Wang, 30-Aug-2024.) |
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ∧ 𝜓))) & ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ 𝜓)) → (𝜒 ↔ 𝜃))) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥 ∈ 𝐵 (𝜓 → 𝜃))) | ||
Theorem | r19.41dv 46877* | A complex deduction form of r19.41v 3185. (Contributed by Zhi Wang, 6-Sep-2024.) |
⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) ⇒ ⊢ ((𝜑 ∧ 𝜒) → ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒)) | ||
Theorem | rspceb2dv 46878* | Restricted existential specialization, using implicit substitution in both directions. (Contributed by Zhi Wang, 28-Sep-2024.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 → 𝜒)) & ⊢ ((𝜑 ∧ 𝜒) → 𝐴 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝜒) → 𝜃) & ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ 𝜒)) | ||
Theorem | rmotru 46879 | Two ways of expressing "at most one" element. (Contributed by Zhi Wang, 19-Sep-2024.) (Proof shortened by BJ, 23-Sep-2024.) |
⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃*𝑥 ∈ 𝐴 ⊤) | ||
Theorem | reutru 46880 | Two ways of expressing "exactly one" element. (Contributed by Zhi Wang, 23-Sep-2024.) |
⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 ⊤) | ||
Theorem | reutruALT 46881 | Alternate proof for reutru 46880. (Contributed by Zhi Wang, 23-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 ⊤) | ||
Theorem | ssdisjd 46882 | Subset preserves disjointness. Deduction form of ssdisj 4419. (Contributed by Zhi Wang, 7-Sep-2024.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → (𝐵 ∩ 𝐶) = ∅) ⇒ ⊢ (𝜑 → (𝐴 ∩ 𝐶) = ∅) | ||
Theorem | ssdisjdr 46883 | Subset preserves disjointness. Deduction form of ssdisj 4419. Alternatively this could be proved with ineqcom 4162 in tandem with ssdisjd 46882. (Contributed by Zhi Wang, 7-Sep-2024.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → (𝐶 ∩ 𝐵) = ∅) ⇒ ⊢ (𝜑 → (𝐶 ∩ 𝐴) = ∅) | ||
Theorem | disjdifb 46884 | Relative complement is anticommutative regarding intersection. (Contributed by Zhi Wang, 5-Sep-2024.) |
⊢ ((𝐴 ∖ 𝐵) ∩ (𝐵 ∖ 𝐴)) = ∅ | ||
Theorem | predisj 46885 | Preimages of disjoint sets are disjoint. (Contributed by Zhi Wang, 9-Sep-2024.) |
⊢ (𝜑 → Fun 𝐹) & ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) & ⊢ (𝜑 → 𝑆 ⊆ (◡𝐹 “ 𝐴)) & ⊢ (𝜑 → 𝑇 ⊆ (◡𝐹 “ 𝐵)) ⇒ ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅) | ||
Theorem | vsn 46886 | The singleton of the universal class is the empty set. (Contributed by Zhi Wang, 19-Sep-2024.) |
⊢ {V} = ∅ | ||
Theorem | mosn 46887* | "At most one" element in a singleton. (Contributed by Zhi Wang, 19-Sep-2024.) |
⊢ (𝐴 = {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴) | ||
Theorem | mo0 46888* | "At most one" element in an empty set. (Contributed by Zhi Wang, 19-Sep-2024.) |
⊢ (𝐴 = ∅ → ∃*𝑥 𝑥 ∈ 𝐴) | ||
Theorem | mosssn 46889* | "At most one" element in a subclass of a singleton. (Contributed by Zhi Wang, 23-Sep-2024.) |
⊢ (𝐴 ⊆ {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴) | ||
Theorem | mo0sn 46890* | Two ways of expressing "at most one" element in a class. (Contributed by Zhi Wang, 19-Sep-2024.) |
⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦})) | ||
Theorem | mosssn2 46891* | Two ways of expressing "at most one" element in a class. (Contributed by Zhi Wang, 23-Sep-2024.) |
⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑦 𝐴 ⊆ {𝑦}) | ||
Theorem | unilbss 46892* | Superclass of the greatest lower bound. A dual statement of ssintub 4927. (Contributed by Zhi Wang, 29-Sep-2024.) |
⊢ ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐴 | ||
Theorem | inpw 46893* | Two ways of expressing a collection of subsets as seen in df-ntr 22371, unimax 4905, and others (Contributed by Zhi Wang, 27-Sep-2024.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∩ 𝒫 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ⊆ 𝐵}) | ||
Theorem | mof0 46894 | There is at most one function into the empty set. (Contributed by Zhi Wang, 19-Sep-2024.) |
⊢ ∃*𝑓 𝑓:𝐴⟶∅ | ||
Theorem | mof02 46895* | A variant of mof0 46894. (Contributed by Zhi Wang, 20-Sep-2024.) |
⊢ (𝐵 = ∅ → ∃*𝑓 𝑓:𝐴⟶𝐵) | ||
Theorem | mof0ALT 46896* | Alternate proof for mof0 46894 with stronger requirements on distinct variables. Uses mo4 2564. (Contributed by Zhi Wang, 19-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ∃*𝑓 𝑓:𝐴⟶∅ | ||
Theorem | eufsnlem 46897* | There is exactly one function into a singleton. For a simpler hypothesis, see eufsn 46898 assuming ax-rep 5242, or eufsn2 46899 assuming ax-pow 5320 and ax-un 7672. (Contributed by Zhi Wang, 19-Sep-2024.) |
⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → (𝐴 × {𝐵}) ∈ 𝑉) ⇒ ⊢ (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵}) | ||
Theorem | eufsn 46898* | There is exactly one function into a singleton, assuming ax-rep 5242. See eufsn2 46899 for different axiom requirements. If existence is not needed, use mofsn 46900 or mofsn2 46901 for fewer axiom assumptions. (Contributed by Zhi Wang, 19-Sep-2024.) |
⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵}) | ||
Theorem | eufsn2 46899* | There is exactly one function into a singleton, assuming ax-pow 5320 and ax-un 7672. Variant of eufsn 46898. If existence is not needed, use mofsn 46900 or mofsn2 46901 for fewer axiom assumptions. (Contributed by Zhi Wang, 19-Sep-2024.) |
⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵}) | ||
Theorem | mofsn 46900* | There is at most one function into a singleton, with fewer axioms than eufsn 46898 and eufsn2 46899. See also mofsn2 46901. (Contributed by Zhi Wang, 19-Sep-2024.) |
⊢ (𝐵 ∈ 𝑉 → ∃*𝑓 𝑓:𝐴⟶{𝐵}) |
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