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Theorem List for Metamath Proof Explorer - 44701-44800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrrxline 44701* The line passing through the two different points 𝑋 and 𝑌 in a generalized real Euclidean space of finite dimension. (Contributed by AV, 14-Jan-2023.)
𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝐿 = (LineM𝐸)    &    · = ( ·𝑠𝐸)    &    + = (+g𝐸)       ((𝐼 ∈ Fin ∧ (𝑋𝑃𝑌𝑃𝑋𝑌)) → (𝑋𝐿𝑌) = {𝑝𝑃 ∣ ∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡) · 𝑋) + (𝑡 · 𝑌))})

Theoremrrxlinesc 44702* Definition of lines passing through two different points in a generalized real Euclidean space of finite dimension, expressed by their coordinates. (Contributed by AV, 13-Feb-2023.)
𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝐿 = (LineM𝐸)       (𝐼 ∈ Fin → 𝐿 = (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑝𝑃 ∣ ∃𝑡 ∈ ℝ ∀𝑖𝐼 (𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))}))

Theoremrrxlinec 44703* 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 44702. (Contributed by AV, 13-Feb-2023.)
𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝐿 = (LineM𝐸)       ((𝐼 ∈ Fin ∧ (𝑋𝑃𝑌𝑃𝑋𝑌)) → (𝑋𝐿𝑌) = {𝑝𝑃 ∣ ∃𝑡 ∈ ℝ ∀𝑖𝐼 (𝑝𝑖) = (((1 − 𝑡) · (𝑋𝑖)) + (𝑡 · (𝑌𝑖)))})

Theoremeenglngeehlnmlem1 44704* Lemma 1 for eenglngeehlnm 44706. (Contributed by AV, 15-Feb-2023.)
(((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) → ((∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖)))) → ∃𝑡 ∈ ℝ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))))

Theoremeenglngeehlnmlem2 44705* Lemma 2 for eenglngeehlnm 44706. (Contributed by AV, 15-Feb-2023.)
(((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) → (∃𝑡 ∈ ℝ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))) → (∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))))))

Theoremeenglngeehlnm 44706 The line definition in the Tarski structure for the Euclidean geometry (see elntg 26762) corresponds to the definition of lines passing through two different points in a left module (see rrxlines 44700). (Contributed by AV, 16-Feb-2023.)
(𝑁 ∈ ℕ → (LineG‘(EEG‘𝑁)) = (LineM‘(𝔼hil𝑁)))

Theoremrrx2line 44707* The line passing through the two different points 𝑋 and 𝑌 in a real Euclidean space of dimension 2. (Contributed by AV, 22-Jan-2023.) (Proof shortened by AV, 13-Feb-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝐿 = (LineM𝐸)       ((𝑋𝑃𝑌𝑃𝑋𝑌) → (𝑋𝐿𝑌) = {𝑝𝑃 ∣ ∃𝑡 ∈ ℝ ((𝑝‘1) = (((1 − 𝑡) · (𝑋‘1)) + (𝑡 · (𝑌‘1))) ∧ (𝑝‘2) = (((1 − 𝑡) · (𝑋‘2)) + (𝑡 · (𝑌‘2))))})

Theoremrrx2vlinest 44708* The vertical line passing through the two different points 𝑋 and 𝑌 in a real Euclidean space of dimension 2 in "standard form". (Contributed by AV, 2-Feb-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝐿 = (LineM𝐸)       ((𝑋𝑃𝑌𝑃 ∧ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) ≠ (𝑌‘2))) → (𝑋𝐿𝑌) = {𝑝𝑃 ∣ (𝑝‘1) = (𝑋‘1)})

Theoremrrx2linest 44709* The line passing through the two different points 𝑋 and 𝑌 in a real Euclidean space of dimension 2 in "standard form". (Contributed by AV, 2-Feb-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝐿 = (LineM𝐸)    &   𝐴 = ((𝑌‘1) − (𝑋‘1))    &   𝐵 = ((𝑌‘2) − (𝑋‘2))    &   𝐶 = (((𝑋‘2) · (𝑌‘1)) − ((𝑋‘1) · (𝑌‘2)))       ((𝑋𝑃𝑌𝑃𝑋𝑌) → (𝑋𝐿𝑌) = {𝑝𝑃 ∣ (𝐴 · (𝑝‘2)) = ((𝐵 · (𝑝‘1)) + 𝐶)})

Theoremrrx2linesl 44710* The line passing through the two different points 𝑋 and 𝑌 in a real Euclidean space of dimension 2, expressed by the slope 𝑆 between the two points ("point-slope form"), sometimes also written as ((𝑝‘2) − (𝑋‘2)) = (𝑆 · ((𝑝‘1) − (𝑋‘1))). (Contributed by AV, 22-Jan-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝐿 = (LineM𝐸)    &   𝑆 = (((𝑌‘2) − (𝑋‘2)) / ((𝑌‘1) − (𝑋‘1)))       ((𝑋𝑃𝑌𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) → (𝑋𝐿𝑌) = {𝑝𝑃 ∣ (𝑝‘2) = ((𝑆 · ((𝑝‘1) − (𝑋‘1))) + (𝑋‘2))})

Theoremrrx2linest2 44711* The line passing through the two different points 𝑋 and 𝑌 in a real Euclidean space of dimension 2 in another "standard form" (usually with (𝑝‘1) = 𝑥 and (𝑝‘2) = 𝑦). (Contributed by AV, 23-Feb-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝐿 = (LineM𝐸)    &   𝐴 = ((𝑋‘2) − (𝑌‘2))    &   𝐵 = ((𝑌‘1) − (𝑋‘1))    &   𝐶 = (((𝑋‘2) · (𝑌‘1)) − ((𝑋‘1) · (𝑌‘2)))       ((𝑋𝑃𝑌𝑃𝑋𝑌) → (𝑋𝐿𝑌) = {𝑝𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶})

Theoremelrrx2linest2 44712 The line passing through the two different points 𝑋 and 𝑌 in a real Euclidean space of dimension 2 in another "standard form" (usually with (𝑝‘1) = 𝑥 and (𝑝‘2) = 𝑦). (Contributed by AV, 23-Feb-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝐿 = (LineM𝐸)    &   𝐴 = ((𝑋‘2) − (𝑌‘2))    &   𝐵 = ((𝑌‘1) − (𝑋‘1))    &   𝐶 = (((𝑋‘2) · (𝑌‘1)) − ((𝑋‘1) · (𝑌‘2)))       ((𝑋𝑃𝑌𝑃𝑋𝑌) → (𝐺 ∈ (𝑋𝐿𝑌) ↔ (𝐺𝑃 ∧ ((𝐴 · (𝐺‘1)) + (𝐵 · (𝐺‘2))) = 𝐶)))

Theoremspheres 44713* The spheres for given centers and radii in a metric space (or any extensible structure having a base set and a distance function). (Contributed by AV, 22-Jan-2023.)
𝐵 = (Base‘𝑊)    &   𝑆 = (Sphere‘𝑊)    &   𝐷 = (dist‘𝑊)       (𝑊𝑉𝑆 = (𝑥𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}))

Theoremsphere 44714* A sphere with center 𝑋 and radius 𝑅 in a metric space (or any extensible structure having a base set and a distance function). (Contributed by AV, 22-Jan-2023.)
𝐵 = (Base‘𝑊)    &   𝑆 = (Sphere‘𝑊)    &   𝐷 = (dist‘𝑊)       ((𝑊𝑉𝑋𝐵𝑅 ∈ (0[,]+∞)) → (𝑋𝑆𝑅) = {𝑝𝐵 ∣ (𝑝𝐷𝑋) = 𝑅})

Theoremrrxsphere 44715* The sphere with center 𝑀 and radius 𝑅 in a generalized real Euclidean space of finite dimension. Remark: this theorem holds also for the degenerate case 𝑅 < 0 (negative radius): in this case, (𝑀𝑆𝑅) is empty. (Contributed by AV, 5-Feb-2023.)
𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝐷 = (dist‘𝐸)    &   𝑆 = (Sphere‘𝐸)       ((𝐼 ∈ Fin ∧ 𝑀𝑃𝑅 ∈ ℝ) → (𝑀𝑆𝑅) = {𝑝𝑃 ∣ (𝑝𝐷𝑀) = 𝑅})

Theorem2sphere 44716* The sphere with center 𝑀 and radius 𝑅 in a two dimensional Euclidean space is a circle. (Contributed by AV, 5-Feb-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝑆 = (Sphere‘𝐸)    &   𝐶 = {𝑝𝑃 ∣ ((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2)) = (𝑅↑2)}       ((𝑀𝑃𝑅 ∈ (0[,)+∞)) → (𝑀𝑆𝑅) = 𝐶)

Theorem2sphere0 44717* The sphere around the origin 0 (see rrx0 23992) with radius 𝑅 in a two dimensional Euclidean space is a circle. (Contributed by AV, 5-Feb-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝑆 = (Sphere‘𝐸)    &    0 = (𝐼 × {0})    &   𝐶 = {𝑝𝑃 ∣ (((𝑝‘1)↑2) + ((𝑝‘2)↑2)) = (𝑅↑2)}       (𝑅 ∈ (0[,)+∞) → ( 0 𝑆𝑅) = 𝐶)

Theoremline2ylem 44718* Lemma for line2y 44722. This proof is based on counterexamples for the following cases: 1. 𝐶 ≠ 0: p = (0,0) (LHS of bicondional is false, RHS is true); 2. 𝐶 = 0 ∧ 𝐵 ≠ 0: p = (1,-A/B) (LHS of bicondional is true, RHS is false); 3. 𝐴 = 𝐵 = 𝐶 = 0: p = (1,1) (LHS of bicondional is true, RHS is false). (Contributed by AV, 4-Feb-2023.)
𝐼 = {1, 2}    &   𝑃 = (ℝ ↑m 𝐼)       ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (∀𝑝𝑃 (((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶 ↔ (𝑝‘1) = 0) → (𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0)))

Theoremline2 44719* Example for a line 𝐺 passing through two different points in "standard form". (Contributed by AV, 3-Feb-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝐿 = (LineM𝐸)    &   𝐺 = {𝑝𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶}    &   𝑋 = {⟨1, 0⟩, ⟨2, (𝐶 / 𝐵)⟩}    &   𝑌 = {⟨1, 1⟩, ⟨2, ((𝐶𝐴) / 𝐵)⟩}       ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) ∧ 𝐶 ∈ ℝ) → 𝐺 = (𝑋𝐿𝑌))

Theoremline2xlem 44720* Lemma for line2x 44721. This proof is based on counterexamples for the following cases: 1. 𝑀 ≠ (𝐶 / 𝐵): p = (0,C/B) (LHS of bicondional is true, RHS is false); 2. 𝐴 ≠ 0 ∧ 𝑀 = (𝐶 / 𝐵): p = (1,C/B) (LHS of bicondional is false, RHS is true). (Contributed by AV, 4-Feb-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝐿 = (LineM𝐸)    &   𝐺 = {𝑝𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶}    &   𝑋 = {⟨1, 0⟩, ⟨2, 𝑀⟩}    &   𝑌 = {⟨1, 1⟩, ⟨2, 𝑀⟩}       (((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) ∧ 𝐶 ∈ ℝ) ∧ 𝑀 ∈ ℝ) → (∀𝑝𝑃 (((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶 ↔ (𝑝‘2) = 𝑀) → (𝐴 = 0 ∧ 𝑀 = (𝐶 / 𝐵))))

Theoremline2x 44721* Example for a horizontal line 𝐺 passing through two different points in "standard form". (Contributed by AV, 3-Feb-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝐿 = (LineM𝐸)    &   𝐺 = {𝑝𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶}    &   𝑋 = {⟨1, 0⟩, ⟨2, 𝑀⟩}    &   𝑌 = {⟨1, 1⟩, ⟨2, 𝑀⟩}       (((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) ∧ 𝐶 ∈ ℝ) ∧ 𝑀 ∈ ℝ) → (𝐺 = (𝑋𝐿𝑌) ↔ (𝐴 = 0 ∧ 𝑀 = (𝐶 / 𝐵))))

Theoremline2y 44722* Example for a vertical line 𝐺 passing through two different points in "standard form". (Contributed by AV, 3-Feb-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝐿 = (LineM𝐸)    &   𝐺 = {𝑝𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶}    &   𝑋 = {⟨1, 0⟩, ⟨2, 𝑀⟩}    &   𝑌 = {⟨1, 0⟩, ⟨2, 𝑁⟩}       (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀𝑁)) → (𝐺 = (𝑋𝐿𝑌) ↔ (𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0)))

Theoremitsclc0lem1 44723 Lemma for theorems about intersections of lines and circles in a real Euclidean space of dimension 2 . (Contributed by AV, 2-May-2023.)
(((𝑆 ∈ ℝ ∧ 𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ (𝑉 ∈ ℝ ∧ 0 ≤ 𝑉) ∧ (𝑊 ∈ ℝ ∧ 𝑊 ≠ 0)) → (((𝑆 · 𝑈) + (𝑇 · (√‘𝑉))) / 𝑊) ∈ ℝ)

Theoremitsclc0lem2 44724 Lemma for theorems about intersections of lines and circles in a real Euclidean space of dimension 2 . (Contributed by AV, 3-May-2023.)
(((𝑆 ∈ ℝ ∧ 𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ (𝑉 ∈ ℝ ∧ 0 ≤ 𝑉) ∧ (𝑊 ∈ ℝ ∧ 𝑊 ≠ 0)) → (((𝑆 · 𝑈) − (𝑇 · (√‘𝑉))) / 𝑊) ∈ ℝ)

Theoremitsclc0lem3 44725 Lemma for theorems about intersections of lines and circles in a real Euclidean space of dimension 2 . (Contributed by AV, 2-May-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))       (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝑅 ∈ ℝ) → 𝐷 ∈ ℝ)

Theoremitscnhlc0yqe 44726 Lemma for itsclc0 44738. Quadratic equation for the y-coordinate of the intersection points of a nonhorizontal line and a circle. (Contributed by AV, 6-Feb-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝑇 = -(2 · (𝐵 · 𝐶))    &   𝑈 = ((𝐶↑2) − ((𝐴↑2) · (𝑅↑2)))       ((((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝑅 ∈ ℝ+ ∧ (𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ)) → ((((𝑋↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑋) + (𝐵 · 𝑌)) = 𝐶) → ((𝑄 · (𝑌↑2)) + ((𝑇 · 𝑌) + 𝑈)) = 0))

Theoremitschlc0yqe 44727 Lemma for itsclc0 44738. Quadratic equation for the y-coordinate of the intersection points of a horizontal line and a circle. (Contributed by AV, 25-Feb-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝑇 = -(2 · (𝐵 · 𝐶))    &   𝑈 = ((𝐶↑2) − ((𝐴↑2) · (𝑅↑2)))       ((((𝐴 ∈ ℝ ∧ 𝐴 = 0) ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝑅 ∈ ℝ+ ∧ (𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ)) → ((((𝑋↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑋) + (𝐵 · 𝑌)) = 𝐶) → ((𝑄 · (𝑌↑2)) + ((𝑇 · 𝑌) + 𝑈)) = 0))

Theoremitsclc0yqe 44728 Lemma for itsclc0 44738. Quadratic equation for the y-coordinate of the intersection points of an arbitrary line and a circle. This theorem holds even for degenerate lines (𝐴 = 𝐵 = 0). (Contributed by AV, 25-Feb-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝑇 = -(2 · (𝐵 · 𝐶))    &   𝑈 = ((𝐶↑2) − ((𝐴↑2) · (𝑅↑2)))       (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝑅 ∈ ℝ+ ∧ (𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ)) → ((((𝑋↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑋) + (𝐵 · 𝑌)) = 𝐶) → ((𝑄 · (𝑌↑2)) + ((𝑇 · 𝑌) + 𝑈)) = 0))

Theoremitsclc0yqsollem1 44729 Lemma 1 for itsclc0yqsol 44731. (Contributed by AV, 6-Feb-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝑇 = -(2 · (𝐵 · 𝐶))    &   𝑈 = ((𝐶↑2) − ((𝐴↑2) · (𝑅↑2)))    &   𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))       (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ 𝑅 ∈ ℂ) → ((𝑇↑2) − (4 · (𝑄 · 𝑈))) = ((4 · (𝐴↑2)) · 𝐷))

Theoremitsclc0yqsollem2 44730 Lemma 2 for itsclc0yqsol 44731. (Contributed by AV, 6-Feb-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝑇 = -(2 · (𝐵 · 𝐶))    &   𝑈 = ((𝐶↑2) − ((𝐴↑2) · (𝑅↑2)))    &   𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))       (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝑅 ∈ ℝ ∧ 0 ≤ 𝐷) → (√‘((𝑇↑2) − (4 · (𝑄 · 𝑈)))) = ((2 · (abs‘𝐴)) · (√‘𝐷)))

Theoremitsclc0yqsol 44731 Lemma for itsclc0 44738. Solutions of the quadratic equations for the y-coordinate of the intersection points of a (nondegenerate) line and a circle. (Contributed by AV, 7-Feb-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))       ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 ≠ 0 ∨ 𝐵 ≠ 0)) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷) ∧ (𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ)) → ((((𝑋↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑋) + (𝐵 · 𝑌)) = 𝐶) → (𝑌 = (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄) ∨ 𝑌 = (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄))))

Theoremitscnhlc0xyqsol 44732 Lemma for itsclc0 44738. Solutions of the quadratic equations for the coordinates of the intersection points of a nonhorizontal line and a circle. (Contributed by AV, 8-Feb-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))       ((((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷) ∧ (𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ)) → ((((𝑋↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑋) + (𝐵 · 𝑌)) = 𝐶) → ((𝑋 = (((𝐴 · 𝐶) + (𝐵 · (√‘𝐷))) / 𝑄) ∧ 𝑌 = (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄)) ∨ (𝑋 = (((𝐴 · 𝐶) − (𝐵 · (√‘𝐷))) / 𝑄) ∧ 𝑌 = (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄)))))

Theoremitschlc0xyqsol1 44733 Lemma for itsclc0 44738. Solutions of the quadratic equations for the coordinates of the intersection points of a horizontal line and a circle. (Contributed by AV, 25-Feb-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))       ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 = 0 ∧ 𝐵 ≠ 0)) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷) ∧ (𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ)) → ((((𝑋↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑋) + (𝐵 · 𝑌)) = 𝐶) → (𝑌 = (𝐶 / 𝐵) ∧ (𝑋 = -((√‘𝐷) / 𝐵) ∨ 𝑋 = ((√‘𝐷) / 𝐵)))))

Theoremitschlc0xyqsol 44734 Lemma for itsclc0 44738. Solutions of the quadratic equations for the coordinates of the intersection points of a horizontal line and a circle. (Contributed by AV, 8-Feb-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))       ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 = 0 ∧ 𝐵 ≠ 0)) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷) ∧ (𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ)) → ((((𝑋↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑋) + (𝐵 · 𝑌)) = 𝐶) → ((𝑋 = (((𝐴 · 𝐶) + (𝐵 · (√‘𝐷))) / 𝑄) ∧ 𝑌 = (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄)) ∨ (𝑋 = (((𝐴 · 𝐶) − (𝐵 · (√‘𝐷))) / 𝑄) ∧ 𝑌 = (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄)))))

Theoremitsclc0xyqsol 44735 Lemma for itsclc0 44738. Solutions of the quadratic equations for the coordinates of the intersection points of a (nondegenerate) line and a circle. (Contributed by AV, 25-Feb-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))       ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 ≠ 0 ∨ 𝐵 ≠ 0)) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷) ∧ (𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ)) → ((((𝑋↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑋) + (𝐵 · 𝑌)) = 𝐶) → ((𝑋 = (((𝐴 · 𝐶) + (𝐵 · (√‘𝐷))) / 𝑄) ∧ 𝑌 = (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄)) ∨ (𝑋 = (((𝐴 · 𝐶) − (𝐵 · (√‘𝐷))) / 𝑄) ∧ 𝑌 = (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄)))))

Theoremitsclc0xyqsolr 44736 Lemma for itsclc0 44738. Solutions of the quadratic equations for the coordinates of the intersection points of a (nondegenerate) line and a circle. (Contributed by AV, 2-May-2023.) (Revised by AV, 14-May-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))       (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 ≠ 0 ∨ 𝐵 ≠ 0) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷)) → (((𝑋 = (((𝐴 · 𝐶) + (𝐵 · (√‘𝐷))) / 𝑄) ∧ 𝑌 = (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄)) ∨ (𝑋 = (((𝐴 · 𝐶) − (𝐵 · (√‘𝐷))) / 𝑄) ∧ 𝑌 = (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄))) → (((𝑋↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑋) + (𝐵 · 𝑌)) = 𝐶)))

Theoremitsclc0xyqsolb 44737 Lemma for itsclc0 44738. Solutions of the quadratic equations for the coordinates of the intersection points of a (nondegenerate) line and a circle. (Contributed by AV, 2-May-2023.) (Revised by AV, 14-May-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))       ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 ≠ 0 ∨ 𝐵 ≠ 0)) ∧ ((𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷) ∧ (𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ))) → ((((𝑋↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑋) + (𝐵 · 𝑌)) = 𝐶) ↔ ((𝑋 = (((𝐴 · 𝐶) + (𝐵 · (√‘𝐷))) / 𝑄) ∧ 𝑌 = (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄)) ∨ (𝑋 = (((𝐴 · 𝐶) − (𝐵 · (√‘𝐷))) / 𝑄) ∧ 𝑌 = (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄)))))

Theoremitsclc0 44738* The intersection points of a line 𝐿 and a circle around the origin. (Contributed by AV, 25-Feb-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝑆 = (Sphere‘𝐸)    &    0 = (𝐼 × {0})    &   𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))    &   𝐿 = {𝑝𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶}       (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 ≠ 0 ∨ 𝐵 ≠ 0) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷)) → ((𝑋 ∈ ( 0 𝑆𝑅) ∧ 𝑋𝐿) → (((𝑋‘1) = (((𝐴 · 𝐶) + (𝐵 · (√‘𝐷))) / 𝑄) ∧ (𝑋‘2) = (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄)) ∨ ((𝑋‘1) = (((𝐴 · 𝐶) − (𝐵 · (√‘𝐷))) / 𝑄) ∧ (𝑋‘2) = (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄)))))

Theoremitsclc0b 44739* The intersection points of a (nondegenerate) line through two points and a circle around the origin. (Contributed by AV, 2-May-2023.) (Revised by AV, 14-May-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝑆 = (Sphere‘𝐸)    &    0 = (𝐼 × {0})    &   𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))    &   𝐿 = {𝑝𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶}       (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 ≠ 0 ∨ 𝐵 ≠ 0) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷)) → ((𝑋 ∈ ( 0 𝑆𝑅) ∧ 𝑋𝐿) ↔ (𝑋𝑃 ∧ (((𝑋‘1) = (((𝐴 · 𝐶) + (𝐵 · (√‘𝐷))) / 𝑄) ∧ (𝑋‘2) = (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄)) ∨ ((𝑋‘1) = (((𝐴 · 𝐶) − (𝐵 · (√‘𝐷))) / 𝑄) ∧ (𝑋‘2) = (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄))))))

Theoremitsclinecirc0 44740 The intersection points of a line through two different points 𝑌 and 𝑍 and a circle around the origin, using the definition of a line in a two dimensional Euclidean space. (Contributed by AV, 25-Feb-2023.) (Proof shortened by AV, 16-May-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝑆 = (Sphere‘𝐸)    &    0 = (𝐼 × {0})    &   𝑄 = ((𝐴↑2) + (𝐵↑2))    &   𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))    &   𝐿 = (LineM𝐸)    &   𝐴 = ((𝑌‘2) − (𝑍‘2))    &   𝐵 = ((𝑍‘1) − (𝑌‘1))    &   𝐶 = (((𝑌‘2) · (𝑍‘1)) − ((𝑌‘1) · (𝑍‘2)))       (((𝑌𝑃𝑍𝑃𝑌𝑍) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷)) → ((𝑋 ∈ ( 0 𝑆𝑅) ∧ 𝑋 ∈ (𝑌𝐿𝑍)) → (((𝑋‘1) = (((𝐴 · 𝐶) + (𝐵 · (√‘𝐷))) / 𝑄) ∧ (𝑋‘2) = (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄)) ∨ ((𝑋‘1) = (((𝐴 · 𝐶) − (𝐵 · (√‘𝐷))) / 𝑄) ∧ (𝑋‘2) = (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄)))))

Theoremitsclinecirc0b 44741 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 = (LineM𝐸)    &   𝐴 = ((𝑋‘2) − (𝑌‘2))    &   𝐵 = ((𝑌‘1) − (𝑋‘1))    &   𝐶 = (((𝑋‘2) · (𝑌‘1)) − ((𝑋‘1) · (𝑌‘2)))       (((𝑋𝑃𝑌𝑃𝑋𝑌) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷)) → ((𝑍 ∈ ( 0 𝑆𝑅) ∧ 𝑍 ∈ (𝑋𝐿𝑌)) ↔ (𝑍𝑃 ∧ (((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))))))

Theoremitsclinecirc0in 44742 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 = (LineM𝐸)    &   𝐴 = ((𝑋‘2) − (𝑌‘2))    &   𝐵 = ((𝑌‘1) − (𝑋‘1))    &   𝐶 = (((𝑋‘2) · (𝑌‘1)) − ((𝑋‘1) · (𝑌‘2)))       (((𝑋𝑃𝑌𝑃𝑋𝑌) ∧ (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)⟩}})

Theoremitsclquadb 44743* 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))

Theoremitsclquadeu 44744* 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))

Theorem2itscplem1 44745 Lemma 1 for 2itscp 44748. (Contributed by AV, 4-Mar-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   𝐷 = (𝑋𝐴)    &   𝐸 = (𝐵𝑌)       (𝜑 → ((((𝐸↑2) · (𝐵↑2)) + ((𝐷↑2) · (𝐴↑2))) − (2 · ((𝐷 · 𝐴) · (𝐸 · 𝐵)))) = (((𝐷 · 𝐴) − (𝐸 · 𝐵))↑2))

Theorem2itscplem2 44746 Lemma 2 for 2itscp 44748. (Contributed by AV, 4-Mar-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   𝐷 = (𝑋𝐴)    &   𝐸 = (𝐵𝑌)    &   𝐶 = ((𝐷 · 𝐵) + (𝐸 · 𝐴))       (𝜑 → (𝐶↑2) = ((((𝐷↑2) · (𝐵↑2)) + (2 · ((𝐷 · 𝐴) · (𝐸 · 𝐵)))) + ((𝐸↑2) · (𝐴↑2))))

Theorem2itscplem3 44747 Lemma D for 2itscp 44748. (Contributed by AV, 4-Mar-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   𝐷 = (𝑋𝐴)    &   𝐸 = (𝐵𝑌)    &   𝐶 = ((𝐷 · 𝐵) + (𝐸 · 𝐴))    &   (𝜑𝑅 ∈ ℝ)    &   𝑄 = ((𝐸↑2) + (𝐷↑2))    &   𝑆 = (((𝑅↑2) · 𝑄) − (𝐶↑2))       (𝜑𝑆 = ((((𝐸↑2) · ((𝑅↑2) − (𝐴↑2))) + ((𝐷↑2) · ((𝑅↑2) − (𝐵↑2)))) − (2 · ((𝐷 · 𝐴) · (𝐸 · 𝐵)))))

Theorem2itscp 44748 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.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   𝐷 = (𝑋𝐴)    &   𝐸 = (𝐵𝑌)    &   𝐶 = ((𝐷 · 𝐵) + (𝐸 · 𝐴))    &   (𝜑𝑅 ∈ ℝ)    &   (𝜑 → ((𝐴↑2) + (𝐵↑2)) < (𝑅↑2))    &   (𝜑 → (𝐵𝑌𝐴𝑋))    &   𝑄 = ((𝐸↑2) + (𝐷↑2))    &   𝑆 = (((𝑅↑2) · 𝑄) − (𝐶↑2))       (𝜑 → 0 < 𝑆)

Theoremitscnhlinecirc02plem1 44749 Lemma 1 for itscnhlinecirc02p 44752. (Contributed by AV, 6-Mar-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   𝐷 = (𝑋𝐴)    &   𝐸 = (𝐵𝑌)    &   𝐶 = ((𝐷 · 𝐵) + (𝐸 · 𝐴))    &   (𝜑𝑅 ∈ ℝ)    &   (𝜑 → ((𝐴↑2) + (𝐵↑2)) < (𝑅↑2))    &   (𝜑𝐵𝑌)       (𝜑 → 0 < ((-(2 · (𝐷 · 𝐶))↑2) − (4 · (((𝐸↑2) + (𝐷↑2)) · ((𝐶↑2) − ((𝐸↑2) · (𝑅↑2)))))))

Theoremitscnhlinecirc02plem2 44750 Lemma 2 for itscnhlinecirc02p 44752. (Contributed by AV, 10-Mar-2023.)
𝐷 = (𝑋𝐴)    &   𝐸 = (𝐵𝑌)    &   𝐶 = ((𝐵 · 𝑋) − (𝐴 · 𝑌))       ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) ∧ 𝐵𝑌) ∧ (𝑅 ∈ ℝ ∧ ((𝐴↑2) + (𝐵↑2)) < (𝑅↑2))) → 0 < ((-(2 · (𝐷 · 𝐶))↑2) − (4 · (((𝐸↑2) + (𝐷↑2)) · ((𝐶↑2) − ((𝐸↑2) · (𝑅↑2)))))))

Theoremitscnhlinecirc02plem3 44751 Lemma 3 for itscnhlinecirc02p 44752. (Contributed by AV, 10-Mar-2023.)
𝐼 = {1, 2}    &   𝐸 = (ℝ^‘𝐼)    &   𝑃 = (ℝ ↑m 𝐼)    &   𝑆 = (Sphere‘𝐸)    &    0 = (𝐼 × {0})    &   𝐿 = (LineM𝐸)    &   𝐷 = (dist‘𝐸)       (((𝑋𝑃𝑌𝑃 ∧ (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)))))))

Theoremitscnhlinecirc02p 44752* 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 = (LineM𝐸)    &   𝐷 = (dist‘𝐸)    &   𝑍 = {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}       (((𝑋𝑃𝑌𝑃 ∧ (𝑋‘2) ≠ (𝑌‘2)) ∧ (𝑅 ∈ ℝ+ ∧ (𝑋𝐷 0 ) < 𝑅)) → ∃!𝑠 ∈ 𝒫 ℝ((♯‘𝑠) = 2 ∧ ∀𝑦𝑠 ∃!𝑥 ∈ ℝ (𝑍 ∈ ( 0 𝑆𝑅) ∧ 𝑍 ∈ (𝑋𝐿𝑌))))

Theoreminlinecirc02plem 44753* Lemma for inlinecirc02p 44754. (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 = (LineM𝐸)    &   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)))       (((𝑋𝑃𝑌𝑃𝑋𝑌) ∧ (𝑅 ∈ ℝ+ ∧ 0 < 𝐷)) → ∃𝑎𝑃𝑏𝑃 ((( 0 𝑆𝑅) ∩ (𝑋𝐿𝑌)) = {𝑎, 𝑏} ∧ 𝑎𝑏))

Theoreminlinecirc02p 44754 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 = (LineM𝐸)    &   𝐷 = (dist‘𝐸)       (((𝑋𝑃𝑌𝑃𝑋𝑌) ∧ (𝑅 ∈ ℝ+ ∧ (𝑋𝐷 0 ) < 𝑅)) → (( 0 𝑆𝑅) ∩ (𝑋𝐿𝑌)) ∈ (Pairsproper𝑃))

Theoreminlinecirc02preu 44755* 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 = (LineM𝐸)    &   𝐷 = (dist‘𝐸)       (((𝑋𝑃𝑌𝑃𝑋𝑌) ∧ (𝑅 ∈ ℝ+ ∧ (𝑋𝐷 0 ) < 𝑅)) → ∃!𝑝 ∈ 𝒫 𝑃((♯‘𝑝) = 2 ∧ 𝑝 = (( 0 𝑆𝑅) ∩ (𝑋𝐿𝑌))))

20.41  Mathbox for Emmett Weisz

20.41.1  Miscellaneous Theorems

Some of these theorems are used in the series of lemmas and theorems proving the defining properties of setrecs.

Theoremnfintd 44756 Bound-variable hypothesis builder for intersection. (Contributed by Emmett Weisz, 16-Jan-2020.)
(𝜑𝑥𝐴)       (𝜑𝑥 𝐴)

Theoremnfiund 44757* Bound-variable hypothesis builder for indexed union. (Contributed by Emmett Weisz, 6-Dec-2019.) Add disjoint variable condition to avoid ax-13 2383. See nfiundg 44758 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
𝑥𝜑    &   (𝜑𝑦𝐴)    &   (𝜑𝑦𝐵)       (𝜑𝑦 𝑥𝐴 𝐵)

Theoremnfiundg 44758 Bound-variable hypothesis builder for indexed union. Usage of this theorem is discouraged because it depends on ax-13 2383, see nfiund 44757 for a weaker version that does not require it. (Contributed by Emmett Weisz, 6-Dec-2019.) (New usage is discouraged.)
𝑥𝜑    &   (𝜑𝑦𝐴)    &   (𝜑𝑦𝐵)       (𝜑𝑦 𝑥𝐴 𝐵)

Theoremiunord 44759* The indexed union of a collection of ordinal numbers 𝐵(𝑥) is ordinal. This proof is based on the proof of ssorduni 7492, but does not use it directly, since ssorduni 7492 does not work when 𝐵 is a proper class. (Contributed by Emmett Weisz, 3-Nov-2019.)
(∀𝑥𝐴 Ord 𝐵 → Ord 𝑥𝐴 𝐵)

Theoremiunordi 44760* The indexed union of a collection of ordinal numbers 𝐵(𝑥) is ordinal. (Contributed by Emmett Weisz, 3-Nov-2019.)
Ord 𝐵       Ord 𝑥𝐴 𝐵

Theoremspd 44761 Specialization deduction, using implicit substitution. Based on the proof of spimed 2399. (Contributed by Emmett Weisz, 17-Jan-2020.)
(𝜒 → Ⅎ𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜒 → (∀𝑥𝜑𝜓))

Theoremspcdvw 44762* A version of spcdv 3591 where 𝜓 and 𝜒 are direct substitutions of each other. This theorem is useful because it does not require 𝜑 and 𝑥 to be distinct variables. (Contributed by Emmett Weisz, 12-Apr-2020.)
(𝜑𝐴𝐵)    &   (𝑥 = 𝐴 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓𝜒))

Theoremtfis2d 44763* Transfinite Induction Schema, using implicit substitution. (Contributed by Emmett Weisz, 3-May-2020.)
(𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))    &   (𝜑 → (𝑥 ∈ On → (∀𝑦𝑥 𝜒𝜓)))       (𝜑 → (𝑥 ∈ On → 𝜓))

Theorembnd2d 44764* Deduction form of bnd2 9314. (Contributed by Emmett Weisz, 19-Jan-2021.)
(𝜑𝐴 ∈ V)    &   (𝜑 → ∀𝑥𝐴𝑦𝐵 𝜓)       (𝜑 → ∃𝑧(𝑧𝐵 ∧ ∀𝑥𝐴𝑦𝑧 𝜓))

Theoremdffun3f 44765* Alternate definition of function, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Emmett Weisz, 14-Mar-2021.)
𝑥𝐴    &   𝑦𝐴    &   𝑧𝐴       (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧)))

20.41.2  Set Recursion

20.41.2.1  Basic Properties of Set Recursion

Symbols in this section:

All the symbols used in the definition of setrecs(𝐹) are explained in the comment of df-setrecs 44767. The class 𝑌 is explained in the comment of setrec1lem1 44770. Glossaries of symbols used in individual proofs, or used differently in different proofs, are in the comments of those proofs.

Syntaxcsetrecs 44766 Extend class notation to include a set defined by transfinite recursion.
class setrecs(𝐹)

Definitiondf-setrecs 44767* Define a class setrecs(𝐹) by transfinite recursion, where (𝐹𝑥) is the set of new elements to add to the class given the set 𝑥 of elements in the class so far. We do not need a base case, because we can start with the empty set, which is vacuously a subset of setrecs(𝐹). The goal of this definition is to construct a class fulfilling theorems setrec1 44774 and setrec2v 44779, which give a more intuitive idea of the meaning of setrecs. Unlike wrecs, setrecs is well-defined for any 𝐹 and meaningful for any function 𝐹.

For example, see theorem onsetrec 44790 for how the class On is defined recursively using the successor function.

The definition works by building subsets of the desired class and taking the union of those subsets. To find such a collection of subsets, consider an arbitrary set 𝑧, and consider the result when applying 𝐹 to any subset 𝑤𝑧. Remember that 𝐹 can be any function, and in general we are interested in functions that give outputs that are larger than their inputs, so we have no reason to expect the outputs to be within 𝑧. However, if we restrict the domain of 𝐹 to a given set 𝑦, the resulting range will be a set. Therefore, with this restricted 𝐹, it makes sense to consider sets 𝑧 that are closed under 𝐹 applied to its subsets. Now we can test whether a given set 𝑦 is recursively generated by 𝐹. If every set 𝑧 that is closed under 𝐹 contains 𝑦, that means that every member of 𝑦 must eventually be generated by 𝐹. On the other hand, if some such 𝑧 does not contain a certain element of 𝑦, then that element can be avoided even if we apply 𝐹 in every possible way to previously generated elements.

Note that such an omitted element might be eventually recursively generated by 𝐹, but not through the elements of 𝑦. In this case, 𝑦 would fail the condition in the definition, but the omitted element would still be included in some larger 𝑦. For example, if 𝐹 is the successor function, the set {∅, 2o} would fail the condition since 2o is not an element of the successor of or {∅}. Remember that we are applying 𝐹 to subsets of 𝑦, not elements of 𝑦. In fact, even the set {1o} fails the condition, since the only subset of previously generated elements is , and suc ∅ does not have 1o as an element. However, we can let 𝑦 be any ordinal, since each of its elements is generated by starting with and repeatedly applying the successor function.

A similar definition I initially used for setrecs(𝐹) was setrecs(𝐹) = ran recs((𝑔 ∈ V ↦ (𝐹 ran 𝑔))). I had initially tried and failed to find an elementary definition, and I had proven theorems analogous to setrec1 44774 and setrec2v 44779 using the old definition before I found the new one. I decided to change definitions for two reasons. First, as John Horton Conway noted in the Appendix to Part Zero of On Numbers and Games, mathematicians should not be caught up in any particular formalization, such as ZF set theory. Instead, they should work under whatever framework best suits the problem, and the formal bases used for different problems can be shown to be equivalent. Thus, Conway preferred defining surreal numbers as equivalence classes of surreal number forms, rather than sign-expansions. Although sign-expansions are easier to implement in ZF set theory, Conway argued that "formalisation within some particular axiomatic set theory is irrelevant". Furthermore, one of the most remarkable properties of the theory of surreal numbers is that it generates so much from almost nothing. Using sign-expansions as the formal definition destroys the beauty of surreal numbers, because ordinals are already built in. For this reason, I replaced the old definition of setrecs, which also relied heavily on ordinal numbers. On the other hand, both surreal numbers and the elementary definition of setrecs immediately generate the ordinal numbers from a (relatively) very simple set-theoretical basis.

Second, although it is still complicated to formalize the theory of recursively generated sets within ZF set theory, it is actually simpler and more natural to do so with set theory directly than with the theory of ordinal numbers. As Conway wrote, indexing the "birthdays" of sets is and should be unnecessary. Using an elementary definition for setrecs removes the reliance on the previously developed theory of ordinal numbers, allowing proofs to be simpler and more direct.

Formalizing surreal numbers within Metamath is probably still not in the spirit of Conway. He said that "attempts to force arbitrary theories into a single formal straitjacket... produce unnecessarily cumbrous and inelegant contortions." Nevertheless, Metamath has proven to be much more versatile than it seems at first, and I think the theory of surreal numbers can be natural while fitting well into the Metamath framework.

The difficulty in writing a definition in Metamath for setrecs(𝐹) is that the necessary properties to prove are self-referential (see setrec1 44774 and setrec2v 44779), so we cannot simply write the properties we want inside a class abstraction as with most definitions. As noted in the comment of df-rdg 8038, this is not actually a requirement of the Metamath language, but we would like to be able to eliminate all definitions by direct mechanical substitution.

We cannot define setrecs using a class abstraction directly, because nothing about its individual elements tells us whether they are in the set. We need to know about previous elements first. One way of getting around this problem without indexing is by defining setrecs(𝐹) as a union or intersection of suitable sets. Thus, instead of using a class abstraction for the elements of setrecs(𝐹), which seems to be impossible, we can use a class abstraction for supersets or subsets of setrecs(𝐹), which "know" about multiple individual elements at a time.

Note that we cannot define setrecs(𝐹) as an intersection of sets, because in general it is a proper class, so any supersets would also be proper classes. However, a proper class can be a union of sets, as long as the collection of such sets is a proper class. Therefore, it is feasible to define setrecs(𝐹) as a union of a class abstraction.

If setrecs(𝐹) = 𝐴, the elements of A must be subsets of setrecs(𝐹) which together include everything recursively generated by 𝐹. We can do this by letting 𝐴 be the class of sets 𝑥 whose elements are all recursively generated by 𝐹.

One necessary condition is that each element of a given 𝑥𝐴 must be generated by 𝐹 when applied to a previous element 𝑦𝐴. In symbols, 𝑥𝐴𝑦𝐴(𝑦𝑥𝑥 ⊆ (𝐹𝑦))}. However, this is not sufficient. All fixed points 𝑥 of 𝐹 will satisfy this condition whether they should be in setrecs(𝐹) or not. If we replace the subset relation with the proper subset relation, 𝑥 cannot be the empty set, even though the empty set should be in 𝐴. Therefore this condition cannot be used in the definition, even if we can find a way to avoid making it circular.

A better strategy is to find a necessary and sufficient condition for all the elements of a set 𝑦𝐴 to be generated by 𝐹 when applied only to sets of previously generated elements within 𝑦. For example, taking 𝐹 to be the successor function, we can let 𝐴 = On rather than 𝒫 On, and we will still have 𝐴 = On as required. This gets rid of the circularity of the definition, since we should have a condition to test whether a given set 𝑦 is in 𝐴 without knowing about any of the other elements of 𝐴.

The definition I ended up using accomplishes this using induction: 𝐴 is defined as the class of sets 𝑦 for which a sort of induction on the elements of 𝑦 holds. However, when creating a definition for setrecs that did not rely on ordinal numbers, I tried at first to write a definition using the well-founded relation predicate, Fr. I thought that this would be simple to do once I found a suitable definition using induction, just as the least- element principle is equivalent to induction on the positive integers. If we let 𝑅 = {⟨𝑎, 𝑏⟩ ∣ (𝐹𝑎) ⊆ 𝑏}, then (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥¬ (𝐹𝑧) ⊆ 𝑦)).

On 22-Jul-2020 I came up with the following definition (Version 1) phrased in terms of induction: {𝑦 ∣ ∀𝑧 (∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ∈ 𝑧)) → 𝑦𝑧)}

In Aug-2020 I came up with an equivalent definition with the goal of phrasing it in terms of the relation Fr. It is the contrapositive of the previous one with 𝑧 replaced by its complement. {𝑦 ∣ ∀𝑧 (𝑦𝑧 → ∃𝑤(𝑤𝑦 ∧ (𝐹𝑤) ∈ 𝑧 ∧ ¬ 𝑤𝑧))}

These definitions didn't work because the induction didn't "get off the ground." If 𝑧 does not contain the empty set, the condition (∀𝑤...𝑦𝑧 fails, so 𝑦 = ∅ doesn't get included in 𝐴 even though it should. This could be fixed by adding the base case as a separate requirement, but the subtler problem would remain that rather than a set of "acceptable" sets, what we really need is a collection 𝑧 of all individuals that have been generated so far. So one approach is to replace every occurrence of 𝑧 with 𝑧, making 𝑧 a set of individuals rather than a family of sets. That solves this problem, but it complicates the foundedness version of the definition, which looked cleaner in Version 1.

There was another problem with Version 1. If we let 𝐹 be the power set function, then the induction in the inductive version works for 𝑧 being the class of transitive sets, restricted to subsets of 𝑦. Therefore, 𝑦 must be transitive by definition of 𝑧. This doesn't affect the union of all such 𝑦, but it may or may not be desirable. The problem is that 𝐹 is only applied to transitive sets, because of the strong requirement 𝑤𝑧, so the definition requires the additional constraint (𝑎𝑏 → (𝐹𝑎) ⊆ (𝐹𝑏)) in order to work. This issue can also be avoided by replacing 𝑧 with 𝑧. The induction version of the result is used in the final definition.

Version 2: (18-Aug-2020) Induction: {𝑦 ∣ ∀𝑧 (∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)} Foundedness: {𝑦 ∣ ∀𝑧(𝑦𝑧 ≠ ∅ → 𝑤(𝑤𝑦𝑤𝑧 = ∅ ∧ (𝐹𝑤) ∩ 𝑧 ≠ ∅))}

In the induction version, not only does 𝑧 include all the elements of 𝑦, but it must include the elements of (𝐹𝑤) for 𝑤 ⊆ (𝑦𝑧) even if those elements of (𝐹𝑤) are not in 𝑦. We shouldn't care about any of the elements of 𝑧 outside 𝑦, but this detail doesn't affect the correctness of the definition. If we replaced (𝐹𝑤) in the definition by ((𝐹𝑤) ∩ 𝑦), we would get the same class for setrecs(𝐹). Suppose we could find a 𝑧 for which the condition fails for a given 𝑦 under the changed definition. Then the antecedent would be true, but 𝑦𝑧 would be false. We could then simply add all elements of (𝐹𝑤) outside of 𝑦 for any 𝑤𝑦, which we can do because all the classes involved are sets. This is not trivial and requires the axioms of union, power set, and replacement. However, the expanded 𝑧 fails the condition under the Metamath definition. The other direction is easier. If a certain 𝑧 fails the Metamath definition, then all (𝐹𝑤) ⊆ 𝑧 for 𝑤 ⊆ (𝑦𝑧), and in particular ((𝐹𝑤) ∩ 𝑦) ⊆ 𝑧.

The foundedness version is starting to look more like ax-reg 9048! We want to take advantage of the preexisting relation Fr, which seems closely related to our foundedness definition. Since we only care about the elements of 𝑧 which are subsets of 𝑦, we can restrict 𝑧 to 𝑦 in the foundedness definition. Furthermore, instead of quantifying over 𝑤, quantify over the elements 𝑣𝑧 overlapping with 𝑤. Versions 3, 4, and 5 are all equivalent to Version 2.

Version 3 - Foundedness (5-Sep-2020): {𝑦 ∣ ∀𝑧((𝑧𝑦𝑧 ≠ ∅) → ∃𝑣𝑧𝑤(𝑤𝑦𝑤𝑧 = ∅ ∧ 𝑣 ∈ (𝐹𝑤)))}

Now, if we replace (𝐹𝑤) by ((𝐹𝑤) ∩ 𝑦), we do not change the definition. We already know that 𝑣𝑦 since 𝑣𝑧 and 𝑧𝑦. All we need to show in order to prove that this change leads to an equivalent definition is to find

To make our definition look exactly like df-fr 5507, we add another variable 𝑢 representing the nonexistent element of 𝑤 in 𝑧.

Version 4 - Foundedness (6-Sep-2020): {𝑦 ∣ ∀𝑧((𝑧𝑦𝑧 ≠ ∅) → 𝑣𝑧𝑤𝑢𝑧(𝑤𝑦 ∧ ¬ 𝑢𝑤𝑣 ∈ (𝐹𝑤))

This is so close to df-fr 5507; the only change needed is to switch 𝑤 with 𝑢𝑧. Unfortunately, I couldn't find any way to switch the quantifiers without interfering with the definition. Maybe there is a definition equivalent to this one that uses Fr, but I couldn't find one. Yet, we can still find a remarkable similarity between Foundedness Version 2 and ax-reg 9048. Rather than a disjoint element of 𝑧, there's a disjoint coverer of an element of 𝑧.

Finally, here's a different dead end I followed:

To clean up our foundedness definition, we keep 𝑧 as a family of sets 𝑦 but allow 𝑤 to be any subset of 𝑧 in the induction. With this stronger induction, we can also allow for the stronger requirement 𝒫 𝑦𝑧 rather than only 𝑦𝑧. This will help improve the foundedness version.

Version 1.1 (28-Aug-2020) Induction: {𝑦 ∣ ∀𝑧(∀𝑤 (𝑤𝑦 → (𝑤 𝑧 → (𝐹𝑤) ∈ 𝑧)) → 𝒫 𝑦𝑧)} Foundedness: {𝑦 ∣ ∀𝑧(∃𝑎(𝑎𝑦𝑎𝑧) → ∃𝑤(𝑤𝑦𝑤 𝑧 = ∅ ∧ (𝐹𝑤) ∈ 𝑧))}

( Edit (Aug 31) - this isn't true! Nothing forces the subset of an element of 𝑧 to be in 𝑧. Version 2 does not have this issue. )

Similarly, we could allow 𝑤 to be any subset of any element of 𝑧 rather than any subset of 𝑧. I think this has the same problem.

We want to take advantage of the preexisting relation Fr, which seems closely related to our foundedness definition. Since we only care about the elements of 𝑧 which are subsets of 𝑦, we can restrict 𝑧 to 𝒫 𝑦 in the foundedness definition:

Version 1.2 (31-Aug-2020) Foundedness: {𝑦 ∣ ∀𝑧((𝑧 ⊆ 𝒫 𝑦𝑧 ≠ ∅) → ∃𝑤(𝑤 ∈ 𝒫 𝑦𝑤 𝑧 = ∅ ∧ (𝐹𝑤) ∈ 𝑧))}

Now this looks more like df-fr 5507! The last step necessary to be able to use Fr directly in our definition is to replace (𝐹𝑤) with its own setvar variable, corresponding to 𝑦 in df-fr 5507.

This definition is incorrect, though, since there's nothing forcing the subset of an element of 𝑧 to be in 𝑧.

Version 1.3 (31-Aug-2020) Induction: {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤 𝑧 → (𝑤𝑧 ∧ (𝐹𝑤) ∈ 𝑧))) → 𝒫 𝑦𝑧)} Foundedness: {𝑦 ∣ ∀𝑧((𝑧 ⊆ 𝒫 𝑦𝑧 ≠ ∅) → ∃𝑤(𝑤 ∈ 𝒫 𝑦 𝑤 𝑧 = ∅ ∧ (𝑤𝑧 ∨ (𝐹𝑤) ∈ 𝑧)))}

𝑧 must contain the supersets of each of its elements in the foundedness version, and we can't make any restrictions on 𝑧 or 𝐹, so this doesn't work.

Let's try letting R be the covering relation 𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑏 ∈ (𝐹𝑎)} to solve the transitivity issue (i.e. that if 𝐹 is the power set relation, 𝐴 consists only of transitive sets). The set (𝐹𝑤) corresponds to the variable 𝑦 in df-fr 5507. Thus, in our case, df-fr 5507 is equivalent to (𝑅 Fr 𝐴 ↔ ∀𝑧((𝑧𝐴𝑧 ≠ ∅) → ∃𝑤((𝐹𝑤) ∈ 𝑧 ∧ ¬ ∃𝑣𝑧𝑣𝑅(𝐹𝑤))). Substituting our relation 𝑅 gives (𝑅 Fr 𝐴 ↔ ∀𝑧((𝑧𝐴𝑧 ≠ ∅) → 𝑤((𝐹𝑤) ∈ 𝑧 ∧ ¬ ∃𝑣𝑧(𝐹𝑤) ∈ (𝐹𝑣)))

This doesn't work for non-injective 𝐹 because we need all 𝑧 to be straddlers, but we don't necessarily need all-straddlers; loops within z are fine for non-injective F.

Consider the foundedness form of Version 1. We want to show ¬ 𝑤𝑧 ↔ ∀𝑣𝑧¬ 𝑣𝑅(𝐹𝑤) so we can replace one with the other. Negate both sides: 𝑤𝑧 ↔ ∃𝑣𝑧𝑣𝑅(𝐹𝑤)

If 𝐹 is injective, then we should be able to pick a suitable R, being careful about the above problem for some F (for example z = transitivity) when changing the antecedent y e. z' to z =/= (/). If we're clever, we can get rid of the injectivity requirement. The forward direction of the above equivalence always holds, but the key is that although the backwards direction doesn't hold in general, we can always find some z' where it doesn't work for 𝑤 itself. If there exists a z' where the version with the w condition fails, then there exists a z' where the version with the v condition also fails. However, Version 1 is not a correct definition, so this doesn't work either. (Contributed by Emmett Weisz, 18-Aug-2020.) (New usage is discouraged.)

setrecs(𝐹) = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}

Theoremsetrecseq 44768 Equality theorem for set recursion. (Contributed by Emmett Weisz, 17-Feb-2021.)
(𝐹 = 𝐺 → setrecs(𝐹) = setrecs(𝐺))

Theoremnfsetrecs 44769 Bound-variable hypothesis builder for setrecs. (Contributed by Emmett Weisz, 21-Oct-2021.)
𝑥𝐹       𝑥setrecs(𝐹)

Theoremsetrec1lem1 44770* Lemma for setrec1 44774. This is a utility theorem showing the equivalence of the statement 𝑋𝑌 and its expanded form. The proof uses elabg 3664 and equivalence theorems.

Variable 𝑌 is the class of sets 𝑦 that are recursively generated by the function 𝐹. In other words, 𝑦𝑌 iff by starting with the empty set and repeatedly applying 𝐹 to subsets 𝑤 of our set, we will eventually generate all the elements of 𝑌. In this theorem, 𝑋 is any element of 𝑌, and 𝑉 is any class. (Contributed by Emmett Weisz, 16-Oct-2020.) (New usage is discouraged.)

𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}    &   (𝜑𝑋𝑉)       (𝜑 → (𝑋𝑌 ↔ ∀𝑧(∀𝑤(𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑋𝑧)))

Theoremsetrec1lem2 44771* Lemma for setrec1 44774. If a family of sets are all recursively generated by 𝐹, so is their union. In this theorem, 𝑋 is a family of sets which are all elements of 𝑌, and 𝑉 is any class. Use dfss3 3954, equivalence and equality theorems, and unissb at the end. Sandwich with applications of setrec1lem1. (Contributed by Emmett Weisz, 24-Jan-2021.) (New usage is discouraged.)
𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}    &   (𝜑𝑋𝑉)    &   (𝜑𝑋𝑌)       (𝜑 𝑋𝑌)

Theoremsetrec1lem3 44772* Lemma for setrec1 44774. If each element 𝑎 of 𝐴 is covered by a set 𝑥 recursively generated by 𝐹, then there is a single such set covering all of 𝐴. The set is constructed explicitly using setrec1lem2 44771. It turns out that 𝑥 = 𝐴 also works, i.e., given the hypotheses it is possible to prove that 𝐴𝑌. I don't know if proving this fact directly using setrec1lem1 44770 would be any easier than the current proof using setrec1lem2 44771, and it would only slightly simplify the proof of setrec1 44774. Other than the use of bnd2d 44764, this is a purely technical theorem for rearranging notation from that of setrec1lem2 44771 to that of setrec1 44774. (Contributed by Emmett Weisz, 20-Jan-2021.) (New usage is discouraged.)
𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}    &   (𝜑𝐴 ∈ V)    &   (𝜑 → ∀𝑎𝐴𝑥(𝑎𝑥𝑥𝑌))       (𝜑 → ∃𝑥(𝐴𝑥𝑥𝑌))

Theoremsetrec1lem4 44773* Lemma for setrec1 44774. If 𝑋 is recursively generated by 𝐹, then so is 𝑋 ∪ (𝐹𝐴).

In the proof of setrec1 44774, the following is substituted for this theorem's 𝜑: (𝜑 ∧ (𝐴𝑥𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤 (𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)})) Therefore, we cannot declare 𝑧 to be a distinct variable from 𝜑, since we need it to appear as a bound variable in 𝜑. This theorem can be proven without the hypothesis 𝑧𝜑, but the proof would be harder to read because theorems in deduction form would be interrupted by theorems like eximi 1828, making the antecedent of each line something more complicated than 𝜑. The proof of setrec1lem2 44771 could similarly be made easier to read by adding the hypothesis 𝑧𝜑, but I had already finished the proof and decided to leave it as is. (Contributed by Emmett Weisz, 26-Nov-2020.) (New usage is discouraged.)

𝑧𝜑    &   𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐴𝑋)    &   (𝜑𝑋𝑌)       (𝜑 → (𝑋 ∪ (𝐹𝐴)) ∈ 𝑌)

Theoremsetrec1 44774 This is the first of two fundamental theorems about set recursion from which all other facts will be derived. It states that the class setrecs(𝐹) is closed under 𝐹. This effectively sets the actual value of setrecs(𝐹) as a lower bound for setrecs(𝐹), as it implies that any set generated by successive applications of 𝐹 is a member of 𝐵. This theorem "gets off the ground" because we can start by letting 𝐴 = ∅, and the hypotheses of the theorem will hold trivially.

Variable 𝐵 represents an abbreviation of setrecs(𝐹) or another name of setrecs(𝐹) (for an example of the latter, see theorem setrecon).

Proof summary: Assume that 𝐴𝐵, meaning that all elements of 𝐴 are in some set recursively generated by 𝐹. Then by setrec1lem3 44772, 𝐴 is a subset of some set recursively generated by 𝐹. (It turns out that 𝐴 itself is recursively generated by 𝐹, but we don't need this fact. See the comment to setrec1lem3 44772.) Therefore, by setrec1lem4 44773, (𝐹𝐴) is a subset of some set recursively generated by 𝐹. Thus, by ssuni 4852, it is a subset of the union of all sets recursively generated by 𝐹.

See df-setrecs 44767 for a detailed description of how the setrecs definition works.

(Contributed by Emmett Weisz, 9-Oct-2020.)

𝐵 = setrecs(𝐹)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐹𝐴) ⊆ 𝐵)

Theoremsetrec2fun 44775* This is the second of two fundamental theorems about set recursion from which all other facts will be derived. It states that the class setrecs(𝐹) is a subclass of all classes 𝐶 that are closed under 𝐹. Taken together, theorems setrec1 44774 and setrec2v 44779 say that setrecs(𝐹) is the minimal class closed under 𝐹.

We express this by saying that if 𝐹 respects the relation and 𝐶 is closed under 𝐹, then 𝐵𝐶. By substituting strategically constructed classes for 𝐶, we can easily prove many useful properties. Although this theorem cannot show equality between 𝐵 and 𝐶, if we intend to prove equality between 𝐵 and some particular class (such as On), we first apply this theorem, then the relevant induction theorem (such as tfi 7560) to the other class. (Contributed by Emmett Weisz, 15-Feb-2021.) (New usage is discouraged.)

𝑎𝐹    &   𝐵 = setrecs(𝐹)    &   Fun 𝐹    &   (𝜑 → ∀𝑎(𝑎𝐶 → (𝐹𝑎) ⊆ 𝐶))       (𝜑𝐵𝐶)

Theoremsetrec2lem1 44776* Lemma for setrec2 44778. The functional part of 𝐹 has the same values as 𝐹. (Contributed by Emmett Weisz, 4-Mar-2021.) (New usage is discouraged.)
((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹𝑎)

Theoremsetrec2lem2 44777* Lemma for setrec2 44778. The functional part of 𝐹 is a function. (Contributed by Emmett Weisz, 6-Mar-2021.) (New usage is discouraged.)
Fun (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})

Theoremsetrec2 44778* This is the second of two fundamental theorems about set recursion from which all other facts will be derived. It states that the class setrecs(𝐹) is a subclass of all classes 𝐶 that are closed under 𝐹. Taken together, theorems setrec1 44774 and setrec2v 44779 uniquely determine setrecs(𝐹) to be the minimal class closed under 𝐹.

We express this by saying that if 𝐹 respects the relation and 𝐶 is closed under 𝐹, then 𝐵𝐶. By substituting strategically constructed classes for 𝐶, we can easily prove many useful properties. Although this theorem cannot show equality between 𝐵 and 𝐶, if we intend to prove equality between 𝐵 and some particular class (such as On), we first apply this theorem, then the relevant induction theorem (such as tfi 7560) to the other class.

(Contributed by Emmett Weisz, 2-Sep-2021.)

𝑎𝐹    &   𝐵 = setrecs(𝐹)    &   (𝜑 → ∀𝑎(𝑎𝐶 → (𝐹𝑎) ⊆ 𝐶))       (𝜑𝐵𝐶)

Theoremsetrec2v 44779* Version of setrec2 44778 with a disjoint variable condition instead of a non-freeness hypothesis. (Contributed by Emmett Weisz, 6-Mar-2021.)
𝐵 = setrecs(𝐹)    &   (𝜑 → ∀𝑎(𝑎𝐶 → (𝐹𝑎) ⊆ 𝐶))       (𝜑𝐵𝐶)

Theoremsetis 44780* Version of setrec2 44778 expressed as an induction schema. This theorem is a generalization of tfis3 7564. (Contributed by Emmett Weisz, 27-Feb-2022.)
𝐵 = setrecs(𝐹)    &   (𝑏 = 𝐴 → (𝜓𝜒))    &   (𝜑 → ∀𝑎(∀𝑏𝑎 𝜓 → ∀𝑏 ∈ (𝐹𝑎)𝜓))       (𝜑 → (𝐴𝐵𝜒))

20.41.2.2  Examples and properties of set recursion

Theoremelsetrecslem 44781* Lemma for elsetrecs 44782. Any element of setrecs(𝐹) is generated by some subset of setrecs(𝐹). This is much weaker than setrec2v 44779. To see why this lemma also requires setrec1 44774, consider what would happen if we replaced 𝐵 with {𝐴}. The antecedent would still hold, but the consequent would fail in general. Consider dispensing with the deduction form. (Contributed by Emmett Weisz, 11-Jul-2021.) (New usage is discouraged.)
𝐵 = setrecs(𝐹)       (𝐴𝐵 → ∃𝑥(𝑥𝐵𝐴 ∈ (𝐹𝑥)))

Theoremelsetrecs 44782* A set 𝐴 is an element of setrecs(𝐹) iff 𝐴 is generated by some subset of setrecs(𝐹). The proof requires both setrec1 44774 and setrec2 44778, but this theorem is not strong enough to uniquely determine setrecs(𝐹). If 𝐹 respects the subset relation, the theorem still holds if both occurrences of are replaced by for a stronger version of the theorem. (Contributed by Emmett Weisz, 12-Jul-2021.)
𝐵 = setrecs(𝐹)       (𝐴𝐵 ↔ ∃𝑥(𝑥𝐵𝐴 ∈ (𝐹𝑥)))

Theoremsetrecsss 44783 The setrecs operator respects the subset relation between two functions 𝐹 and 𝐺. (Contributed by Emmett Weisz, 13-Mar-2022.)
(𝜑 → Fun 𝐺)    &   (𝜑𝐹𝐺)       (𝜑 → setrecs(𝐹) ⊆ setrecs(𝐺))

Theoremsetrecsres 44784 A recursively generated class is unaffected when its input function is restricted to subsets of the class. (Contributed by Emmett Weisz, 14-Mar-2022.)
𝐵 = setrecs(𝐹)    &   (𝜑 → Fun 𝐹)       (𝜑𝐵 = setrecs((𝐹 ↾ 𝒫 𝐵)))

Theoremvsetrec 44785 Construct V using set recursion. The proof indirectly uses trcl 9162, which relies on rec, but theoretically 𝐶 in trcl 9162 could be constructed using setrecs instead. The proof of this theorem uses the dummy variable 𝑎 rather than 𝑥 to avoid a distinct variable requirement between 𝐹 and 𝑥. (Contributed by Emmett Weisz, 23-Jun-2021.)
𝐹 = (𝑥 ∈ V ↦ 𝒫 𝑥)       setrecs(𝐹) = V

Theorem0setrec 44786 If a function sends the empty set to itself, the function will not recursively generate any sets, regardless of its other values. (Contributed by Emmett Weisz, 23-Jun-2021.)
(𝜑 → (𝐹‘∅) = ∅)       (𝜑 → setrecs(𝐹) = ∅)

Theoremonsetreclem1 44787* Lemma for onsetrec 44790. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.)
𝐹 = (𝑥 ∈ V ↦ { 𝑥, suc 𝑥})       (𝐹𝑎) = { 𝑎, suc 𝑎}

Theoremonsetreclem2 44788* Lemma for onsetrec 44790. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.)
𝐹 = (𝑥 ∈ V ↦ { 𝑥, suc 𝑥})       (𝑎 ⊆ On → (𝐹𝑎) ⊆ On)

Theoremonsetreclem3 44789* Lemma for onsetrec 44790. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.)
𝐹 = (𝑥 ∈ V ↦ { 𝑥, suc 𝑥})       (𝑎 ∈ On → 𝑎 ∈ (𝐹𝑎))

Theoremonsetrec 44790 Construct On using set recursion. When 𝑥 ∈ On, the function 𝐹 constructs the least ordinal greater than any of the elements of 𝑥, which is 𝑥 for a limit ordinal and suc 𝑥 for a successor ordinal.

For example, (𝐹‘{1o, 2o}) = { {1o, 2o}, suc {1o, 2o}} = {2o, 3o} which contains 3o, and (𝐹‘ω) = { ω, suc ω} = {ω, ω +o 1o}, which contains ω. If we start with the empty set and keep applying 𝐹 transfinitely many times, all ordinal numbers will be generated.

Any function 𝐹 fulfilling lemmas onsetreclem2 44788 and onsetreclem3 44789 will recursively generate On; for example, 𝐹 = (𝑥 ∈ V ↦ suc suc 𝑥}) also works. Whether this function or the function in the theorem is used, taking this theorem as a definition of On is unsatisfying because it relies on the different properties of limit and successor ordinals. A different approach could be to let 𝐹 = (𝑥 ∈ V ↦ {𝑦 ∈ 𝒫 𝑥 ∣ Tr 𝑦}), based on dfon2 33025.

The proof of this theorem uses the dummy variable 𝑎 rather than 𝑥 to avoid a distinct variable condition between 𝐹 and 𝑥. (Contributed by Emmett Weisz, 22-Jun-2021.)

𝐹 = (𝑥 ∈ V ↦ { 𝑥, suc 𝑥})       setrecs(𝐹) = On

20.41.3  Construction of Games and Surreal Numbers

Model organization after organization of reals - see TOC

Syntaxcpg 44791 Extend class notation to include the class of partisan game forms.
class Pg

Definitiondf-pg 44792 Define the class of partisan games. More precisely, this is the class of partisan game forms, many of which represent equal partisan games. In Metamath, equality between partisan games is represented by a different equivalence relation than class equality. (Contributed by Emmett Weisz, 22-Aug-2021.)
Pg = setrecs((𝑥 ∈ V ↦ (𝒫 𝑥 × 𝒫 𝑥)))

Theoremelpglem1 44793* Lemma for elpg 44796. (Contributed by Emmett Weisz, 28-Aug-2021.)
(∃𝑥(𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥)) → ((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg))

Theoremelpglem2 44794* Lemma for elpg 44796. (Contributed by Emmett Weisz, 28-Aug-2021.)
(((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg) → ∃𝑥(𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥)))

Theoremelpglem3 44795* Lemma for elpg 44796. (Contributed by Emmett Weisz, 28-Aug-2021.)
(∃𝑥(𝑥 ⊆ Pg ∧ 𝐴 ∈ ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥)) ↔ (𝐴 ∈ (V × V) ∧ ∃𝑥(𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥))))

Theoremelpg 44796 Membership in the class of partisan games. In John Horton Conway's On Numbers and Games, this is stated as "If 𝐿 and 𝑅 are any two sets of games, then there is a game {𝐿𝑅}. All games are constructed in this way." The first sentence corresponds to the backward direction of our theorem, and the second to the forward direction. (Contributed by Emmett Weisz, 27-Aug-2021.)
(𝐴 ∈ Pg ↔ (𝐴 ∈ (V × V) ∧ (1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg))

20.42  Mathbox for David A. Wheeler

This is the mathbox of David A. Wheeler, dwheeler at dwheeler dot com. Among other things, I have added a number of formal definitions for widely-used functions, e.g., those defined in ISO 80000-2:2009(E) Quantities and units - Part 2: Mathematical signs and symbols used in the natural sciences and technology and the NIST Digital Library of Mathematical Functions http://dlmf.nist.gov/.

20.42.1  Natural deduction

Theoremsbidd 44797 An identity theorem for substitution. See sbid 2249. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.)
(𝜑 → [𝑥 / 𝑥]𝜓)       (𝜑𝜓)

Theoremsbidd-misc 44798 An identity theorem for substitution. See sbid 2249. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.)
((𝜑 → [𝑥 / 𝑥]𝜓) ↔ (𝜑𝜓))

20.42.2  Greater than, greater than or equal to.

As a stylistic issue, set.mm prefers 'less than' instead of 'greater than' to reduce the number of conversion steps. Here we formally define the widely-used relations 'greater than' and 'greater than or equal to', so that we have formal definitions of them, as well as a few related theorems.

Syntaxcge-real 44799 Extend wff notation to include the 'greater than or equal to' relation, see df-gte 44801.
class

Syntaxcgt 44800 Extend wff notation to include the 'greater than' relation, see df-gt 44802.
class >

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