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Theorem List for Metamath Proof Explorer - 45101-45200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrrxsphere 45101* 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 45102* 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 45103* The sphere around the origin 0 (see rrx0 23999) 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 45104* Lemma for line2y 45108. 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 45105* 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 45106* Lemma for line2x 45107. 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 45107* 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 45108* 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 45109 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 45110 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 45111 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 45112 Lemma for itsclc0 45124. 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 45113 Lemma for itsclc0 45124. 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 45114 Lemma for itsclc0 45124. 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 45115 Lemma 1 for itsclc0yqsol 45117. (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 45116 Lemma 2 for itsclc0yqsol 45117. (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 45117 Lemma for itsclc0 45124. 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 45118 Lemma for itsclc0 45124. 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 45119 Lemma for itsclc0 45124. 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 45120 Lemma for itsclc0 45124. 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 45121 Lemma for itsclc0 45124. 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 45122 Lemma for itsclc0 45124. 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 45123 Lemma for itsclc0 45124. 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 45124* 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 45125* 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 45126 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 45127 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 45128 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 45129* 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 45130* 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 45131 Lemma 1 for 2itscp 45134. (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 45132 Lemma 2 for 2itscp 45134. (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 45133 Lemma D for 2itscp 45134. (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 45134 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 45135 Lemma 1 for itscnhlinecirc02p 45138. (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 45136 Lemma 2 for itscnhlinecirc02p 45138. (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 45137 Lemma 3 for itscnhlinecirc02p 45138. (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 45138* 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 45139* Lemma for inlinecirc02p 45140. (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 45140 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 45141* 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))))
 
20.42  Mathbox for Emmett Weisz
 
20.42.1  Miscellaneous Theorems

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

 
Theoremnfintd 45142 Bound-variable hypothesis builder for intersection. (Contributed by Emmett Weisz, 16-Jan-2020.)
(𝜑𝑥𝐴)       (𝜑𝑥 𝐴)
 
Theoremnfiund 45143* Bound-variable hypothesis builder for indexed union. (Contributed by Emmett Weisz, 6-Dec-2019.) Add disjoint variable condition to avoid ax-13 2391. See nfiundg 45144 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
𝑥𝜑    &   (𝜑𝑦𝐴)    &   (𝜑𝑦𝐵)       (𝜑𝑦 𝑥𝐴 𝐵)
 
Theoremnfiundg 45144 Bound-variable hypothesis builder for indexed union. Usage of this theorem is discouraged because it depends on ax-13 2391, see nfiund 45143 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 45145* The indexed union of a collection of ordinal numbers 𝐵(𝑥) is ordinal. This proof is based on the proof of ssorduni 7485, but does not use it directly, since ssorduni 7485 does not work when 𝐵 is a proper class. (Contributed by Emmett Weisz, 3-Nov-2019.)
(∀𝑥𝐴 Ord 𝐵 → Ord 𝑥𝐴 𝐵)
 
Theoremiunordi 45146* 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 45147 Specialization deduction, using implicit substitution. Based on the proof of spimed 2407. (Contributed by Emmett Weisz, 17-Jan-2020.)
(𝜒 → Ⅎ𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜒 → (∀𝑥𝜑𝜓))
 
Theoremspcdvw 45148* A version of spcdv 3568 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 45149* 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 45150* Deduction form of bnd2 9310. (Contributed by Emmett Weisz, 19-Jan-2021.)
(𝜑𝐴 ∈ V)    &   (𝜑 → ∀𝑥𝐴𝑦𝐵 𝜓)       (𝜑 → ∃𝑧(𝑧𝐵 ∧ ∀𝑥𝐴𝑦𝑧 𝜓))
 
Theoremdffun3f 45151* 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)))
 
20.42.2  Set Recursion
 
20.42.2.1  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 45153. The class 𝑌 is explained in the comment of setrec1lem1 45156. Glossaries of symbols used in individual proofs, or used differently in different proofs, are in the comments of those proofs.

 
Syntaxcsetrecs 45152 Extend class notation to include a set defined by transfinite recursion.
class setrecs(𝐹)
 
Definitiondf-setrecs 45153* 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 45160 and setrec2v 45165, 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 45176 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 45160 and setrec2v 45165 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 45160 and setrec2v 45165), 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 8033, 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 9044! 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 5491, 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 5491; 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 9044. 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 5491! 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 5491.

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 5491. Thus, in our case, df-fr 5491 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 45154 Equality theorem for set recursion. (Contributed by Emmett Weisz, 17-Feb-2021.)
(𝐹 = 𝐺 → setrecs(𝐹) = setrecs(𝐺))
 
Theoremnfsetrecs 45155 Bound-variable hypothesis builder for setrecs. (Contributed by Emmett Weisz, 21-Oct-2021.)
𝑥𝐹       𝑥setrecs(𝐹)
 
Theoremsetrec1lem1 45156* Lemma for setrec1 45160. This is a utility theorem showing the equivalence of the statement 𝑋𝑌 and its expanded form. The proof uses elabg 3641 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 45157* Lemma for setrec1 45160. 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 3930, 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 45158* Lemma for setrec1 45160. 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 45157. 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 45156 would be any easier than the current proof using setrec1lem2 45157, and it would only slightly simplify the proof of setrec1 45160. Other than the use of bnd2d 45150, this is a purely technical theorem for rearranging notation from that of setrec1lem2 45157 to that of setrec1 45160. (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 45159* Lemma for setrec1 45160. If 𝑋 is recursively generated by 𝐹, then so is 𝑋 ∪ (𝐹𝐴).

In the proof of setrec1 45160, 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 1836, making the antecedent of each line something more complicated than 𝜑. The proof of setrec1lem2 45157 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 45160 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 45158, 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 45158.) Therefore, by setrec1lem4 45159, (𝐹𝐴) is a subset of some set recursively generated by 𝐹. Thus, by ssuni 4838, it is a subset of the union of all sets recursively generated by 𝐹.

See df-setrecs 45153 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 45161* 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 45160 and setrec2v 45165 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 7553) 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 45162* Lemma for setrec2 45164. 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 45163* Lemma for setrec2 45164. 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 45164* 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 45160 and setrec2v 45165 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 7553) to the other class.

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

𝑎𝐹    &   𝐵 = setrecs(𝐹)    &   (𝜑 → ∀𝑎(𝑎𝐶 → (𝐹𝑎) ⊆ 𝐶))       (𝜑𝐵𝐶)
 
Theoremsetrec2v 45165* Version of setrec2 45164 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 45166* Version of setrec2 45164 expressed as an induction schema. This theorem is a generalization of tfis3 7557. (Contributed by Emmett Weisz, 27-Feb-2022.)
𝐵 = setrecs(𝐹)    &   (𝑏 = 𝐴 → (𝜓𝜒))    &   (𝜑 → ∀𝑎(∀𝑏𝑎 𝜓 → ∀𝑏 ∈ (𝐹𝑎)𝜓))       (𝜑 → (𝐴𝐵𝜒))
 
20.42.2.2  Examples and properties of set recursion
 
Theoremelsetrecslem 45167* Lemma for elsetrecs 45168. Any element of setrecs(𝐹) is generated by some subset of setrecs(𝐹). This is much weaker than setrec2v 45165. To see why this lemma also requires setrec1 45160, 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 45168* A set 𝐴 is an element of setrecs(𝐹) iff 𝐴 is generated by some subset of setrecs(𝐹). The proof requires both setrec1 45160 and setrec2 45164, 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 45169 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 45170 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 45171 Construct V using set recursion. The proof indirectly uses trcl 9158, which relies on rec, but theoretically 𝐶 in trcl 9158 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 45172 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 45173* Lemma for onsetrec 45176. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.)
𝐹 = (𝑥 ∈ V ↦ { 𝑥, suc 𝑥})       (𝐹𝑎) = { 𝑎, suc 𝑎}
 
Theoremonsetreclem2 45174* Lemma for onsetrec 45176. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.)
𝐹 = (𝑥 ∈ V ↦ { 𝑥, suc 𝑥})       (𝑎 ⊆ On → (𝐹𝑎) ⊆ On)
 
Theoremonsetreclem3 45175* Lemma for onsetrec 45176. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.)
𝐹 = (𝑥 ∈ V ↦ { 𝑥, suc 𝑥})       (𝑎 ∈ On → 𝑎 ∈ (𝐹𝑎))
 
Theoremonsetrec 45176 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 45174 and onsetreclem3 45175 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 33111.

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
 
20.42.3  Construction of Games and Surreal Numbers

Model organization after organization of reals - see TOC

 
Syntaxcpg 45177 Extend class notation to include the class of partisan game forms.
class Pg
 
Definitiondf-pg 45178 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 45179* Lemma for elpg 45182. (Contributed by Emmett Weisz, 28-Aug-2021.)
(∃𝑥(𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥)) → ((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg))
 
Theoremelpglem2 45180* Lemma for elpg 45182. (Contributed by Emmett Weisz, 28-Aug-2021.)
(((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg) → ∃𝑥(𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥)))
 
Theoremelpglem3 45181* Lemma for elpg 45182. (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 45182 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))
 
20.43  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/.

 
20.43.1  Natural deduction
 
Theoremsbidd 45183 An identity theorem for substitution. See sbid 2258. 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 45184 An identity theorem for substitution. See sbid 2258. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.)
((𝜑 → [𝑥 / 𝑥]𝜓) ↔ (𝜑𝜓))
 
20.43.2  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 45185 Extend wff notation to include the 'greater than or equal to' relation, see df-gte 45187.
class
 
Syntaxcgt 45186 Extend wff notation to include the 'greater than' relation, see df-gt 45188.
class >
 
Definitiondf-gte 45187 Define the 'greater than or equal' predicate over the reals. Defined in ISO 80000-2:2009(E) operation 2-7.10. It is used as a primitive in the "NIST Digital Library of Mathematical Functions" , front introduction, "Common Notations and Definitions" section at http://dlmf.nist.gov/front/introduction#Sx4. This relation is merely the converse of the 'less than or equal to' relation defined by df-le 10670.

We do not write this as (𝑥𝑦𝑦𝑥), and similarly we do not write ` > ` as (𝑥 > 𝑦𝑦 < 𝑥), because these are not definitional axioms as understood by mmj2 (those definitions will be flagged as being "potentially non-conservative"). We could write them this way: > = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑦 < 𝑥)} and ≥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑦𝑥)} but these are very complicated. This definition of , and the similar one for > (df-gt 45188), are a bit strange when you see them for the first time, but these definitions are much simpler for us to process and are clearly conservative definitions. (My thanks to Mario Carneiro for pointing out this simpler approach.) See gte-lte 45189 for a more conventional expression of the relationship between < and >. As a stylistic issue, set.mm prefers 'less than' instead of 'greater than' to reduce the number of conversion steps. Thus, we discourage its use, but include its definition so that there is a formal definition of this symbol.

(Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.)

≥ =
 
Definitiondf-gt 45188 The 'greater than' relation is merely the converse of the 'less than or equal to' relation defined by df-lt 10539. Defined in ISO 80000-2:2009(E) operation 2-7.12. See df-gte 45187 for a discussion on why this approach is used for the definition. See gt-lt 45190 and gt-lth 45192 for more conventional expression of the relationship between < and >.

As a stylistic issue, set.mm prefers 'less than or equal' instead of 'greater than or equal' to reduce the number of conversion steps. Thus, we discourage its use, but include its definition so that there is a formal definition of this symbol.

(Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.)

> = <
 
Theoremgte-lte 45189 Simple relationship between and . (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.)
((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵𝐵𝐴))
 
Theoremgt-lt 45190 Simple relationship between < and >. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.)
((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 > 𝐵𝐵 < 𝐴))
 
Theoremgte-lteh 45191 Relationship between and using hypotheses. (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝐵𝐵𝐴)
 
Theoremgt-lth 45192 Relationship between < and > using hypotheses. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴 > 𝐵𝐵 < 𝐴)
 
Theoremex-gt 45193 Simple example of >, in this case, 0 is not greater than 0. This is useful as an example, and helps us gain confidence that we've correctly defined the symbol. (Contributed by David A. Wheeler, 1-Jan-2017.) (New usage is discouraged.)
¬ 0 > 0
 
Theoremex-gte 45194 Simple example of , in this case, 0 is greater than or equal to 0. This is useful as an example, and helps us gain confidence that we've correctly defined the symbol. (Contributed by David A. Wheeler, 1-Jan-2017.) (New usage is discouraged.)
0 ≥ 0
 
20.43.3  Hyperbolic trigonometric functions

It is a convention of set.mm to not use sinh and so on directly, and instead of use expansions such as (cos‘(i · 𝑥)). However, I believe it's important to give formal definitions for these conventional functions as they are typically used, so here they are. A few related identities are also proved.

 
Syntaxcsinh 45195 Extend class notation to include the hyperbolic sine function, see df-sinh 45198.
class sinh
 
Syntaxccosh 45196 Extend class notation to include the hyperbolic cosine function. see df-cosh 45199.
class cosh
 
Syntaxctanh 45197 Extend class notation to include the hyperbolic tangent function, see df-tanh 45200.
class tanh
 
Definitiondf-sinh 45198 Define the hyperbolic sine function (sinh). We define it this way for cmpt 5122, which requires the form (𝑥𝐴𝐵). See sinhval-named 45201 for a simple way to evaluate it. We define this function by dividing by i, which uses fewer operations than many conventional definitions (and thus is more convenient to use in set.mm). See sinh-conventional 45204 for a justification that our definition is the same as the conventional definition of sinh used in other sources. (Contributed by David A. Wheeler, 20-Apr-2015.)
sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
 
Definitiondf-cosh 45199 Define the hyperbolic cosine function (cosh). We define it this way for cmpt 5122, which requires the form (𝑥𝐴𝐵). (Contributed by David A. Wheeler, 10-May-2015.)
cosh = (𝑥 ∈ ℂ ↦ (cos‘(i · 𝑥)))
 
Definitiondf-tanh 45200 Define the hyperbolic tangent function (tanh). We define it this way for cmpt 5122, which requires the form (𝑥𝐴𝐵). (Contributed by David A. Wheeler, 10-May-2015.)
tanh = (𝑥 ∈ (cosh “ (ℂ ∖ {0})) ↦ ((tan‘(i · 𝑥)) / i))
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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