P. 477 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 47601-47700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	itsclc0yqe 47601	Lemma for itsclc0 47611. Quadratic equation for the y-coordinate of the intersection points of an arbitrary line and a circle. This theorem holds even for degenerate lines (𝐴 = 𝐵 = 0). (Contributed by AV, 25-Feb-2023.)
⊢ 𝑄 = ((𝐴↑2) + (𝐵↑2))    &   ⊢ 𝑇 = -(2 · (𝐵 · 𝐶))    &   ⊢ 𝑈 = ((𝐶↑2) − ((𝐴↑2) · (𝑅↑2)))    ⇒   ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝑅 ∈ ℝ+ ∧ (𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ)) → ((((𝑋↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑋) + (𝐵 · 𝑌)) = 𝐶) → ((𝑄 · (𝑌↑2)) + ((𝑇 · 𝑌) + 𝑈)) = 0))
Theorem	itsclc0yqsollem1 47602	Lemma 1 for itsclc0yqsol 47604. (Contributed by AV, 6-Feb-2023.)
⊢ 𝑄 = ((𝐴↑2) + (𝐵↑2))    &   ⊢ 𝑇 = -(2 · (𝐵 · 𝐶))    &   ⊢ 𝑈 = ((𝐶↑2) − ((𝐴↑2) · (𝑅↑2)))    &   ⊢ 𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))    ⇒   ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ 𝑅 ∈ ℂ) → ((𝑇↑2) − (4 · (𝑄 · 𝑈))) = ((4 · (𝐴↑2)) · 𝐷))
Theorem	itsclc0yqsollem2 47603	Lemma 2 for itsclc0yqsol 47604. (Contributed by AV, 6-Feb-2023.)
⊢ 𝑄 = ((𝐴↑2) + (𝐵↑2))    &   ⊢ 𝑇 = -(2 · (𝐵 · 𝐶))    &   ⊢ 𝑈 = ((𝐶↑2) − ((𝐴↑2) · (𝑅↑2)))    &   ⊢ 𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))    ⇒   ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝑅 ∈ ℝ ∧ 0 ≤ 𝐷) → (√‘((𝑇↑2) − (4 · (𝑄 · 𝑈)))) = ((2 · (abs‘𝐴)) · (√‘𝐷)))
Theorem	itsclc0yqsol 47604	Lemma for itsclc0 47611. Solutions of the quadratic equations for the y-coordinate of the intersection points of a (nondegenerate) line and a circle. (Contributed by AV, 7-Feb-2023.)
⊢ 𝑄 = ((𝐴↑2) + (𝐵↑2))    &   ⊢ 𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))    ⇒   ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 ≠ 0 ∨ 𝐵 ≠ 0)) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷) ∧ (𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ)) → ((((𝑋↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑋) + (𝐵 · 𝑌)) = 𝐶) → (𝑌 = (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄) ∨ 𝑌 = (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄))))
Theorem	itscnhlc0xyqsol 47605	Lemma for itsclc0 47611. Solutions of the quadratic equations for the coordinates of the intersection points of a nonhorizontal line and a circle. (Contributed by AV, 8-Feb-2023.)
⊢ 𝑄 = ((𝐴↑2) + (𝐵↑2))    &   ⊢ 𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))    ⇒   ⊢ ((((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷) ∧ (𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ)) → ((((𝑋↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑋) + (𝐵 · 𝑌)) = 𝐶) → ((𝑋 = (((𝐴 · 𝐶) + (𝐵 · (√‘𝐷))) / 𝑄) ∧ 𝑌 = (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄)) ∨ (𝑋 = (((𝐴 · 𝐶) − (𝐵 · (√‘𝐷))) / 𝑄) ∧ 𝑌 = (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄)))))
Theorem	itschlc0xyqsol1 47606	Lemma for itsclc0 47611. Solutions of the quadratic equations for the coordinates of the intersection points of a horizontal line and a circle. (Contributed by AV, 25-Feb-2023.)
⊢ 𝑄 = ((𝐴↑2) + (𝐵↑2))    &   ⊢ 𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))    ⇒   ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 = 0 ∧ 𝐵 ≠ 0)) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷) ∧ (𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ)) → ((((𝑋↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑋) + (𝐵 · 𝑌)) = 𝐶) → (𝑌 = (𝐶 / 𝐵) ∧ (𝑋 = -((√‘𝐷) / 𝐵) ∨ 𝑋 = ((√‘𝐷) / 𝐵)))))
Theorem	itschlc0xyqsol 47607	Lemma for itsclc0 47611. Solutions of the quadratic equations for the coordinates of the intersection points of a horizontal line and a circle. (Contributed by AV, 8-Feb-2023.)
⊢ 𝑄 = ((𝐴↑2) + (𝐵↑2))    &   ⊢ 𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))    ⇒   ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 = 0 ∧ 𝐵 ≠ 0)) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷) ∧ (𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ)) → ((((𝑋↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑋) + (𝐵 · 𝑌)) = 𝐶) → ((𝑋 = (((𝐴 · 𝐶) + (𝐵 · (√‘𝐷))) / 𝑄) ∧ 𝑌 = (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄)) ∨ (𝑋 = (((𝐴 · 𝐶) − (𝐵 · (√‘𝐷))) / 𝑄) ∧ 𝑌 = (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄)))))
Theorem	itsclc0xyqsol 47608	Lemma for itsclc0 47611. Solutions of the quadratic equations for the coordinates of the intersection points of a (nondegenerate) line and a circle. (Contributed by AV, 25-Feb-2023.)
⊢ 𝑄 = ((𝐴↑2) + (𝐵↑2))    &   ⊢ 𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))    ⇒   ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 ≠ 0 ∨ 𝐵 ≠ 0)) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷) ∧ (𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ)) → ((((𝑋↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑋) + (𝐵 · 𝑌)) = 𝐶) → ((𝑋 = (((𝐴 · 𝐶) + (𝐵 · (√‘𝐷))) / 𝑄) ∧ 𝑌 = (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄)) ∨ (𝑋 = (((𝐴 · 𝐶) − (𝐵 · (√‘𝐷))) / 𝑄) ∧ 𝑌 = (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄)))))
Theorem	itsclc0xyqsolr 47609	Lemma for itsclc0 47611. Solutions of the quadratic equations for the coordinates of the intersection points of a (nondegenerate) line and a circle. (Contributed by AV, 2-May-2023.) (Revised by AV, 14-May-2023.)
⊢ 𝑄 = ((𝐴↑2) + (𝐵↑2))    &   ⊢ 𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))    ⇒   ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 ≠ 0 ∨ 𝐵 ≠ 0) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷)) → (((𝑋 = (((𝐴 · 𝐶) + (𝐵 · (√‘𝐷))) / 𝑄) ∧ 𝑌 = (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄)) ∨ (𝑋 = (((𝐴 · 𝐶) − (𝐵 · (√‘𝐷))) / 𝑄) ∧ 𝑌 = (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄))) → (((𝑋↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑋) + (𝐵 · 𝑌)) = 𝐶)))
Theorem	itsclc0xyqsolb 47610	Lemma for itsclc0 47611. Solutions of the quadratic equations for the coordinates of the intersection points of a (nondegenerate) line and a circle. (Contributed by AV, 2-May-2023.) (Revised by AV, 14-May-2023.)
⊢ 𝑄 = ((𝐴↑2) + (𝐵↑2))    &   ⊢ 𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))    ⇒   ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 ≠ 0 ∨ 𝐵 ≠ 0)) ∧ ((𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷) ∧ (𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ))) → ((((𝑋↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑋) + (𝐵 · 𝑌)) = 𝐶) ↔ ((𝑋 = (((𝐴 · 𝐶) + (𝐵 · (√‘𝐷))) / 𝑄) ∧ 𝑌 = (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄)) ∨ (𝑋 = (((𝐴 · 𝐶) − (𝐵 · (√‘𝐷))) / 𝑄) ∧ 𝑌 = (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄)))))
Theorem	itsclc0 47611*	The intersection points of a line 𝐿 and a circle around the origin. (Contributed by AV, 25-Feb-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝑆 = (Sphere‘𝐸)    &   ⊢ 0 = (𝐼 × {0})    &   ⊢ 𝑄 = ((𝐴↑2) + (𝐵↑2))    &   ⊢ 𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))    &   ⊢ 𝐿 = {𝑝 ∈ 𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶}    ⇒   ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 ≠ 0 ∨ 𝐵 ≠ 0) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷)) → ((𝑋 ∈ ( 0 𝑆𝑅) ∧ 𝑋 ∈ 𝐿) → (((𝑋‘1) = (((𝐴 · 𝐶) + (𝐵 · (√‘𝐷))) / 𝑄) ∧ (𝑋‘2) = (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄)) ∨ ((𝑋‘1) = (((𝐴 · 𝐶) − (𝐵 · (√‘𝐷))) / 𝑄) ∧ (𝑋‘2) = (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄)))))
Theorem	itsclc0b 47612*	The intersection points of a (nondegenerate) line through two points and a circle around the origin. (Contributed by AV, 2-May-2023.) (Revised by AV, 14-May-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝑆 = (Sphere‘𝐸)    &   ⊢ 0 = (𝐼 × {0})    &   ⊢ 𝑄 = ((𝐴↑2) + (𝐵↑2))    &   ⊢ 𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))    &   ⊢ 𝐿 = {𝑝 ∈ 𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶}    ⇒   ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 ≠ 0 ∨ 𝐵 ≠ 0) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷)) → ((𝑋 ∈ ( 0 𝑆𝑅) ∧ 𝑋 ∈ 𝐿) ↔ (𝑋 ∈ 𝑃 ∧ (((𝑋‘1) = (((𝐴 · 𝐶) + (𝐵 · (√‘𝐷))) / 𝑄) ∧ (𝑋‘2) = (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄)) ∨ ((𝑋‘1) = (((𝐴 · 𝐶) − (𝐵 · (√‘𝐷))) / 𝑄) ∧ (𝑋‘2) = (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄))))))
Theorem	itsclinecirc0 47613	The intersection points of a line through two different points 𝑌 and 𝑍 and a circle around the origin, using the definition of a line in a two dimensional Euclidean space. (Contributed by AV, 25-Feb-2023.) (Proof shortened by AV, 16-May-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝑆 = (Sphere‘𝐸)    &   ⊢ 0 = (𝐼 × {0})    &   ⊢ 𝑄 = ((𝐴↑2) + (𝐵↑2))    &   ⊢ 𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))    &   ⊢ 𝐿 = (LineM‘𝐸)    &   ⊢ 𝐴 = ((𝑌‘2) − (𝑍‘2))    &   ⊢ 𝐵 = ((𝑍‘1) − (𝑌‘1))    &   ⊢ 𝐶 = (((𝑌‘2) · (𝑍‘1)) − ((𝑌‘1) · (𝑍‘2)))    ⇒   ⊢ (((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷)) → ((𝑋 ∈ ( 0 𝑆𝑅) ∧ 𝑋 ∈ (𝑌𝐿𝑍)) → (((𝑋‘1) = (((𝐴 · 𝐶) + (𝐵 · (√‘𝐷))) / 𝑄) ∧ (𝑋‘2) = (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄)) ∨ ((𝑋‘1) = (((𝐴 · 𝐶) − (𝐵 · (√‘𝐷))) / 𝑄) ∧ (𝑋‘2) = (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄)))))
Theorem	itsclinecirc0b 47614	The intersection points of a line through two different points and a circle around the origin, using the definition of a line in a two dimensional Euclidean space. (Contributed by AV, 2-May-2023.) (Revised by AV, 14-May-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝑆 = (Sphere‘𝐸)    &   ⊢ 0 = (𝐼 × {0})    &   ⊢ 𝑄 = ((𝐴↑2) + (𝐵↑2))    &   ⊢ 𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))    &   ⊢ 𝐿 = (LineM‘𝐸)    &   ⊢ 𝐴 = ((𝑋‘2) − (𝑌‘2))    &   ⊢ 𝐵 = ((𝑌‘1) − (𝑋‘1))    &   ⊢ 𝐶 = (((𝑋‘2) · (𝑌‘1)) − ((𝑋‘1) · (𝑌‘2)))    ⇒   ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷)) → ((𝑍 ∈ ( 0 𝑆𝑅) ∧ 𝑍 ∈ (𝑋𝐿𝑌)) ↔ (𝑍 ∈ 𝑃 ∧ (((𝑍‘1) = (((𝐴 · 𝐶) + (𝐵 · (√‘𝐷))) / 𝑄) ∧ (𝑍‘2) = (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄)) ∨ ((𝑍‘1) = (((𝐴 · 𝐶) − (𝐵 · (√‘𝐷))) / 𝑄) ∧ (𝑍‘2) = (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄))))))
Theorem	itsclinecirc0in 47615	The intersection points of a line through two different points and a circle around the origin, using the definition of a line in a two dimensional Euclidean space, expressed as intersection. (Contributed by AV, 7-May-2023.) (Revised by AV, 14-May-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝑆 = (Sphere‘𝐸)    &   ⊢ 0 = (𝐼 × {0})    &   ⊢ 𝑄 = ((𝐴↑2) + (𝐵↑2))    &   ⊢ 𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))    &   ⊢ 𝐿 = (LineM‘𝐸)    &   ⊢ 𝐴 = ((𝑋‘2) − (𝑌‘2))    &   ⊢ 𝐵 = ((𝑌‘1) − (𝑋‘1))    &   ⊢ 𝐶 = (((𝑋‘2) · (𝑌‘1)) − ((𝑋‘1) · (𝑌‘2)))    ⇒   ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷)) → (( 0 𝑆𝑅) ∩ (𝑋𝐿𝑌)) = {{⟨1, (((𝐴 · 𝐶) + (𝐵 · (√‘𝐷))) / 𝑄)⟩, ⟨2, (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄)⟩}, {⟨1, (((𝐴 · 𝐶) − (𝐵 · (√‘𝐷))) / 𝑄)⟩, ⟨2, (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄)⟩}})
Theorem	itsclquadb 47616*	Quadratic equation for the y-coordinate of the intersection points of a line and a circle. (Contributed by AV, 22-Feb-2023.)
⊢ 𝑄 = ((𝐴↑2) + (𝐵↑2))    &   ⊢ 𝑇 = -(2 · (𝐵 · 𝐶))    &   ⊢ 𝑈 = ((𝐶↑2) − ((𝐴↑2) · (𝑅↑2)))    ⇒   ⊢ ((((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝑅 ∈ ℝ+ ∧ 𝑌 ∈ ℝ) → (∃𝑥 ∈ ℝ (((𝑥↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑥) + (𝐵 · 𝑌)) = 𝐶) ↔ ((𝑄 · (𝑌↑2)) + ((𝑇 · 𝑌) + 𝑈)) = 0))
Theorem	itsclquadeu 47617*	Quadratic equation for the y-coordinate of the intersection points of a line and a circle. (Contributed by AV, 23-Feb-2023.)
⊢ 𝑄 = ((𝐴↑2) + (𝐵↑2))    &   ⊢ 𝑇 = -(2 · (𝐵 · 𝐶))    &   ⊢ 𝑈 = ((𝐶↑2) − ((𝐴↑2) · (𝑅↑2)))    ⇒   ⊢ ((((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝑅 ∈ ℝ+ ∧ 𝑌 ∈ ℝ) → (∃!𝑥 ∈ ℝ (((𝑥↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑥) + (𝐵 · 𝑌)) = 𝐶) ↔ ((𝑄 · (𝑌↑2)) + ((𝑇 · 𝑌) + 𝑈)) = 0))
Theorem	2itscplem1 47618	Lemma 1 for 2itscp 47621. (Contributed by AV, 4-Mar-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ∈ ℝ)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    &   ⊢ 𝐷 = (𝑋 − 𝐴)    &   ⊢ 𝐸 = (𝐵 − 𝑌)    ⇒   ⊢ (𝜑 → ((((𝐸↑2) · (𝐵↑2)) + ((𝐷↑2) · (𝐴↑2))) − (2 · ((𝐷 · 𝐴) · (𝐸 · 𝐵)))) = (((𝐷 · 𝐴) − (𝐸 · 𝐵))↑2))
Theorem	2itscplem2 47619	Lemma 2 for 2itscp 47621. (Contributed by AV, 4-Mar-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ∈ ℝ)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    &   ⊢ 𝐷 = (𝑋 − 𝐴)    &   ⊢ 𝐸 = (𝐵 − 𝑌)    &   ⊢ 𝐶 = ((𝐷 · 𝐵) + (𝐸 · 𝐴))    ⇒   ⊢ (𝜑 → (𝐶↑2) = ((((𝐷↑2) · (𝐵↑2)) + (2 · ((𝐷 · 𝐴) · (𝐸 · 𝐵)))) + ((𝐸↑2) · (𝐴↑2))))
Theorem	2itscplem3 47620	Lemma D for 2itscp 47621. (Contributed by AV, 4-Mar-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ∈ ℝ)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    &   ⊢ 𝐷 = (𝑋 − 𝐴)    &   ⊢ 𝐸 = (𝐵 − 𝑌)    &   ⊢ 𝐶 = ((𝐷 · 𝐵) + (𝐸 · 𝐴))    &   ⊢ (𝜑 → 𝑅 ∈ ℝ)    &   ⊢ 𝑄 = ((𝐸↑2) + (𝐷↑2))    &   ⊢ 𝑆 = (((𝑅↑2) · 𝑄) − (𝐶↑2))    ⇒   ⊢ (𝜑 → 𝑆 = ((((𝐸↑2) · ((𝑅↑2) − (𝐴↑2))) + ((𝐷↑2) · ((𝑅↑2) − (𝐵↑2)))) − (2 · ((𝐷 · 𝐴) · (𝐸 · 𝐵)))))
Theorem	2itscp 47621	A condition for a quadratic equation with real coefficients (for the intersection points of a line with a circle) to have (exactly) two different real solutions. (Contributed by AV, 5-Mar-2023.) (Revised by AV, 16-May-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ∈ ℝ)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    &   ⊢ 𝐷 = (𝑋 − 𝐴)    &   ⊢ 𝐸 = (𝐵 − 𝑌)    &   ⊢ 𝐶 = ((𝐷 · 𝐵) + (𝐸 · 𝐴))    &   ⊢ (𝜑 → 𝑅 ∈ ℝ)    &   ⊢ (𝜑 → ((𝐴↑2) + (𝐵↑2)) < (𝑅↑2))    &   ⊢ (𝜑 → (𝐵 ≠ 𝑌 ∨ 𝐴 ≠ 𝑋))    &   ⊢ 𝑄 = ((𝐸↑2) + (𝐷↑2))    &   ⊢ 𝑆 = (((𝑅↑2) · 𝑄) − (𝐶↑2))    ⇒   ⊢ (𝜑 → 0 < 𝑆)
Theorem	itscnhlinecirc02plem1 47622	Lemma 1 for itscnhlinecirc02p 47625. (Contributed by AV, 6-Mar-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ∈ ℝ)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    &   ⊢ 𝐷 = (𝑋 − 𝐴)    &   ⊢ 𝐸 = (𝐵 − 𝑌)    &   ⊢ 𝐶 = ((𝐷 · 𝐵) + (𝐸 · 𝐴))    &   ⊢ (𝜑 → 𝑅 ∈ ℝ)    &   ⊢ (𝜑 → ((𝐴↑2) + (𝐵↑2)) < (𝑅↑2))    &   ⊢ (𝜑 → 𝐵 ≠ 𝑌)    ⇒   ⊢ (𝜑 → 0 < ((-(2 · (𝐷 · 𝐶))↑2) − (4 · (((𝐸↑2) + (𝐷↑2)) · ((𝐶↑2) − ((𝐸↑2) · (𝑅↑2)))))))
Theorem	itscnhlinecirc02plem2 47623	Lemma 2 for itscnhlinecirc02p 47625. (Contributed by AV, 10-Mar-2023.)
⊢ 𝐷 = (𝑋 − 𝐴)    &   ⊢ 𝐸 = (𝐵 − 𝑌)    &   ⊢ 𝐶 = ((𝐵 · 𝑋) − (𝐴 · 𝑌))    ⇒   ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) ∧ 𝐵 ≠ 𝑌) ∧ (𝑅 ∈ ℝ ∧ ((𝐴↑2) + (𝐵↑2)) < (𝑅↑2))) → 0 < ((-(2 · (𝐷 · 𝐶))↑2) − (4 · (((𝐸↑2) + (𝐷↑2)) · ((𝐶↑2) − ((𝐸↑2) · (𝑅↑2)))))))
Theorem	itscnhlinecirc02plem3 47624	Lemma 3 for itscnhlinecirc02p 47625. (Contributed by AV, 10-Mar-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝑆 = (Sphere‘𝐸)    &   ⊢ 0 = (𝐼 × {0})    &   ⊢ 𝐿 = (LineM‘𝐸)    &   ⊢ 𝐷 = (dist‘𝐸)    ⇒   ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘2) ≠ (𝑌‘2)) ∧ (𝑅 ∈ ℝ+ ∧ (𝑋𝐷 0 ) < 𝑅)) → 0 < ((-(2 · (((𝑌‘1) − (𝑋‘1)) · (((𝑋‘2) · (𝑌‘1)) − ((𝑋‘1) · (𝑌‘2)))))↑2) − (4 · (((((𝑋‘2) − (𝑌‘2))↑2) + (((𝑌‘1) − (𝑋‘1))↑2)) · (((((𝑋‘2) · (𝑌‘1)) − ((𝑋‘1) · (𝑌‘2)))↑2) − ((((𝑋‘2) − (𝑌‘2))↑2) · (𝑅↑2)))))))
Theorem	itscnhlinecirc02p 47625*	Intersection of a nonhorizontal line with a circle: A nonhorizontal line passing through a point within a circle around the origin intersects the circle at exactly two different points. (Contributed by AV, 28-Jan-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝑆 = (Sphere‘𝐸)    &   ⊢ 0 = (𝐼 × {0})    &   ⊢ 𝐿 = (LineM‘𝐸)    &   ⊢ 𝐷 = (dist‘𝐸)    &   ⊢ 𝑍 = {⟨1, 𝑥⟩, ⟨2, 𝑦⟩}    ⇒   ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘2) ≠ (𝑌‘2)) ∧ (𝑅 ∈ ℝ+ ∧ (𝑋𝐷 0 ) < 𝑅)) → ∃!𝑠 ∈ 𝒫 ℝ((♯‘𝑠) = 2 ∧ ∀𝑦 ∈ 𝑠 ∃!𝑥 ∈ ℝ (𝑍 ∈ ( 0 𝑆𝑅) ∧ 𝑍 ∈ (𝑋𝐿𝑌))))
Theorem	inlinecirc02plem 47626*	Lemma for inlinecirc02p 47627. (Contributed by AV, 7-May-2023.) (Revised by AV, 15-May-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝑆 = (Sphere‘𝐸)    &   ⊢ 0 = (𝐼 × {0})    &   ⊢ 𝐿 = (LineM‘𝐸)    &   ⊢ 𝑄 = ((𝐴↑2) + (𝐵↑2))    &   ⊢ 𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))    &   ⊢ 𝐴 = ((𝑋‘2) − (𝑌‘2))    &   ⊢ 𝐵 = ((𝑌‘1) − (𝑋‘1))    &   ⊢ 𝐶 = (((𝑋‘2) · (𝑌‘1)) − ((𝑋‘1) · (𝑌‘2)))    ⇒   ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) ∧ (𝑅 ∈ ℝ+ ∧ 0 < 𝐷)) → ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ((( 0 𝑆𝑅) ∩ (𝑋𝐿𝑌)) = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏))
Theorem	inlinecirc02p 47627	Intersection of a line with a circle: A line passing through a point within a circle around the origin intersects the circle at exactly two different points. (Contributed by AV, 9-May-2023.) (Revised by AV, 16-May-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝑆 = (Sphere‘𝐸)    &   ⊢ 0 = (𝐼 × {0})    &   ⊢ 𝐿 = (LineM‘𝐸)    &   ⊢ 𝐷 = (dist‘𝐸)    ⇒   ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) ∧ (𝑅 ∈ ℝ+ ∧ (𝑋𝐷 0 ) < 𝑅)) → (( 0 𝑆𝑅) ∩ (𝑋𝐿𝑌)) ∈ (Pairsproper‘𝑃))
Theorem	inlinecirc02preu 47628*	Intersection of a line with a circle: A line passing through a point within a circle around the origin intersects the circle at exactly two different points, expressed with restricted uniqueness (and without the definition of proper pairs). (Contributed by AV, 16-May-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝑆 = (Sphere‘𝐸)    &   ⊢ 0 = (𝐼 × {0})    &   ⊢ 𝐿 = (LineM‘𝐸)    &   ⊢ 𝐷 = (dist‘𝐸)    ⇒   ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) ∧ (𝑅 ∈ ℝ+ ∧ (𝑋𝐷 0 ) < 𝑅)) → ∃!𝑝 ∈ 𝒫 𝑃((♯‘𝑝) = 2 ∧ 𝑝 = (( 0 𝑆𝑅) ∩ (𝑋𝐿𝑌))))
21.46  Mathbox for Zhi Wang
21.46.1  Propositional calculus
Theorem	pm4.71da 47629	Deduction converting a biconditional to a biconditional with conjunction. Variant of pm4.71d 561. (Contributed by Zhi Wang, 30-Aug-2024.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜒)))
Theorem	logic1 47630	Distribution of implication over biconditional with replacement (deduction form). (Contributed by Zhi Wang, 30-Aug-2024.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜓 → (𝜃 ↔ 𝜏)))    ⇒   ⊢ (𝜑 → ((𝜓 → 𝜃) ↔ (𝜒 → 𝜏)))
Theorem	logic1a 47631	Variant of logic1 47630. (Contributed by Zhi Wang, 30-Aug-2024.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ ((𝜑 ∧ 𝜓) → (𝜃 ↔ 𝜏))    ⇒   ⊢ (𝜑 → ((𝜓 → 𝜃) ↔ (𝜒 → 𝜏)))
Theorem	logic2 47632	Variant of logic1 47630. (Contributed by Zhi Wang, 30-Aug-2024.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)))    ⇒   ⊢ (𝜑 → ((𝜓 → 𝜃) ↔ (𝜒 → 𝜏)))
Theorem	pm5.32dav 47633	Distribution of implication over biconditional (deduction form). Variant of pm5.32da 578. (Contributed by Zhi Wang, 30-Aug-2024.)
⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))    ⇒   ⊢ (𝜑 → ((𝜒 ∧ 𝜓) ↔ (𝜃 ∧ 𝜓)))
Theorem	pm5.32dra 47634	Reverse distribution of implication over biconditional (deduction form). (Contributed by Zhi Wang, 6-Sep-2024.)
⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜃)))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))
Theorem	exp12bd 47635	The import-export theorem (impexp 450) for biconditionals (deduction form). (Contributed by Zhi Wang, 3-Sep-2024.)
⊢ (𝜑 → (((𝜓 ∧ 𝜒) → 𝜃) ↔ ((𝜏 ∧ 𝜂) → 𝜁)))    ⇒   ⊢ (𝜑 → ((𝜓 → (𝜒 → 𝜃)) ↔ (𝜏 → (𝜂 → 𝜁))))
Theorem	mpbiran3d 47636	Equivalence with a conjunction one of whose conjuncts is a consequence of the other. Deduction form. (Contributed by Zhi Wang, 24-Sep-2024.)
⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))    &   ⊢ ((𝜑 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))
Theorem	mpbiran4d 47637	Equivalence with a conjunction one of whose conjuncts is a consequence of the other. Deduction form. (Contributed by Zhi Wang, 27-Sep-2024.)
⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))    &   ⊢ ((𝜑 ∧ 𝜃) → 𝜒)    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜃))
21.46.2  Predicate calculus with equality
21.46.2.1  Axiom scheme ax-5 (Distinctness)
Theorem	dtrucor3 47638*	An example of how ax-5 1905 without a distinct variable condition causes paradox in models of at least two objects. The hypothesis "dtrucor3.1" is provable from dtru 5426 in the ZF set theory. axc16nf 2246 and euae 2647 demonstrate that the violation of dtru 5426 leads to a model with only one object assuming its existence (ax-6 1963). The conclusion is also provable in the empty model ( see emptyal 1903). See also nf5 2270 and nf5i 2134 for the relation between unconditional ax-5 1905 and being not free. (Contributed by Zhi Wang, 23-Sep-2024.)
⊢ ¬ ∀𝑥 𝑥 = 𝑦    &   ⊢ (𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑦)    ⇒   ⊢ ∀𝑥 𝑥 = 𝑦
21.46.3  ZF Set Theory - start with the Axiom of Extensionality
21.46.3.1  Restricted quantification
Theorem	ralbidb 47639*	Formula-building rule for restricted universal quantifier and additional condition (deduction form). See ralbidc 47640 for a more generalized form. (Contributed by Zhi Wang, 6-Sep-2024.)
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ∧ 𝜓)))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜒 ↔ 𝜃))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥 ∈ 𝐵 (𝜓 → 𝜃)))
Theorem	ralbidc 47640*	Formula-building rule for restricted universal quantifier and additional condition (deduction form). A variant of ralbidb 47639. (Contributed by Zhi Wang, 30-Aug-2024.)
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ∧ 𝜓)))    &   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ 𝜓)) → (𝜒 ↔ 𝜃)))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥 ∈ 𝐵 (𝜓 → 𝜃)))
Theorem	r19.41dv 47641*	A complex deduction form of r19.41v 3180. (Contributed by Zhi Wang, 6-Sep-2024.)
⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)    ⇒   ⊢ ((𝜑 ∧ 𝜒) → ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒))
Theorem	rspceb2dv 47642*	Restricted existential specialization, using implicit substitution in both directions. (Contributed by Zhi Wang, 28-Sep-2024.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 → 𝜒))    &   ⊢ ((𝜑 ∧ 𝜒) → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝜒) → 𝜃)    &   ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ 𝜒))
Theorem	rmotru 47643	Two ways of expressing "at most one" element. (Contributed by Zhi Wang, 19-Sep-2024.) (Proof shortened by BJ, 23-Sep-2024.)
⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃*𝑥 ∈ 𝐴 ⊤)
Theorem	reutru 47644	Two ways of expressing "exactly one" element. (Contributed by Zhi Wang, 23-Sep-2024.)
⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 ⊤)
Theorem	reutruALT 47645	Alternate proof for reutru 47644. (Contributed by Zhi Wang, 23-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 ⊤)
21.46.3.2  The empty set
Theorem	ssdisjd 47646	Subset preserves disjointness. Deduction form of ssdisj 4451. (Contributed by Zhi Wang, 7-Sep-2024.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → (𝐵 ∩ 𝐶) = ∅)    ⇒   ⊢ (𝜑 → (𝐴 ∩ 𝐶) = ∅)
Theorem	ssdisjdr 47647	Subset preserves disjointness. Deduction form of ssdisj 4451. Alternatively this could be proved with ineqcom 4194 in tandem with ssdisjd 47646. (Contributed by Zhi Wang, 7-Sep-2024.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → (𝐶 ∩ 𝐵) = ∅)    ⇒   ⊢ (𝜑 → (𝐶 ∩ 𝐴) = ∅)
Theorem	disjdifb 47648	Relative complement is anticommutative regarding intersection. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ ((𝐴 ∖ 𝐵) ∩ (𝐵 ∖ 𝐴)) = ∅
Theorem	predisj 47649	Preimages of disjoint sets are disjoint. (Contributed by Zhi Wang, 9-Sep-2024.)
⊢ (𝜑 → Fun 𝐹)    &   ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)    &   ⊢ (𝜑 → 𝑆 ⊆ (◡𝐹 “ 𝐴))    &   ⊢ (𝜑 → 𝑇 ⊆ (◡𝐹 “ 𝐵))    ⇒   ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)
21.46.3.3  Unordered and ordered pairs
Theorem	vsn 47650	The singleton of the universal class is the empty set. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ {V} = ∅
Theorem	mosn 47651*	"At most one" element in a singleton. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (𝐴 = {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴)
Theorem	mo0 47652*	"At most one" element in an empty set. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (𝐴 = ∅ → ∃*𝑥 𝑥 ∈ 𝐴)
Theorem	mosssn 47653*	"At most one" element in a subclass of a singleton. (Contributed by Zhi Wang, 23-Sep-2024.)
⊢ (𝐴 ⊆ {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴)
Theorem	mo0sn 47654*	Two ways of expressing "at most one" element in a class. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}))
Theorem	mosssn2 47655*	Two ways of expressing "at most one" element in a class. (Contributed by Zhi Wang, 23-Sep-2024.)
⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑦 𝐴 ⊆ {𝑦})
21.46.3.4  The union of a class
Theorem	unilbss 47656*	Superclass of the greatest lower bound. A dual statement of ssintub 4960. (Contributed by Zhi Wang, 29-Sep-2024.)
⊢ ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐴
21.46.4  ZF Set Theory - add the Axiom of Replacement
21.46.4.1  Theorems requiring subset and intersection existence
Theorem	inpw 47657*	Two ways of expressing a collection of subsets as seen in df-ntr 22845, unimax 4938, and others (Contributed by Zhi Wang, 27-Sep-2024.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∩ 𝒫 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ⊆ 𝐵})
21.46.5  ZF Set Theory - add the Axiom of Power Sets
21.46.5.1  Functions
Theorem	mof0 47658	There is at most one function into the empty set. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ ∃*𝑓 𝑓:𝐴⟶∅
Theorem	mof02 47659*	A variant of mof0 47658. (Contributed by Zhi Wang, 20-Sep-2024.)
⊢ (𝐵 = ∅ → ∃*𝑓 𝑓:𝐴⟶𝐵)
Theorem	mof0ALT 47660*	Alternate proof for mof0 47658 with stronger requirements on distinct variables. Uses mo4 2552. (Contributed by Zhi Wang, 19-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃*𝑓 𝑓:𝐴⟶∅
Theorem	eufsnlem 47661*	There is exactly one function into a singleton. For a simpler hypothesis, see eufsn 47662 assuming ax-rep 5275, or eufsn2 47663 assuming ax-pow 5353 and ax-un 7718. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → (𝐴 × {𝐵}) ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})
Theorem	eufsn 47662*	There is exactly one function into a singleton, assuming ax-rep 5275. See eufsn2 47663 for different axiom requirements. If existence is not needed, use mofsn 47664 or mofsn2 47665 for fewer axiom assumptions. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})
Theorem	eufsn2 47663*	There is exactly one function into a singleton, assuming ax-pow 5353 and ax-un 7718. Variant of eufsn 47662. If existence is not needed, use mofsn 47664 or mofsn2 47665 for fewer axiom assumptions. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})
Theorem	mofsn 47664*	There is at most one function into a singleton, with fewer axioms than eufsn 47662 and eufsn2 47663. See also mofsn2 47665. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (𝐵 ∈ 𝑉 → ∃*𝑓 𝑓:𝐴⟶{𝐵})
Theorem	mofsn2 47665*	There is at most one function into a singleton. An unconditional variant of mofsn 47664, i.e., the singleton could be empty if 𝑌 is a proper class. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (𝐵 = {𝑌} → ∃*𝑓 𝑓:𝐴⟶𝐵)
Theorem	mofsssn 47666*	There is at most one function into a subclass of a singleton. (Contributed by Zhi Wang, 24-Sep-2024.)
⊢ (𝐵 ⊆ {𝑌} → ∃*𝑓 𝑓:𝐴⟶𝐵)
Theorem	mofmo 47667*	There is at most one function into a class containing at most one element. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (∃*𝑥 𝑥 ∈ 𝐵 → ∃*𝑓 𝑓:𝐴⟶𝐵)
Theorem	mofeu 47668*	The uniqueness of a function into a set with at most one element. (Contributed by Zhi Wang, 1-Oct-2024.)
⊢ 𝐺 = (𝐴 × 𝐵)    &   ⊢ (𝜑 → (𝐵 = ∅ → 𝐴 = ∅))    &   ⊢ (𝜑 → ∃*𝑥 𝑥 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹 = 𝐺))
Theorem	elfvne0 47669	If a function value has a member, then the function is not an empty set (An artifact of our function value definition.) (Contributed by Zhi Wang, 16-Sep-2024.)
⊢ (𝐴 ∈ (𝐹‘𝐵) → 𝐹 ≠ ∅)
Theorem	fdomne0 47670	A function with non-empty domain is non-empty and has non-empty codomain. (Contributed by Zhi Wang, 1-Oct-2024.)
⊢ ((𝐹:𝑋⟶𝑌 ∧ 𝑋 ≠ ∅) → (𝐹 ≠ ∅ ∧ 𝑌 ≠ ∅))
Theorem	f1sn2g 47671	A function that maps a singleton to a class is injective. (Contributed by Zhi Wang, 1-Oct-2024.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:{𝐴}⟶𝐵) → 𝐹:{𝐴}–1-1→𝐵)
Theorem	f102g 47672	A function that maps the empty set to a class is injective. (Contributed by Zhi Wang, 1-Oct-2024.)
⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵)
Theorem	f1mo 47673*	A function that maps a set with at most one element to a class is injective. (Contributed by Zhi Wang, 1-Oct-2024.)
⊢ ((∃*𝑥 𝑥 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵)
Theorem	f002 47674	A function with an empty codomain must have empty domain. (Contributed by Zhi Wang, 1-Oct-2024.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → (𝐵 = ∅ → 𝐴 = ∅))
Theorem	map0cor 47675*	A function exists iff an empty codomain is accompanied with an empty domain. (Contributed by Zhi Wang, 1-Oct-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ((𝐵 = ∅ → 𝐴 = ∅) ↔ ∃𝑓 𝑓:𝐴⟶𝐵))
21.46.5.2  Operations
Theorem	fvconstr 47676	Two ways of expressing 𝐴𝑅𝐵. (Contributed by Zhi Wang, 18-Sep-2024.)
⊢ (𝜑 → 𝐹 = (𝑅 × {𝑌}))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ≠ ∅)    ⇒   ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐵) = 𝑌))
Theorem	fvconstrn0 47677	Two ways of expressing 𝐴𝑅𝐵. (Contributed by Zhi Wang, 20-Sep-2024.)
⊢ (𝜑 → 𝐹 = (𝑅 × {𝑌}))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ≠ ∅)    ⇒   ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐵) ≠ ∅))
Theorem	fvconstr2 47678	Two ways of expressing 𝐴𝑅𝐵. (Contributed by Zhi Wang, 18-Sep-2024.)
⊢ (𝜑 → 𝐹 = (𝑅 × {𝑌}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐴𝐹𝐵))    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐵)
21.46.6  ZF Set Theory - add the Axiom of Union
21.46.6.1  Equinumerosity
Theorem	fvconst0ci 47679	A constant function's value is either the constant or the empty set. (An artifact of our function value definition.) (Contributed by Zhi Wang, 18-Sep-2024.)
⊢ 𝐵 ∈ V    &   ⊢ 𝑌 = ((𝐴 × {𝐵})‘𝑋)    ⇒   ⊢ (𝑌 = ∅ ∨ 𝑌 = 𝐵)
Theorem	fvconstdomi 47680	A constant function's value is dominated by the constant. (An artifact of our function value definition.) (Contributed by Zhi Wang, 18-Sep-2024.)
⊢ 𝐵 ∈ V    ⇒   ⊢ ((𝐴 × {𝐵})‘𝑋) ≼ 𝐵
Theorem	f1omo 47681*	There is at most one element in the function value of a constant function whose output is 1o. (An artifact of our function value definition.) Proof could be significantly shortened by fvconstdomi 47680 assuming ax-un 7718 (see f1omoALT 47682). (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (𝜑 → 𝐹 = (𝐴 × {1o}))    ⇒   ⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋))
Theorem	f1omoALT 47682*	There is at most one element in the function value of a constant function whose output is 1o. (An artifact of our function value definition.) Use f1omo 47681 without assuming ax-un 7718. (Contributed by Zhi Wang, 18-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐹 = (𝐴 × {1o}))    ⇒   ⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋))
21.46.7  Order sets
21.46.7.1  Real number intervals
Theorem	iccin 47683	Intersection of two closed intervals of extended reals. (Contributed by Zhi Wang, 9-Sep-2024.)
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) → ((𝐴[,]𝐵) ∩ (𝐶[,]𝐷)) = (if(𝐴 ≤ 𝐶, 𝐶, 𝐴)[,]if(𝐵 ≤ 𝐷, 𝐵, 𝐷)))
Theorem	iccdisj2 47684	If the upper bound of one closed interval is less than the lower bound of the other, the intervals are disjoint. (Contributed by Zhi Wang, 9-Sep-2024.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐷 ∈ ℝ* ∧ 𝐵 < 𝐶) → ((𝐴[,]𝐵) ∩ (𝐶[,]𝐷)) = ∅)
Theorem	iccdisj 47685	If the upper bound of one closed interval is less than the lower bound of the other, the intervals are disjoint. (Contributed by Zhi Wang, 9-Sep-2024.)
⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) ∧ 𝐵 < 𝐶) → ((𝐴[,]𝐵) ∩ (𝐶[,]𝐷)) = ∅)
21.46.8  Moore spaces
Theorem	mreuniss 47686	The union of a collection of closed sets is a subset. (Contributed by Zhi Wang, 29-Sep-2024.)
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) → ∪ 𝑆 ⊆ 𝑋)
21.46.9  Topology Additional contents for topology.
21.46.9.1  Closure and interior
Theorem	clduni 47687	The union of closed sets is the underlying set of the topology (the union of open sets). (Contributed by Zhi Wang, 6-Sep-2024.)
⊢ (𝐽 ∈ Top → ∪ (Clsd‘𝐽) = ∪ 𝐽)
Theorem	opncldeqv 47688*	Conditions on open sets are equivalent to conditions on closed sets. (Contributed by Zhi Wang, 30-Aug-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ ((𝜑 ∧ 𝑥 = (∪ 𝐽 ∖ 𝑦)) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐽 𝜓 ↔ ∀𝑦 ∈ (Clsd‘𝐽)𝜒))
Theorem	opndisj 47689	Two ways of saying that two open sets are disjoint, if 𝐽 is a topology and 𝑋 is an open set. (Contributed by Zhi Wang, 6-Sep-2024.)
⊢ (𝑍 = (∪ 𝐽 ∖ 𝑋) → (𝑌 ∈ (𝐽 ∩ 𝒫 𝑍) ↔ (𝑌 ∈ 𝐽 ∧ (𝑋 ∩ 𝑌) = ∅)))
Theorem	clddisj 47690	Two ways of saying that two closed sets are disjoint, if 𝐽 is a topology and 𝑋 is a closed set. An alternative proof is similar to that of opndisj 47689 with elssuni 4931 replaced by the combination of cldss 22854 and eqid 2724. (Contributed by Zhi Wang, 6-Sep-2024.)
⊢ (𝑍 = (∪ 𝐽 ∖ 𝑋) → (𝑌 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ (𝑋 ∩ 𝑌) = ∅)))
21.46.9.2  Neighborhoods
Theorem	neircl 47691	Reverse closure of the neighborhood operation. (This theorem depends on a function's value being empty outside of its domain, but it will make later theorems simpler to state.) (Contributed by Zhi Wang, 16-Sep-2024.)
⊢ (𝑁 ∈ ((nei‘𝐽)‘𝑆) → 𝐽 ∈ Top)
Theorem	opnneilem 47692*	Lemma factoring out common proof steps of opnneil 47696 and opnneirv 47694. (Contributed by Zhi Wang, 31-Aug-2024.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒)))
Theorem	opnneir 47693*	If something is true for an open neighborhood, it must be true for a neighborhood. (Contributed by Zhi Wang, 31-Aug-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝜓) → ∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓))
Theorem	opnneirv 47694*	A variant of opnneir 47693 with different dummy variables. (Contributed by Zhi Wang, 31-Aug-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝜓) → ∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝜒))
Theorem	opnneilv 47695*	The converse of opnneir 47693 with different dummy variables. Note that the second hypothesis could be generalized by adding 𝑦 ∈ 𝐽 to the antecedent. See the proof for details. Although 𝐽 ∈ Top might be redundant here (see neircl 47691), it is listed for explicitness. (Contributed by Zhi Wang, 31-Aug-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝑥) → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 → ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒)))
Theorem	opnneil 47696*	A variant of opnneilv 47695. (Contributed by Zhi Wang, 31-Aug-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝑥) → (𝜓 → 𝜒))    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 → ∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝜓)))
Theorem	opnneieqv 47697*	The equivalence between neighborhood and open neighborhood. See opnneieqvv 47698 for different dummy variables. (Contributed by Zhi Wang, 31-Aug-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝑥) → (𝜓 → 𝜒))    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 ↔ ∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝜓)))
Theorem	opnneieqvv 47698*	The equivalence between neighborhood and open neighborhood. A variant of opnneieqv 47697 with two dummy variables. (Contributed by Zhi Wang, 31-Aug-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝑥) → (𝜓 → 𝜒))    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 ↔ ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒)))
21.46.9.3  Subspace topologies
Theorem	restcls2lem 47699	A closed set in a subspace topology is a subset of the subspace. (Contributed by Zhi Wang, 2-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑋 = ∪ 𝐽)    &   ⊢ (𝜑 → 𝑌 ⊆ 𝑋)    &   ⊢ (𝜑 → 𝐾 = (𝐽 ↾t 𝑌))    &   ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐾))    ⇒   ⊢ (𝜑 → 𝑆 ⊆ 𝑌)
Theorem	restcls2 47700	A closed set in a subspace topology is the closure in the original topology intersecting with the subspace. (Contributed by Zhi Wang, 2-Sep-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑋 = ∪ 𝐽)    &   ⊢ (𝜑 → 𝑌 ⊆ 𝑋)    &   ⊢ (𝜑 → 𝐾 = (𝐽 ↾t 𝑌))    &   ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐾))    ⇒   ⊢ (𝜑 → 𝑆 = (((cls‘𝐽)‘𝑆) ∩ 𝑌))