P. 476 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 47501-47600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	itcoval 47501*	The value of the function that returns the n-th iterate of a class (usually a function) with regard to composition. (Contributed by AV, 2-May-2024.)
⊢ (𝐹 ∈ 𝑉 → (IterComp‘𝐹) = seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹))))
Theorem	itcoval0 47502	A function iterated zero times (defined as identity function). (Contributed by AV, 2-May-2024.)
⊢ (𝐹 ∈ 𝑉 → ((IterComp‘𝐹)‘0) = ( I ↾ dom 𝐹))
Theorem	itcoval1 47503	A function iterated once. (Contributed by AV, 2-May-2024.)
⊢ ((Rel 𝐹 ∧ 𝐹 ∈ 𝑉) → ((IterComp‘𝐹)‘1) = 𝐹)
Theorem	itcoval2 47504	A function iterated twice. (Contributed by AV, 2-May-2024.)
⊢ ((Rel 𝐹 ∧ 𝐹 ∈ 𝑉) → ((IterComp‘𝐹)‘2) = (𝐹 ∘ 𝐹))
Theorem	itcoval3 47505	A function iterated three times. (Contributed by AV, 2-May-2024.)
⊢ ((Rel 𝐹 ∧ 𝐹 ∈ 𝑉) → ((IterComp‘𝐹)‘3) = (𝐹 ∘ (𝐹 ∘ 𝐹)))
Theorem	itcoval0mpt 47506*	A mapping iterated zero times (defined as identity function). (Contributed by AV, 4-May-2024.)
⊢ 𝐹 = (𝑛 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ 𝑊) → ((IterComp‘𝐹)‘0) = (𝑛 ∈ 𝐴 ↦ 𝑛))
Theorem	itcovalsuc 47507*	The value of the function that returns the n-th iterate of a function with regard to composition at a successor. (Contributed by AV, 4-May-2024.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ∧ ((IterComp‘𝐹)‘𝑌) = 𝐺) → ((IterComp‘𝐹)‘(𝑌 + 1)) = (𝐺(𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔))𝐹))
Theorem	itcovalsucov 47508	The value of the function that returns the n-th iterate of a function with regard to composition at a successor. (Contributed by AV, 4-May-2024.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ∧ ((IterComp‘𝐹)‘𝑌) = 𝐺) → ((IterComp‘𝐹)‘(𝑌 + 1)) = (𝐹 ∘ 𝐺))
Theorem	itcovalendof 47509	The n-th iterate of an endofunction is an endofunction. (Contributed by AV, 7-May-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐴)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ((IterComp‘𝐹)‘𝑁):𝐴⟶𝐴)
Theorem	itcovalpclem1 47510*	Lemma 1 for itcovalpc 47512: induction basis. (Contributed by AV, 4-May-2024.)
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶))    ⇒   ⊢ (𝐶 ∈ ℕ0 → ((IterComp‘𝐹)‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 0))))
Theorem	itcovalpclem2 47511*	Lemma 2 for itcovalpc 47512: induction step. (Contributed by AV, 4-May-2024.)
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶))    ⇒   ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦))) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1))))))
Theorem	itcovalpc 47512*	The value of the function that returns the n-th iterate of the "plus a constant" function with regard to composition. (Contributed by AV, 4-May-2024.)
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶))    ⇒   ⊢ ((𝐼 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → ((IterComp‘𝐹)‘𝐼) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝐼))))
Theorem	itcovalt2lem2lem1 47513	Lemma 1 for itcovalt2lem2 47516. (Contributed by AV, 6-May-2024.)
⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → (((𝑁 + 𝐶) · 𝑌) − 𝐶) ∈ ℕ0)
Theorem	itcovalt2lem2lem2 47514	Lemma 2 for itcovalt2lem2 47516. (Contributed by AV, 7-May-2024.)
⊢ (((𝑌 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → ((2 · (((𝑁 + 𝐶) · (2↑𝑌)) − 𝐶)) + 𝐶) = (((𝑁 + 𝐶) · (2↑(𝑌 + 1))) − 𝐶))
Theorem	itcovalt2lem1 47515*	Lemma 1 for itcovalt2 47517: induction basis. (Contributed by AV, 5-May-2024.)
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 𝐶))    ⇒   ⊢ (𝐶 ∈ ℕ0 → ((IterComp‘𝐹)‘0) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑0)) − 𝐶)))
Theorem	itcovalt2lem2 47516*	Lemma 2 for itcovalt2 47517: induction step. (Contributed by AV, 7-May-2024.)
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 𝐶))    ⇒   ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶)) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑(𝑦 + 1))) − 𝐶))))
Theorem	itcovalt2 47517*	The value of the function that returns the n-th iterate of the "times 2 plus a constant" function with regard to composition. (Contributed by AV, 7-May-2024.)
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 𝐶))    ⇒   ⊢ ((𝐼 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → ((IterComp‘𝐹)‘𝐼) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝐼)) − 𝐶)))
Theorem	ackvalsuc1mpt 47518*	The Ackermann function at a successor of the first argument as a mapping of the second argument. (Contributed by Thierry Arnoux, 28-Apr-2024.) (Revised by AV, 4-May-2024.)
⊢ (𝑀 ∈ ℕ0 → (Ack‘(𝑀 + 1)) = (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘𝑀))‘(𝑛 + 1))‘1)))
Theorem	ackvalsuc1 47519	The Ackermann function at a successor of the first argument and an arbitrary second argument. (Contributed by Thierry Arnoux, 28-Apr-2024.) (Revised by AV, 4-May-2024.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((Ack‘(𝑀 + 1))‘𝑁) = (((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))‘1))
Theorem	ackval0 47520	The Ackermann function at 0. (Contributed by AV, 2-May-2024.)
⊢ (Ack‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 1))
Theorem	ackval1 47521	The Ackermann function at 1. (Contributed by AV, 4-May-2024.)
⊢ (Ack‘1) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2))
Theorem	ackval2 47522	The Ackermann function at 2. (Contributed by AV, 4-May-2024.)
⊢ (Ack‘2) = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 3))
Theorem	ackval3 47523	The Ackermann function at 3. (Contributed by AV, 7-May-2024.)
⊢ (Ack‘3) = (𝑛 ∈ ℕ0 ↦ ((2↑(𝑛 + 3)) − 3))
Theorem	ackendofnn0 47524	The Ackermann function at any nonnegative integer is an endofunction on the nonnegative integers. (Contributed by AV, 8-May-2024.)
⊢ (𝑀 ∈ ℕ0 → (Ack‘𝑀):ℕ0⟶ℕ0)
Theorem	ackfnnn0 47525	The Ackermann function at any nonnegative integer is a function on the nonnegative integers. (Contributed by AV, 4-May-2024.) (Proof shortened by AV, 8-May-2024.)
⊢ (𝑀 ∈ ℕ0 → (Ack‘𝑀) Fn ℕ0)
Theorem	ackval0val 47526	The Ackermann function at 0 (for the first argument). This is the first equation of Péter's definition of the Ackermann function. (Contributed by AV, 4-May-2024.)
⊢ (𝑀 ∈ ℕ0 → ((Ack‘0)‘𝑀) = (𝑀 + 1))
Theorem	ackvalsuc0val 47527	The Ackermann function at a successor (of the first argument). This is the second equation of Péter's definition of the Ackermann function. (Contributed by AV, 4-May-2024.)
⊢ (𝑀 ∈ ℕ0 → ((Ack‘(𝑀 + 1))‘0) = ((Ack‘𝑀)‘1))
Theorem	ackvalsucsucval 47528	The Ackermann function at the successors. This is the third equation of Péter's definition of the Ackermann function. (Contributed by AV, 8-May-2024.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((Ack‘(𝑀 + 1))‘(𝑁 + 1)) = ((Ack‘𝑀)‘((Ack‘(𝑀 + 1))‘𝑁)))
Theorem	ackval0012 47529	The Ackermann function at (0,0), (0,1), (0,2). (Contributed by AV, 2-May-2024.)
⊢ ⟨((Ack‘0)‘0), ((Ack‘0)‘1), ((Ack‘0)‘2)⟩ = ⟨1, 2, 3⟩
Theorem	ackval1012 47530	The Ackermann function at (1,0), (1,1), (1,2). (Contributed by AV, 4-May-2024.)
⊢ ⟨((Ack‘1)‘0), ((Ack‘1)‘1), ((Ack‘1)‘2)⟩ = ⟨2, 3, 4⟩
Theorem	ackval2012 47531	The Ackermann function at (2,0), (2,1), (2,2). (Contributed by AV, 4-May-2024.)
⊢ ⟨((Ack‘2)‘0), ((Ack‘2)‘1), ((Ack‘2)‘2)⟩ = ⟨3, 5, 7⟩
Theorem	ackval3012 47532	The Ackermann function at (3,0), (3,1), (3,2). (Contributed by AV, 7-May-2024.)
⊢ ⟨((Ack‘3)‘0), ((Ack‘3)‘1), ((Ack‘3)‘2)⟩ = ⟨5, ;13, ;29⟩
Theorem	ackval40 47533	The Ackermann function at (4,0). (Contributed by AV, 9-May-2024.)
⊢ ((Ack‘4)‘0) = ;13
Theorem	ackval41a 47534	The Ackermann function at (4,1). (Contributed by AV, 9-May-2024.)
⊢ ((Ack‘4)‘1) = ((2↑;16) − 3)
Theorem	ackval41 47535	The Ackermann function at (4,1). (Contributed by AV, 9-May-2024.)
⊢ ((Ack‘4)‘1) = ;;;;65533
Theorem	ackval42 47536	The Ackermann function at (4,2). (Contributed by AV, 9-May-2024.)
⊢ ((Ack‘4)‘2) = ((2↑;;;;65536) − 3)
Theorem	ackval42a 47537	The Ackermann function at (4,2), expressed with powers of 2. (Contributed by AV, 9-May-2024.)
⊢ ((Ack‘4)‘2) = ((2↑(2↑(2↑(2↑2)))) − 3)
Theorem	ackval50 47538	The Ackermann function at (5,0). (Contributed by AV, 9-May-2024.)
⊢ ((Ack‘5)‘0) = ;;;;65533
21.45.22  Elementary geometry (extension)
21.45.22.1  Auxiliary theorems
Theorem	fv1prop 47539	The function value of unordered pair of ordered pairs with first components 1 and 2 at 1. (Contributed by AV, 4-Feb-2023.)
⊢ (𝐴 ∈ 𝑉 → ({⟨1, 𝐴⟩, ⟨2, 𝐵⟩}‘1) = 𝐴)
Theorem	fv2prop 47540	The function value of unordered pair of ordered pairs with first components 1 and 2 at 1. (Contributed by AV, 4-Feb-2023.)
⊢ (𝐵 ∈ 𝑉 → ({⟨1, 𝐴⟩, ⟨2, 𝐵⟩}‘2) = 𝐵)
Theorem	submuladdmuld 47541	Transformation of a sum of a product of a difference and a product with the subtrahend of the difference. (Contributed by AV, 2-Feb-2023.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐷 ∈ ℂ)    ⇒   ⊢ (𝜑 → (((𝐴 − 𝐵) · 𝐶) + (𝐵 · 𝐷)) = ((𝐴 · 𝐶) + (𝐵 · (𝐷 − 𝐶))))
Theorem	affinecomb1 47542*	Combination of two real affine combinations, one class variable resolved. (Contributed by AV, 22-Jan-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    &   ⊢ (𝜑 → 𝐸 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ ℝ)    &   ⊢ (𝜑 → 𝐺 ∈ ℝ)    &   ⊢ 𝑆 = ((𝐺 − 𝐹) / (𝐶 − 𝐵))    ⇒   ⊢ (𝜑 → (∃𝑡 ∈ ℝ (𝐴 = (((1 − 𝑡) · 𝐵) + (𝑡 · 𝐶)) ∧ 𝐸 = (((1 − 𝑡) · 𝐹) + (𝑡 · 𝐺))) ↔ 𝐸 = ((𝑆 · (𝐴 − 𝐵)) + 𝐹)))
Theorem	affinecomb2 47543*	Combination of two real affine combinations, presented without fraction. (Contributed by AV, 22-Jan-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    &   ⊢ (𝜑 → 𝐸 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ ℝ)    &   ⊢ (𝜑 → 𝐺 ∈ ℝ)    ⇒   ⊢ (𝜑 → (∃𝑡 ∈ ℝ (𝐴 = (((1 − 𝑡) · 𝐵) + (𝑡 · 𝐶)) ∧ 𝐸 = (((1 − 𝑡) · 𝐹) + (𝑡 · 𝐺))) ↔ ((𝐶 − 𝐵) · 𝐸) = (((𝐺 − 𝐹) · 𝐴) + ((𝐹 · 𝐶) − (𝐵 · 𝐺)))))
Theorem	affineid 47544	Identity of an affine combination. (Contributed by AV, 2-Feb-2023.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝑇 ∈ ℂ)    ⇒   ⊢ (𝜑 → (((1 − 𝑇) · 𝐴) + (𝑇 · 𝐴)) = 𝐴)
Theorem	1subrec1sub 47545	Subtract the reciprocal of 1 minus a number from 1 results in the number divided by the number minus 1. (Contributed by AV, 15-Feb-2023.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 1) → (1 − (1 / (1 − 𝐴))) = (𝐴 / (𝐴 − 1)))
Theorem	resum2sqcl 47546	The sum of two squares of real numbers is a real number. (Contributed by AV, 7-Feb-2023.)
⊢ 𝑄 = ((𝐴↑2) + (𝐵↑2))    ⇒   ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝑄 ∈ ℝ)
Theorem	resum2sqgt0 47547	The sum of the square of a nonzero real number and the square of another real number is greater than zero. (Contributed by AV, 7-Feb-2023.)
⊢ 𝑄 = ((𝐴↑2) + (𝐵↑2))    ⇒   ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℝ) → 0 < 𝑄)
Theorem	resum2sqrp 47548	The sum of the square of a nonzero real number and the square of another real number is a positive real number. (Contributed by AV, 2-May-2023.)
⊢ 𝑄 = ((𝐴↑2) + (𝐵↑2))    ⇒   ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℝ) → 𝑄 ∈ ℝ+)
Theorem	resum2sqorgt0 47549	The sum of the square of two real numbers is greater than zero if at least one of the real numbers is nonzero. (Contributed by AV, 26-Feb-2023.)
⊢ 𝑄 = ((𝐴↑2) + (𝐵↑2))    ⇒   ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐴 ≠ 0 ∨ 𝐵 ≠ 0)) → 0 < 𝑄)
Theorem	reorelicc 47550	Membership in and outside of a closed real interval. (Contributed by AV, 15-Feb-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 < 𝐴 ∨ 𝐶 ∈ (𝐴[,]𝐵) ∨ 𝐵 < 𝐶))
21.45.22.2  Real euclidean space of dimension 2
Theorem	rrx2pxel 47551	The x-coordinate of a point in a real Euclidean space of dimension 2 is a real number. (Contributed by AV, 2-Feb-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    ⇒   ⊢ (𝑋 ∈ 𝑃 → (𝑋‘1) ∈ ℝ)
Theorem	rrx2pyel 47552	The y-coordinate of a point in a real Euclidean space of dimension 2 is a real number. (Contributed by AV, 2-Feb-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    ⇒   ⊢ (𝑋 ∈ 𝑃 → (𝑋‘2) ∈ ℝ)
Theorem	prelrrx2 47553	An unordered pair of ordered pairs with first components 1 and 2 and real numbers as second components is a point in a real Euclidean space of dimension 2. (Contributed by AV, 4-Feb-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    ⇒   ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → {⟨1, 𝐴⟩, ⟨2, 𝐵⟩} ∈ 𝑃)
Theorem	prelrrx2b 47554	An unordered pair of ordered pairs with first components 1 and 2 and real numbers as second components is a point in a real Euclidean space of dimension 2, determined by its coordinates. (Contributed by AV, 7-May-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    ⇒   ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ)) → ((𝑍 ∈ 𝑃 ∧ (((𝑍‘1) = 𝐴 ∧ (𝑍‘2) = 𝐵) ∨ ((𝑍‘1) = 𝑋 ∧ (𝑍‘2) = 𝑌))) ↔ 𝑍 ∈ {{⟨1, 𝐴⟩, ⟨2, 𝐵⟩}, {⟨1, 𝑋⟩, ⟨2, 𝑌⟩}}))
Theorem	rrx2pnecoorneor 47555	If two different points 𝑋 and 𝑌 in a real Euclidean space of dimension 2 are different, then they are different at least at one coordinate. (Contributed by AV, 26-Feb-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    ⇒   ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → ((𝑋‘1) ≠ (𝑌‘1) ∨ (𝑋‘2) ≠ (𝑌‘2)))
Theorem	rrx2pnedifcoorneor 47556	If two different points 𝑋 and 𝑌 in a real Euclidean space of dimension 2 are different, then at least one difference of two corresponding coordinates is not 0. (Contributed by AV, 26-Feb-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝐴 = ((𝑌‘1) − (𝑋‘1))    &   ⊢ 𝐵 = ((𝑌‘2) − (𝑋‘2))    ⇒   ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (𝐴 ≠ 0 ∨ 𝐵 ≠ 0))
Theorem	rrx2pnedifcoorneorr 47557	If two different points 𝑋 and 𝑌 in a real Euclidean space of dimension 2 are different, then at least one difference of two corresponding coordinates is not 0. (Contributed by AV, 26-Feb-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝐴 = ((𝑌‘1) − (𝑋‘1))    &   ⊢ 𝐵 = ((𝑋‘2) − (𝑌‘2))    ⇒   ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (𝐴 ≠ 0 ∨ 𝐵 ≠ 0))
Theorem	rrx2xpref1o 47558*	There is a bijection between the set of ordered pairs of real numbers (the cartesian product of the real numbers) and the set of points in the two dimensional Euclidean plane (represented as mappings from {1, 2} to the real numbers). (Contributed by AV, 12-Mar-2023.)
⊢ 𝑅 = (ℝ ↑m {1, 2})    &   ⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})    ⇒   ⊢ 𝐹:(ℝ × ℝ)–1-1-onto→𝑅
Theorem	rrx2xpreen 47559	The set of points in the two dimensional Euclidean plane and the set of ordered pairs of real numbers (the cartesian product of the real numbers) are equinumerous. (Contributed by AV, 12-Mar-2023.)
⊢ 𝑅 = (ℝ ↑m {1, 2})    ⇒   ⊢ 𝑅 ≈ (ℝ × ℝ)
Theorem	rrx2plord 47560*	The lexicographical ordering for points in the two dimensional Euclidean plane: a point is less than another point iff its first coordinate is less than the first coordinate of the other point, or the first coordinates of both points are equal and the second coordinate of the first point is less than the second coordinate of the other point: ⟨𝑎, 𝑏⟩ ≤ ⟨𝑥, 𝑦⟩ iff (𝑎 < 𝑥 ∨ (𝑎 = 𝑥 ∧ 𝑏 ≤ 𝑦)). (Contributed by AV, 12-Mar-2023.)
⊢ 𝑂 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))}    ⇒   ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (𝑋𝑂𝑌 ↔ ((𝑋‘1) < (𝑌‘1) ∨ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) < (𝑌‘2)))))
Theorem	rrx2plord1 47561*	The lexicographical ordering for points in the two dimensional Euclidean plane: a point is less than another point if its first coordinate is less than the first coordinate of the other point. (Contributed by AV, 12-Mar-2023.)
⊢ 𝑂 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))}    ⇒   ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ∧ (𝑋‘1) < (𝑌‘1)) → 𝑋𝑂𝑌)
Theorem	rrx2plord2 47562*	The lexicographical ordering for points in the two dimensional Euclidean plane: if the first coordinates of two points are equal, a point is less than another point iff the second coordinate of the point is less than the second coordinate of the other point. (Contributed by AV, 12-Mar-2023.)
⊢ 𝑂 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))}    &   ⊢ 𝑅 = (ℝ ↑m {1, 2})    ⇒   ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅 ∧ (𝑋‘1) = (𝑌‘1)) → (𝑋𝑂𝑌 ↔ (𝑋‘2) < (𝑌‘2)))
Theorem	rrx2plordisom 47563*	The set of points in the two dimensional Euclidean plane with the lexicographical ordering is isomorphic to the cartesian product of the real numbers with the lexicographical ordering implied by the ordering of the real numbers. (Contributed by AV, 12-Mar-2023.)
⊢ 𝑂 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))}    &   ⊢ 𝑅 = (ℝ ↑m {1, 2})    &   ⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ℝ × ℝ) ∧ 𝑦 ∈ (ℝ × ℝ)) ∧ ((1st ‘𝑥) < (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) < (2nd ‘𝑦))))}    ⇒   ⊢ 𝐹 Isom 𝑇, 𝑂 ((ℝ × ℝ), 𝑅)
Theorem	rrx2plordso 47564*	The lexicographical ordering for points in the two dimensional Euclidean plane is a strict total ordering. (Contributed by AV, 12-Mar-2023.)
⊢ 𝑂 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))}    &   ⊢ 𝑅 = (ℝ ↑m {1, 2})    ⇒   ⊢ 𝑂 Or 𝑅
Theorem	ehl2eudisval0 47565	The Euclidean distance of a point to the origin in a real Euclidean space of dimension 2. (Contributed by AV, 26-Feb-2023.)
⊢ 𝐸 = (𝔼hil‘2)    &   ⊢ 𝑋 = (ℝ ↑m {1, 2})    &   ⊢ 𝐷 = (dist‘𝐸)    &   ⊢ 0 = ({1, 2} × {0})    ⇒   ⊢ (𝐹 ∈ 𝑋 → (𝐹𝐷 0 ) = (√‘(((𝐹‘1)↑2) + ((𝐹‘2)↑2))))
Theorem	ehl2eudis0lt 47566	An upper bound of the Euclidean distance of a point to the origin in a real Euclidean space of dimension 2. (Contributed by AV, 9-May-2023.)
⊢ 𝐸 = (𝔼hil‘2)    &   ⊢ 𝑋 = (ℝ ↑m {1, 2})    &   ⊢ 𝐷 = (dist‘𝐸)    &   ⊢ 0 = ({1, 2} × {0})    ⇒   ⊢ ((𝐹 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+) → ((𝐹𝐷 0 ) < 𝑅 ↔ (((𝐹‘1)↑2) + ((𝐹‘2)↑2)) < (𝑅↑2)))
21.45.22.3  Spheres and lines in real Euclidean spaces
Syntax	cline 47567	Declare the syntax for lines in generalized real Euclidean spaces.
class LineM
Syntax	csph 47568	Declare the syntax for spheres in generalized real Euclidean spaces.
class Sphere
Definition	df-line 47569*	Definition of lines passing through two different points in a left module (or any extended structure having a base set, an addition, and a scalar multiplication). (Contributed by AV, 14-Jan-2023.)
⊢ LineM = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤), 𝑦 ∈ ((Base‘𝑤) ∖ {𝑥}) ↦ {𝑝 ∈ (Base‘𝑤) ∣ ∃𝑡 ∈ (Base‘(Scalar‘𝑤))𝑝 = ((((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡)( ·𝑠 ‘𝑤)𝑥)(+g‘𝑤)(𝑡( ·𝑠 ‘𝑤)𝑦))}))
Definition	df-sph 47570*	Definition of spheres for given centers and radii in a metric space (or more generally, in a distance space, see distspace 24143, or even in any extended structure having a base set and a distance function into the real numbers. (Contributed by AV, 14-Jan-2023.)
⊢ Sphere = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤), 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟}))
Theorem	lines 47571*	The lines passing through two different points in a left module (or any extended structure having a base set, an addition, and a scalar multiplication). (Contributed by AV, 14-Jan-2023.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐿 = (LineM‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ − = (-g‘𝑆)    &   ⊢ 1 = (1r‘𝑆)    ⇒   ⊢ (𝑊 ∈ 𝑉 → 𝐿 = (𝑥 ∈ 𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))}))
Theorem	line 47572*	The line passing through the two different points 𝑋 and 𝑌 in a left module (or any extended structure having a base set, an addition, and a scalar multiplication). (Contributed by AV, 14-Jan-2023.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐿 = (LineM‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ − = (-g‘𝑆)    &   ⊢ 1 = (1r‘𝑆)    ⇒   ⊢ ((𝑊 ∈ 𝑉 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ≠ 𝑌)) → (𝑋𝐿𝑌) = {𝑝 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝐾 𝑝 = ((( 1 − 𝑡) · 𝑋) + (𝑡 · 𝑌))})
Theorem	rrxlines 47573*	Definition of lines passing through two different points in a generalized real Euclidean space of finite dimension. (Contributed by AV, 14-Jan-2023.)
⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝐿 = (LineM‘𝐸)    &   ⊢ · = ( ·𝑠 ‘𝐸)    &   ⊢ + = (+g‘𝐸)    ⇒   ⊢ (𝐼 ∈ Fin → 𝐿 = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))}))
Theorem	rrxline 47574*	The line passing through the two different points 𝑋 and 𝑌 in a generalized real Euclidean space of finite dimension. (Contributed by AV, 14-Jan-2023.)
⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝐿 = (LineM‘𝐸)    &   ⊢ · = ( ·𝑠 ‘𝐸)    &   ⊢ + = (+g‘𝐸)    ⇒   ⊢ ((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) → (𝑋𝐿𝑌) = {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡) · 𝑋) + (𝑡 · 𝑌))})
Theorem	rrxlinesc 47575*	Definition of lines passing through two different points in a generalized real Euclidean space of finite dimension, expressed by their coordinates. (Contributed by AV, 13-Feb-2023.)
⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝐿 = (LineM‘𝐸)    ⇒   ⊢ (𝐼 ∈ Fin → 𝐿 = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ ∀𝑖 ∈ 𝐼 (𝑝‘𝑖) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖)))}))
Theorem	rrxlinec 47576*	The line passing through the two different points 𝑋 and 𝑌 in a generalized real Euclidean space of finite dimension, expressed by its coordinates. Remark: This proof is shorter and requires less distinct variables than the proof using rrxlinesc 47575. (Contributed by AV, 13-Feb-2023.)
⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝐿 = (LineM‘𝐸)    ⇒   ⊢ ((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) → (𝑋𝐿𝑌) = {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ ∀𝑖 ∈ 𝐼 (𝑝‘𝑖) = (((1 − 𝑡) · (𝑋‘𝑖)) + (𝑡 · (𝑌‘𝑖)))})
Theorem	eenglngeehlnmlem1 47577*	Lemma 1 for eenglngeehlnm 47579. (Contributed by AV, 15-Feb-2023.)
⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) → ((∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − 𝑘) · (𝑥‘𝑖)) + (𝑘 · (𝑦‘𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖)))) → ∃𝑡 ∈ ℝ ∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖)))))
Theorem	eenglngeehlnmlem2 47578*	Lemma 2 for eenglngeehlnm 47579. (Contributed by AV, 15-Feb-2023.)
⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) → (∃𝑡 ∈ ℝ ∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖))) → (∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − 𝑘) · (𝑥‘𝑖)) + (𝑘 · (𝑦‘𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))))))
Theorem	eenglngeehlnm 47579	The line definition in the Tarski structure for the Euclidean geometry (see elntg 28677) corresponds to the definition of lines passing through two different points in a left module (see rrxlines 47573). (Contributed by AV, 16-Feb-2023.)
⊢ (𝑁 ∈ ℕ → (LineG‘(EEG‘𝑁)) = (LineM‘(𝔼hil‘𝑁)))
Theorem	rrx2line 47580*	The line passing through the two different points 𝑋 and 𝑌 in a real Euclidean space of dimension 2. (Contributed by AV, 22-Jan-2023.) (Proof shortened by AV, 13-Feb-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝐿 = (LineM‘𝐸)    ⇒   ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (𝑋𝐿𝑌) = {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ ((𝑝‘1) = (((1 − 𝑡) · (𝑋‘1)) + (𝑡 · (𝑌‘1))) ∧ (𝑝‘2) = (((1 − 𝑡) · (𝑋‘2)) + (𝑡 · (𝑌‘2))))})
Theorem	rrx2vlinest 47581*	The vertical line passing through the two different points 𝑋 and 𝑌 in a real Euclidean space of dimension 2 in "standard form". (Contributed by AV, 2-Feb-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝐿 = (LineM‘𝐸)    ⇒   ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) ≠ (𝑌‘2))) → (𝑋𝐿𝑌) = {𝑝 ∈ 𝑃 ∣ (𝑝‘1) = (𝑋‘1)})
Theorem	rrx2linest 47582*	The line passing through the two different points 𝑋 and 𝑌 in a real Euclidean space of dimension 2 in "standard form". (Contributed by AV, 2-Feb-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝐿 = (LineM‘𝐸)    &   ⊢ 𝐴 = ((𝑌‘1) − (𝑋‘1))    &   ⊢ 𝐵 = ((𝑌‘2) − (𝑋‘2))    &   ⊢ 𝐶 = (((𝑋‘2) · (𝑌‘1)) − ((𝑋‘1) · (𝑌‘2)))    ⇒   ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (𝑋𝐿𝑌) = {𝑝 ∈ 𝑃 ∣ (𝐴 · (𝑝‘2)) = ((𝐵 · (𝑝‘1)) + 𝐶)})
Theorem	rrx2linesl 47583*	The line passing through the two different points 𝑋 and 𝑌 in a real Euclidean space of dimension 2, expressed by the slope 𝑆 between the two points ("point-slope form"), sometimes also written as ((𝑝‘2) − (𝑋‘2)) = (𝑆 · ((𝑝‘1) − (𝑋‘1))). (Contributed by AV, 22-Jan-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝐿 = (LineM‘𝐸)    &   ⊢ 𝑆 = (((𝑌‘2) − (𝑋‘2)) / ((𝑌‘1) − (𝑋‘1)))    ⇒   ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) → (𝑋𝐿𝑌) = {𝑝 ∈ 𝑃 ∣ (𝑝‘2) = ((𝑆 · ((𝑝‘1) − (𝑋‘1))) + (𝑋‘2))})
Theorem	rrx2linest2 47584*	The line passing through the two different points 𝑋 and 𝑌 in a real Euclidean space of dimension 2 in another "standard form" (usually with (𝑝‘1) = 𝑥 and (𝑝‘2) = 𝑦). (Contributed by AV, 23-Feb-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝐿 = (LineM‘𝐸)    &   ⊢ 𝐴 = ((𝑋‘2) − (𝑌‘2))    &   ⊢ 𝐵 = ((𝑌‘1) − (𝑋‘1))    &   ⊢ 𝐶 = (((𝑋‘2) · (𝑌‘1)) − ((𝑋‘1) · (𝑌‘2)))    ⇒   ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (𝑋𝐿𝑌) = {𝑝 ∈ 𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶})
Theorem	elrrx2linest2 47585	The line passing through the two different points 𝑋 and 𝑌 in a real Euclidean space of dimension 2 in another "standard form" (usually with (𝑝‘1) = 𝑥 and (𝑝‘2) = 𝑦). (Contributed by AV, 23-Feb-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝐿 = (LineM‘𝐸)    &   ⊢ 𝐴 = ((𝑋‘2) − (𝑌‘2))    &   ⊢ 𝐵 = ((𝑌‘1) − (𝑋‘1))    &   ⊢ 𝐶 = (((𝑋‘2) · (𝑌‘1)) − ((𝑋‘1) · (𝑌‘2)))    ⇒   ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (𝐺 ∈ (𝑋𝐿𝑌) ↔ (𝐺 ∈ 𝑃 ∧ ((𝐴 · (𝐺‘1)) + (𝐵 · (𝐺‘2))) = 𝐶)))
Theorem	spheres 47586*	The spheres for given centers and radii in a metric space (or any extensible structure having a base set and a distance function). (Contributed by AV, 22-Jan-2023.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝑆 = (Sphere‘𝑊)    &   ⊢ 𝐷 = (dist‘𝑊)    ⇒   ⊢ (𝑊 ∈ 𝑉 → 𝑆 = (𝑥 ∈ 𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}))
Theorem	sphere 47587*	A sphere with center 𝑋 and radius 𝑅 in a metric space (or any extensible structure having a base set and a distance function). (Contributed by AV, 22-Jan-2023.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝑆 = (Sphere‘𝑊)    &   ⊢ 𝐷 = (dist‘𝑊)    ⇒   ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑅 ∈ (0[,]+∞)) → (𝑋𝑆𝑅) = {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑋) = 𝑅})
Theorem	rrxsphere 47588*	The sphere with center 𝑀 and radius 𝑅 in a generalized real Euclidean space of finite dimension. Remark: this theorem holds also for the degenerate case 𝑅 < 0 (negative radius): in this case, (𝑀𝑆𝑅) is empty. (Contributed by AV, 5-Feb-2023.)
⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝐷 = (dist‘𝐸)    &   ⊢ 𝑆 = (Sphere‘𝐸)    ⇒   ⊢ ((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) → (𝑀𝑆𝑅) = {𝑝 ∈ 𝑃 ∣ (𝑝𝐷𝑀) = 𝑅})
Theorem	2sphere 47589*	The sphere with center 𝑀 and radius 𝑅 in a two dimensional Euclidean space is a circle. (Contributed by AV, 5-Feb-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝑆 = (Sphere‘𝐸)    &   ⊢ 𝐶 = {𝑝 ∈ 𝑃 ∣ ((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2)) = (𝑅↑2)}    ⇒   ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) → (𝑀𝑆𝑅) = 𝐶)
Theorem	2sphere0 47590*	The sphere around the origin 0 (see rrx0 25246) with radius 𝑅 in a two dimensional Euclidean space is a circle. (Contributed by AV, 5-Feb-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝑆 = (Sphere‘𝐸)    &   ⊢ 0 = (𝐼 × {0})    &   ⊢ 𝐶 = {𝑝 ∈ 𝑃 ∣ (((𝑝‘1)↑2) + ((𝑝‘2)↑2)) = (𝑅↑2)}    ⇒   ⊢ (𝑅 ∈ (0[,)+∞) → ( 0 𝑆𝑅) = 𝐶)
Theorem	line2ylem 47591*	Lemma for line2y 47595. This proof is based on counterexamples for the following cases: 1. 𝐶 ≠ 0: p = (0,0) (LHS of bicondional is false, RHS is true); 2. 𝐶 = 0 ∧ 𝐵 ≠ 0: p = (1,-A/B) (LHS of bicondional is true, RHS is false); 3. 𝐴 = 𝐵 = 𝐶 = 0: p = (1,1) (LHS of bicondional is true, RHS is false). (Contributed by AV, 4-Feb-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    ⇒   ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (∀𝑝 ∈ 𝑃 (((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶 ↔ (𝑝‘1) = 0) → (𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0)))
Theorem	line2 47592*	Example for a line 𝐺 passing through two different points in "standard form". (Contributed by AV, 3-Feb-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝐿 = (LineM‘𝐸)    &   ⊢ 𝐺 = {𝑝 ∈ 𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶}    &   ⊢ 𝑋 = {⟨1, 0⟩, ⟨2, (𝐶 / 𝐵)⟩}    &   ⊢ 𝑌 = {⟨1, 1⟩, ⟨2, ((𝐶 − 𝐴) / 𝐵)⟩}    ⇒   ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) ∧ 𝐶 ∈ ℝ) → 𝐺 = (𝑋𝐿𝑌))
Theorem	line2xlem 47593*	Lemma for line2x 47594. This proof is based on counterexamples for the following cases: 1. 𝑀 ≠ (𝐶 / 𝐵): p = (0,C/B) (LHS of bicondional is true, RHS is false); 2. 𝐴 ≠ 0 ∧ 𝑀 = (𝐶 / 𝐵): p = (1,C/B) (LHS of bicondional is false, RHS is true). (Contributed by AV, 4-Feb-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝐿 = (LineM‘𝐸)    &   ⊢ 𝐺 = {𝑝 ∈ 𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶}    &   ⊢ 𝑋 = {⟨1, 0⟩, ⟨2, 𝑀⟩}    &   ⊢ 𝑌 = {⟨1, 1⟩, ⟨2, 𝑀⟩}    ⇒   ⊢ (((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) ∧ 𝐶 ∈ ℝ) ∧ 𝑀 ∈ ℝ) → (∀𝑝 ∈ 𝑃 (((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶 ↔ (𝑝‘2) = 𝑀) → (𝐴 = 0 ∧ 𝑀 = (𝐶 / 𝐵))))
Theorem	line2x 47594*	Example for a horizontal line 𝐺 passing through two different points in "standard form". (Contributed by AV, 3-Feb-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝐿 = (LineM‘𝐸)    &   ⊢ 𝐺 = {𝑝 ∈ 𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶}    &   ⊢ 𝑋 = {⟨1, 0⟩, ⟨2, 𝑀⟩}    &   ⊢ 𝑌 = {⟨1, 1⟩, ⟨2, 𝑀⟩}    ⇒   ⊢ (((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) ∧ 𝐶 ∈ ℝ) ∧ 𝑀 ∈ ℝ) → (𝐺 = (𝑋𝐿𝑌) ↔ (𝐴 = 0 ∧ 𝑀 = (𝐶 / 𝐵))))
Theorem	line2y 47595*	Example for a vertical line 𝐺 passing through two different points in "standard form". (Contributed by AV, 3-Feb-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝐿 = (LineM‘𝐸)    &   ⊢ 𝐺 = {𝑝 ∈ 𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶}    &   ⊢ 𝑋 = {⟨1, 0⟩, ⟨2, 𝑀⟩}    &   ⊢ 𝑌 = {⟨1, 0⟩, ⟨2, 𝑁⟩}    ⇒   ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) → (𝐺 = (𝑋𝐿𝑌) ↔ (𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0)))
Theorem	itsclc0lem1 47596	Lemma for theorems about intersections of lines and circles in a real Euclidean space of dimension 2 . (Contributed by AV, 2-May-2023.)
⊢ (((𝑆 ∈ ℝ ∧ 𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ (𝑉 ∈ ℝ ∧ 0 ≤ 𝑉) ∧ (𝑊 ∈ ℝ ∧ 𝑊 ≠ 0)) → (((𝑆 · 𝑈) + (𝑇 · (√‘𝑉))) / 𝑊) ∈ ℝ)
Theorem	itsclc0lem2 47597	Lemma for theorems about intersections of lines and circles in a real Euclidean space of dimension 2 . (Contributed by AV, 3-May-2023.)
⊢ (((𝑆 ∈ ℝ ∧ 𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ) ∧ (𝑉 ∈ ℝ ∧ 0 ≤ 𝑉) ∧ (𝑊 ∈ ℝ ∧ 𝑊 ≠ 0)) → (((𝑆 · 𝑈) − (𝑇 · (√‘𝑉))) / 𝑊) ∈ ℝ)
Theorem	itsclc0lem3 47598	Lemma for theorems about intersections of lines and circles in a real Euclidean space of dimension 2 . (Contributed by AV, 2-May-2023.)
⊢ 𝑄 = ((𝐴↑2) + (𝐵↑2))    &   ⊢ 𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))    ⇒   ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝑅 ∈ ℝ) → 𝐷 ∈ ℝ)
Theorem	itscnhlc0yqe 47599	Lemma for itsclc0 47611. Quadratic equation for the y-coordinate of the intersection points of a nonhorizontal line and a circle. (Contributed by AV, 6-Feb-2023.)
⊢ 𝑄 = ((𝐴↑2) + (𝐵↑2))    &   ⊢ 𝑇 = -(2 · (𝐵 · 𝐶))    &   ⊢ 𝑈 = ((𝐶↑2) − ((𝐴↑2) · (𝑅↑2)))    ⇒   ⊢ ((((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝑅 ∈ ℝ+ ∧ (𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ)) → ((((𝑋↑2) + (𝑌↑2)) = (𝑅↑2) ∧ ((𝐴 · 𝑋) + (𝐵 · 𝑌)) = 𝐶) → ((𝑄 · (𝑌↑2)) + ((𝑇 · 𝑌) + 𝑈)) = 0))
Theorem	itschlc0yqe 47600	Lemma for itsclc0 47611. 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))