P. 461 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 46001-46100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	1arymaptf1o 46001*	The mapping of unary (endo)functions is a one-to-one function onto the set of endofunctions. (Contributed by AV, 19-May-2024.)
⊢ 𝐻 = (ℎ ∈ (1-aryF 𝑋) ↦ (𝑥 ∈ 𝑋 ↦ (ℎ‘{⟨0, 𝑥⟩})))    ⇒   ⊢ (𝑋 ∈ 𝑉 → 𝐻:(1-aryF 𝑋)–1-1-onto→(𝑋 ↑m 𝑋))
Theorem	1aryenef 46002	The set of unary (endo)functions and the set of endofunctions are equinumerous. (Contributed by AV, 19-May-2024.)
⊢ (1-aryF 𝑋) ≈ (𝑋 ↑m 𝑋)
Theorem	1aryenefmnd 46003	The set of unary (endo)functions and the base set of the monoid of endofunctions are equinumerous. (Contributed by AV, 19-May-2024.)
⊢ (1-aryF 𝑋) ≈ (Base‘(EndoFMnd‘𝑋))
Theorem	2aryfvalel 46004	A binary (endo)function on a set 𝑋. (Contributed by AV, 20-May-2024.)
⊢ (𝑋 ∈ 𝑉 → (𝐹 ∈ (2-aryF 𝑋) ↔ 𝐹:(𝑋 ↑m {0, 1})⟶𝑋))
Theorem	fv2arycl 46005	Closure of a binary (endo)function. (Contributed by AV, 20-May-2024.)
⊢ ((𝐺 ∈ (2-aryF 𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐺‘{⟨0, 𝐴⟩, ⟨1, 𝐵⟩}) ∈ 𝑋)
Theorem	2arympt 46006*	A binary (endo)function in maps-to notation. (Contributed by AV, 20-May-2024.)
⊢ 𝐹 = (𝑥 ∈ (𝑋 ↑m {0, 1}) ↦ ((𝑥‘0)𝑂(𝑥‘1)))    ⇒   ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑂:(𝑋 × 𝑋)⟶𝑋) → 𝐹 ∈ (2-aryF 𝑋))
Theorem	2arymptfv 46007*	The value of a binary (endo)function in maps-to notation. (Contributed by AV, 20-May-2024.)
⊢ 𝐹 = (𝑥 ∈ (𝑋 ↑m {0, 1}) ↦ ((𝑥‘0)𝑂(𝑥‘1)))    ⇒   ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐹‘{⟨0, 𝐴⟩, ⟨1, 𝐵⟩}) = (𝐴𝑂𝐵))
Theorem	2arymaptfv 46008*	The value of the mapping of binary (endo)functions. (Contributed by AV, 21-May-2024.)
⊢ 𝐻 = (ℎ ∈ (2-aryF 𝑋) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{⟨0, 𝑥⟩, ⟨1, 𝑦⟩})))    ⇒   ⊢ (𝐹 ∈ (2-aryF 𝑋) → (𝐻‘𝐹) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝐹‘{⟨0, 𝑥⟩, ⟨1, 𝑦⟩})))
Theorem	2arymaptf 46009*	The mapping of binary (endo)functions is a function into the set of binary operations. (Contributed by AV, 21-May-2024.)
⊢ 𝐻 = (ℎ ∈ (2-aryF 𝑋) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{⟨0, 𝑥⟩, ⟨1, 𝑦⟩})))    ⇒   ⊢ (𝑋 ∈ 𝑉 → 𝐻:(2-aryF 𝑋)⟶(𝑋 ↑m (𝑋 × 𝑋)))
Theorem	2arymaptf1 46010*	The mapping of binary (endo)functions is a one-to-one function into the set of binary operations. (Contributed by AV, 22-May-2024.)
⊢ 𝐻 = (ℎ ∈ (2-aryF 𝑋) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{⟨0, 𝑥⟩, ⟨1, 𝑦⟩})))    ⇒   ⊢ (𝑋 ∈ 𝑉 → 𝐻:(2-aryF 𝑋)–1-1→(𝑋 ↑m (𝑋 × 𝑋)))
Theorem	2arymaptfo 46011*	The mapping of binary (endo)functions is a function onto the set of binary operations. (Contributed by AV, 23-May-2024.)
⊢ 𝐻 = (ℎ ∈ (2-aryF 𝑋) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{⟨0, 𝑥⟩, ⟨1, 𝑦⟩})))    ⇒   ⊢ (𝑋 ∈ 𝑉 → 𝐻:(2-aryF 𝑋)–onto→(𝑋 ↑m (𝑋 × 𝑋)))
Theorem	2arymaptf1o 46012*	The mapping of binary (endo)functions is a one-to-one function onto the set of binary operations. (Contributed by AV, 23-May-2024.)
⊢ 𝐻 = (ℎ ∈ (2-aryF 𝑋) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{⟨0, 𝑥⟩, ⟨1, 𝑦⟩})))    ⇒   ⊢ (𝑋 ∈ 𝑉 → 𝐻:(2-aryF 𝑋)–1-1-onto→(𝑋 ↑m (𝑋 × 𝑋)))
Theorem	2aryenef 46013	The set of binary (endo)functions and the set of binary operations are equinumerous. (Contributed by AV, 19-May-2024.)
⊢ (2-aryF 𝑋) ≈ (𝑋 ↑m (𝑋 × 𝑋))
20.41.22.14  The Ackermann function According to Wikipedia ("Ackermann function", 8-May-2024, https://en.wikipedia.org/wiki/Ackermann_function): "In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive. ... One common version is the two-argument Ackermann-Péter function developed by Rózsa Péter and Raphael Robinson. Its value grows very rapidly; for example, A(4,2) results in 2^65536-3 [see ackval42 46053)], an integer of 19,729 decimal digits." In the following, the Ackermann function is defined as iterated 1-ary function (also mentioned in Wikipedia), see df-ack 46017, based on a definition IterComp of "the n-th iterate of (a class/function) f", see df-itco 46016. As an illustration, we have ((IterComp‘𝐹)‘3) = (𝐹 ∘ (𝐹 ∘ 𝐹))) (see itcoval3 46022). The following recursive definition of the Ackermann function follows immediately from Definition df-ack 46017: ((Ack‘(𝑀 + 1))‘𝑁) = (((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))‘1)). That Definition df-ack 46017 is equivalent to Péter's definition is proven by the following three theorems: ackval0val 46043: ((Ack‘0)‘𝑀) = (𝑀 + 1); ackvalsuc0val 46044: ((Ack‘(𝑀 + 1))‘0) = ((Ack‘𝑀)‘1); ackvalsucsucval 46045: ((Ack‘(𝑀 + 1))‘(𝑁 + 1)) = ((Ack‘𝑀)‘((Ack‘(𝑀 + 1))‘𝑁)). The initial values of the Ackermann function are calculated in the following four theorems: ackval0012 46046: 𝐴(0, 0) = 1, 𝐴(0, 1) = 2, 𝐴(0, 2) = 3; ackval1012 46047: 𝐴(1, 0) = 2, 𝐴(1, 1) = 3, 𝐴(1, 3) = 4; ackval2012 46048: 𝐴(2, 0) = 3, 𝐴(2, 1) = 5, 𝐴(2, 3) = 7; ackval3012 46049: 𝐴(3, 0) = 5, 𝐴(3, 1) = ;13, 𝐴(3, 3) = ;29.
Syntax	citco 46014	Extend the definition of a class to include iterated functions.
class IterComp
Syntax	cack 46015	Extend the definition of a class to include the Ackermann function operator.
class Ack
Definition	df-itco 46016*	Define a function (recursively) that returns the n-th iterate of a class (usually a function) with regard to composition. (Contributed by Thierry Arnoux, 28-Apr-2024.) (Revised by AV, 2-May-2024.)
⊢ IterComp = (𝑓 ∈ V ↦ seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝑓 ∘ 𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝑓), 𝑓))))
Definition	df-ack 46017*	Define the Ackermann function (recursively). (Contributed by Thierry Arnoux, 28-Apr-2024.) (Revised by AV, 2-May-2024.)
⊢ Ack = seq0((𝑓 ∈ V, 𝑗 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ (((IterComp‘𝑓)‘(𝑛 + 1))‘1))), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)), 𝑖)))
Theorem	itcoval 46018*	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 46019	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 46020	A function iterated once. (Contributed by AV, 2-May-2024.)
⊢ ((Rel 𝐹 ∧ 𝐹 ∈ 𝑉) → ((IterComp‘𝐹)‘1) = 𝐹)
Theorem	itcoval2 46021	A function iterated twice. (Contributed by AV, 2-May-2024.)
⊢ ((Rel 𝐹 ∧ 𝐹 ∈ 𝑉) → ((IterComp‘𝐹)‘2) = (𝐹 ∘ 𝐹))
Theorem	itcoval3 46022	A function iterated three times. (Contributed by AV, 2-May-2024.)
⊢ ((Rel 𝐹 ∧ 𝐹 ∈ 𝑉) → ((IterComp‘𝐹)‘3) = (𝐹 ∘ (𝐹 ∘ 𝐹)))
Theorem	itcoval0mpt 46023*	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 46024*	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 46025	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 46026	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 46027*	Lemma 1 for itcovalpc 46029: 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 46028*	Lemma 2 for itcovalpc 46029: 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 46029*	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 46030	Lemma 1 for itcovalt2lem2 46033. (Contributed by AV, 6-May-2024.)
⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → (((𝑁 + 𝐶) · 𝑌) − 𝐶) ∈ ℕ0)
Theorem	itcovalt2lem2lem2 46031	Lemma 2 for itcovalt2lem2 46033. (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 46032*	Lemma 1 for itcovalt2 46034: 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 46033*	Lemma 2 for itcovalt2 46034: 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 46034*	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 46035*	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 46036	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 46037	The Ackermann function at 0. (Contributed by AV, 2-May-2024.)
⊢ (Ack‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 1))
Theorem	ackval1 46038	The Ackermann function at 1. (Contributed by AV, 4-May-2024.)
⊢ (Ack‘1) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2))
Theorem	ackval2 46039	The Ackermann function at 2. (Contributed by AV, 4-May-2024.)
⊢ (Ack‘2) = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 3))
Theorem	ackval3 46040	The Ackermann function at 3. (Contributed by AV, 7-May-2024.)
⊢ (Ack‘3) = (𝑛 ∈ ℕ0 ↦ ((2↑(𝑛 + 3)) − 3))
Theorem	ackendofnn0 46041	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 46042	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 46043	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 46044	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 46045	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 46046	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 46047	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 46048	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 46049	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 46050	The Ackermann function at (4,0). (Contributed by AV, 9-May-2024.)
⊢ ((Ack‘4)‘0) = ;13
Theorem	ackval41a 46051	The Ackermann function at (4,1). (Contributed by AV, 9-May-2024.)
⊢ ((Ack‘4)‘1) = ((2↑;16) − 3)
Theorem	ackval41 46052	The Ackermann function at (4,1). (Contributed by AV, 9-May-2024.)
⊢ ((Ack‘4)‘1) = ;;;;65533
Theorem	ackval42 46053	The Ackermann function at (4,2). (Contributed by AV, 9-May-2024.)
⊢ ((Ack‘4)‘2) = ((2↑;;;;65536) − 3)
Theorem	ackval42a 46054	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 46055	The Ackermann function at (5,0). (Contributed by AV, 9-May-2024.)
⊢ ((Ack‘5)‘0) = ;;;;65533
20.41.23  Elementary geometry (extension)
20.41.23.1  Auxiliary theorems
Theorem	fv1prop 46056	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 46057	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 46058	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 46059*	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 46060*	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 46061	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 46062	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 46063	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 46064	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 46065	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 46066	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 46067	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))
20.41.23.2  Real euclidean space of dimension 2
Theorem	rrx2pxel 46068	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 46069	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 46070	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 46071	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 46072	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 46073	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 46074	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 46075*	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 46076	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 46077*	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 46078*	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 46079*	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 46080*	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 46081*	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 46082	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 46083	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)))
20.41.23.3  Spheres and lines in real Euclidean spaces
Syntax	cline 46084	Declare the syntax for lines in generalized real Euclidean spaces.
class LineM
Syntax	csph 46085	Declare the syntax for spheres in generalized real Euclidean spaces.
class Sphere
Definition	df-line 46086*	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 46087*	Definition of spheres for given centers and radii in a metric space (or more generally, in a distance space, see distspace 23478, 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 46088*	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 46089*	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 46090*	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 46091*	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 46092*	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 46093*	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 46092. (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 46094*	Lemma 1 for eenglngeehlnm 46096. (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 46095*	Lemma 2 for eenglngeehlnm 46096. (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 46096	The line definition in the Tarski structure for the Euclidean geometry (see elntg 27361) corresponds to the definition of lines passing through two different points in a left module (see rrxlines 46090). (Contributed by AV, 16-Feb-2023.)
⊢ (𝑁 ∈ ℕ → (LineG‘(EEG‘𝑁)) = (LineM‘(𝔼hil‘𝑁)))
Theorem	rrx2line 46097*	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 46098*	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 46099*	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 46100*	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))})