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Theorem List for Metamath Proof Explorer - 44801-44900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnn0digval 44801 The 𝐾 th digit of a nonnegative real number 𝑅 in the positional system with base 𝐵. (Contributed by AV, 23-May-2020.)
((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℕ0𝑅 ∈ (0[,)+∞)) → (𝐾(digit‘𝐵)𝑅) = ((⌊‘(𝑅 / (𝐵𝐾))) mod 𝐵))

Theoremdignn0fr 44802 The digits of the fractional part of a nonnegative integer are 0. (Contributed by AV, 23-May-2020.)
((𝐵 ∈ ℕ ∧ 𝐾 ∈ (ℤ ∖ ℕ0) ∧ 𝑁 ∈ ℕ0) → (𝐾(digit‘𝐵)𝑁) = 0)

Theoremdignn0ldlem 44803 Lemma for dignnld 44804. (Contributed by AV, 25-May-2020.)
((𝐵 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ‘((⌊‘(𝐵 logb 𝑁)) + 1))) → 𝑁 < (𝐵𝐾))

Theoremdignnld 44804 The leading digits of a positive integer are 0. (Contributed by AV, 25-May-2020.)
((𝐵 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ‘((⌊‘(𝐵 logb 𝑁)) + 1))) → (𝐾(digit‘𝐵)𝑁) = 0)

Theoremdig2nn0ld 44805 The leading digits of a positive integer in a binary system are 0. (Contributed by AV, 25-May-2020.)
((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ‘(#b𝑁))) → (𝐾(digit‘2)𝑁) = 0)

Theoremdig2nn1st 44806 The first (relevant) digit of a positive integer in a binary system is 1. (Contributed by AV, 26-May-2020.)
(𝑁 ∈ ℕ → (((#b𝑁) − 1)(digit‘2)𝑁) = 1)

Theoremdig0 44807 All digits of 0 are 0. (Contributed by AV, 24-May-2020.)
((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (𝐾(digit‘𝐵)0) = 0)

Theoremdigexp 44808 The 𝐾 th digit of a power to the base is either 1 or 0. (Contributed by AV, 24-May-2020.)
((𝐵 ∈ (ℤ‘2) ∧ 𝐾 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐾(digit‘𝐵)(𝐵𝑁)) = if(𝐾 = 𝑁, 1, 0))

Theoremdig1 44809 All but one digits of 1 are 0. (Contributed by AV, 24-May-2020.)
((𝐵 ∈ (ℤ‘2) ∧ 𝐾 ∈ ℤ) → (𝐾(digit‘𝐵)1) = if(𝐾 = 0, 1, 0))

Theorem0dig1 44810 The 0 th digit of 1 is 1 in any positional system. (Contributed by AV, 28-May-2020.)
(𝐵 ∈ (ℤ‘2) → (0(digit‘𝐵)1) = 1)

Theorem0dig2pr01 44811 The integers 0 and 1 correspond to their last bit. (Contributed by AV, 28-May-2010.)
(𝑁 ∈ {0, 1} → (0(digit‘2)𝑁) = 𝑁)

Theoremdig2nn0 44812 A digit of a nonnegative integer 𝑁 in a binary system is either 0 or 1. (Contributed by AV, 24-May-2020.)
((𝑁 ∈ ℕ0𝐾 ∈ ℤ) → (𝐾(digit‘2)𝑁) ∈ {0, 1})

Theorem0dig2nn0e 44813 The last bit of an even integer is 0. (Contributed by AV, 3-Jun-2010.)
((𝑁 ∈ ℕ0 ∧ (𝑁 / 2) ∈ ℕ0) → (0(digit‘2)𝑁) = 0)

Theorem0dig2nn0o 44814 The last bit of an odd integer is 1. (Contributed by AV, 3-Jun-2010.)
((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → (0(digit‘2)𝑁) = 1)

Theoremdig2bits 44815 The 𝐾 th digit of a nonnegative integer 𝑁 in a binary system is its 𝐾 th bit. (Contributed by AV, 24-May-2020.)
((𝑁 ∈ ℕ0𝐾 ∈ ℕ0) → ((𝐾(digit‘2)𝑁) = 1 ↔ 𝐾 ∈ (bits‘𝑁)))

20.41.22.11  Nonnegative integer as sum of its shifted digits

Theoremdignn0flhalflem1 44816 Lemma 1 for dignn0flhalf 44819. (Contributed by AV, 7-Jun-2012.)
((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ) → (⌊‘((𝐴 / (2↑𝑁)) − 1)) < (⌊‘((𝐴 − 1) / (2↑𝑁))))

Theoremdignn0flhalflem2 44817 Lemma 2 for dignn0flhalf 44819. (Contributed by AV, 7-Jun-2012.)
((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (⌊‘(𝐴 / (2↑(𝑁 + 1)))) = (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))))

Theoremdignn0ehalf 44818 The digits of the half of an even nonnegative integer are the digits of the integer shifted by 1. (Contributed by AV, 3-Jun-2010.)
(((𝐴 / 2) ∈ ℕ0𝐴 ∈ ℕ0𝐼 ∈ ℕ0) → ((𝐼 + 1)(digit‘2)𝐴) = (𝐼(digit‘2)(𝐴 / 2)))

Theoremdignn0flhalf 44819 The digits of the rounded half of a nonnegative integer are the digits of the integer shifted by 1. (Contributed by AV, 7-Jun-2010.)
((𝐴 ∈ (ℤ‘2) ∧ 𝐼 ∈ ℕ0) → ((𝐼 + 1)(digit‘2)𝐴) = (𝐼(digit‘2)(⌊‘(𝐴 / 2))))

Theoremnn0sumshdiglemA 44820* Lemma for nn0sumshdig 44824 (induction step, even multiplier). (Contributed by AV, 3-Jun-2020.)
(((𝑎 ∈ ℕ ∧ (𝑎 / 2) ∈ ℕ) ∧ 𝑦 ∈ ℕ) → (∀𝑥 ∈ ℕ0 ((#b𝑥) = 𝑦𝑥 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑥) · (2↑𝑘))) → ((#b𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)))))

Theoremnn0sumshdiglemB 44821* Lemma for nn0sumshdig 44824 (induction step, odd multiplier). (Contributed by AV, 7-Jun-2020.)
(((𝑎 ∈ ℕ ∧ ((𝑎 − 1) / 2) ∈ ℕ0) ∧ 𝑦 ∈ ℕ) → (∀𝑥 ∈ ℕ0 ((#b𝑥) = 𝑦𝑥 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑥) · (2↑𝑘))) → ((#b𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)))))

Theoremnn0sumshdiglem1 44822* Lemma 1 for nn0sumshdig 44824 (induction step). (Contributed by AV, 7-Jun-2020.)
(𝑦 ∈ ℕ → (∀𝑎 ∈ ℕ0 ((#b𝑎) = 𝑦𝑎 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑎) · (2↑𝑘))) → ∀𝑎 ∈ ℕ0 ((#b𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)))))

Theoremnn0sumshdiglem2 44823* Lemma 2 for nn0sumshdig 44824. (Contributed by AV, 7-Jun-2020.)
(𝐿 ∈ ℕ → ∀𝑎 ∈ ℕ0 ((#b𝑎) = 𝐿𝑎 = Σ𝑘 ∈ (0..^𝐿)((𝑘(digit‘2)𝑎) · (2↑𝑘))))

Theoremnn0sumshdig 44824* A nonnegative integer can be represented as sum of its shifted bits. (Contributed by AV, 7-Jun-2020.)
(𝐴 ∈ ℕ0𝐴 = Σ𝑘 ∈ (0..^(#b𝐴))((𝑘(digit‘2)𝐴) · (2↑𝑘)))

20.41.22.12  Algorithms for the multiplication of nonnegative integers

Theoremnn0mulfsum 44825* Trivial algorithm to calculate the product of two nonnegative integers 𝑎 and 𝑏 by adding up 𝑏 𝑎 times. (Contributed by AV, 17-May-2020.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = Σ𝑘 ∈ (1...𝐴)𝐵)

Theoremnn0mullong 44826* Standard algorithm (also known as "long multiplication" or "grade-school multiplication") to calculate the product of two nonnegative integers 𝑎 and 𝑏 by multiplying the multiplicand 𝑏 by each digit of the multiplier 𝑎 and then add up all the properly shifted results. Here, the binary representation of the multiplier 𝑎 is used, i.e. the above mentioned "digits" are 0 or 1. This is a similar result as provided by smumul 15816. (Contributed by AV, 7-Jun-2020.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = Σ𝑘 ∈ (0..^(#b𝐴))(((𝑘(digit‘2)𝐴) · (2↑𝑘)) · 𝐵))

20.41.22.13  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 44866)], 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 44830, based on a definition IterComp of "the n-th iterate of (a class/function) f", see df-itco 44829. As an illustration, we have ((IterComp‘𝐹)‘3) = (𝐹 ∘ (𝐹𝐹))) (see itcoval3 44835).

The following recursive definition of the Ackermann function follows immediately from the definition df-ack 44830: ((Ack‘(𝑀 + 1))‘𝑁) = (((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))‘1)).

That the definition df-ack 44830 is equivalent to Péter's definition is proven by the following three theorems:

ackval0val 44856: ((Ack‘0)‘𝑀) = (𝑀 + 1) ackvalsuc0val 44857: ((Ack‘(𝑀 + 1))‘0) = ((Ack‘𝑀)‘1) ackvalsucsucval 44858: ((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 44859: 𝐴(0, 0) = 1, 𝐴(0, 1) = 2, 𝐴(0, 2) = 3 ackval1012 44860: 𝐴(1, 0) = 2, 𝐴(1, 1) = 3, 𝐴(1, 3) = 4 ackval2012 44861: 𝐴(2, 0) = 3, 𝐴(2, 1) = 5, 𝐴(2, 3) = 7 ackval3012 44862: 𝐴(3, 0) = 5, 𝐴(3, 1) = 13, 𝐴(3, 3) = 29

Syntaxcitco 44827 Extend the definition of a class to include iterated functions.
class IterComp

Syntaxcack 44828 Extend the definition of a class to include the Ackermann function operator.
class Ack

Definitiondf-itco 44829* 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 ↦ (𝑓𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝑓), 𝑓))))

Definitiondf-ack 44830* 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)))

Theoremitcoval 44831* 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.)
(𝐹𝑉 → (IterComp‘𝐹) = seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹))))

Theoremitcoval0 44832 A function iterated zero times (defined as identity function). (Contributed by AV, 2-May-2024.)
(𝐹𝑉 → ((IterComp‘𝐹)‘0) = ( I ↾ dom 𝐹))

Theoremitcoval1 44833 A function iterated once. (Contributed by AV, 2-May-2024.)
((Rel 𝐹𝐹𝑉) → ((IterComp‘𝐹)‘1) = 𝐹)

Theoremitcoval2 44834 A function iterated twice. (Contributed by AV, 2-May-2024.)
((Rel 𝐹𝐹𝑉) → ((IterComp‘𝐹)‘2) = (𝐹𝐹))

Theoremitcoval3 44835 A function iterated three times. (Contributed by AV, 2-May-2024.)
((Rel 𝐹𝐹𝑉) → ((IterComp‘𝐹)‘3) = (𝐹 ∘ (𝐹𝐹)))

Theoremitcoval0mpt 44836* A mapping iterated zero times (defined as identity function). (Contributed by AV, 4-May-2024.)
𝐹 = (𝑛𝐴𝐵)       ((𝐴𝑉 ∧ ∀𝑛𝐴 𝐵𝑊) → ((IterComp‘𝐹)‘0) = (𝑛𝐴𝑛))

Theoremitcovalsuc 44837* 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.)
((𝐹𝑉𝑌 ∈ ℕ0 ∧ ((IterComp‘𝐹)‘𝑌) = 𝐺) → ((IterComp‘𝐹)‘(𝑌 + 1)) = (𝐺(𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔))𝐹))

Theoremitcovalsucov 44838 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.)
((𝐹𝑉𝑌 ∈ ℕ0 ∧ ((IterComp‘𝐹)‘𝑌) = 𝐺) → ((IterComp‘𝐹)‘(𝑌 + 1)) = (𝐹𝐺))

Theoremitcovalendof 44839 The n-th iterate of an endofunction is an endofunction. (Contributed by AV, 7-May-2024.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐴)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → ((IterComp‘𝐹)‘𝑁):𝐴𝐴)

Theoremitcovalpclem1 44840* Lemma 1 for itcovalpc 44842: induction basis. (Contributed by AV, 4-May-2024.)
𝐹 = (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶))       (𝐶 ∈ ℕ0 → ((IterComp‘𝐹)‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 0))))

Theoremitcovalpclem2 44841* Lemma 2 for itcovalpc 44842: induction step. (Contributed by AV, 4-May-2024.)
𝐹 = (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶))       ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → (((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦))) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1))))))

Theoremitcovalpc 44842* 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 ∈ ℕ0𝐶 ∈ ℕ0) → ((IterComp‘𝐹)‘𝐼) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝐼))))

Theoremitcovalt2lem2lem1 44843 Lemma 1 for itcovalt2lem2 44846. (Contributed by AV, 6-May-2024.)
(((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → (((𝑁 + 𝐶) · 𝑌) − 𝐶) ∈ ℕ0)

Theoremitcovalt2lem2lem2 44844 Lemma 2 for itcovalt2lem2 44846. (Contributed by AV, 7-May-2024.)
(((𝑌 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → ((2 · (((𝑁 + 𝐶) · (2↑𝑌)) − 𝐶)) + 𝐶) = (((𝑁 + 𝐶) · (2↑(𝑌 + 1))) − 𝐶))

Theoremitcovalt2lem1 44845* Lemma 1 for itcovalt2 44847: 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)))

Theoremitcovalt2lem2 44846* Lemma 2 for itcovalt2 44847: induction step. (Contributed by AV, 7-May-2024.)
𝐹 = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 𝐶))       ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → (((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶)) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑(𝑦 + 1))) − 𝐶))))

Theoremitcovalt2 44847* 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 ∈ ℕ0𝐶 ∈ ℕ0) → ((IterComp‘𝐹)‘𝐼) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝐼)) − 𝐶)))

Theoremackvalsuc1mpt 44848* 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)))

Theoremackvalsuc1 44849 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 ∈ ℕ0𝑁 ∈ ℕ0) → ((Ack‘(𝑀 + 1))‘𝑁) = (((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))‘1))

Theoremackval0 44850 The Ackermann function at 0. (Contributed by AV, 2-May-2024.)
(Ack‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 1))

Theoremackval1 44851 The Ackermann function at 1. (Contributed by AV, 4-May-2024.)
(Ack‘1) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2))

Theoremackval2 44852 The Ackermann function at 2. (Contributed by AV, 4-May-2024.)
(Ack‘2) = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 3))

Theoremackval3 44853 The Ackermann function at 3. (Contributed by AV, 7-May-2024.)
(Ack‘3) = (𝑛 ∈ ℕ0 ↦ ((2↑(𝑛 + 3)) − 3))

Theoremackendofnn0 44854 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)

Theoremackfnnn0 44855 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)

Theoremackval0val 44856 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))

Theoremackvalsuc0val 44857 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))

Theoremackvalsucsucval 44858 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 ∈ ℕ0𝑁 ∈ ℕ0) → ((Ack‘(𝑀 + 1))‘(𝑁 + 1)) = ((Ack‘𝑀)‘((Ack‘(𝑀 + 1))‘𝑁)))

Theoremackval0012 44859 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⟩

Theoremackval1012 44860 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⟩

Theoremackval2012 44861 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⟩

Theoremackval3012 44862 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⟩

Theoremackval40 44863 The Ackermann function at (4,0). (Contributed by AV, 9-May-2024.)
((Ack‘4)‘0) = 13

Theoremackval41a 44864 The Ackermann function at (4,1). (Contributed by AV, 9-May-2024.)
((Ack‘4)‘1) = ((2↑16) − 3)

Theoremackval41 44865 The Ackermann function at (4,1). (Contributed by AV, 9-May-2024.)
((Ack‘4)‘1) = 65533

Theoremackval42 44866 The Ackermann function at (4,2). (Contributed by AV, 9-May-2024.)
((Ack‘4)‘2) = ((2↑65536) − 3)

Theoremackval42a 44867 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)

Theoremackval50 44868 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

Theoremfv1prop 44869 The function value of unordered pair of ordered pairs with first components 1 and 2 at 1. (Contributed by AV, 4-Feb-2023.)
(𝐴𝑉 → ({⟨1, 𝐴⟩, ⟨2, 𝐵⟩}‘1) = 𝐴)

Theoremfv2prop 44870 The function value of unordered pair of ordered pairs with first components 1 and 2 at 1. (Contributed by AV, 4-Feb-2023.)
(𝐵𝑉 → ({⟨1, 𝐴⟩, ⟨2, 𝐵⟩}‘2) = 𝐵)

Theoremsubmuladdmuld 44871 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.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)       (𝜑 → (((𝐴𝐵) · 𝐶) + (𝐵 · 𝐷)) = ((𝐴 · 𝐶) + (𝐵 · (𝐷𝐶))))

Theoremaffinecomb1 44872* Combination of two real affine combinations, one class variable resolved. (Contributed by AV, 22-Jan-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵𝐶)    &   (𝜑𝐸 ∈ ℝ)    &   (𝜑𝐹 ∈ ℝ)    &   (𝜑𝐺 ∈ ℝ)    &   𝑆 = ((𝐺𝐹) / (𝐶𝐵))       (𝜑 → (∃𝑡 ∈ ℝ (𝐴 = (((1 − 𝑡) · 𝐵) + (𝑡 · 𝐶)) ∧ 𝐸 = (((1 − 𝑡) · 𝐹) + (𝑡 · 𝐺))) ↔ 𝐸 = ((𝑆 · (𝐴𝐵)) + 𝐹)))

Theoremaffinecomb2 44873* Combination of two real affine combinations, presented without fraction. (Contributed by AV, 22-Jan-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵𝐶)    &   (𝜑𝐸 ∈ ℝ)    &   (𝜑𝐹 ∈ ℝ)    &   (𝜑𝐺 ∈ ℝ)       (𝜑 → (∃𝑡 ∈ ℝ (𝐴 = (((1 − 𝑡) · 𝐵) + (𝑡 · 𝐶)) ∧ 𝐸 = (((1 − 𝑡) · 𝐹) + (𝑡 · 𝐺))) ↔ ((𝐶𝐵) · 𝐸) = (((𝐺𝐹) · 𝐴) + ((𝐹 · 𝐶) − (𝐵 · 𝐺)))))

Theoremaffineid 44874 Identity of an affine combination. (Contributed by AV, 2-Feb-2023.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝑇 ∈ ℂ)       (𝜑 → (((1 − 𝑇) · 𝐴) + (𝑇 · 𝐴)) = 𝐴)

Theorem1subrec1sub 44875 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)))

Theoremresum2sqcl 44876 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)

Theoremresum2sqgt0 44877 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)

Theoremresum2sqrp 44878 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+)

Theoremresum2sqorgt0 44879 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)

Theoremreorelicc 44880 Membership in and outside of a closed real interval. (Contributed by AV, 15-Feb-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 < 𝐴𝐶 ∈ (𝐴[,]𝐵) ∨ 𝐵 < 𝐶))

20.41.23.2  Real euclidean space of dimension 2

Theoremrrx2pxel 44881 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)       (𝑋𝑃 → (𝑋‘1) ∈ ℝ)

Theoremrrx2pyel 44882 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)       (𝑋𝑃 → (𝑋‘2) ∈ ℝ)

Theoremprelrrx2 44883 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)

Theoremprelrrx2b 44884 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)) → ((𝑍𝑃 ∧ (((𝑍‘1) = 𝐴 ∧ (𝑍‘2) = 𝐵) ∨ ((𝑍‘1) = 𝑋 ∧ (𝑍‘2) = 𝑌))) ↔ 𝑍 ∈ {{⟨1, 𝐴⟩, ⟨2, 𝐵⟩}, {⟨1, 𝑋⟩, ⟨2, 𝑌⟩}}))

Theoremrrx2pnecoorneor 44885 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)       ((𝑋𝑃𝑌𝑃𝑋𝑌) → ((𝑋‘1) ≠ (𝑌‘1) ∨ (𝑋‘2) ≠ (𝑌‘2)))

Theoremrrx2pnedifcoorneor 44886 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))       ((𝑋𝑃𝑌𝑃𝑋𝑌) → (𝐴 ≠ 0 ∨ 𝐵 ≠ 0))

Theoremrrx2pnedifcoorneorr 44887 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))       ((𝑋𝑃𝑌𝑃𝑋𝑌) → (𝐴 ≠ 0 ∨ 𝐵 ≠ 0))

Theoremrrx2xpref1o 44888* 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-onto𝑅

Theoremrrx2xpreen 44889 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)

Theoremrrx2plord 44890* 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 = 𝑥𝑏𝑦)). (Contributed by AV, 12-Mar-2023.)
𝑂 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑅𝑦𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))}       ((𝑋𝑅𝑌𝑅) → (𝑋𝑂𝑌 ↔ ((𝑋‘1) < (𝑌‘1) ∨ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) < (𝑌‘2)))))

Theoremrrx2plord1 44891* 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⟩ ∣ ((𝑥𝑅𝑦𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))}       ((𝑋𝑅𝑌𝑅 ∧ (𝑋‘1) < (𝑌‘1)) → 𝑋𝑂𝑌)

Theoremrrx2plord2 44892* 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⟩ ∣ ((𝑥𝑅𝑦𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))}    &   𝑅 = (ℝ ↑m {1, 2})       ((𝑋𝑅𝑌𝑅 ∧ (𝑋‘1) = (𝑌‘1)) → (𝑋𝑂𝑌 ↔ (𝑋‘2) < (𝑌‘2)))

Theoremrrx2plordisom 44893* 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⟩ ∣ ((𝑥𝑅𝑦𝑅) ∧ ((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)) ∧ ((1st𝑥) < (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) < (2nd𝑦))))}       𝐹 Isom 𝑇, 𝑂 ((ℝ × ℝ), 𝑅)

Theoremrrx2plordso 44894* 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⟩ ∣ ((𝑥𝑅𝑦𝑅) ∧ ((𝑥‘1) < (𝑦‘1) ∨ ((𝑥‘1) = (𝑦‘1) ∧ (𝑥‘2) < (𝑦‘2))))}    &   𝑅 = (ℝ ↑m {1, 2})       𝑂 Or 𝑅

Theoremehl2eudisval0 44895 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})       (𝐹𝑋 → (𝐹𝐷 0 ) = (√‘(((𝐹‘1)↑2) + ((𝐹‘2)↑2))))

Theoremehl2eudis0lt 44896 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})       ((𝐹𝑋𝑅 ∈ ℝ+) → ((𝐹𝐷 0 ) < 𝑅 ↔ (((𝐹‘1)↑2) + ((𝐹‘2)↑2)) < (𝑅↑2)))

20.41.23.3  Spheres and lines in real Euclidean spaces

Syntaxcline 44897 Declare the syntax for lines in generalized real Euclidean spaces.
class LineM

Syntaxcsph 44898 Declare the syntax for spheres in generalized real Euclidean spaces.
class Sphere

Definitiondf-line 44899* 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)( ·𝑠𝑤)𝑥)(+g𝑤)(𝑡( ·𝑠𝑤)𝑦))}))

Definitiondf-sph 44900* Definition of spheres for given centers and radii in a metric space (or more generally, in a distance space, see distspace 22898, 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}))

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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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