P. 111 of Theorem List - Intuitionistic Logic Explorer

Type

Label

Description

Statement

Theorem

bcp1nk 11001

The proportion of one binomial coefficient to another with 𝑁 and 𝐾 increased by 1. (Contributed by Mario Carneiro, 16-Jan-2015.)

⊢ (𝐾 ∈ (0...𝑁) → ((𝑁 + 1)C(𝐾 + 1)) = ((𝑁C𝐾) · ((𝑁 + 1) / (𝐾 + 1))))

Theorem

bcval5 11002

Write out the top and bottom parts of the binomial coefficient (𝑁C𝐾) = (𝑁 · (𝑁 − 1) · ... · ((𝑁 − 𝐾) + 1)) / 𝐾! explicitly. In this form, it is valid even for 𝑁 < 𝐾, although it is no longer valid for nonpositive 𝐾. (Contributed by Mario Carneiro, 22-May-2014.) (Revised by Jim Kingdon, 23-Apr-2023.)

⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ ℕ) → (𝑁C𝐾) = ((seq((𝑁 − 𝐾) + 1)( · , I )‘𝑁) / (!‘𝐾)))

Theorem

bcn2 11003

Binomial coefficient: 𝑁 choose 2. (Contributed by Mario Carneiro, 22-May-2014.)

⊢ (𝑁 ∈ ℕ₀ → (𝑁C2) = ((𝑁 · (𝑁 − 1)) / 2))

Theorem

bcp1m1 11004

Compute the binomial coefficient of (𝑁 + 1) over (𝑁 − 1) (Contributed by Scott Fenton, 11-May-2014.) (Revised by Mario Carneiro, 22-May-2014.)

⊢ (𝑁 ∈ ℕ₀ → ((𝑁 + 1)C(𝑁 − 1)) = (((𝑁 + 1) · 𝑁) / 2))

Theorem

bcpasc 11005

Pascal's rule for the binomial coefficient, generalized to all integers 𝐾. Equation 2 of [Gleason] p. 295. (Contributed by NM, 13-Jul-2005.) (Revised by Mario Carneiro, 10-Mar-2014.)

⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ ℤ) → ((𝑁C𝐾) + (𝑁C(𝐾 − 1))) = ((𝑁 + 1)C𝐾))

Theorem

bccl 11006

A binomial coefficient, in its extended domain, is a nonnegative integer. (Contributed by NM, 10-Jul-2005.) (Revised by Mario Carneiro, 9-Nov-2013.)

⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ ℤ) → (𝑁C𝐾) ∈ ℕ₀)

Theorem

bccl2 11007

A binomial coefficient, in its standard domain, is a positive integer. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 10-Mar-2014.)

⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) ∈ ℕ)

Theorem

bcn2m1 11008

Compute the binomial coefficient "𝑁 choose 2 " from "(𝑁 − 1) choose 2 ": (N-1) + ( (N-1) 2 ) = ( N 2 ). (Contributed by Alexander van der Vekens, 7-Jan-2018.)

⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) + ((𝑁 − 1)C2)) = (𝑁C2))

Theorem

bcn2p1 11009

Compute the binomial coefficient "(𝑁 + 1) choose 2 " from "𝑁 choose 2 ": N + ( N 2 ) = ( (N+1) 2 ). (Contributed by Alexander van der Vekens, 8-Jan-2018.)

⊢ (𝑁 ∈ ℕ₀ → (𝑁 + (𝑁C2)) = ((𝑁 + 1)C2))

Theorem

permnn 11010

The number of permutations of 𝑁 − 𝑅 objects from a collection of 𝑁 objects is a positive integer. (Contributed by Jason Orendorff, 24-Jan-2007.)

⊢ (𝑅 ∈ (0...𝑁) → ((!‘𝑁) / (!‘𝑅)) ∈ ℕ)

Theorem

bcnm1 11011

The binomial coefficent of (𝑁 − 1) is 𝑁. (Contributed by Scott Fenton, 16-May-2014.)

⊢ (𝑁 ∈ ℕ₀ → (𝑁C(𝑁 − 1)) = 𝑁)

Theorem

4bc3eq4 11012

The value of four choose three. (Contributed by Scott Fenton, 11-Jun-2016.)

⊢ (4C3) = 4

Theorem

4bc2eq6 11013

The value of four choose two. (Contributed by Scott Fenton, 9-Jan-2017.)

⊢ (4C2) = 6

4.6.10  The ` # ` (set size) function

Syntax

chash 11014

Extend the definition of a class to include the set size function.

class ♯

Definition

df-ihash 11015*

Define the set size function ♯, which gives the cardinality of a finite set as a member of ℕ₀, and assigns all infinite sets the value +∞. For example, (♯‘{0, 1, 2}) = 3.

Since we don't know that an arbitrary set is either finite or infinite (by inffiexmid 7084), the behavior beyond finite sets is not as useful as it might appear. For example, we wouldn't expect to be able to define this function in a meaningful way on 𝒫 1_o, which cannot be shown to be finite (per pw1fin 7088).

Note that we use the sharp sign (♯) for this function and we use the different character octothorpe (#) for the apartness relation (see df-ap 8745). We adopt the former notation from Corollary 8.2.4 of [AczelRathjen], p. 80 (although that work only defines it for finite sets).

This definition (in terms of ∪ and ≼) is not taken directly from the literature, but for finite sets should be equivalent to the conventional definition that the size of a finite set is the unique natural number which is equinumerous to the given set. (Contributed by Jim Kingdon, 19-Feb-2022.)

⊢ ♯ = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))

Theorem

hashinfuni 11016*

The ordinal size of an infinite set is ω. (Contributed by Jim Kingdon, 20-Feb-2022.)

⊢ (ω ≼ 𝐴 → ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} = ω)

Theorem

hashinfom 11017

The value of the ♯ function on an infinite set. (Contributed by Jim Kingdon, 20-Feb-2022.)

⊢ (ω ≼ 𝐴 → (♯‘𝐴) = +∞)

Theorem

hashennnuni 11018*

The ordinal size of a set equinumerous to an element of ω is that element of ω. (Contributed by Jim Kingdon, 20-Feb-2022.)

⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} = 𝑁)

Theorem

hashennn 11019*

The size of a set equinumerous to an element of ω. (Contributed by Jim Kingdon, 21-Feb-2022.)

⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → (♯‘𝐴) = (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘𝑁))

Theorem

hashcl 11020

Closure of the ♯ function. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 13-Jul-2014.)

⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ₀)

Theorem

hashfiv01gt1 11021

The size of a finite set is either 0 or 1 or greater than 1. (Contributed by Jim Kingdon, 21-Feb-2022.)

⊢ (𝑀 ∈ Fin → ((♯‘𝑀) = 0 ∨ (♯‘𝑀) = 1 ∨ 1 < (♯‘𝑀)))

Theorem

hashfz1 11022

The set (1...𝑁) has 𝑁 elements. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.)

⊢ (𝑁 ∈ ℕ₀ → (♯‘(1...𝑁)) = 𝑁)

Theorem

hashen 11023

Two finite sets have the same number of elements iff they are equinumerous. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.)

⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) = (♯‘𝐵) ↔ 𝐴 ≈ 𝐵))

Theorem

hasheqf1o 11024*

The size of two finite sets is equal if and only if there is a bijection mapping one of the sets onto the other. (Contributed by Alexander van der Vekens, 17-Dec-2017.)

⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) = (♯‘𝐵) ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵))

Theorem

fiinfnf1o 11025*

There is no bijection between a finite set and an infinite set. By infnfi 7070 the theorem would also hold if "infinite" were expressed as ω ≼ 𝐵. (Contributed by Alexander van der Vekens, 25-Dec-2017.)

⊢ ((𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → ¬ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)

Theorem

fihasheqf1oi 11026

The size of two finite sets is equal if there is a bijection mapping one of the sets onto the other. (Contributed by Jim Kingdon, 21-Feb-2022.)

⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1-onto→𝐵) → (♯‘𝐴) = (♯‘𝐵))

Theorem

fihashf1rn 11027

The size of a finite set which is a one-to-one function is equal to the size of the function's range. (Contributed by Jim Kingdon, 21-Feb-2022.)

⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → (♯‘𝐹) = (♯‘ran 𝐹))

Theorem

fihasheqf1od 11028

The size of two finite sets is equal if there is a bijection mapping one of the sets onto the other. (Contributed by Jim Kingdon, 21-Feb-2022.)

⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)    ⇒   ⊢ (𝜑 → (♯‘𝐴) = (♯‘𝐵))

Theorem

fz1eqb 11029

Two possibly-empty 1-based finite sets of sequential integers are equal iff their endpoints are equal. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 29-Mar-2014.)

⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((1...𝑀) = (1...𝑁) ↔ 𝑀 = 𝑁))

Theorem

filtinf 11030

The size of an infinite set is greater than the size of a finite set. (Contributed by Jim Kingdon, 21-Feb-2022.)

⊢ ((𝐴 ∈ Fin ∧ ω ≼ 𝐵) → (♯‘𝐴) < (♯‘𝐵))

Theorem

isfinite4im 11031

A finite set is equinumerous to the range of integers from one up to the hash value of the set. (Contributed by Jim Kingdon, 22-Feb-2022.)

⊢ (𝐴 ∈ Fin → (1...(♯‘𝐴)) ≈ 𝐴)

Theorem

fihasheq0 11032

Two ways of saying a finite set is empty. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 27-Jul-2014.) (Intuitionized by Jim Kingdon, 23-Feb-2022.)

⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))

Theorem

fihashneq0 11033

Two ways of saying a finite set is not empty. Also, "A is inhabited" would be equivalent by fin0 7060. (Contributed by Alexander van der Vekens, 23-Sep-2018.) (Intuitionized by Jim Kingdon, 23-Feb-2022.)

⊢ (𝐴 ∈ Fin → (0 < (♯‘𝐴) ↔ 𝐴 ≠ ∅))

Theorem

hashnncl 11034

Positive natural closure of the hash function. (Contributed by Mario Carneiro, 16-Jan-2015.)

⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ ↔ 𝐴 ≠ ∅))

Theorem

hash0 11035

The empty set has size zero. (Contributed by Mario Carneiro, 8-Jul-2014.)

⊢ (♯‘∅) = 0

Theorem

fihashelne0d 11036

A finite set with an element has nonzero size. (Contributed by Rohan Ridenour, 3-Aug-2023.)

⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    ⇒   ⊢ (𝜑 → ¬ (♯‘𝐴) = 0)

Theorem

hashsng 11037

The size of a singleton. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 13-Feb-2013.)

⊢ (𝐴 ∈ 𝑉 → (♯‘{𝐴}) = 1)

Theorem

fihashen1 11038

A finite set has size 1 if and only if it is equinumerous to the ordinal 1. (Contributed by AV, 14-Apr-2019.) (Intuitionized by Jim Kingdon, 23-Feb-2022.)

⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = 1 ↔ 𝐴 ≈ 1_o))

Theorem

fihashfn 11039

A function on a finite set is equinumerous to its domain. (Contributed by Mario Carneiro, 12-Mar-2015.) (Intuitionized by Jim Kingdon, 24-Feb-2022.)

⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (♯‘𝐹) = (♯‘𝐴))

Theorem

fseq1hash 11040

The value of the size function on a finite 1-based sequence. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 12-Mar-2015.)

⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹 Fn (1...𝑁)) → (♯‘𝐹) = 𝑁)

Theorem

omgadd 11041

Mapping ordinal addition to integer addition. (Contributed by Jim Kingdon, 24-Feb-2022.)

⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    ⇒   ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐺‘(𝐴 +_o 𝐵)) = ((𝐺‘𝐴) + (𝐺‘𝐵)))

Theorem

fihashdom 11042

Dominance relation for the size function. (Contributed by Jim Kingdon, 24-Feb-2022.)

⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) ≤ (♯‘𝐵) ↔ 𝐴 ≼ 𝐵))

Theorem

hashunlem 11043

Lemma for hashun 11044. Ordinal size of the union. (Contributed by Jim Kingdon, 25-Feb-2022.)

⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)    &   ⊢ (𝜑 → 𝑁 ∈ ω)    &   ⊢ (𝜑 → 𝑀 ∈ ω)    &   ⊢ (𝜑 → 𝐴 ≈ 𝑁)    &   ⊢ (𝜑 → 𝐵 ≈ 𝑀)    ⇒   ⊢ (𝜑 → (𝐴 ∪ 𝐵) ≈ (𝑁 +_o 𝑀))

Theorem

hashun 11044

The size of the union of disjoint finite sets is the sum of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.) (Revised by Mario Carneiro, 15-Sep-2013.)

⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))

Theorem

fihashgt0 11045

The cardinality of a finite nonempty set is greater than zero. (Contributed by Thierry Arnoux, 2-Mar-2017.)

⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → 0 < (♯‘𝐴))

Theorem

1elfz0hash 11046

1 is an element of the finite set of sequential nonnegative integers bounded by the size of a nonempty finite set. (Contributed by AV, 9-May-2020.)

⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → 1 ∈ (0...(♯‘𝐴)))

Theorem

hashunsng 11047

The size of the union of a finite set with a disjoint singleton is one more than the size of the set. (Contributed by Paul Chapman, 30-Nov-2012.)

⊢ (𝐵 ∈ 𝑉 → ((𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ 𝐴) → (♯‘(𝐴 ∪ {𝐵})) = ((♯‘𝐴) + 1)))

Theorem

hashprg 11048

The size of an unordered pair. (Contributed by Mario Carneiro, 27-Sep-2013.) (Revised by Mario Carneiro, 5-May-2016.) (Revised by AV, 18-Sep-2021.)

⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≠ 𝐵 ↔ (♯‘{𝐴, 𝐵}) = 2))

Theorem

prhash2ex 11049

There is (at least) one set with two different elements: the unordered pair containing 0 and 1. In contrast to pr0hash2ex 11055, numbers are used instead of sets because their representation is shorter (and more comprehensive). (Contributed by AV, 29-Jan-2020.)

⊢ (♯‘{0, 1}) = 2

Theorem

hashp1i 11050

Size of a natural number ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)

⊢ 𝐴 ∈ ω    &   ⊢ 𝐵 = suc 𝐴    &   ⊢ (♯‘𝐴) = 𝑀    &   ⊢ (𝑀 + 1) = 𝑁    ⇒   ⊢ (♯‘𝐵) = 𝑁

Theorem

hash1 11051

Size of a natural number ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)

⊢ (♯‘1_o) = 1

Theorem

hash2 11052

Size of a natural number ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)

⊢ (♯‘2_o) = 2

Theorem

hash3 11053

Size of a natural number ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)

⊢ (♯‘3_o) = 3

Theorem

hash4 11054

Size of a natural number ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)

⊢ (♯‘4_o) = 4

Theorem

pr0hash2ex 11055

There is (at least) one set with two different elements: the unordered pair containing the empty set and the singleton containing the empty set. (Contributed by AV, 29-Jan-2020.)

⊢ (♯‘{∅, {∅}}) = 2

Theorem

fihashss 11056

The size of a subset is less than or equal to the size of its superset. (Contributed by Alexander van der Vekens, 14-Jul-2018.)

⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → (♯‘𝐵) ≤ (♯‘𝐴))

Theorem

fiprsshashgt1 11057

The size of a superset of a proper unordered pair is greater than 1. (Contributed by AV, 6-Feb-2021.)

⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ 𝐶 ∈ Fin) → ({𝐴, 𝐵} ⊆ 𝐶 → 2 ≤ (♯‘𝐶)))

Theorem

fihashssdif 11058

The size of the difference of a finite set and a finite subset is the set's size minus the subset's. (Contributed by Jim Kingdon, 31-May-2022.)

⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → (♯‘(𝐴 ∖ 𝐵)) = ((♯‘𝐴) − (♯‘𝐵)))

Theorem

hashdifsn 11059

The size of the difference of a finite set and a singleton subset is the set's size minus 1. (Contributed by Alexander van der Vekens, 6-Jan-2018.)

⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴) → (♯‘(𝐴 ∖ {𝐵})) = ((♯‘𝐴) − 1))

Theorem

hashdifpr 11060

The size of the difference of a finite set and a proper ordered pair subset is the set's size minus 2. (Contributed by AV, 16-Dec-2020.)

⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → (♯‘(𝐴 ∖ {𝐵, 𝐶})) = ((♯‘𝐴) − 2))

Theorem

hashfz 11061

Value of the numeric cardinality of a nonempty integer range. (Contributed by Stefan O'Rear, 12-Sep-2014.) (Proof shortened by Mario Carneiro, 15-Apr-2015.)

⊢ (𝐵 ∈ (ℤ_≥‘𝐴) → (♯‘(𝐴...𝐵)) = ((𝐵 − 𝐴) + 1))

Theorem

hashfzo 11062

Cardinality of a half-open set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)

⊢ (𝐵 ∈ (ℤ_≥‘𝐴) → (♯‘(𝐴..^𝐵)) = (𝐵 − 𝐴))

Theorem

hashfzo0 11063

Cardinality of a half-open set of integers based at zero. (Contributed by Stefan O'Rear, 15-Aug-2015.)

⊢ (𝐵 ∈ ℕ₀ → (♯‘(0..^𝐵)) = 𝐵)

Theorem

hashfzp1 11064

Value of the numeric cardinality of a (possibly empty) integer range. (Contributed by AV, 19-Jun-2021.)

⊢ (𝐵 ∈ (ℤ_≥‘𝐴) → (♯‘((𝐴 + 1)...𝐵)) = (𝐵 − 𝐴))

Theorem

hashfz0 11065

Value of the numeric cardinality of a nonempty range of nonnegative integers. (Contributed by Alexander van der Vekens, 21-Jul-2018.)

⊢ (𝐵 ∈ ℕ₀ → (♯‘(0...𝐵)) = (𝐵 + 1))

Theorem

hashxp 11066

The size of the Cartesian product of two finite sets is the product of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.)

⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴 × 𝐵)) = ((♯‘𝐴) · (♯‘𝐵)))

Theorem

fimaxq 11067*

A finite set of rational numbers has a maximum. (Contributed by Jim Kingdon, 6-Sep-2022.)

⊢ ((𝐴 ⊆ ℚ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)

Theorem

fiubm 11068*

Lemma for fiubz 11069 and fiubnn 11070. A general form of those theorems. (Contributed by Jim Kingdon, 29-Oct-2024.)

⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐵 ⊆ ℚ)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)

Theorem

fiubz 11069*

A finite set of integers has an upper bound which is an integer. (Contributed by Jim Kingdon, 29-Oct-2024.)

⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)

Theorem

fiubnn 11070*

A finite set of natural numbers has an upper bound which is a a natural number. (Contributed by Jim Kingdon, 29-Oct-2024.)

⊢ ((𝐴 ⊆ ℕ ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ ℕ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)

Theorem

resunimafz0 11071

The union of a restriction by an image over an open range of nonnegative integers and a singleton of an ordered pair is a restriction by an image over an interval of nonnegative integers. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 20-Feb-2021.)

⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)    &   ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))    ⇒   ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩}))

Theorem

fnfz0hash 11072

The size of a function on a finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 25-Jun-2018.)

⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹 Fn (0...𝑁)) → (♯‘𝐹) = (𝑁 + 1))

Theorem

ffz0hash 11073

The size of a function on a finite set of sequential nonnegative integers equals the upper bound of the sequence increased by 1. (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Proof shortened by AV, 11-Apr-2021.)

⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(0...𝑁)⟶𝐵) → (♯‘𝐹) = (𝑁 + 1))

Theorem

ffzo0hash 11074

The size of a function on a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 25-Mar-2018.)

⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹 Fn (0..^𝑁)) → (♯‘𝐹) = 𝑁)

Theorem

fnfzo0hash 11075

The size of a function on a half-open range of nonnegative integers equals the upper bound of this range. (Contributed by Alexander van der Vekens, 26-Jan-2018.) (Proof shortened by AV, 11-Apr-2021.)

⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(0..^𝑁)⟶𝐵) → (♯‘𝐹) = 𝑁)

Theorem

hashfacen 11076*

The number of bijections between two sets is a cardinal invariant. (Contributed by Mario Carneiro, 21-Jan-2015.)

⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐶} ≈ {𝑓 ∣ 𝑓:𝐵–1-1-onto→𝐷})

Theorem

leisorel 11077

Version of isorel 5941 for strictly increasing functions on the reals. (Contributed by Mario Carneiro, 6-Apr-2015.) (Revised by Mario Carneiro, 9-Sep-2015.)

⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶 ≤ 𝐷 ↔ (𝐹‘𝐶) ≤ (𝐹‘𝐷)))

Theorem

zfz1isolemsplit 11078

Lemma for zfz1iso 11081. Removing one element from an integer range. (Contributed by Jim Kingdon, 8-Sep-2022.)

⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (1...(♯‘𝑋)) = ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)}))

Theorem

zfz1isolemiso 11079*

Lemma for zfz1iso 11081. Adding one element to the order isomorphism. (Contributed by Jim Kingdon, 8-Sep-2022.)

⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ⊆ ℤ)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑋)    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝑋 𝑧 ≤ 𝑀)    &   ⊢ (𝜑 → 𝐺 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})))    &   ⊢ (𝜑 → 𝐴 ∈ (1...(♯‘𝑋)))    &   ⊢ (𝜑 → 𝐵 ∈ (1...(♯‘𝑋)))    ⇒   ⊢ (𝜑 → (𝐴 < 𝐵 ↔ ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) < ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵)))

Theorem

zfz1isolem1 11080*

Lemma for zfz1iso 11081. Existence of an order isomorphism given the existence of shorter isomorphisms. (Contributed by Jim Kingdon, 7-Sep-2022.)

⊢ (𝜑 → 𝐾 ∈ ω)    &   ⊢ (𝜑 → ∀𝑦(((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑦 ≈ 𝐾) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑦)), 𝑦)))    &   ⊢ (𝜑 → 𝑋 ⊆ ℤ)    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ≈ suc 𝐾)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑋)    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝑋 𝑧 ≤ 𝑀)    ⇒   ⊢ (𝜑 → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑋)), 𝑋))

Theorem

zfz1iso 11081*

A finite set of integers has an order isomorphism to a one-based finite sequence. (Contributed by Jim Kingdon, 3-Sep-2022.)

⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝐴))

Theorem

seq3coll 11082*

The function 𝐹 contains a sparse set of nonzero values to be summed. The function 𝐺 is an order isomorphism from the set of nonzero values of 𝐹 to a 1-based finite sequence, and 𝐻 collects these nonzero values together. Under these conditions, the sum over the values in 𝐻 yields the same result as the sum over the original set 𝐹. (Contributed by Mario Carneiro, 2-Apr-2014.) (Revised by Jim Kingdon, 9-Apr-2023.)

⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → (𝑍 + 𝑘) = 𝑘)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → (𝑘 + 𝑍) = 𝑘)    &   ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑛 ∈ 𝑆)) → (𝑘 + 𝑛) ∈ 𝑆)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐺 Isom < , < ((1...(♯‘𝐴)), 𝐴))    &   ⊢ (𝜑 → 𝑁 ∈ (1...(♯‘𝐴)))    &   ⊢ (𝜑 → 𝐴 ⊆ (ℤ_≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → (𝐻‘𝑘) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀...(𝐺‘(♯‘𝐴))) ∖ 𝐴)) → (𝐹‘𝑘) = 𝑍)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (𝐻‘𝑛) = (𝐹‘(𝐺‘𝑛)))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘𝑁)) = (seq1( + , 𝐻)‘𝑁))

4.6.10.1  Proper unordered pairs and triples (sets of size 2 and 3)

Theorem

hash2en 11083

Two equivalent ways to say a set has two elements. (Contributed by Jim Kingdon, 4-Dec-2025.)

⊢ (𝑉 ≈ 2_o ↔ (𝑉 ∈ Fin ∧ (♯‘𝑉) = 2))

Theorem

hashdmprop2dom 11084

A class which contains two ordered pairs with different first components has at least two elements. (Contributed by AV, 12-Nov-2021.)

⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑍)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} ⊆ 𝐹)    ⇒   ⊢ (𝜑 → 2_o ≼ dom 𝐹)

4.6.10.2  Functions with a domain containing at least two different elements

Theorem

fundm2domnop0 11085

A function with a domain containing (at least) two different elements is not an ordered pair. This theorem (which requires that (𝐺 ∖ {∅}) needs to be a function instead of 𝐺) is useful for proofs for extensible structures, see structn0fun 13066. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 7-Jun-2021.) (Proof shortened by AV, 15-Nov-2021.)

⊢ ((Fun (𝐺 ∖ {∅}) ∧ 2_o ≼ dom 𝐺) → ¬ 𝐺 ∈ (V × V))

Theorem

fundm2domnop 11086

A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 12-Oct-2020.) (Proof shortened by AV, 9-Jun-2021.)

⊢ ((Fun 𝐺 ∧ 2_o ≼ dom 𝐺) → ¬ 𝐺 ∈ (V × V))

Theorem

fun2dmnop0 11087

A function with a domain containing (at least) two different elements is not an ordered pair. This stronger version of fun2dmnop 11088 (with the less restrictive requirement that (𝐺 ∖ {∅}) needs to be a function instead of 𝐺) is useful for proofs for extensible structures, see structn0fun 13066. (Contributed by AV, 21-Sep-2020.) (Revised by AV, 7-Jun-2021.)

⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → ¬ 𝐺 ∈ (V × V))

Theorem

fun2dmnop 11088

A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 21-Sep-2020.) (Proof shortened by AV, 9-Jun-2021.)

⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((Fun 𝐺 ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → ¬ 𝐺 ∈ (V × V))

4.7  Words over a set

This section is about words (or strings) over a set (alphabet) defined as finite sequences of symbols (or characters) being elements of the alphabet. Although it is often required that the underlying set/alphabet be nonempty, finite and not a proper class, these restrictions are not made in the current definition df-word 11090. Note that the empty word ∅ (i.e., the empty set) is the only word over an empty alphabet, see 0wrd0 11115. The set Word 𝑆 of words over 𝑆 is the free monoid over 𝑆, where the monoid law is concatenation and the monoid unit is the empty word. Besides the definition of words themselves, several operations on words are defined in this section:

Name	Reference	Explanation	Example	Remarks
Length (or size) of a word	df-ihash 11015: (♯‘𝑊)	Operation which determines the number of the symbols within the word.	𝑊:(0..^𝑁)⟶𝑆 → (𝑊 ∈ Word 𝑆 ∧ (♯‘𝑊) = 𝑁	This is not a special definition for words, but for arbitrary sets.
First symbol of a word	df-fv 5329: (𝑊‘0)	Operation which extracts the first symbol of a word. The result is the symbol at the first place in the sequence representing the word.	𝑊:(0..^1)⟶𝑆 → (𝑊 ∈ Word 𝑆 ∧ (𝑊‘0) ∈ 𝑆	This is not a special definition for words, but for arbitrary functions with a half-open range of nonnegative integers as domain.
Last symbol of a word	df-lsw 11135: (lastS‘𝑊)	Operation which extracts the last symbol of a word. The result is the symbol at the last place in the sequence representing the word.	𝑊:(0..^3)⟶𝑆 → (𝑊 ∈ Word 𝑆 ∧ (lastS‘𝑊) = (𝑊‘2)	Note that the index of the last symbol is less by 1 than the length of the word.
Subword (or substring) of a word	df-substr 11199: (𝑊 substr ⟨𝐼, 𝐽⟩)	Operation which extracts a portion of a word. The result is a consecutive, reindexed part of the sequence representing the word.	𝑊:(0..^3)⟶𝑆 → (𝑊 ∈ Word 𝑆 ∧ (𝑊 substr ⟨1, 2⟩) ∈ Word 𝑆 ∧ (♯‘(𝑊 substr ⟨1, 2⟩)) = 1	Note that the last index of the range defining the subword is greater by 1 than the index of the last symbol of the subword in the sequence of the original word.
Concatenation of two words	df-concat 11144: (𝑊 ++ 𝑈)	Operation combining two words to one new word. The result is a combined, reindexed sequence build from the sequences representing the two words.	(𝑊 ∈ Word 𝑆 ∧ 𝑈 ∈ Word 𝑆) → (♯‘(𝑊 ++ 𝑈)) = ((♯‘𝑊) + (♯‘𝑈))	Note that the index of the first symbol of the second concatenated word is the length of the first word in the concatenation.
Singleton word	df-s1 11169: ⟨“𝑆”⟩	Constructor building a word of length 1 from a symbol.	(♯‘⟨“𝑆”⟩) = 1

Conventions:

• Words are usually represented by class variable 𝑊, or if two words are involved, by 𝑊 and 𝑈 or by 𝐴 and 𝐵.
• The alphabets are usually represented by class variable 𝑉 (because any arbitrary set can be an alphabet), sometimes also by 𝑆 (especially as codomain of the function representing a word, because the alphabet is the set of symbols).
• The symbols are usually represented by class variables 𝑆 or 𝐴, or if two symbols are involved, by 𝑆 and 𝑇 or by 𝐴 and 𝐵.
• The indices of the sequence representing a word are usually represented by class variable 𝐼, if two indices are involved (especially for subwords) by 𝐼 and 𝐽, or by 𝑀 and 𝑁.
• The length of a word is usually represented by class variables 𝑁 or 𝐿.
• The number of positions by which to cyclically shift a word is usually represented by class variables 𝑁 or 𝐿.

4.7.1  Definitions and basic theorems

Syntax

cword 11089

Syntax for the Word operator.

class Word 𝑆

Definition

df-word 11090*

Define the class of words over a set. A word (sometimes also called a string) is a finite sequence of symbols from a set (alphabet) 𝑆. Definition in Section 9.1 of [AhoHopUll] p. 318. The domain is forced to be an initial segment of ℕ₀ so that two words with the same symbols in the same order be equal. The set Word 𝑆 is sometimes denoted by S*, using the Kleene star, although the Kleene star, or Kleene closure, is sometimes reserved to denote an operation on languages. The set Word 𝑆 equipped with concatenation is the free monoid over 𝑆, and the monoid unit is the empty word. (Contributed by FL, 14-Jan-2014.) (Revised by Stefan O'Rear, 14-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ₀ 𝑤:(0..^𝑙)⟶𝑆}

Theorem

iswrd 11091*

Property of being a word over a set with an existential quantifier over the length. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Proof shortened by AV, 13-May-2020.)

⊢ (𝑊 ∈ Word 𝑆 ↔ ∃𝑙 ∈ ℕ₀ 𝑊:(0..^𝑙)⟶𝑆)

Theorem

wrdval 11092*

Value of the set of words over a set. (Contributed by Stefan O'Rear, 10-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

⊢ (𝑆 ∈ 𝑉 → Word 𝑆 = ∪ 𝑙 ∈ ℕ₀ (𝑆 ↑_𝑚 (0..^𝑙)))

Theorem

lencl 11093

The length of a word is a nonnegative integer. This corresponds to the definition in Section 9.1 of [AhoHopUll] p. 318. (Contributed by Stefan O'Rear, 27-Aug-2015.)

⊢ (𝑊 ∈ Word 𝑆 → (♯‘𝑊) ∈ ℕ₀)

Theorem

iswrdinn0 11094

A zero-based sequence is a word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Revised by Jim Kingdon, 16-Aug-2025.)

⊢ ((𝑊:(0..^𝐿)⟶𝑆 ∧ 𝐿 ∈ ℕ₀) → 𝑊 ∈ Word 𝑆)

Theorem

wrdf 11095

A word is a zero-based sequence with a recoverable upper limit. (Contributed by Stefan O'Rear, 15-Aug-2015.)

⊢ (𝑊 ∈ Word 𝑆 → 𝑊:(0..^(♯‘𝑊))⟶𝑆)

Theorem

iswrdiz 11096

A zero-based sequence is a word. In iswrdinn0 11094 we can specify a length as an nonnegative integer. However, it will occasionally be helpful to allow a negative length, as well as zero, to specify an empty sequence. (Contributed by Jim Kingdon, 18-Aug-2025.)

⊢ ((𝑊:(0..^𝐿)⟶𝑆 ∧ 𝐿 ∈ ℤ) → 𝑊 ∈ Word 𝑆)

Theorem

wrddm 11097

The indices of a word (i.e. its domain regarded as function) are elements of an open range of nonnegative integers (of length equal to the length of the word). (Contributed by AV, 2-May-2020.)

⊢ (𝑊 ∈ Word 𝑆 → dom 𝑊 = (0..^(♯‘𝑊)))

Theorem

sswrd 11098

The set of words respects ordering on the base set. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Proof shortened by AV, 13-May-2020.)

⊢ (𝑆 ⊆ 𝑇 → Word 𝑆 ⊆ Word 𝑇)

Theorem

snopiswrd 11099

A singleton of an ordered pair (with 0 as first component) is a word. (Contributed by AV, 23-Nov-2018.) (Proof shortened by AV, 18-Apr-2021.)

⊢ (𝑆 ∈ 𝑉 → {⟨0, 𝑆⟩} ∈ Word 𝑉)

Theorem

wrdexg 11100

The set of words over a set is a set. (Contributed by Mario Carneiro, 26-Feb-2016.) (Proof shortened by JJ, 18-Nov-2022.)

⊢ (𝑆 ∈ 𝑉 → Word 𝑆 ∈ V)