P. 105 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10401-10500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fzopth 10401	A finite set of sequential integers can represent an ordered pair. (Contributed by NM, 31-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) ↔ (𝑀 = 𝐽 ∧ 𝑁 = 𝐾)))
Theorem	fzass4 10402	Two ways to express a nondecreasing sequence of four integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
⊢ ((𝐵 ∈ (𝐴...𝐷) ∧ 𝐶 ∈ (𝐵...𝐷)) ↔ (𝐵 ∈ (𝐴...𝐶) ∧ 𝐶 ∈ (𝐴...𝐷)))
Theorem	fzss1 10403	Subset relationship for finite sets of sequential integers. (Contributed by NM, 28-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (𝐾...𝑁) ⊆ (𝑀...𝑁))
Theorem	fzss2 10404	Subset relationship for finite sets of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (𝑀...𝐾) ⊆ (𝑀...𝑁))
Theorem	fzssuz 10405	A finite set of sequential integers is a subset of an upper set of integers. (Contributed by NM, 28-Oct-2005.)
⊢ (𝑀...𝑁) ⊆ (ℤ≥‘𝑀)
Theorem	fzsn 10406	A finite interval of integers with one element. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
Theorem	fzssp1 10407	Subset relationship for finite sets of sequential integers. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1))
Theorem	fzssnn 10408	Finite sets of sequential integers starting from a natural are a subset of the positive integers. (Contributed by Thierry Arnoux, 4-Aug-2017.)
⊢ (𝑀 ∈ ℕ → (𝑀...𝑁) ⊆ ℕ)
Theorem	fzsuc 10409	Join a successor to the end of a finite set of sequential integers. (Contributed by NM, 19-Jul-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)}))
Theorem	fzspl 10410	Split the last element of a finite set of sequential integers. More generic than fzsuc 10409. (Contributed by Thierry Arnoux, 7-Nov-2016.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...𝑁) = ((𝑀...(𝑁 − 1)) ∪ {𝑁}))
Theorem	fzpred 10411	Join a predecessor to the beginning of a finite set of sequential integers. (Contributed by AV, 24-Aug-2019.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...𝑁) = ({𝑀} ∪ ((𝑀 + 1)...𝑁)))
Theorem	fzpreddisj 10412	A finite set of sequential integers is disjoint with its predecessor. (Contributed by AV, 24-Aug-2019.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ({𝑀} ∩ ((𝑀 + 1)...𝑁)) = ∅)
Theorem	elfzp1 10413	Append an element to a finite set of sequential integers. (Contributed by NM, 19-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝐾 ∈ (𝑀...(𝑁 + 1)) ↔ (𝐾 ∈ (𝑀...𝑁) ∨ 𝐾 = (𝑁 + 1))))
Theorem	fzp1ss 10414	Subset relationship for finite sets of sequential integers. (Contributed by NM, 26-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝑀 ∈ ℤ → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁))
Theorem	fzelp1 10415	Membership in a set of sequential integers with an appended element. (Contributed by NM, 7-Dec-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (𝑀...(𝑁 + 1)))
Theorem	fzp1elp1 10416	Add one to an element of a finite set of integers. (Contributed by Jeff Madsen, 6-Jun-2010.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (𝑀...𝑁) → (𝐾 + 1) ∈ (𝑀...(𝑁 + 1)))
Theorem	fznatpl1 10417	Shift membership in a finite sequence of naturals. (Contributed by Scott Fenton, 17-Jul-2013.)
⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → (𝐼 + 1) ∈ (1...𝑁))
Theorem	fzpr 10418	A finite interval of integers with two elements. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝑀 ∈ ℤ → (𝑀...(𝑀 + 1)) = {𝑀, (𝑀 + 1)})
Theorem	fztp 10419	A finite interval of integers with three elements. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 7-Mar-2014.)
⊢ (𝑀 ∈ ℤ → (𝑀...(𝑀 + 2)) = {𝑀, (𝑀 + 1), (𝑀 + 2)})
Theorem	fzsuc2 10420	Join a successor to the end of a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Mar-2014.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 − 1))) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)}))
Theorem	fzp1disj 10421	(𝑀...(𝑁 + 1)) is the disjoint union of (𝑀...𝑁) with {(𝑁 + 1)}. (Contributed by Mario Carneiro, 7-Mar-2014.)
⊢ ((𝑀...𝑁) ∩ {(𝑁 + 1)}) = ∅
Theorem	fzdifsuc 10422	Remove a successor from the end of a finite set of sequential integers. (Contributed by AV, 4-Sep-2019.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...𝑁) = ((𝑀...(𝑁 + 1)) ∖ {(𝑁 + 1)}))
Theorem	fzprval 10423*	Two ways of defining the first two values of a sequence on ℕ. (Contributed by NM, 5-Sep-2011.)
⊢ (∀𝑥 ∈ (1...2)(𝐹‘𝑥) = if(𝑥 = 1, 𝐴, 𝐵) ↔ ((𝐹‘1) = 𝐴 ∧ (𝐹‘2) = 𝐵))
Theorem	fztpval 10424*	Two ways of defining the first three values of a sequence on ℕ. (Contributed by NM, 13-Sep-2011.)
⊢ (∀𝑥 ∈ (1...3)(𝐹‘𝑥) = if(𝑥 = 1, 𝐴, if(𝑥 = 2, 𝐵, 𝐶)) ↔ ((𝐹‘1) = 𝐴 ∧ (𝐹‘2) = 𝐵 ∧ (𝐹‘3) = 𝐶))
Theorem	fzrev 10425	Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝐾 ∈ ((𝐽 − 𝑁)...(𝐽 − 𝑀)) ↔ (𝐽 − 𝐾) ∈ (𝑀...𝑁)))
Theorem	fzrev2 10426	Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝐽 − 𝐾) ∈ ((𝐽 − 𝑁)...(𝐽 − 𝑀))))
Theorem	fzrev2i 10427	Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
⊢ ((𝐽 ∈ ℤ ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐽 − 𝐾) ∈ ((𝐽 − 𝑁)...(𝐽 − 𝑀)))
Theorem	fzrev3 10428	The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)
⊢ (𝐾 ∈ ℤ → (𝐾 ∈ (𝑀...𝑁) ↔ ((𝑀 + 𝑁) − 𝐾) ∈ (𝑀...𝑁)))
Theorem	fzrev3i 10429	The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)
⊢ (𝐾 ∈ (𝑀...𝑁) → ((𝑀 + 𝑁) − 𝐾) ∈ (𝑀...𝑁))
Theorem	fznn 10430	Finite set of sequential integers starting at 1. (Contributed by NM, 31-Aug-2011.) (Revised by Mario Carneiro, 18-Jun-2015.)
⊢ (𝑁 ∈ ℤ → (𝐾 ∈ (1...𝑁) ↔ (𝐾 ∈ ℕ ∧ 𝐾 ≤ 𝑁)))
Theorem	elfz1b 10431	Membership in a 1 based finite set of sequential integers. (Contributed by AV, 30-Oct-2018.)
⊢ (𝑁 ∈ (1...𝑀) ↔ (𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ≤ 𝑀))
Theorem	elfzm11 10432	Membership in a finite set of sequential integers. (Contributed by Paul Chapman, 21-Mar-2011.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...(𝑁 − 1)) ↔ (𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾 ∧ 𝐾 < 𝑁)))
Theorem	uzsplit 10433	Express an upper integer set as the disjoint (see uzdisj 10434) union of the first 𝑁 values and the rest. (Contributed by Mario Carneiro, 24-Apr-2014.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑀) = ((𝑀...(𝑁 − 1)) ∪ (ℤ≥‘𝑁)))
Theorem	uzdisj 10434	The first 𝑁 elements of an upper integer set are distinct from any later members. (Contributed by Mario Carneiro, 24-Apr-2014.)
⊢ ((𝑀...(𝑁 − 1)) ∩ (ℤ≥‘𝑁)) = ∅
Theorem	fseq1p1m1 10435	Add/remove an item to/from the end of a finite sequence. (Contributed by Paul Chapman, 17-Nov-2012.) (Revised by Mario Carneiro, 7-Mar-2014.)
⊢ 𝐻 = {⟨(𝑁 + 1), 𝐵⟩}    ⇒   ⊢ (𝑁 ∈ ℕ0 → ((𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻)) ↔ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))))
Theorem	fseq1m1p1 10436	Add/remove an item to/from the end of a finite sequence. (Contributed by Paul Chapman, 17-Nov-2012.)
⊢ 𝐻 = {⟨𝑁, 𝐵⟩}    ⇒   ⊢ (𝑁 ∈ ℕ → ((𝐹:(1...(𝑁 − 1))⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻)) ↔ (𝐺:(1...𝑁)⟶𝐴 ∧ (𝐺‘𝑁) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...(𝑁 − 1))))))
Theorem	fz1sbc 10437*	Quantification over a one-member finite set of sequential integers in terms of substitution. (Contributed by NM, 28-Nov-2005.)
⊢ (𝑁 ∈ ℤ → (∀𝑘 ∈ (𝑁...𝑁)𝜑 ↔ [𝑁 / 𝑘]𝜑))
Theorem	elfzp1b 10438	An integer is a member of a 0-based finite set of sequential integers iff its successor is a member of the corresponding 1-based set. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (0...(𝑁 − 1)) ↔ (𝐾 + 1) ∈ (1...𝑁)))
Theorem	elfzm1b 10439	An integer is a member of a 1-based finite set of sequential integers iff its predecessor is a member of the corresponding 0-based set. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (1...𝑁) ↔ (𝐾 − 1) ∈ (0...(𝑁 − 1))))
Theorem	elfzp12 10440	Options for membership in a finite interval of integers. (Contributed by Jeff Madsen, 18-Jun-2010.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)...𝑁))))
Theorem	fzm1 10441	Choices for an element of a finite interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 ∈ (𝑀...(𝑁 − 1)) ∨ 𝐾 = 𝑁)))
Theorem	fzneuz 10442	No finite set of sequential integers equals an upper set of integers. (Contributed by NM, 11-Dec-2005.)
⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℤ) → ¬ (𝑀...𝑁) = (ℤ≥‘𝐾))
Theorem	fznuz 10443	Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 30-Jun-2013.) (Revised by Mario Carneiro, 24-Aug-2013.)
⊢ (𝐾 ∈ (𝑀...𝑁) → ¬ 𝐾 ∈ (ℤ≥‘(𝑁 + 1)))
Theorem	uznfz 10444	Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 24-Aug-2013.)
⊢ (𝐾 ∈ (ℤ≥‘𝑁) → ¬ 𝐾 ∈ (𝑀...(𝑁 − 1)))
Theorem	fzp1nel 10445	One plus the upper bound of a finite set of integers is not a member of that set. (Contributed by Scott Fenton, 16-Dec-2017.)
⊢ ¬ (𝑁 + 1) ∈ (𝑀...𝑁)
Theorem	fzrevral 10446*	Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (∀𝑗 ∈ (𝑀...𝑁)𝜑 ↔ ∀𝑘 ∈ ((𝐾 − 𝑁)...(𝐾 − 𝑀))[(𝐾 − 𝑘) / 𝑗]𝜑))
Theorem	fzrevral2 10447*	Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (∀𝑗 ∈ ((𝐾 − 𝑁)...(𝐾 − 𝑀))𝜑 ↔ ∀𝑘 ∈ (𝑀...𝑁)[(𝐾 − 𝑘) / 𝑗]𝜑))
Theorem	fzrevral3 10448*	Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (∀𝑗 ∈ (𝑀...𝑁)𝜑 ↔ ∀𝑘 ∈ (𝑀...𝑁)[((𝑀 + 𝑁) − 𝑘) / 𝑗]𝜑))
Theorem	fzshftral 10449*	Shift the scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 27-Nov-2005.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (∀𝑗 ∈ (𝑀...𝑁)𝜑 ↔ ∀𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))[(𝑘 − 𝐾) / 𝑗]𝜑))
Theorem	ige2m1fz1 10450	Membership of an integer greater than 1 decreased by 1 in a 1 based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
⊢ (𝑁 ∈ (ℤ≥‘2) → (𝑁 − 1) ∈ (1...𝑁))
Theorem	ige2m1fz 10451	Membership in a 0 based finite set of sequential integers. (Contributed by Alexander van der Vekens, 18-Jun-2018.) (Proof shortened by Alexander van der Vekens, 15-Sep-2018.)
⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ (0...𝑁))
Theorem	fz01or 10452	An integer is in the integer range from zero to one iff it is either zero or one. (Contributed by Jim Kingdon, 11-Nov-2021.)
⊢ (𝐴 ∈ (0...1) ↔ (𝐴 = 0 ∨ 𝐴 = 1))
4.5.5  Finite intervals of nonnegative integers Finite intervals of nonnegative integers (or "finite sets of sequential nonnegative integers") are finite intervals of integers with 0 as lower bound: (0...𝑁), usually abbreviated by "fz0".
Theorem	elfz2nn0 10453	Membership in a finite set of sequential nonnegative integers. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (0...𝑁) ↔ (𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝐾 ≤ 𝑁))
Theorem	fznn0 10454	Characterization of a finite set of sequential nonnegative integers. (Contributed by NM, 1-Aug-2005.)
⊢ (𝑁 ∈ ℕ0 → (𝐾 ∈ (0...𝑁) ↔ (𝐾 ∈ ℕ0 ∧ 𝐾 ≤ 𝑁)))
Theorem	elfznn0 10455	A member of a finite set of sequential nonnegative integers is a nonnegative integer. (Contributed by NM, 5-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℕ0)
Theorem	elfz3nn0 10456	The upper bound of a nonempty finite set of sequential nonnegative integers is a nonnegative integer. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (0...𝑁) → 𝑁 ∈ ℕ0)
Theorem	fz0ssnn0 10457	Finite sets of sequential nonnegative integers starting with 0 are subsets of NN0. (Contributed by JJ, 1-Jun-2021.)
⊢ (0...𝑁) ⊆ ℕ0
Theorem	fz1ssfz0 10458	Subset relationship for finite sets of sequential integers. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (1...𝑁) ⊆ (0...𝑁)
Theorem	0elfz 10459	0 is an element of a finite set of sequential nonnegative integers with a nonnegative integer as upper bound. (Contributed by AV, 6-Apr-2018.)
⊢ (𝑁 ∈ ℕ0 → 0 ∈ (0...𝑁))
Theorem	nn0fz0 10460	A nonnegative integer is always part of the finite set of sequential nonnegative integers with this integer as upper bound. (Contributed by Scott Fenton, 21-Mar-2018.)
⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (0...𝑁))
Theorem	elfz0add 10461	An element of a finite set of sequential nonnegative integers is an element of an extended finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 28-Mar-2018.) (Proof shortened by OpenAI, 25-Mar-2020.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝑁 ∈ (0...𝐴) → 𝑁 ∈ (0...(𝐴 + 𝐵))))
Theorem	fz0sn 10462	An integer range from 0 to 0 is a singleton. (Contributed by AV, 18-Apr-2021.)
⊢ (0...0) = {0}
Theorem	fz0tp 10463	An integer range from 0 to 2 is an unordered triple. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
⊢ (0...2) = {0, 1, 2}
Theorem	fz0to3un2pr 10464	An integer range from 0 to 3 is the union of two unordered pairs. (Contributed by AV, 7-Feb-2021.)
⊢ (0...3) = ({0, 1} ∪ {2, 3})
Theorem	fz0to4untppr 10465	An integer range from 0 to 4 is the union of a triple and a pair. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
⊢ (0...4) = ({0, 1, 2} ∪ {3, 4})
Theorem	elfz0ubfz0 10466	An element of a finite set of sequential nonnegative integers is an element of a finite set of sequential nonnegative integers with the upper bound being an element of the finite set of sequential nonnegative integers with the same lower bound as for the first interval and the element under consideration as upper bound. (Contributed by Alexander van der Vekens, 3-Apr-2018.)
⊢ ((𝐾 ∈ (0...𝑁) ∧ 𝐿 ∈ (𝐾...𝑁)) → 𝐾 ∈ (0...𝐿))
Theorem	elfz0fzfz0 10467	A member of a finite set of sequential nonnegative integers is a member of a finite set of sequential nonnegative integers with a member of a finite set of sequential nonnegative integers starting at the upper bound of the first interval. (Contributed by Alexander van der Vekens, 27-May-2018.)
⊢ ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...𝑋)) → 𝑀 ∈ (0...𝑁))
Theorem	fz0fzelfz0 10468	If a member of a finite set of sequential integers with a lower bound being a member of a finite set of sequential nonnegative integers with the same upper bound, this member is also a member of the finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 21-Apr-2018.)
⊢ ((𝑁 ∈ (0...𝑅) ∧ 𝑀 ∈ (𝑁...𝑅)) → 𝑀 ∈ (0...𝑅))
Theorem	fznn0sub2 10469	Subtraction closure for a member of a finite set of sequential nonnegative integers. (Contributed by NM, 26-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐾 ∈ (0...𝑁) → (𝑁 − 𝐾) ∈ (0...𝑁))
Theorem	uzsubfz0 10470	Membership of an integer greater than L decreased by L in a finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
⊢ ((𝐿 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝐿)) → (𝑁 − 𝐿) ∈ (0...𝑁))
Theorem	fz0fzdiffz0 10471	The difference of an integer in a finite set of sequential nonnegative integers and and an integer of a finite set of sequential integers with the same upper bound and the nonnegative integer as lower bound is a member of the finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 6-Jun-2018.)
⊢ ((𝑀 ∈ (0...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐾 − 𝑀) ∈ (0...𝑁))
Theorem	elfzmlbm 10472	Subtracting the lower bound of a finite set of sequential integers from an element of this set. (Contributed by Alexander van der Vekens, 29-Mar-2018.) (Proof shortened by OpenAI, 25-Mar-2020.)
⊢ (𝐾 ∈ (𝑀...𝑁) → (𝐾 − 𝑀) ∈ (0...(𝑁 − 𝑀)))
Theorem	elfzmlbp 10473	Subtracting the lower bound of a finite set of sequential integers from an element of this set. (Contributed by Alexander van der Vekens, 29-Mar-2018.)
⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ (𝑀...(𝑀 + 𝑁))) → (𝐾 − 𝑀) ∈ (0...𝑁))
Theorem	fzctr 10474	Lemma for theorems about the central binomial coefficient. (Contributed by Mario Carneiro, 8-Mar-2014.) (Revised by Mario Carneiro, 2-Aug-2014.)
⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (0...(2 · 𝑁)))
Theorem	difelfzle 10475	The difference of two integers from a finite set of sequential nonnegative integers is also element of this finite set of sequential integers. (Contributed by Alexander van der Vekens, 12-Jun-2018.)
⊢ ((𝐾 ∈ (0...𝑁) ∧ 𝑀 ∈ (0...𝑁) ∧ 𝐾 ≤ 𝑀) → (𝑀 − 𝐾) ∈ (0...𝑁))
Theorem	difelfznle 10476	The difference of two integers from a finite set of sequential nonnegative integers increased by the upper bound is also element of this finite set of sequential integers. (Contributed by Alexander van der Vekens, 12-Jun-2018.)
⊢ ((𝐾 ∈ (0...𝑁) ∧ 𝑀 ∈ (0...𝑁) ∧ ¬ 𝐾 ≤ 𝑀) → ((𝑀 + 𝑁) − 𝐾) ∈ (0...𝑁))
Theorem	nn0split 10477	Express the set of nonnegative integers as the disjoint (see nn0disj 10479) union of the first 𝑁 + 1 values and the rest. (Contributed by AV, 8-Nov-2019.)
⊢ (𝑁 ∈ ℕ0 → ℕ0 = ((0...𝑁) ∪ (ℤ≥‘(𝑁 + 1))))
Theorem	nnsplit 10478	Express the set of positive integers as the disjoint union of the first 𝑁 values and the rest. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ (𝑁 ∈ ℕ → ℕ = ((1...𝑁) ∪ (ℤ≥‘(𝑁 + 1))))
Theorem	nn0disj 10479	The first 𝑁 + 1 elements of the set of nonnegative integers are distinct from any later members. (Contributed by AV, 8-Nov-2019.)
⊢ ((0...𝑁) ∩ (ℤ≥‘(𝑁 + 1))) = ∅
Theorem	1fv 10480	A function on a singleton. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
⊢ ((𝑁 ∈ 𝑉 ∧ 𝑃 = {⟨0, 𝑁⟩}) → (𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁))
Theorem	4fvwrd4 10481*	The first four function values of a word of length at least 4. (Contributed by Alexander van der Vekens, 18-Nov-2017.)
⊢ ((𝐿 ∈ (ℤ≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)))
Theorem	2ffzeq 10482*	Two functions over 0 based finite set of sequential integers are equal if and only if their domains have the same length and the function values are the same at each position. (Contributed by Alexander van der Vekens, 30-Jun-2018.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝐹:(0...𝑀)⟶𝑋 ∧ 𝑃:(0...𝑁)⟶𝑌) → (𝐹 = 𝑃 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0...𝑀)(𝐹‘𝑖) = (𝑃‘𝑖))))
4.5.6  Half-open integer ranges
Syntax	cfzo 10483	Syntax for half-open integer ranges.
class ..^
Definition	df-fzo 10484*	Define a function generating sets of integers using a half-open range. Read (𝑀..^𝑁) as the integers from 𝑀 up to, but not including, 𝑁; contrast with (𝑀...𝑁) df-fz 10349, which includes 𝑁. Not including the endpoint simplifies a number of formulas related to cardinality and splitting; contrast fzosplit 10520 with fzsplit 10391, for instance. (Contributed by Stefan O'Rear, 14-Aug-2015.)
⊢ ..^ = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1)))
Theorem	fzof 10485	Functionality of the half-open integer set function. (Contributed by Stefan O'Rear, 14-Aug-2015.)
⊢ ..^:(ℤ × ℤ)⟶𝒫 ℤ
Theorem	elfzoel1 10486	Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐵 ∈ ℤ)
Theorem	elfzoel2 10487	Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐶 ∈ ℤ)
Theorem	elfzoelz 10488	Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐴 ∈ ℤ)
Theorem	fzoval 10489	Value of the half-open integer set in terms of the closed integer set. (Contributed by Stefan O'Rear, 14-Aug-2015.)
⊢ (𝑁 ∈ ℤ → (𝑀..^𝑁) = (𝑀...(𝑁 − 1)))
Theorem	elfzo 10490	Membership in a half-open finite set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀..^𝑁) ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 < 𝑁)))
Theorem	elfzo2 10491	Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
⊢ (𝐾 ∈ (𝑀..^𝑁) ↔ (𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ ℤ ∧ 𝐾 < 𝑁))
Theorem	elfzouz 10492	Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 ∈ (ℤ≥‘𝑀))
Theorem	nelfzo 10493	An integer not being a member of a half-open finite set of integers. (Contributed by AV, 29-Apr-2020.)
⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∉ (𝑀..^𝑁) ↔ (𝐾 < 𝑀 ∨ 𝑁 ≤ 𝐾)))
Theorem	fzodcel 10494	Decidability of membership in a half-open integer interval. (Contributed by Jim Kingdon, 25-Aug-2022.)
⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝐾 ∈ (𝑀..^𝑁))
Theorem	fzolb 10495	The left endpoint of a half-open integer interval is in the set iff the two arguments are integers with 𝑀 < 𝑁. This provides an alternate notation for the "strict upper integer" predicate by analogy to the "weak upper integer" predicate 𝑀 ∈ (ℤ≥‘𝑁). (Contributed by Mario Carneiro, 29-Sep-2015.)
⊢ (𝑀 ∈ (𝑀..^𝑁) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁))
Theorem	fzolb2 10496	The left endpoint of a half-open integer interval is in the set iff the two arguments are integers with 𝑀 < 𝑁. This provides an alternate notation for the "strict upper integer" predicate by analogy to the "weak upper integer" predicate 𝑀 ∈ (ℤ≥‘𝑁). (Contributed by Mario Carneiro, 29-Sep-2015.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∈ (𝑀..^𝑁) ↔ 𝑀 < 𝑁))
Theorem	elfzole1 10497	A member in a half-open integer interval is greater than or equal to the lower bound. (Contributed by Stefan O'Rear, 15-Aug-2015.)
⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝑀 ≤ 𝐾)
Theorem	elfzolt2 10498	A member in a half-open integer interval is less than the upper bound. (Contributed by Stefan O'Rear, 15-Aug-2015.)
⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 < 𝑁)
Theorem	elfzolt3 10499	Membership in a half-open integer interval implies that the bounds are unequal. (Contributed by Stefan O'Rear, 15-Aug-2015.)
⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝑀 < 𝑁)
Theorem	elfzolt2b 10500	A member in a half-open integer interval is less than the upper bound. (Contributed by Mario Carneiro, 29-Sep-2015.)
⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 ∈ (𝐾..^𝑁))