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Theorem List for Metamath Proof Explorer - 12101-12200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfz10 12101 There are no integers between 1 and 0. (Contributed by Jeff Madsen, 16-Jun-2010.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
(1...0) = ∅

Theoremuzsubsubfz 12102 Membership of an integer greater than L decreased by ( L - M ) in an M based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
((𝐿 ∈ (ℤ𝑀) ∧ 𝑁 ∈ (ℤ𝐿)) → (𝑁 − (𝐿𝑀)) ∈ (𝑀...𝑁))

Theoremuzsubsubfz1 12103 Membership of an integer greater than L decreased by ( L - 1 ) in a 1 based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
((𝐿 ∈ ℕ ∧ 𝑁 ∈ (ℤ𝐿)) → (𝑁 − (𝐿 − 1)) ∈ (1...𝑁))

Theoremige3m2fz 12104 Membership of an integer greater than 2 decreased by 2 in a 1 based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
(𝑁 ∈ (ℤ‘3) → (𝑁 − 2) ∈ (1...𝑁))

Theoremfzsplit2 12105 Split a finite interval of integers into two parts. (Contributed by Mario Carneiro, 13-Apr-2016.)
(((𝐾 + 1) ∈ (ℤ𝑀) ∧ 𝑁 ∈ (ℤ𝐾)) → (𝑀...𝑁) = ((𝑀...𝐾) ∪ ((𝐾 + 1)...𝑁)))

Theoremfzsplit 12106 Split a finite interval of integers into two parts. (Contributed by Jeff Madsen, 17-Jun-2010.) (Revised by Mario Carneiro, 13-Apr-2016.)
(𝐾 ∈ (𝑀...𝑁) → (𝑀...𝑁) = ((𝑀...𝐾) ∪ ((𝐾 + 1)...𝑁)))

Theoremfzdisj 12107 Condition for two finite intervals of integers to be disjoint. (Contributed by Jeff Madsen, 17-Jun-2010.)
(𝐾 < 𝑀 → ((𝐽...𝐾) ∩ (𝑀...𝑁)) = ∅)

Theoremfz01en 12108 0-based and 1-based finite sets of sequential integers are equinumerous. (Contributed by Paul Chapman, 11-Apr-2009.)
(𝑁 ∈ ℤ → (0...(𝑁 − 1)) ≈ (1...𝑁))

Theoremelfznn 12109 A member of a finite set of sequential integers starting at 1 is a positive integer. (Contributed by NM, 24-Aug-2005.)
(𝐾 ∈ (1...𝑁) → 𝐾 ∈ ℕ)

Theoremelfz1end 12110 A nonempty finite range of integers contains its end point. (Contributed by Stefan O'Rear, 10-Oct-2014.)
(𝐴 ∈ ℕ ↔ 𝐴 ∈ (1...𝐴))

Theoremfz1ssnn 12111 A finite set of positive integers is a set of positive integers. (Contributed by Stefan O'Rear, 16-Oct-2014.)
(1...𝐴) ⊆ ℕ

Theoremfznn0sub 12112 Subtraction closure for a member of a finite set of sequential integers. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐾 ∈ (𝑀...𝑁) → (𝑁𝐾) ∈ ℕ0)

Theoremfzmmmeqm 12113 Subtracting the difference of a member of a finite range of integers and the lower bound of the range from the difference of the upper bound and the lower bound of the range results in the difference of the upper bound of the range and the member. (Contributed by Alexander van der Vekens, 27-May-2018.)
(𝑀 ∈ (𝐿...𝑁) → ((𝑁𝐿) − (𝑀𝐿)) = (𝑁𝑀))

Theoremfzaddel 12114 Membership of a sum in a finite set of sequential integers. (Contributed by NM, 30-Jul-2005.)
(((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝐽 ∈ (𝑀...𝑁) ↔ (𝐽 + 𝐾) ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))))

Theoremfzadd2 12115 Membership of a sum in a finite interval of integers. (Contributed by Jeff Madsen, 17-Jun-2010.)
(((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑂 ∈ ℤ ∧ 𝑃 ∈ ℤ)) → ((𝐽 ∈ (𝑀...𝑁) ∧ 𝐾 ∈ (𝑂...𝑃)) → (𝐽 + 𝐾) ∈ ((𝑀 + 𝑂)...(𝑁 + 𝑃))))

Theoremfzsubel 12116 Membership of a difference in a finite set of sequential integers. (Contributed by NM, 30-Jul-2005.)
(((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝐽 ∈ (𝑀...𝑁) ↔ (𝐽𝐾) ∈ ((𝑀𝐾)...(𝑁𝐾))))

Theoremfzopth 12117 A finite set of sequential integers has the ordered pair property (compare opth 4769) under certain conditions. (Contributed by NM, 31-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝑁 ∈ (ℤ𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) ↔ (𝑀 = 𝐽𝑁 = 𝐾)))

Theoremfzass4 12118 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)))

Theoremfzss1 12119 Subset relationship for finite sets of sequential integers. (Contributed by NM, 28-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
(𝐾 ∈ (ℤ𝑀) → (𝐾...𝑁) ⊆ (𝑀...𝑁))

Theoremfzss2 12120 Subset relationship for finite sets of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 30-Apr-2015.)
(𝑁 ∈ (ℤ𝐾) → (𝑀...𝐾) ⊆ (𝑀...𝑁))

Theoremfzssuz 12121 A finite set of sequential integers is a subset of an upper set of integers. (Contributed by NM, 28-Oct-2005.)
(𝑀...𝑁) ⊆ (ℤ𝑀)

Theoremfzsn 12122 A finite interval of integers with one element. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})

Theoremfzssp1 12123 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))

Theoremfzssnn 12124 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)

Theoremssfzunsn 12125 A subset of a finite sequence of integers extended by an integer is a subset of a (possibly extended) finite sequence of integers. (Contributed by AV, 8-Jun-2021.)
((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ𝑀)) → (𝑆 ∪ {𝐼}) ⊆ (𝑀...if(𝐼𝑁, 𝑁, 𝐼)))

Theoremfzsuc 12126 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 ∈ (ℤ𝑀) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)}))

Theoremfzpred 12127 Join a predecessor to the beginning of a finite set of sequential integers. (Contributed by AV, 24-Aug-2019.)
(𝑁 ∈ (ℤ𝑀) → (𝑀...𝑁) = ({𝑀} ∪ ((𝑀 + 1)...𝑁)))

Theoremfzpreddisj 12128 A finite set of sequential integers is disjoint with its predecessor. (Contributed by AV, 24-Aug-2019.)
(𝑁 ∈ (ℤ𝑀) → ({𝑀} ∩ ((𝑀 + 1)...𝑁)) = ∅)

Theoremelfzp1 12129 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 ∈ (ℤ𝑀) → (𝐾 ∈ (𝑀...(𝑁 + 1)) ↔ (𝐾 ∈ (𝑀...𝑁) ∨ 𝐾 = (𝑁 + 1))))

Theoremfzp1ss 12130 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))

Theoremfzelp1 12131 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)))

Theoremfzp1elp1 12132 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)))

Theoremfznatpl1 12133 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))

Theoremfzpr 12134 A finite interval of integers with two elements. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑀 ∈ ℤ → (𝑀...(𝑀 + 1)) = {𝑀, (𝑀 + 1)})

Theoremfztp 12135 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)})

Theoremfzsuc2 12136 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)}))

Theoremfzp1disj 12137 (𝑀...(𝑁 + 1)) is the disjoint union of (𝑀...𝑁) with {(𝑁 + 1)}. (Contributed by Mario Carneiro, 7-Mar-2014.)
((𝑀...𝑁) ∩ {(𝑁 + 1)}) = ∅

Theoremfzdifsuc 12138 Remove a successor from the end of a finite set of sequential integers. (Contributed by AV, 4-Sep-2019.)
(𝑁 ∈ (ℤ𝑀) → (𝑀...𝑁) = ((𝑀...(𝑁 + 1)) ∖ {(𝑁 + 1)}))

Theoremfzprval 12139* Two ways of defining the first two values of a sequence on . (Contributed by NM, 5-Sep-2011.)
(∀𝑥 ∈ (1...2)(𝐹𝑥) = if(𝑥 = 1, 𝐴, 𝐵) ↔ ((𝐹‘1) = 𝐴 ∧ (𝐹‘2) = 𝐵))

Theoremfztpval 12140* Two ways of defining the first three values of a sequence on . (Contributed by NM, 13-Sep-2011.)
(∀𝑥 ∈ (1...3)(𝐹𝑥) = if(𝑥 = 1, 𝐴, if(𝑥 = 2, 𝐵, 𝐶)) ↔ ((𝐹‘1) = 𝐴 ∧ (𝐹‘2) = 𝐵 ∧ (𝐹‘3) = 𝐶))

Theoremfzrev 12141 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 ∈ ((𝐽𝑁)...(𝐽𝑀)) ↔ (𝐽𝐾) ∈ (𝑀...𝑁)))

Theoremfzrev2 12142 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) ↔ (𝐽𝐾) ∈ ((𝐽𝑁)...(𝐽𝑀))))

Theoremfzrev2i 12143 Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
((𝐽 ∈ ℤ ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐽𝐾) ∈ ((𝐽𝑁)...(𝐽𝑀)))

Theoremfzrev3 12144 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)))

Theoremfzrev3i 12145 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))

Theoremfznn 12146 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 ∧ 𝐾𝑁)))

Theoremelfz1b 12147 Membership in a 1 based finite set of sequential integers. (Contributed by AV, 30-Oct-2018.)
(𝑁 ∈ (1...𝑀) ↔ (𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁𝑀))

Theoremelfzm11 12148 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 ∧ 𝑀𝐾𝐾 < 𝑁)))

Theoremuzsplit 12149 Express an upper integer set as the disjoint (see uzdisj 12150) union of the first 𝑁 values and the rest. (Contributed by Mario Carneiro, 24-Apr-2014.)
(𝑁 ∈ (ℤ𝑀) → (ℤ𝑀) = ((𝑀...(𝑁 − 1)) ∪ (ℤ𝑁)))

Theoremuzdisj 12150 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)) ∩ (ℤ𝑁)) = ∅

Theoremfseq1p1m1 12151 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)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻)) ↔ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))))

Theoremfseq1m1p1 12152 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))⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻)) ↔ (𝐺:(1...𝑁)⟶𝐴 ∧ (𝐺𝑁) = 𝐵𝐹 = (𝐺 ↾ (1...(𝑁 − 1))))))

Theoremfz1sbc 12153* 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))

Theoremelfzp1b 12154 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)))

Theoremelfzm1b 12155 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))))

Theoremelfzp12 12156 Options for membership in a finite interval of integers. (Contributed by Jeff Madsen, 18-Jun-2010.)
(𝑁 ∈ (ℤ𝑀) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 = 𝑀𝐾 ∈ ((𝑀 + 1)...𝑁))))

Theoremfzm1 12157 Choices for an element of a finite interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑁 ∈ (ℤ𝑀) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 ∈ (𝑀...(𝑁 − 1)) ∨ 𝐾 = 𝑁)))

Theoremfzneuz 12158 No finite set of sequential integers equals an upper set of integers. (Contributed by NM, 11-Dec-2005.)
((𝑁 ∈ (ℤ𝑀) ∧ 𝐾 ∈ ℤ) → ¬ (𝑀...𝑁) = (ℤ𝐾))

Theoremfznuz 12159 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)))

Theoremuznfz 12160 Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 24-Aug-2013.)
(𝐾 ∈ (ℤ𝑁) → ¬ 𝐾 ∈ (𝑀...(𝑁 − 1)))

Theoremfzp1nel 12161 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)

Theoremfzrevral 12162* 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 ∈ ((𝐾𝑁)...(𝐾𝑀))[(𝐾𝑘) / 𝑗]𝜑))

Theoremfzrevral2 12163* 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 ∈ ((𝐾𝑁)...(𝐾𝑀))𝜑 ↔ ∀𝑘 ∈ (𝑀...𝑁)[(𝐾𝑘) / 𝑗]𝜑))

Theoremfzrevral3 12164* 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))

Theoremfzshftral 12165* 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))[(𝑘𝐾) / 𝑗]𝜑))

Theoremige2m1fz1 12166 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))

Theoremige2m1fz 12167 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))

5.5.6  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".

Theoremelfz2nn0 12168 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 ∈ ℕ0𝑁 ∈ ℕ0𝐾𝑁))

Theoremfznn0 12169 Characterization of a finite set of sequential nonnegative integers. (Contributed by NM, 1-Aug-2005.)
(𝑁 ∈ ℕ0 → (𝐾 ∈ (0...𝑁) ↔ (𝐾 ∈ ℕ0𝐾𝑁)))

Theoremelfznn0 12170 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)

Theoremelfz3nn0 12171 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)

Theoremfz0ssnn0 12172 Finite sets of sequential nonnegative integers starting with 0 are subsets of NN0. (Contributed by JJ, 1-Jun-2021.)
(0...𝑁) ⊆ ℕ0

Theorem0elfz 12173 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))

Theoremnn0fz0 12174 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 ∈ ℕ0𝑁 ∈ (0...𝑁))

Theoremelfz0add 12175 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 ∈ ℕ0𝐵 ∈ ℕ0) → (𝑁 ∈ (0...𝐴) → 𝑁 ∈ (0...(𝐴 + 𝐵))))

Theoremfz0sn 12176 An integer range from 0 to 0 is a singleton. (Contributed by AV, 18-Apr-2021.)
(0...0) = {0}

Theoremfz0tp 12177 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}

Theoremfz0to3un2pr 12178 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})

Theoremfz0to4untppr 12179 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})

Theoremelfz0ubfz0 12180 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))

Theoremelfz0fzfz0 12181 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))

Theoremfz0fzelfz0 12182 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))

Theoremfznn0sub2 12183 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) → (𝑁𝐾) ∈ (0...𝑁))

Theoremuzsubfz0 12184 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 ∈ ℕ0𝑁 ∈ (ℤ𝐿)) → (𝑁𝐿) ∈ (0...𝑁))

Theoremfz0fzdiffz0 12185 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)) → (𝐾𝑀) ∈ (0...𝑁))

Theoremelfzmlbm 12186 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) → (𝐾𝑀) ∈ (0...(𝑁𝑀)))

Theoremelfzmlbp 12187 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))) → (𝐾𝑀) ∈ (0...𝑁))

Theoremfzctr 12188 Lemma for theorems about the central binomial coefficient. (Contributed by Mario Carneiro, 8-Mar-2014.) (Revised by Mario Carneiro, 2-Aug-2014.)
(𝑁 ∈ ℕ0𝑁 ∈ (0...(2 · 𝑁)))

Theoremdifelfzle 12189 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) ∧ 𝐾𝑀) → (𝑀𝐾) ∈ (0...𝑁))

Theoremdifelfznle 12190 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) ∧ ¬ 𝐾𝑀) → ((𝑀 + 𝑁) − 𝐾) ∈ (0...𝑁))

Theoremnn0split 12191 Express the set of nonnegative integers as the disjoint (see nn0disj 12192) union of the first 𝑁 + 1 values and the rest. (Contributed by AV, 8-Nov-2019.)
(𝑁 ∈ ℕ0 → ℕ0 = ((0...𝑁) ∪ (ℤ‘(𝑁 + 1))))

Theoremnn0disj 12192 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))) = ∅

Theoremfz0sn0fz1 12193 A finite set of sequential nonnegative integers is the union of the singleton containing 0 and a finite set of sequential positive integers. (Contributed by AV, 20-Mar-2021.)
(𝑁 ∈ ℕ0 → (0...𝑁) = ({0} ∪ (1...𝑁)))

Theoremfvffz0 12194 The function value of a function from a finite interval of nonnegative integers. (Contributed by AV, 13-Feb-2021.)
(((𝑁 ∈ ℕ0𝐼 ∈ ℕ0𝐼 < 𝑁) ∧ 𝑃:(0...𝑁)⟶𝑉) → (𝑃𝐼) ∈ 𝑉)

Theorem1fv 12195 A one value function. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Proof shortened by AV, 18-Apr-2021.)
((𝑁𝑉𝑃 = {⟨0, 𝑁⟩}) → (𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁))

Theorem4fvwrd4 12196* 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) → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)))

Theorem2ffzeq 12197* 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 ∈ ℕ0𝐹:(0...𝑀)⟶𝑋𝑃:(0...𝑁)⟶𝑌) → (𝐹 = 𝑃 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0...𝑀)(𝐹𝑖) = (𝑃𝑖))))

Theorempreduz 12198 The value of the predecessor class over an upper integer set. (Contributed by Scott Fenton, 16-May-2014.)
(𝑁 ∈ (ℤ𝑀) → Pred( < , (ℤ𝑀), 𝑁) = (𝑀...(𝑁 − 1)))

Theoremprednn 12199 The value of the predecessor class over the naturals. (Contributed by Scott Fenton, 6-Aug-2013.)
(𝑁 ∈ ℕ → Pred( < , ℕ, 𝑁) = (1...(𝑁 − 1)))

Theoremprednn0 12200 The value of the predecessor class over 0. (Contributed by Scott Fenton, 9-May-2014.)
(𝑁 ∈ ℕ0 → Pred( < , ℕ0, 𝑁) = (0...(𝑁 − 1)))

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