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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | peano2fzr 8901 | A Peano-postulate-like theorem for downward closure of a finite set of sequential integers. (Contributed by Mario Carneiro, 27-May-2014.) |
⊢ ((𝐾 ∈ (ℤ_{≥}‘𝑀) ∧ (𝐾 + 1) ∈ (𝑀...𝑁)) → 𝐾 ∈ (𝑀...𝑁)) | ||
Theorem | fzm 8902* | Properties of a finite interval of integers which is inhabited. (Contributed by Jim Kingdon, 15-Apr-2020.) |
⊢ (∃𝑥 𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (ℤ_{≥}‘𝑀)) | ||
Theorem | fztri3or 8903 | Trichotomy in terms of a finite interval of integers. (Contributed by Jim Kingdon, 1-Jun-2020.) |
⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 < 𝑀 ∨ 𝐾 ∈ (𝑀...𝑁) ∨ 𝑁 < 𝐾)) | ||
Theorem | fzdcel 8904 | Decidability of membership in a finite interval of integers. (Contributed by Jim Kingdon, 1-Jun-2020.) |
⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝐾 ∈ (𝑀...𝑁)) | ||
Theorem | fznlem 8905 | A finite set of sequential integers is empty if the bounds are reversed. (Contributed by Jim Kingdon, 16-Apr-2020.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < 𝑀 → (𝑀...𝑁) = ∅)) | ||
Theorem | fzn 8906 | A finite set of sequential integers is empty if the bounds are reversed. (Contributed by NM, 22-Aug-2005.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < 𝑀 ↔ (𝑀...𝑁) = ∅)) | ||
Theorem | fzen 8907 | A shifted finite set of sequential integers is equinumerous to the original set. (Contributed by Paul Chapman, 11-Apr-2009.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀...𝑁) ≈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) | ||
Theorem | fz1n 8908 | A 1-based finite set of sequential integers is empty iff it ends at index 0. (Contributed by Paul Chapman, 22-Jun-2011.) |
⊢ (𝑁 ∈ ℕ_{0} → ((1...𝑁) = ∅ ↔ 𝑁 = 0)) | ||
Theorem | 0fz1 8909 | Two ways to say a finite 1-based sequence is empty. (Contributed by Paul Chapman, 26-Oct-2012.) |
⊢ ((𝑁 ∈ ℕ_{0} ∧ 𝐹 Fn (1...𝑁)) → (𝐹 = ∅ ↔ 𝑁 = 0)) | ||
Theorem | fz10 8910 | 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) = ∅ | ||
Theorem | uzsubsubfz 8911 | 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 ∈ (ℤ_{≥}‘𝑀) ∧ 𝑁 ∈ (ℤ_{≥}‘𝐿)) → (𝑁 − (𝐿 − 𝑀)) ∈ (𝑀...𝑁)) | ||
Theorem | uzsubsubfz1 8912 | 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 ∈ (ℤ_{≥}‘𝐿)) → (𝑁 − (𝐿 − 1)) ∈ (1...𝑁)) | ||
Theorem | ige3m2fz 8913 | 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)) | ||
Theorem | fzsplit2 8914 | Split a finite interval of integers into two parts. (Contributed by Mario Carneiro, 13-Apr-2016.) |
⊢ (((𝐾 + 1) ∈ (ℤ_{≥}‘𝑀) ∧ 𝑁 ∈ (ℤ_{≥}‘𝐾)) → (𝑀...𝑁) = ((𝑀...𝐾) ∪ ((𝐾 + 1)...𝑁))) | ||
Theorem | fzsplit 8915 | 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))) | ||
Theorem | fzdisj 8916 | Condition for two finite intervals of integers to be disjoint. (Contributed by Jeff Madsen, 17-Jun-2010.) |
⊢ (𝐾 < 𝑀 → ((𝐽...𝐾) ∩ (𝑀...𝑁)) = ∅) | ||
Theorem | fz01en 8917 | 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)) | ||
Theorem | elfznn 8918 | 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) | ||
Theorem | elfz1end 8919 | A nonempty finite range of integers contains its end point. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ (𝐴 ∈ ℕ ↔ 𝐴 ∈ (1...𝐴)) | ||
Theorem | fznn0sub 8920 | 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) → (𝑁 − 𝐾) ∈ ℕ_{0}) | ||
Theorem | fzmmmeqm 8921 | 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) → ((𝑁 − 𝐿) − (𝑀 − 𝐿)) = (𝑁 − 𝑀)) | ||
Theorem | fzaddel 8922 | 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)))) | ||
Theorem | fzsubel 8923 | 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) ↔ (𝐽 − 𝐾) ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾)))) | ||
Theorem | fzopth 8924 | 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 8925 | 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 8926 | 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 8927 | 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 8928 | 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 8929 | A finite interval of integers with one element. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀}) | ||
Theorem | fzssp1 8930 | 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 | fzsuc 8931 | 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 | fzpred 8932 | 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 8933 | 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 8934 | 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 8935 | 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 8936 | 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 8937 | 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 8938 | 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 8939 | 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 8940 | 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 8941 | 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 8942 | (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 8943 | 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 8944* | 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 8945* | 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 8946 | 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 8947 | 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 8948 | 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 8949 | 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 8950 | 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 8951 | 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 8952 | 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 8953 | 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 8954 | Express an upper integer set as the disjoint (see uzdisj 8955) 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 8955 | 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 8956 | 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 ∈ ℕ_{0} → ((𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻)) ↔ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁))))) | ||
Theorem | fseq1m1p1 8957 | 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 8958* | 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 8959 | 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 8960 | 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 8961 | 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 8962 | 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 8963 | 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 8964 | 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 8965 | 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 8966 | 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 8967* | 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 8968* | 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 8969* | 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 8970* | 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 8971 | 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 8972 | 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 ∈ ℕ_{0} ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ (0...𝑁)) | ||
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 8973 | 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 ∈ ℕ_{0} ∧ 𝑁 ∈ ℕ_{0} ∧ 𝐾 ≤ 𝑁)) | ||
Theorem | fznn0 8974 | Characterization of a finite set of sequential nonnegative integers. (Contributed by NM, 1-Aug-2005.) |
⊢ (𝑁 ∈ ℕ_{0} → (𝐾 ∈ (0...𝑁) ↔ (𝐾 ∈ ℕ_{0} ∧ 𝐾 ≤ 𝑁))) | ||
Theorem | elfznn0 8975 | 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 ∈ ℕ_{0}) | ||
Theorem | elfz3nn0 8976 | 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 ∈ ℕ_{0}) | ||
Theorem | 0elfz 8977 | 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 ∈ ℕ_{0} → 0 ∈ (0...𝑁)) | ||
Theorem | nn0fz0 8978 | 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 ∈ ℕ_{0} ↔ 𝑁 ∈ (0...𝑁)) | ||
Theorem | elfz0add 8979 | 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 ∈ ℕ_{0} ∧ 𝐵 ∈ ℕ_{0}) → (𝑁 ∈ (0...𝐴) → 𝑁 ∈ (0...(𝐴 + 𝐵)))) | ||
Theorem | elfz0addOLD 8980 | 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.) Obsolete version of elfz0add 8979 as of 25-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ((𝐴 ∈ ℕ_{0} ∧ 𝐵 ∈ ℕ_{0}) → (𝑁 ∈ (0...𝐴) → 𝑁 ∈ (0...(𝐴 + 𝐵)))) | ||
Theorem | fz0tp 8981 | 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 | elfz0ubfz0 8982 | 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 8983 | 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 8984 | 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 8985 | 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 8986 | 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 ∈ ℕ_{0} ∧ 𝑁 ∈ (ℤ_{≥}‘𝐿)) → (𝑁 − 𝐿) ∈ (0...𝑁)) | ||
Theorem | fz0fzdiffz0 8987 | 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 8988 | 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 | elfzmlbmOLD 8989 | 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.) Obsolete version of elfzmlbm 8988 as of 25-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝐾 ∈ (𝑀...𝑁) → (𝐾 − 𝑀) ∈ (0...(𝑁 − 𝑀))) | ||
Theorem | elfzmlbp 8990 | 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 8991 | Lemma for theorems about the central binomial coefficient. (Contributed by Mario Carneiro, 8-Mar-2014.) (Revised by Mario Carneiro, 2-Aug-2014.) |
⊢ (𝑁 ∈ ℕ_{0} → 𝑁 ∈ (0...(2 · 𝑁))) | ||
Theorem | difelfzle 8992 | 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 8993 | 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 8994 | Express the set of nonnegative integers as the disjoint (see nn0disj 8995) union of the first 𝑁 + 1 values and the rest. (Contributed by AV, 8-Nov-2019.) |
⊢ (𝑁 ∈ ℕ_{0} → ℕ_{0} = ((0...𝑁) ∪ (ℤ_{≥}‘(𝑁 + 1)))) | ||
Theorem | nn0disj 8995 | 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 8996 | A one value function. (Contributed by Alexander van der Vekens, 3-Dec-2017.) |
⊢ ((𝑁 ∈ 𝑉 ∧ 𝑃 = {⟨0, 𝑁⟩}) → (𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁)) | ||
Theorem | 4fvwrd4 8997* | 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 8998* | 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 ∈ ℕ_{0} ∧ 𝐹:(0...𝑀)⟶𝑋 ∧ 𝑃:(0...𝑁)⟶𝑌) → (𝐹 = 𝑃 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0...𝑀)(𝐹‘𝑖) = (𝑃‘𝑖)))) | ||
Syntax | cfzo 8999 | Syntax for half-open integer ranges. |
class ..^ | ||
Definition | df-fzo 9000* | 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 8875, which includes 𝑁. Not including the endpoint simplifies a number of formulae related to cardinality and splitting; contrast fzosplit 9033 with fzsplit 8915, for instance. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
⊢ ..^ = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1))) |
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