P. 107 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10601-10700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	elfzomelpfzo 10601	An integer increased by another integer is an element of a half-open integer range if and only if the integer is contained in the half-open integer range with bounds decreased by the other integer. (Contributed by Alexander van der Vekens, 30-Mar-2018.)
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ)) → (𝐾 ∈ ((𝑀 − 𝐿)..^(𝑁 − 𝐿)) ↔ (𝐾 + 𝐿) ∈ (𝑀..^𝑁)))
Theorem	peano2fzor 10602	A Peano-postulate-like theorem for downward closure of a half-open integer range. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ ((𝐾 ∈ (ℤ≥‘𝑀) ∧ (𝐾 + 1) ∈ (𝑀..^𝑁)) → 𝐾 ∈ (𝑀..^𝑁))
Theorem	fzosplitsn 10603	Extending a half-open range by a singleton on the end. (Contributed by Stefan O'Rear, 23-Aug-2015.)
⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐴..^(𝐵 + 1)) = ((𝐴..^𝐵) ∪ {𝐵}))
Theorem	fzosplitpr 10604	Extending a half-open integer range by an unordered pair at the end. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐴..^(𝐵 + 2)) = ((𝐴..^𝐵) ∪ {𝐵, (𝐵 + 1)}))
Theorem	fzosplitprm1 10605	Extending a half-open integer range by an unordered pair at the end. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → (𝐴..^(𝐵 + 1)) = ((𝐴..^(𝐵 − 1)) ∪ {(𝐵 − 1), 𝐵}))
Theorem	fzosplitsni 10606	Membership in a half-open range extended by a singleton. (Contributed by Stefan O'Rear, 23-Aug-2015.)
⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐶 ∈ (𝐴..^(𝐵 + 1)) ↔ (𝐶 ∈ (𝐴..^𝐵) ∨ 𝐶 = 𝐵)))
Theorem	fzisfzounsn 10607	A finite interval of integers as union of a half-open integer range and a singleton. (Contributed by Alexander van der Vekens, 15-Jun-2018.)
⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐴...𝐵) = ((𝐴..^𝐵) ∪ {𝐵}))
Theorem	fzostep1 10608	Two possibilities for a number one greater than a number in a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
⊢ (𝐴 ∈ (𝐵..^𝐶) → ((𝐴 + 1) ∈ (𝐵..^𝐶) ∨ (𝐴 + 1) = 𝐶))
Theorem	fzoshftral 10609*	Shift the scanning order inside of a quantification over a half-open integer range, analogous to fzshftral 10467. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (∀𝑗 ∈ (𝑀..^𝑁)𝜑 ↔ ∀𝑘 ∈ ((𝑀 + 𝐾)..^(𝑁 + 𝐾))[(𝑘 − 𝐾) / 𝑗]𝜑))
Theorem	fzind2 10610*	Induction on the integers from 𝑀 to 𝑁 inclusive. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. Version of fzind 9714 using integer range definitions. (Contributed by Mario Carneiro, 6-Feb-2016.)
⊢ (𝑥 = 𝑀 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐾 → (𝜑 ↔ 𝜏))    &   ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜓)    &   ⊢ (𝑦 ∈ (𝑀..^𝑁) → (𝜒 → 𝜃))    ⇒   ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝜏)
Theorem	exfzdc 10611*	Decidability of the existence of an integer defined by a decidable proposition. (Contributed by Jim Kingdon, 28-Jan-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝑁)) → DECID 𝜓)    ⇒   ⊢ (𝜑 → DECID ∃𝑛 ∈ (𝑀...𝑁)𝜓)
Theorem	fvinim0ffz 10612	The function values for the borders of a finite interval of integers, which is the domain of the function, are not in the image of the interior of the interval iff the intersection of the images of the interior and the borders is empty. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 5-Feb-2021.)
⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ ↔ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹‘𝐾) ∉ (𝐹 “ (1..^𝐾)))))
Theorem	subfzo0 10613	The difference between two elements in a half-open range of nonnegative integers is greater than the negation of the upper bound and less than the upper bound of the range. (Contributed by AV, 20-Mar-2021.)
⊢ ((𝐼 ∈ (0..^𝑁) ∧ 𝐽 ∈ (0..^𝑁)) → (-𝑁 < (𝐼 − 𝐽) ∧ (𝐼 − 𝐽) < 𝑁))
Theorem	zsupcllemstep 10614*	Lemma for zsupcl 10616. Induction step. (Contributed by Jim Kingdon, 7-Dec-2021.)
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → DECID 𝜓)    ⇒   ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (((𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))) → ((𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘(𝐾 + 1)) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))))
Theorem	zsupcllemex 10615*	Lemma for zsupcl 10616. Existence of the supremum. (Contributed by Jim Kingdon, 7-Dec-2021.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝑛 = 𝑀 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → 𝜒)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → DECID 𝜓)    &   ⊢ (𝜑 → ∃𝑗 ∈ (ℤ≥‘𝑀)∀𝑛 ∈ (ℤ≥‘𝑗) ¬ 𝜓)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))
Theorem	zsupcl 10616*	Closure of supremum for decidable integer properties. The property which defines the set we are taking the supremum of must (a) be true at 𝑀 (which corresponds to the nonempty condition of classical supremum theorems), (b) decidable at each value after 𝑀, and (c) be false after 𝑗 (which corresponds to the upper bound condition found in classical supremum theorems). (Contributed by Jim Kingdon, 7-Dec-2021.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝑛 = 𝑀 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → 𝜒)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → DECID 𝜓)    &   ⊢ (𝜑 → ∃𝑗 ∈ (ℤ≥‘𝑀)∀𝑛 ∈ (ℤ≥‘𝑗) ¬ 𝜓)    ⇒   ⊢ (𝜑 → sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < ) ∈ (ℤ≥‘𝑀))
Theorem	zssinfcl 10617*	The infimum of a set of integers is an element of the set. (Contributed by Jim Kingdon, 16-Jan-2022.)
⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐵 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐵 𝑧 < 𝑦)))    &   ⊢ (𝜑 → 𝐵 ⊆ ℤ)    &   ⊢ (𝜑 → inf(𝐵, ℝ, < ) ∈ ℤ)    ⇒   ⊢ (𝜑 → inf(𝐵, ℝ, < ) ∈ 𝐵)
Theorem	infssuzex 10618*	Existence of the infimum of a subset of an upper set of integers. (Contributed by Jim Kingdon, 13-Jan-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑆 = {𝑛 ∈ (ℤ≥‘𝑀) ∣ 𝜓}    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝐴)) → DECID 𝜓)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝑆 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝑆 𝑧 < 𝑦)))
Theorem	infssuzledc 10619*	The infimum of a subset of an upper set of integers is less than or equal to all members of the subset. (Contributed by Jim Kingdon, 13-Jan-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑆 = {𝑛 ∈ (ℤ≥‘𝑀) ∣ 𝜓}    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝐴)) → DECID 𝜓)    ⇒   ⊢ (𝜑 → inf(𝑆, ℝ, < ) ≤ 𝐴)
Theorem	infssuzcldc 10620*	The infimum of a subset of an upper set of integers belongs to the subset. (Contributed by Jim Kingdon, 20-Jan-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑆 = {𝑛 ∈ (ℤ≥‘𝑀) ∣ 𝜓}    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝐴)) → DECID 𝜓)    ⇒   ⊢ (𝜑 → inf(𝑆, ℝ, < ) ∈ 𝑆)
Theorem	infssfzcldc 10621*	The infimum of a decidable inhabited subset of an integer range is a member of the set. (Contributed by Jim Kingdon, 12-Jun-2026.)
⊢ 𝑆 = {𝑛 ∈ (𝑀...𝑁) ∣ 𝜓}    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝐴)) → DECID 𝜓)    ⇒   ⊢ (𝜑 → inf(𝑆, ℝ, < ) ∈ 𝑆)
Theorem	infssfzledc 10622*	The infimum of a decidable inhabited subset of an integer range is a lower bound for that set. (Contributed by Jim Kingdon, 12-Jun-2026.)
⊢ 𝑆 = {𝑛 ∈ (𝑀...𝑁) ∣ 𝜓}    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝐴)) → DECID 𝜓)    ⇒   ⊢ (𝜑 → inf(𝑆, ℝ, < ) ≤ 𝐴)
Theorem	suprzubdc 10623*	The supremum of a bounded-above decidable set of integers is greater than any member of the set. (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Jim Kingdon, 5-Oct-2024.)
⊢ (𝜑 → 𝐴 ⊆ ℤ)    &   ⊢ (𝜑 → ∀𝑥 ∈ ℤ DECID 𝑥 ∈ 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐵 ≤ sup(𝐴, ℝ, < ))
Theorem	nninfdcex 10624*	A decidable set of natural numbers has an infimum. (Contributed by Jim Kingdon, 28-Sep-2024.)
⊢ (𝜑 → 𝐴 ⊆ ℕ)    &   ⊢ (𝜑 → ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴)    &   ⊢ (𝜑 → ∃𝑦 𝑦 ∈ 𝐴)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
Theorem	zsupssdc 10625*	An inhabited decidable bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-suploc 8264.) (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Jim Kingdon, 5-Oct-2024.)
⊢ (𝜑 → 𝐴 ⊆ ℤ)    &   ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    &   ⊢ (𝜑 → ∀𝑥 ∈ ℤ DECID 𝑥 ∈ 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
Theorem	suprzcl2dc 10626*	The supremum of a bounded-above decidable set of integers is a member of the set. (This theorem avoids ax-pre-suploc 8264.) (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Jim Kingdon, 6-Oct-2024.)
⊢ (𝜑 → 𝐴 ⊆ ℤ)    &   ⊢ (𝜑 → ∀𝑥 ∈ ℤ DECID 𝑥 ∈ 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)    &   ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    ⇒   ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ 𝐴)
4.5.7  Rational numbers (cont.)
Theorem	qtri3or 10627	Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)
⊢ ((𝑀 ∈ ℚ ∧ 𝑁 ∈ ℚ) → (𝑀 < 𝑁 ∨ 𝑀 = 𝑁 ∨ 𝑁 < 𝑀))
Theorem	qletric 10628	Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴))
Theorem	qlelttric 10629	Rational trichotomy. (Contributed by Jim Kingdon, 7-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 ≤ 𝐵 ∨ 𝐵 < 𝐴))
Theorem	qltnle 10630	'Less than' expressed in terms of 'less than or equal to'. (Contributed by Jim Kingdon, 8-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
Theorem	qdceq 10631	Equality of rationals is decidable. (Contributed by Jim Kingdon, 11-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → DECID 𝐴 = 𝐵)
Theorem	qdclt 10632	Rational < is decidable. (Contributed by Jim Kingdon, 7-Aug-2025.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → DECID 𝐴 < 𝐵)
Theorem	qdcle 10633	Rational ≤ is decidable. (Contributed by Jim Kingdon, 28-Oct-2025.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → DECID 𝐴 ≤ 𝐵)
Theorem	exbtwnzlemstep 10634*	Lemma for exbtwnzlemex 10636. Induction step. (Contributed by Jim Kingdon, 10-May-2022.)
⊢ (𝜑 → 𝐾 ∈ ℕ)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℤ) → (𝑛 ≤ 𝐴 ∨ 𝐴 < 𝑛))    ⇒   ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + (𝐾 + 1)))) → ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝐾)))
Theorem	exbtwnzlemshrink 10635*	Lemma for exbtwnzlemex 10636. Shrinking the range around 𝐴. (Contributed by Jim Kingdon, 10-May-2022.)
⊢ (𝜑 → 𝐽 ∈ ℕ)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℤ) → (𝑛 ≤ 𝐴 ∨ 𝐴 < 𝑛))    ⇒   ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝐽))) → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))
Theorem	exbtwnzlemex 10636*	Existence of an integer so that a given real number is between the integer and its successor. The real number must satisfy the 𝑛 ≤ 𝐴 ∨ 𝐴 < 𝑛 hypothesis. For example either a rational number or a number which is irrational (in the sense of being apart from any rational number) will meet this condition. The proof starts by finding two integers which are less than and greater than 𝐴. Then this range can be shrunk by choosing an integer in between the endpoints of the range and then deciding which half of the range to keep based on the 𝑛 ≤ 𝐴 ∨ 𝐴 < 𝑛 hypothesis, and iterating until the range consists of two consecutive integers. (Contributed by Jim Kingdon, 8-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℤ) → (𝑛 ≤ 𝐴 ∨ 𝐴 < 𝑛))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))
Theorem	exbtwnz 10637*	If a real number is between an integer and its successor, there is a unique greatest integer less than or equal to the real number. (Contributed by Jim Kingdon, 10-May-2022.)
⊢ (𝜑 → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → ∃!𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))
Theorem	qbtwnz 10638*	There is a unique greatest integer less than or equal to a rational number. (Contributed by Jim Kingdon, 8-Oct-2021.)
⊢ (𝐴 ∈ ℚ → ∃!𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))
Theorem	rebtwn2zlemstep 10639*	Lemma for rebtwn2z 10641. Induction step. (Contributed by Jim Kingdon, 13-Oct-2021.)
⊢ ((𝐾 ∈ (ℤ≥‘2) ∧ 𝐴 ∈ ℝ ∧ ∃𝑚 ∈ ℤ (𝑚 < 𝐴 ∧ 𝐴 < (𝑚 + (𝐾 + 1)))) → ∃𝑚 ∈ ℤ (𝑚 < 𝐴 ∧ 𝐴 < (𝑚 + 𝐾)))
Theorem	rebtwn2zlemshrink 10640*	Lemma for rebtwn2z 10641. Shrinking the range around the given real number. (Contributed by Jim Kingdon, 13-Oct-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐽 ∈ (ℤ≥‘2) ∧ ∃𝑚 ∈ ℤ (𝑚 < 𝐴 ∧ 𝐴 < (𝑚 + 𝐽))) → ∃𝑥 ∈ ℤ (𝑥 < 𝐴 ∧ 𝐴 < (𝑥 + 2)))
Theorem	rebtwn2z 10641*	A real number can be bounded by integers above and below which are two apart. The proof starts by finding two integers which are less than and greater than the given real number. Then this range can be shrunk by choosing an integer in between the endpoints of the range and then deciding which half of the range to keep based on weak linearity, and iterating until the range consists of integers which are two apart. (Contributed by Jim Kingdon, 13-Oct-2021.)
⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℤ (𝑥 < 𝐴 ∧ 𝐴 < (𝑥 + 2)))
Theorem	qbtwnrelemcalc 10642	Lemma for qbtwnre 10643. Calculations involved in showing the constructed rational number is less than 𝐵. (Contributed by Jim Kingdon, 14-Oct-2021.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝑀 < (𝐴 · (2 · 𝑁)))    &   ⊢ (𝜑 → (1 / 𝑁) < (𝐵 − 𝐴))    ⇒   ⊢ (𝜑 → ((𝑀 + 2) / (2 · 𝑁)) < 𝐵)
Theorem	qbtwnre 10643*	The rational numbers are dense in ℝ: any two real numbers have a rational between them. Exercise 6 of [Apostol] p. 28. (Contributed by NM, 18-Nov-2004.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))
Theorem	qbtwnxr 10644*	The rational numbers are dense in ℝ*: any two extended real numbers have a rational between them. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))
Theorem	qavgle 10645	The average of two rational numbers is less than or equal to at least one of them. (Contributed by Jim Kingdon, 3-Nov-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (((𝐴 + 𝐵) / 2) ≤ 𝐴 ∨ ((𝐴 + 𝐵) / 2) ≤ 𝐵))
Theorem	ioo0 10646	An empty open interval of extended reals. (Contributed by NM, 6-Feb-2007.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴(,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))
Theorem	ioom 10647*	An open interval of extended reals is inhabited iff the lower argument is less than the upper argument. (Contributed by Jim Kingdon, 27-Nov-2021.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (∃𝑥 𝑥 ∈ (𝐴(,)𝐵) ↔ 𝐴 < 𝐵))
Theorem	ico0 10648	An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴[,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))
Theorem	ioc0 10649	An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴(,]𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))
Theorem	dfrp2 10650	Alternate definition of the positive real numbers. (Contributed by Thierry Arnoux, 4-May-2020.)
⊢ ℝ+ = (0(,)+∞)
Theorem	elicod 10651	Membership in a left-closed right-open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐶)    &   ⊢ (𝜑 → 𝐶 < 𝐵)    ⇒   ⊢ (𝜑 → 𝐶 ∈ (𝐴[,)𝐵))
Theorem	icogelb 10652	An element of a left-closed right-open interval is greater than or equal to its lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐴 ≤ 𝐶)
Theorem	elicore 10653	A member of a left-closed right-open interval of reals is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 ∈ ℝ)
Theorem	xqltnle 10654	"Less than" expressed in terms of "less than or equal to", for extended numbers which are rational or +∞. We have not yet had enough usage of such numbers to warrant fully developing the concept, as in ℕ0* or ℝ*, so for now we just have a handful of theorems for what we need. (Contributed by Jim Kingdon, 5-Jun-2025.)
⊢ (((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
4.6  Elementary integer functions
4.6.1  The floor and ceiling functions
Syntax	cfl 10655	Extend class notation with floor (greatest integer) function.
class ⌊
Syntax	cceil 10656	Extend class notation to include the ceiling function.
class ⌈
Definition	df-fl 10657*	Define the floor (greatest integer less than or equal to) function. See flval 10659 for its value, flqlelt 10663 for its basic property, and flqcl 10660 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 16622). Although we define this on real numbers so that notations are similar to the Metamath Proof Explorer, in the absence of excluded middle few theorems will be possible for all real numbers. Imagine a real number which is around 2.99995 or 3.00001 . In order to determine whether its floor is 2 or 3, it would be necessary to compute the number to arbitrary precision. The term "floor" was coined by Ken Iverson. He also invented a mathematical notation for floor, consisting of an L-shaped left bracket and its reflection as a right bracket. In APL, the left-bracket alone is used, and we borrow this idea. (Thanks to Paul Chapman for this information.) (Contributed by NM, 14-Nov-2004.)
⊢ ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
Definition	df-ceil 10658	The ceiling (least integer greater than or equal to) function. Defined in ISO 80000-2:2009(E) operation 2-9.18 and the "NIST Digital Library of Mathematical Functions" , front introduction, "Common Notations and Definitions" section at http://dlmf.nist.gov/front/introduction#Sx4. See ceilqval 10695 for its value, ceilqge 10699 and ceilqm1lt 10701 for its basic properties, and ceilqcl 10697 for its closure. For example, (⌈‘(3 / 2)) = 2 while (⌈‘-(3 / 2)) = -1 (ex-ceil 16623). As described in df-fl 10657 most theorems are only for rationals, not reals. The symbol ⌈ is inspired by the gamma shaped left bracket of the usual notation. (Contributed by David A. Wheeler, 19-May-2015.)
⊢ ⌈ = (𝑥 ∈ ℝ ↦ -(⌊‘-𝑥))
Theorem	flval 10659*	Value of the floor (greatest integer) function. The floor of 𝐴 is the (unique) integer less than or equal to 𝐴 whose successor is strictly greater than 𝐴. (Contributed by NM, 14-Nov-2004.) (Revised by Mario Carneiro, 2-Nov-2013.)
⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) = (℩𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))))
Theorem	flqcl 10660	The floor (greatest integer) function yields an integer when applied to a rational (closure law). For a similar closure law for real numbers apart from any integer, see flapcl 10662. (Contributed by Jim Kingdon, 8-Oct-2021.)
⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ∈ ℤ)
Theorem	apbtwnz 10661*	There is a unique greatest integer less than or equal to a real number which is apart from all integers. (Contributed by Jim Kingdon, 11-May-2022.)
⊢ ((𝐴 ∈ ℝ ∧ ∀𝑛 ∈ ℤ 𝐴 # 𝑛) → ∃!𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))
Theorem	flapcl 10662*	The floor (greatest integer) function yields an integer when applied to a real number apart from any integer. For example, an irrational number (see for example sqrt2irrap 12905) would satisfy this condition. (Contributed by Jim Kingdon, 11-May-2022.)
⊢ ((𝐴 ∈ ℝ ∧ ∀𝑛 ∈ ℤ 𝐴 # 𝑛) → (⌊‘𝐴) ∈ ℤ)
Theorem	flqlelt 10663	A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
⊢ (𝐴 ∈ ℚ → ((⌊‘𝐴) ≤ 𝐴 ∧ 𝐴 < ((⌊‘𝐴) + 1)))
Theorem	flqcld 10664	The floor (greatest integer) function is an integer (closure law). (Contributed by Jim Kingdon, 8-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ ℚ)    ⇒   ⊢ (𝜑 → (⌊‘𝐴) ∈ ℤ)
Theorem	flqle 10665	A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ≤ 𝐴)
Theorem	flqltp1 10666	A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
⊢ (𝐴 ∈ ℚ → 𝐴 < ((⌊‘𝐴) + 1))
Theorem	qfraclt1 10667	The fractional part of a rational number is less than one. (Contributed by Jim Kingdon, 8-Oct-2021.)
⊢ (𝐴 ∈ ℚ → (𝐴 − (⌊‘𝐴)) < 1)
Theorem	qfracge0 10668	The fractional part of a rational number is nonnegative. (Contributed by Jim Kingdon, 8-Oct-2021.)
⊢ (𝐴 ∈ ℚ → 0 ≤ (𝐴 − (⌊‘𝐴)))
Theorem	flqge 10669	The floor function value is the greatest integer less than or equal to its argument. (Contributed by Jim Kingdon, 8-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → (𝐵 ≤ 𝐴 ↔ 𝐵 ≤ (⌊‘𝐴)))
Theorem	flqlt 10670	The floor function value is less than the next integer. (Contributed by Jim Kingdon, 8-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ (⌊‘𝐴) < 𝐵))
Theorem	flid 10671	An integer is its own floor. (Contributed by NM, 15-Nov-2004.)
⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) = 𝐴)
Theorem	flqidm 10672	The floor function is idempotent. (Contributed by Jim Kingdon, 8-Oct-2021.)
⊢ (𝐴 ∈ ℚ → (⌊‘(⌊‘𝐴)) = (⌊‘𝐴))
Theorem	flqidz 10673	A rational number equals its floor iff it is an integer. (Contributed by Jim Kingdon, 9-Oct-2021.)
⊢ (𝐴 ∈ ℚ → ((⌊‘𝐴) = 𝐴 ↔ 𝐴 ∈ ℤ))
Theorem	flqltnz 10674	If A is not an integer, then the floor of A is less than A. (Contributed by Jim Kingdon, 9-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ ¬ 𝐴 ∈ ℤ) → (⌊‘𝐴) < 𝐴)
Theorem	flqwordi 10675	Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐴 ≤ 𝐵) → (⌊‘𝐴) ≤ (⌊‘𝐵))
Theorem	flqword2 10676	Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐴 ≤ 𝐵) → (⌊‘𝐵) ∈ (ℤ≥‘(⌊‘𝐴)))
Theorem	flqbi 10677	A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → ((⌊‘𝐴) = 𝐵 ↔ (𝐵 ≤ 𝐴 ∧ 𝐴 < (𝐵 + 1))))
Theorem	flqbi2 10678	A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.)
⊢ ((𝑁 ∈ ℤ ∧ 𝐹 ∈ ℚ) → ((⌊‘(𝑁 + 𝐹)) = 𝑁 ↔ (0 ≤ 𝐹 ∧ 𝐹 < 1)))
Theorem	adddivflid 10679	The floor of a sum of an integer and a fraction is equal to the integer iff the denominator of the fraction is less than the numerator. (Contributed by AV, 14-Jul-2021.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℕ) → (𝐵 < 𝐶 ↔ (⌊‘(𝐴 + (𝐵 / 𝐶))) = 𝐴))
Theorem	flqge0nn0 10680	The floor of a number greater than or equal to 0 is a nonnegative integer. (Contributed by Jim Kingdon, 10-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ0)
Theorem	flqge1nn 10681	The floor of a number greater than or equal to 1 is a positive integer. (Contributed by Jim Kingdon, 10-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ)
Theorem	fldivnn0 10682	The floor function of a division of a nonnegative integer by a positive integer is a nonnegative integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
⊢ ((𝐾 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → (⌊‘(𝐾 / 𝐿)) ∈ ℕ0)
Theorem	divfl0 10683	The floor of a fraction is 0 iff the denominator is less than the numerator. (Contributed by AV, 8-Jul-2021.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 ↔ (⌊‘(𝐴 / 𝐵)) = 0))
Theorem	flqaddz 10684	An integer can be moved in and out of the floor of a sum. (Contributed by Jim Kingdon, 10-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℤ) → (⌊‘(𝐴 + 𝑁)) = ((⌊‘𝐴) + 𝑁))
Theorem	flqzadd 10685	An integer can be moved in and out of the floor of a sum. (Contributed by Jim Kingdon, 10-Oct-2021.)
⊢ ((𝑁 ∈ ℤ ∧ 𝐴 ∈ ℚ) → (⌊‘(𝑁 + 𝐴)) = (𝑁 + (⌊‘𝐴)))
Theorem	flqmulnn0 10686	Move a nonnegative integer in and out of a floor. (Contributed by Jim Kingdon, 10-Oct-2021.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℚ) → (𝑁 · (⌊‘𝐴)) ≤ (⌊‘(𝑁 · 𝐴)))
Theorem	btwnzge0 10687	A real bounded between an integer and its successor is nonnegative iff the integer is nonnegative. Second half of Lemma 13-4.1 of [Gleason] p. 217. (Contributed by NM, 12-Mar-2005.)
⊢ (((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) ∧ (𝑁 ≤ 𝐴 ∧ 𝐴 < (𝑁 + 1))) → (0 ≤ 𝐴 ↔ 0 ≤ 𝑁))
Theorem	2tnp1ge0ge0 10688	Two times an integer plus one is not negative iff the integer is not negative. (Contributed by AV, 19-Jun-2021.)
⊢ (𝑁 ∈ ℤ → (0 ≤ ((2 · 𝑁) + 1) ↔ 0 ≤ 𝑁))
Theorem	flhalf 10689	Ordering relation for the floor of half of an integer. (Contributed by NM, 1-Jan-2006.) (Proof shortened by Mario Carneiro, 7-Jun-2016.)
⊢ (𝑁 ∈ ℤ → 𝑁 ≤ (2 · (⌊‘((𝑁 + 1) / 2))))
Theorem	fldivnn0le 10690	The floor function of a division of a nonnegative integer by a positive integer is less than or equal to the division. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
⊢ ((𝐾 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → (⌊‘(𝐾 / 𝐿)) ≤ (𝐾 / 𝐿))
Theorem	flltdivnn0lt 10691	The floor function of a division of a nonnegative integer by a positive integer is less than the division of a greater dividend by the same positive integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
⊢ ((𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → (𝐾 < 𝑁 → (⌊‘(𝐾 / 𝐿)) < (𝑁 / 𝐿)))
Theorem	fldiv4p1lem1div2 10692	The floor of an integer equal to 3 or greater than 4, increased by 1, is less than or equal to the half of the integer minus 1. (Contributed by AV, 8-Jul-2021.)
⊢ ((𝑁 = 3 ∨ 𝑁 ∈ (ℤ≥‘5)) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
Theorem	fldiv4lem1div2uz2 10693	The floor of an integer greater than 1, divided by 4 is less than or equal to the half of the integer minus 1. (Contributed by AV, 5-Jul-2021.) (Proof shortened by AV, 9-Jul-2022.)
⊢ (𝑁 ∈ (ℤ≥‘2) → (⌊‘(𝑁 / 4)) ≤ ((𝑁 − 1) / 2))
Theorem	fldiv4lem1div2 10694	The floor of a positive integer divided by 4 is less than or equal to the half of the integer minus 1. (Contributed by AV, 9-Jul-2021.)
⊢ (𝑁 ∈ ℕ → (⌊‘(𝑁 / 4)) ≤ ((𝑁 − 1) / 2))
Theorem	ceilqval 10695	The value of the ceiling function. (Contributed by Jim Kingdon, 10-Oct-2021.)
⊢ (𝐴 ∈ ℚ → (⌈‘𝐴) = -(⌊‘-𝐴))
Theorem	ceiqcl 10696	The ceiling function returns an integer (closure law). (Contributed by Jim Kingdon, 11-Oct-2021.)
⊢ (𝐴 ∈ ℚ → -(⌊‘-𝐴) ∈ ℤ)
Theorem	ceilqcl 10697	Closure of the ceiling function. (Contributed by Jim Kingdon, 11-Oct-2021.)
⊢ (𝐴 ∈ ℚ → (⌈‘𝐴) ∈ ℤ)
Theorem	ceiqge 10698	The ceiling of a real number is greater than or equal to that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
⊢ (𝐴 ∈ ℚ → 𝐴 ≤ -(⌊‘-𝐴))
Theorem	ceilqge 10699	The ceiling of a real number is greater than or equal to that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
⊢ (𝐴 ∈ ℚ → 𝐴 ≤ (⌈‘𝐴))
Theorem	ceiqm1l 10700	One less than the ceiling of a real number is strictly less than that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
⊢ (𝐴 ∈ ℚ → (-(⌊‘-𝐴) − 1) < 𝐴)