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	fzofzp1b 10401	If a point is in a half-open range, the next point is in the closed range. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ (𝐶 ∈ (ℤ≥‘𝐴) → (𝐶 ∈ (𝐴..^𝐵) ↔ (𝐶 + 1) ∈ (𝐴...𝐵)))
Theorem	elfzom1b 10402	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 Mario Carneiro, 27-Sep-2015.)
⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (1..^𝑁) ↔ (𝐾 − 1) ∈ (0..^(𝑁 − 1))))
Theorem	elfzonelfzo 10403	If an element of a half-open integer range is not contained in the lower subrange, it must be in the upper subrange. (Contributed by Alexander van der Vekens, 30-Mar-2018.)
⊢ (𝑁 ∈ ℤ → ((𝐾 ∈ (𝑀..^𝑅) ∧ ¬ 𝐾 ∈ (𝑀..^𝑁)) → 𝐾 ∈ (𝑁..^𝑅)))
Theorem	elfzomelpfzo 10404	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 10405	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 10406	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	fzosplitprm1 10407	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 10408	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 10409	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 10410	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 10411*	Shift the scanning order inside of a quantification over a half-open integer range, analogous to fzshftral 10272. (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 10412*	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 9530 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 10413*	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 10414	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 10415	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 10416*	Lemma for zsupcl 10418. 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 10417*	Lemma for zsupcl 10418. 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 10418*	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 10419*	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 10420*	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 10421*	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 10422*	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	suprzubdc 10423*	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 10424*	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 10425*	An inhabited decidable bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-suploc 8088.) (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 10426*	The supremum of a bounded-above decidable set of integers is a member of the set. (This theorem avoids ax-pre-suploc 8088.) (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 10427	Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)
⊢ ((𝑀 ∈ ℚ ∧ 𝑁 ∈ ℚ) → (𝑀 < 𝑁 ∨ 𝑀 = 𝑁 ∨ 𝑁 < 𝑀))
Theorem	qletric 10428	Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴))
Theorem	qlelttric 10429	Rational trichotomy. (Contributed by Jim Kingdon, 7-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 ≤ 𝐵 ∨ 𝐵 < 𝐴))
Theorem	qltnle 10430	'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 10431	Equality of rationals is decidable. (Contributed by Jim Kingdon, 11-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → DECID 𝐴 = 𝐵)
Theorem	qdclt 10432	Rational < is decidable. (Contributed by Jim Kingdon, 7-Aug-2025.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → DECID 𝐴 < 𝐵)
Theorem	qdcle 10433	Rational ≤ is decidable. (Contributed by Jim Kingdon, 28-Oct-2025.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → DECID 𝐴 ≤ 𝐵)
Theorem	exbtwnzlemstep 10434*	Lemma for exbtwnzlemex 10436. 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 10435*	Lemma for exbtwnzlemex 10436. 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 10436*	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 10437*	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 10438*	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 10439*	Lemma for rebtwn2z 10441. 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 10440*	Lemma for rebtwn2z 10441. 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 10441*	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 10442	Lemma for qbtwnre 10443. 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 10443*	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 10444*	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 10445	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 10446	An empty open interval of extended reals. (Contributed by NM, 6-Feb-2007.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴(,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))
Theorem	ioom 10447*	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 10448	An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴[,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))
Theorem	ioc0 10449	An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴(,]𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))
Theorem	dfrp2 10450	Alternate definition of the positive real numbers. (Contributed by Thierry Arnoux, 4-May-2020.)
⊢ ℝ+ = (0(,)+∞)
Theorem	elicod 10451	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 10452	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 10453	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 10454	"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 10455	Extend class notation with floor (greatest integer) function.
class ⌊
Syntax	cceil 10456	Extend class notation to include the ceiling function.
class ⌈
Definition	df-fl 10457*	Define the floor (greatest integer less than or equal to) function. See flval 10459 for its value, flqlelt 10463 for its basic property, and flqcl 10460 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 15999). 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 10458	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 10495 for its value, ceilqge 10499 and ceilqm1lt 10501 for its basic properties, and ceilqcl 10497 for its closure. For example, (⌈‘(3 / 2)) = 2 while (⌈‘-(3 / 2)) = -1 (ex-ceil 16000). As described in df-fl 10457 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 10459*	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 10460	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 10462. (Contributed by Jim Kingdon, 8-Oct-2021.)
⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ∈ ℤ)
Theorem	apbtwnz 10461*	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 10462*	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 12668) would satisfy this condition. (Contributed by Jim Kingdon, 11-May-2022.)
⊢ ((𝐴 ∈ ℝ ∧ ∀𝑛 ∈ ℤ 𝐴 # 𝑛) → (⌊‘𝐴) ∈ ℤ)
Theorem	flqlelt 10463	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 10464	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 10465	A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ≤ 𝐴)
Theorem	flqltp1 10466	A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
⊢ (𝐴 ∈ ℚ → 𝐴 < ((⌊‘𝐴) + 1))
Theorem	qfraclt1 10467	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 10468	The fractional part of a rational number is nonnegative. (Contributed by Jim Kingdon, 8-Oct-2021.)
⊢ (𝐴 ∈ ℚ → 0 ≤ (𝐴 − (⌊‘𝐴)))
Theorem	flqge 10469	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 10470	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 10471	An integer is its own floor. (Contributed by NM, 15-Nov-2004.)
⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) = 𝐴)
Theorem	flqidm 10472	The floor function is idempotent. (Contributed by Jim Kingdon, 8-Oct-2021.)
⊢ (𝐴 ∈ ℚ → (⌊‘(⌊‘𝐴)) = (⌊‘𝐴))
Theorem	flqidz 10473	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 10474	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 10475	Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐴 ≤ 𝐵) → (⌊‘𝐴) ≤ (⌊‘𝐵))
Theorem	flqword2 10476	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 10477	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 10478	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 10479	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 10480	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 10481	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 10482	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 10483	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 10484	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 10485	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 10486	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 10487	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 10488	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 10489	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 10490	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 10491	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 10492	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 10493	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 10494	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 10495	The value of the ceiling function. (Contributed by Jim Kingdon, 10-Oct-2021.)
⊢ (𝐴 ∈ ℚ → (⌈‘𝐴) = -(⌊‘-𝐴))
Theorem	ceiqcl 10496	The ceiling function returns an integer (closure law). (Contributed by Jim Kingdon, 11-Oct-2021.)
⊢ (𝐴 ∈ ℚ → -(⌊‘-𝐴) ∈ ℤ)
Theorem	ceilqcl 10497	Closure of the ceiling function. (Contributed by Jim Kingdon, 11-Oct-2021.)
⊢ (𝐴 ∈ ℚ → (⌈‘𝐴) ∈ ℤ)
Theorem	ceiqge 10498	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 10499	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 10500	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)