HomeHome Intuitionistic Logic Explorer
Theorem List (p. 83 of 94)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 8201-8300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem9t5e45 8201 9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 5) = 45
 
Theorem9t6e54 8202 9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 6) = 54
 
Theorem9t7e63 8203 9 times 7 equals 63. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 7) = 63
 
Theorem9t8e72 8204 9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 8) = 72
 
Theorem9t9e81 8205 9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 9) = 81
 
Theoremdecbin0 8206 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
A 0       (4 · A) = (2 · (2 · A))
 
Theoremdecbin2 8207 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
A 0       ((4 · A) + 2) = (2 · ((2 · A) + 1))
 
Theoremdecbin3 8208 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
A 0       ((4 · A) + 3) = ((2 · ((2 · A) + 1)) + 1)
 
3.4.10  Upper sets of integers
 
Syntaxcuz 8209 Extend class notation with the upper integer function. Read "𝑀 " as "the set of integers greater than or equal to 𝑀."
class
 
Definitiondf-uz 8210* Define a function whose value at 𝑗 is the semi-infinite set of contiguous integers starting at 𝑗, which we will also call the upper integers starting at 𝑗. Read "𝑀 " as "the set of integers greater than or equal to 𝑀." See uzval 8211 for its value, uzssz 8228 for its relationship to , nnuz 8244 and nn0uz 8243 for its relationships to and 0, and eluz1 8213 and eluz2 8215 for its membership relations. (Contributed by NM, 5-Sep-2005.)
= (𝑗 ℤ ↦ {𝑘 ℤ ∣ 𝑗𝑘})
 
Theoremuzval 8211* The value of the upper integers function. (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
(𝑁 ℤ → (ℤ𝑁) = {𝑘 ℤ ∣ 𝑁𝑘})
 
Theoremuzf 8212 The domain and range of the upper integers function. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 3-Nov-2013.)
:ℤ⟶𝒫 ℤ
 
Theoremeluz1 8213 Membership in the upper set of integers starting at 𝑀. (Contributed by NM, 5-Sep-2005.)
(𝑀 ℤ → (𝑁 (ℤ𝑀) ↔ (𝑁 𝑀𝑁)))
 
Theoremeluzel2 8214 Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
(𝑁 (ℤ𝑀) → 𝑀 ℤ)
 
Theoremeluz2 8215 Membership in an upper set of integers. We use the fact that a function's value (under our function value definition) is empty outside of its domain to show 𝑀 . (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
(𝑁 (ℤ𝑀) ↔ (𝑀 𝑁 𝑀𝑁))
 
Theoremeluz1i 8216 Membership in an upper set of integers. (Contributed by NM, 5-Sep-2005.)
𝑀        (𝑁 (ℤ𝑀) ↔ (𝑁 𝑀𝑁))
 
Theoremeluzuzle 8217 An integer in an upper set of integers is an element of an upper set of integers with a smaller bound. (Contributed by Alexander van der Vekens, 17-Jun-2018.)
((B BA) → (𝐶 (ℤA) → 𝐶 (ℤB)))
 
Theoremeluzelz 8218 A member of an upper set of integers is an integer. (Contributed by NM, 6-Sep-2005.)
(𝑁 (ℤ𝑀) → 𝑁 ℤ)
 
Theoremeluzelre 8219 A member of an upper set of integers is a real. (Contributed by Mario Carneiro, 31-Aug-2013.)
(𝑁 (ℤ𝑀) → 𝑁 ℝ)
 
Theoremeluzelcn 8220 A member of an upper set of integers is a complex number. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
(𝑁 (ℤ𝑀) → 𝑁 ℂ)
 
Theoremeluzle 8221 Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.)
(𝑁 (ℤ𝑀) → 𝑀𝑁)
 
Theoremeluz 8222 Membership in an upper set of integers. (Contributed by NM, 2-Oct-2005.)
((𝑀 𝑁 ℤ) → (𝑁 (ℤ𝑀) ↔ 𝑀𝑁))
 
Theoremuzid 8223 Membership of the least member in an upper set of integers. (Contributed by NM, 2-Sep-2005.)
(𝑀 ℤ → 𝑀 (ℤ𝑀))
 
Theoremuzn0 8224 The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.)
(𝑀 ran ℤ𝑀 ≠ ∅)
 
Theoremuztrn 8225 Transitive law for sets of upper integers. (Contributed by NM, 20-Sep-2005.)
((𝑀 (ℤ𝐾) 𝐾 (ℤ𝑁)) → 𝑀 (ℤ𝑁))
 
Theoremuztrn2 8226 Transitive law for sets of upper integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
𝑍 = (ℤ𝐾)       ((𝑁 𝑍 𝑀 (ℤ𝑁)) → 𝑀 𝑍)
 
Theoremuzneg 8227 Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.)
(𝑁 (ℤ𝑀) → -𝑀 (ℤ‘-𝑁))
 
Theoremuzssz 8228 An upper set of integers is a subset of all integers. (Contributed by NM, 2-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
(ℤ𝑀) ⊆ ℤ
 
Theoremuzss 8229 Subset relationship for two sets of upper integers. (Contributed by NM, 5-Sep-2005.)
(𝑁 (ℤ𝑀) → (ℤ𝑁) ⊆ (ℤ𝑀))
 
Theoremuztric 8230 Trichotomy of the ordering relation on integers, stated in terms of upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 25-Jun-2013.)
((𝑀 𝑁 ℤ) → (𝑁 (ℤ𝑀) 𝑀 (ℤ𝑁)))
 
Theoremuz11 8231 The upper integers function is one-to-one. (Contributed by NM, 12-Dec-2005.)
(𝑀 ℤ → ((ℤ𝑀) = (ℤ𝑁) ↔ 𝑀 = 𝑁))
 
Theoremeluzp1m1 8232 Membership in the next upper set of integers. (Contributed by NM, 12-Sep-2005.)
((𝑀 𝑁 (ℤ‘(𝑀 + 1))) → (𝑁 − 1) (ℤ𝑀))
 
Theoremeluzp1l 8233 Strict ordering implied by membership in the next upper set of integers. (Contributed by NM, 12-Sep-2005.)
((𝑀 𝑁 (ℤ‘(𝑀 + 1))) → 𝑀 < 𝑁)
 
Theoremeluzp1p1 8234 Membership in the next upper set of integers. (Contributed by NM, 5-Oct-2005.)
(𝑁 (ℤ𝑀) → (𝑁 + 1) (ℤ‘(𝑀 + 1)))
 
Theoremeluzaddi 8235 Membership in a later upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.)
𝑀     &   𝐾        (𝑁 (ℤ𝑀) → (𝑁 + 𝐾) (ℤ‘(𝑀 + 𝐾)))
 
Theoremeluzsubi 8236 Membership in an earlier upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.)
𝑀     &   𝐾        (𝑁 (ℤ‘(𝑀 + 𝐾)) → (𝑁𝐾) (ℤ𝑀))
 
Theoremeluzadd 8237 Membership in a later upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝑁 (ℤ𝑀) 𝐾 ℤ) → (𝑁 + 𝐾) (ℤ‘(𝑀 + 𝐾)))
 
Theoremeluzsub 8238 Membership in an earlier upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝑀 𝐾 𝑁 (ℤ‘(𝑀 + 𝐾))) → (𝑁𝐾) (ℤ𝑀))
 
Theoremuzm1 8239 Choices for an element of an upper interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑁 (ℤ𝑀) → (𝑁 = 𝑀 (𝑁 − 1) (ℤ𝑀)))
 
Theoremuznn0sub 8240 The nonnegative difference of integers is a nonnegative integer. (Contributed by NM, 4-Sep-2005.)
(𝑁 (ℤ𝑀) → (𝑁𝑀) 0)
 
Theoremuzin 8241 Intersection of two upper intervals of integers. (Contributed by Mario Carneiro, 24-Dec-2013.)
((𝑀 𝑁 ℤ) → ((ℤ𝑀) ∩ (ℤ𝑁)) = (ℤ‘if(𝑀𝑁, 𝑁, 𝑀)))
 
Theoremuzp1 8242 Choices for an element of an upper interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑁 (ℤ𝑀) → (𝑁 = 𝑀 𝑁 (ℤ‘(𝑀 + 1))))
 
Theoremnn0uz 8243 Nonnegative integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
0 = (ℤ‘0)
 
Theoremnnuz 8244 Positive integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
ℕ = (ℤ‘1)
 
Theoremelnnuz 8245 A positive integer expressed as a member of an upper set of integers. (Contributed by NM, 6-Jun-2006.)
(𝑁 ℕ ↔ 𝑁 (ℤ‘1))
 
Theoremelnn0uz 8246 A nonnegative integer expressed as a member an upper set of integers. (Contributed by NM, 6-Jun-2006.)
(𝑁 0𝑁 (ℤ‘0))
 
Theoremeluz2nn 8247 An integer is greater than or equal to 2 is a positive integer. (Contributed by AV, 3-Nov-2018.)
(A (ℤ‘2) → A ℕ)
 
Theoremeluzge2nn0 8248 If an integer is greater than or equal to 2, then it is a nonnegative integer. (Contributed by AV, 27-Aug-2018.) (Proof shortened by AV, 3-Nov-2018.)
(𝑁 (ℤ‘2) → 𝑁 0)
 
Theoremuzuzle23 8249 An integer in the upper set of integers starting at 3 is element of the upper set of integers starting at 2. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
(A (ℤ‘3) → A (ℤ‘2))
 
Theoremeluzge3nn 8250 If an integer is greater than 3, then it is a positive integer. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
(𝑁 (ℤ‘3) → 𝑁 ℕ)
 
Theoremuz3m2nn 8251 An integer greater than or equal to 3 decreased by 2 is a positive integer. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
(𝑁 (ℤ‘3) → (𝑁 − 2) ℕ)
 
Theorem1eluzge0 8252 1 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
1 (ℤ‘0)
 
Theorem2eluzge0 8253 2 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.) (Proof shortened by OpenAI, 25-Mar-2020.)
2 (ℤ‘0)
 
Theorem2eluzge0OLD 8254 2 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.) Obsolete version of 2eluzge0 8253 as of 25-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.)
2 (ℤ‘0)
 
Theorem2eluzge1 8255 2 is an integer greater than or equal to 1. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
2 (ℤ‘1)
 
Theoremuznnssnn 8256 The upper integers starting from a natural are a subset of the naturals. (Contributed by Scott Fenton, 29-Jun-2013.)
(𝑁 ℕ → (ℤ𝑁) ⊆ ℕ)
 
Theoremraluz 8257* Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(𝑀 ℤ → (𝑛 (ℤ𝑀)φ𝑛 ℤ (𝑀𝑛φ)))
 
Theoremraluz2 8258* Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(𝑛 (ℤ𝑀)φ ↔ (𝑀 ℤ → 𝑛 ℤ (𝑀𝑛φ)))
 
Theoremrexuz 8259* Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(𝑀 ℤ → (𝑛 (ℤ𝑀)φ𝑛 ℤ (𝑀𝑛 φ)))
 
Theoremrexuz2 8260* Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(𝑛 (ℤ𝑀)φ ↔ (𝑀 𝑛 ℤ (𝑀𝑛 φ)))
 
Theorem2rexuz 8261* Double existential quantification in an upper set of integers. (Contributed by NM, 3-Nov-2005.)
(𝑚𝑛 (ℤ𝑚)φ𝑚 𝑛 ℤ (𝑚𝑛 φ))
 
Theorempeano2uz 8262 Second Peano postulate for an upper set of integers. (Contributed by NM, 7-Sep-2005.)
(𝑁 (ℤ𝑀) → (𝑁 + 1) (ℤ𝑀))
 
Theorempeano2uzs 8263 Second Peano postulate for an upper set of integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
𝑍 = (ℤ𝑀)       (𝑁 𝑍 → (𝑁 + 1) 𝑍)
 
Theorempeano2uzr 8264 Reversed second Peano axiom for upper integers. (Contributed by NM, 2-Jan-2006.)
((𝑀 𝑁 (ℤ‘(𝑀 + 1))) → 𝑁 (ℤ𝑀))
 
Theoremuzaddcl 8265 Addition closure law for an upper set of integers. (Contributed by NM, 4-Jun-2006.)
((𝑁 (ℤ𝑀) 𝐾 0) → (𝑁 + 𝐾) (ℤ𝑀))
 
Theoremnn0pzuz 8266 The sum of a nonnegative integer and an integer is an integer greater than or equal to that integer. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
((𝑁 0 𝑍 ℤ) → (𝑁 + 𝑍) (ℤ𝑍))
 
Theoremuzind4 8267* Induction on the upper set of integers that starts at an integer 𝑀. The first four hypotheses give us the substitution instances we need, and the last two are the basis and the induction step. (Contributed by NM, 7-Sep-2005.)
(𝑗 = 𝑀 → (φψ))    &   (𝑗 = 𝑘 → (φχ))    &   (𝑗 = (𝑘 + 1) → (φθ))    &   (𝑗 = 𝑁 → (φτ))    &   (𝑀 ℤ → ψ)    &   (𝑘 (ℤ𝑀) → (χθ))       (𝑁 (ℤ𝑀) → τ)
 
Theoremuzind4ALT 8268* Induction on the upper set of integers that starts at an integer 𝑀. The last four hypotheses give us the substitution instances we need; the first two are the basis and the induction step. Either uzind4 8267 or uzind4ALT 8268 may be used; see comment for nnind 7671. (Contributed by NM, 7-Sep-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝑀 ℤ → ψ)    &   (𝑘 (ℤ𝑀) → (χθ))    &   (𝑗 = 𝑀 → (φψ))    &   (𝑗 = 𝑘 → (φχ))    &   (𝑗 = (𝑘 + 1) → (φθ))    &   (𝑗 = 𝑁 → (φτ))       (𝑁 (ℤ𝑀) → τ)
 
Theoremuzind4s 8269* Induction on the upper set of integers that starts at an integer 𝑀, using explicit substitution. The hypotheses are the basis and the induction step. (Contributed by NM, 4-Nov-2005.)
(𝑀 ℤ → [𝑀 / 𝑘]φ)    &   (𝑘 (ℤ𝑀) → (φ[(𝑘 + 1) / 𝑘]φ))       (𝑁 (ℤ𝑀) → [𝑁 / 𝑘]φ)
 
Theoremuzind4s2 8270* Induction on the upper set of integers that starts at an integer 𝑀, using explicit substitution. The hypotheses are the basis and the induction step. Use this instead of uzind4s 8269 when 𝑗 and 𝑘 must be distinct in [(𝑘 + 1) / 𝑗]φ. (Contributed by NM, 16-Nov-2005.)
(𝑀 ℤ → [𝑀 / 𝑗]φ)    &   (𝑘 (ℤ𝑀) → ([𝑘 / 𝑗]φ[(𝑘 + 1) / 𝑗]φ))       (𝑁 (ℤ𝑀) → [𝑁 / 𝑗]φ)
 
Theoremuzind4i 8271* Induction on the upper integers that start at 𝑀. The first hypothesis specifies the lower bound, the next four give us the substitution instances we need, and the last two are the basis and the induction step. (Contributed by NM, 4-Sep-2005.)
𝑀     &   (𝑗 = 𝑀 → (φψ))    &   (𝑗 = 𝑘 → (φχ))    &   (𝑗 = (𝑘 + 1) → (φθ))    &   (𝑗 = 𝑁 → (φτ))    &   ψ    &   (𝑘 (ℤ𝑀) → (χθ))       (𝑁 (ℤ𝑀) → τ)
 
Theoremindstr 8272* Strong Mathematical Induction for positive integers (inference schema). (Contributed by NM, 17-Aug-2001.)
(x = y → (φψ))    &   (x ℕ → (y ℕ (y < xψ) → φ))       (x ℕ → φ)
 
Theoremeluznn0 8273 Membership in a nonnegative upper set of integers implies membership in 0. (Contributed by Paul Chapman, 22-Jun-2011.)
((𝑁 0 𝑀 (ℤ𝑁)) → 𝑀 0)
 
Theoremeluznn 8274 Membership in a positive upper set of integers implies membership in . (Contributed by JJ, 1-Oct-2018.)
((𝑁 𝑀 (ℤ𝑁)) → 𝑀 ℕ)
 
Theoremeluz2b1 8275 Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.)
(𝑁 (ℤ‘2) ↔ (𝑁 1 < 𝑁))
 
Theoremeluz2b2 8276 Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.)
(𝑁 (ℤ‘2) ↔ (𝑁 1 < 𝑁))
 
Theoremeluz2b3 8277 Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.)
(𝑁 (ℤ‘2) ↔ (𝑁 𝑁 ≠ 1))
 
Theoremuz2m1nn 8278 One less than an integer greater than or equal to 2 is a positive integer. (Contributed by Paul Chapman, 17-Nov-2012.)
(𝑁 (ℤ‘2) → (𝑁 − 1) ℕ)
 
Theorem1nuz2 8279 1 is not in (ℤ‘2). (Contributed by Paul Chapman, 21-Nov-2012.)
¬ 1 (ℤ‘2)
 
Theoremelnn1uz2 8280 A positive integer is either 1 or greater than or equal to 2. (Contributed by Paul Chapman, 17-Nov-2012.)
(𝑁 ℕ ↔ (𝑁 = 1 𝑁 (ℤ‘2)))
 
Theoremuz2mulcl 8281 Closure of multiplication of integers greater than or equal to 2. (Contributed by Paul Chapman, 26-Oct-2012.)
((𝑀 (ℤ‘2) 𝑁 (ℤ‘2)) → (𝑀 · 𝑁) (ℤ‘2))
 
Theoremindstr2 8282* Strong Mathematical Induction for positive integers (inference schema). The first two hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by Paul Chapman, 21-Nov-2012.)
(x = 1 → (φχ))    &   (x = y → (φψ))    &   χ    &   (x (ℤ‘2) → (y ℕ (y < xψ) → φ))       (x ℕ → φ)
 
Theoremeluzdc 8283 Membership of an integer in an upper set of integers is decidable. (Contributed by Jim Kingdon, 18-Apr-2020.)
((𝑀 𝑁 ℤ) → DECID 𝑁 (ℤ𝑀))
 
Theoremublbneg 8284* The image under negation of a bounded-above set of reals is bounded below. (Contributed by Paul Chapman, 21-Mar-2011.)
(x y A yxx y {z ℝ ∣ -z A}xy)
 
Theoremeqreznegel 8285* Two ways to express the image under negation of a set of integers. (Contributed by Paul Chapman, 21-Mar-2011.)
(A ⊆ ℤ → {z ℝ ∣ -z A} = {z ℤ ∣ -z A})
 
Theoremnegm 8286* The image under negation of an inhabited set of reals is inhabited. (Contributed by Jim Kingdon, 10-Apr-2020.)
((A ⊆ ℝ x x A) → y y {z ℝ ∣ -z A})
 
Theoremlbzbi 8287* If a set of reals is bounded below, it is bounded below by an integer. (Contributed by Paul Chapman, 21-Mar-2011.)
(A ⊆ ℝ → (x y A xyx y A xy))
 
Theoremnn01to3 8288 A (nonnegative) integer between 1 and 3 must be 1, 2 or 3. (Contributed by Alexander van der Vekens, 13-Sep-2018.)
((𝑁 0 1 ≤ 𝑁 𝑁 ≤ 3) → (𝑁 = 1 𝑁 = 2 𝑁 = 3))
 
Theoremnn0ge2m1nnALT 8289 Alternate proof of nn0ge2m1nn 7978: If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is a positive integer. This version is proved using eluz2 8215, a theorem for upper sets of integers, which are defined later than the positive and nonnegative integers. This proof is, however, much shorter than the proof of nn0ge2m1nn 7978. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝑁 0 2 ≤ 𝑁) → (𝑁 − 1) ℕ)
 
3.4.11  Rational numbers (as a subset of complex numbers)
 
Syntaxcq 8290 Extend class notation to include the class of rationals.
class
 
Definitiondf-q 8291 Define the set of rational numbers. Based on definition of rationals in [Apostol] p. 22. See elq 8293 for the relation "is rational." (Contributed by NM, 8-Jan-2002.)
ℚ = ( / “ (ℤ × ℕ))
 
Theoremdivfnzn 8292 Division restricted to ℤ × ℕ is a function. Given excluded middle, it would be easy to prove this for ℂ × (ℂ ∖ {0}). The key difference is that an element of is apart from zero, whereas being an element of ℂ ∖ {0} implies being not equal to zero. (Contributed by Jim Kingdon, 19-Mar-2020.)
( / ↾ (ℤ × ℕ)) Fn (ℤ × ℕ)
 
Theoremelq 8293* Membership in the set of rationals. (Contributed by NM, 8-Jan-2002.) (Revised by Mario Carneiro, 28-Jan-2014.)
(A ℚ ↔ x y A = (x / y))
 
Theoremqmulz 8294* If A is rational, then some integer multiple of it is an integer. (Contributed by NM, 7-Nov-2008.) (Revised by Mario Carneiro, 22-Jul-2014.)
(A ℚ → x ℕ (A · x) ℤ)
 
Theoremznq 8295 The ratio of an integer and a positive integer is a rational number. (Contributed by NM, 12-Jan-2002.)
((A B ℕ) → (A / B) ℚ)
 
Theoremqre 8296 A rational number is a real number. (Contributed by NM, 14-Nov-2002.)
(A ℚ → A ℝ)
 
Theoremzq 8297 An integer is a rational number. (Contributed by NM, 9-Jan-2002.)
(A ℤ → A ℚ)
 
Theoremzssq 8298 The integers are a subset of the rationals. (Contributed by NM, 9-Jan-2002.)
ℤ ⊆ ℚ
 
Theoremnn0ssq 8299 The nonnegative integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
0 ⊆ ℚ
 
Theoremnnssq 8300 The positive integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
ℕ ⊆ ℚ
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9381
  Copyright terms: Public domain < Previous  Next >