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Theorem List for Intuitionistic Logic Explorer - 8201-8300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem6t3e18 8201 6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
(6 · 3) = 18

Theorem6t4e24 8202 6 times 4 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
(6 · 4) = 24

Theorem6t5e30 8203 6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.)
(6 · 5) = 30

Theorem6t6e36 8204 6 times 6 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
(6 · 6) = 36

Theorem7t2e14 8205 7 times 2 equals 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 2) = 14

Theorem7t3e21 8206 7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 3) = 21

Theorem7t4e28 8207 7 times 4 equals 28. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 4) = 28

Theorem7t5e35 8208 7 times 5 equals 35. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 5) = 35

Theorem7t6e42 8209 7 times 6 equals 42. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 6) = 42

Theorem7t7e49 8210 7 times 7 equals 49. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 7) = 49

Theorem8t2e16 8211 8 times 2 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 · 2) = 16

Theorem8t3e24 8212 8 times 3 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 · 3) = 24

Theorem8t4e32 8213 8 times 4 equals 32. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 · 4) = 32

Theorem8t5e40 8214 8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 · 5) = 40

Theorem8t6e48 8215 8 times 6 equals 48. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 · 6) = 48

Theorem8t7e56 8216 8 times 7 equals 56. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 · 7) = 56

Theorem8t8e64 8217 8 times 8 equals 64. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 · 8) = 64

Theorem9t2e18 8218 9 times 2 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 2) = 18

Theorem9t3e27 8219 9 times 3 equals 27. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 3) = 27

Theorem9t4e36 8220 9 times 4 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 4) = 36

Theorem9t5e45 8221 9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 5) = 45

Theorem9t6e54 8222 9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 6) = 54

Theorem9t7e63 8223 9 times 7 equals 63. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 7) = 63

Theorem9t8e72 8224 9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 8) = 72

Theorem9t9e81 8225 9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 9) = 81

Theoremdecbin0 8226 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
A 0       (4 · A) = (2 · (2 · A))

Theoremdecbin2 8227 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
A 0       ((4 · A) + 2) = (2 · ((2 · A) + 1))

Theoremdecbin3 8228 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 8229 Extend class notation with the upper integer function. Read "𝑀 " as "the set of integers greater than or equal to 𝑀."
class

Definitiondf-uz 8230* 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 8231 for its value, uzssz 8248 for its relationship to , nnuz 8264 and nn0uz 8263 for its relationships to and 0, and eluz1 8233 and eluz2 8235 for its membership relations. (Contributed by NM, 5-Sep-2005.)
= (𝑗 ℤ ↦ {𝑘 ℤ ∣ 𝑗𝑘})

Theoremuzval 8231* 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 8232 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 8233 Membership in the upper set of integers starting at 𝑀. (Contributed by NM, 5-Sep-2005.)
(𝑀 ℤ → (𝑁 (ℤ𝑀) ↔ (𝑁 𝑀𝑁)))

Theoremeluzel2 8234 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 8235 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 8236 Membership in an upper set of integers. (Contributed by NM, 5-Sep-2005.)
𝑀        (𝑁 (ℤ𝑀) ↔ (𝑁 𝑀𝑁))

Theoremeluzuzle 8237 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 8238 A member of an upper set of integers is an integer. (Contributed by NM, 6-Sep-2005.)
(𝑁 (ℤ𝑀) → 𝑁 ℤ)

Theoremeluzelre 8239 A member of an upper set of integers is a real. (Contributed by Mario Carneiro, 31-Aug-2013.)
(𝑁 (ℤ𝑀) → 𝑁 ℝ)

Theoremeluzelcn 8240 A member of an upper set of integers is a complex number. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
(𝑁 (ℤ𝑀) → 𝑁 ℂ)

Theoremeluzle 8241 Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.)
(𝑁 (ℤ𝑀) → 𝑀𝑁)

Theoremeluz 8242 Membership in an upper set of integers. (Contributed by NM, 2-Oct-2005.)
((𝑀 𝑁 ℤ) → (𝑁 (ℤ𝑀) ↔ 𝑀𝑁))

Theoremuzid 8243 Membership of the least member in an upper set of integers. (Contributed by NM, 2-Sep-2005.)
(𝑀 ℤ → 𝑀 (ℤ𝑀))

Theoremuzn0 8244 The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.)
(𝑀 ran ℤ𝑀 ≠ ∅)

Theoremuztrn 8245 Transitive law for sets of upper integers. (Contributed by NM, 20-Sep-2005.)
((𝑀 (ℤ𝐾) 𝐾 (ℤ𝑁)) → 𝑀 (ℤ𝑁))

Theoremuztrn2 8246 Transitive law for sets of upper integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
𝑍 = (ℤ𝐾)       ((𝑁 𝑍 𝑀 (ℤ𝑁)) → 𝑀 𝑍)

Theoremuzneg 8247 Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.)
(𝑁 (ℤ𝑀) → -𝑀 (ℤ‘-𝑁))

Theoremuzssz 8248 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 8249 Subset relationship for two sets of upper integers. (Contributed by NM, 5-Sep-2005.)
(𝑁 (ℤ𝑀) → (ℤ𝑁) ⊆ (ℤ𝑀))

Theoremuztric 8250 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 8251 The upper integers function is one-to-one. (Contributed by NM, 12-Dec-2005.)
(𝑀 ℤ → ((ℤ𝑀) = (ℤ𝑁) ↔ 𝑀 = 𝑁))

Theoremeluzp1m1 8252 Membership in the next upper set of integers. (Contributed by NM, 12-Sep-2005.)
((𝑀 𝑁 (ℤ‘(𝑀 + 1))) → (𝑁 − 1) (ℤ𝑀))

Theoremeluzp1l 8253 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 8254 Membership in the next upper set of integers. (Contributed by NM, 5-Oct-2005.)
(𝑁 (ℤ𝑀) → (𝑁 + 1) (ℤ‘(𝑀 + 1)))

Theoremeluzaddi 8255 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 8256 Membership in an earlier upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.)
𝑀     &   𝐾        (𝑁 (ℤ‘(𝑀 + 𝐾)) → (𝑁𝐾) (ℤ𝑀))

Theoremeluzadd 8257 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 8258 Membership in an earlier upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝑀 𝐾 𝑁 (ℤ‘(𝑀 + 𝐾))) → (𝑁𝐾) (ℤ𝑀))

Theoremuzm1 8259 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 8260 The nonnegative difference of integers is a nonnegative integer. (Contributed by NM, 4-Sep-2005.)
(𝑁 (ℤ𝑀) → (𝑁𝑀) 0)

Theoremuzin 8261 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 8262 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 8263 Nonnegative integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
0 = (ℤ‘0)

Theoremnnuz 8264 Positive integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
ℕ = (ℤ‘1)

Theoremelnnuz 8265 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 8266 A nonnegative integer expressed as a member an upper set of integers. (Contributed by NM, 6-Jun-2006.)
(𝑁 0𝑁 (ℤ‘0))

Theoremeluz2nn 8267 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 8268 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 8269 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 8270 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 8271 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 8272 1 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
1 (ℤ‘0)

Theorem2eluzge0 8273 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 8274 2 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.) Obsolete version of 2eluzge0 8273 as of 25-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.)
2 (ℤ‘0)

Theorem2eluzge1 8275 2 is an integer greater than or equal to 1. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
2 (ℤ‘1)

Theoremuznnssnn 8276 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 8277* Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(𝑀 ℤ → (𝑛 (ℤ𝑀)φ𝑛 ℤ (𝑀𝑛φ)))

Theoremraluz2 8278* Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(𝑛 (ℤ𝑀)φ ↔ (𝑀 ℤ → 𝑛 ℤ (𝑀𝑛φ)))

Theoremrexuz 8279* Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(𝑀 ℤ → (𝑛 (ℤ𝑀)φ𝑛 ℤ (𝑀𝑛 φ)))

Theoremrexuz2 8280* Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(𝑛 (ℤ𝑀)φ ↔ (𝑀 𝑛 ℤ (𝑀𝑛 φ)))

Theorem2rexuz 8281* Double existential quantification in an upper set of integers. (Contributed by NM, 3-Nov-2005.)
(𝑚𝑛 (ℤ𝑚)φ𝑚 𝑛 ℤ (𝑚𝑛 φ))

Theorempeano2uz 8282 Second Peano postulate for an upper set of integers. (Contributed by NM, 7-Sep-2005.)
(𝑁 (ℤ𝑀) → (𝑁 + 1) (ℤ𝑀))

Theorempeano2uzs 8283 Second Peano postulate for an upper set of integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
𝑍 = (ℤ𝑀)       (𝑁 𝑍 → (𝑁 + 1) 𝑍)

Theorempeano2uzr 8284 Reversed second Peano axiom for upper integers. (Contributed by NM, 2-Jan-2006.)
((𝑀 𝑁 (ℤ‘(𝑀 + 1))) → 𝑁 (ℤ𝑀))

Theoremuzaddcl 8285 Addition closure law for an upper set of integers. (Contributed by NM, 4-Jun-2006.)
((𝑁 (ℤ𝑀) 𝐾 0) → (𝑁 + 𝐾) (ℤ𝑀))

Theoremnn0pzuz 8286 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 8287* 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 8288* 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 8287 or uzind4ALT 8288 may be used; see comment for nnind 7691. (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 8289* 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 8290* 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 8289 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 8291* 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 8292* 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 8293 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 8294 Membership in a positive upper set of integers implies membership in . (Contributed by JJ, 1-Oct-2018.)
((𝑁 𝑀 (ℤ𝑁)) → 𝑀 ℕ)

Theoremeluz2b1 8295 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 8296 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 8297 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 8298 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 8299 1 is not in (ℤ‘2). (Contributed by Paul Chapman, 21-Nov-2012.)
¬ 1 (ℤ‘2)

Theoremelnn1uz2 8300 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)))

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