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Theorem List for Intuitionistic Logic Explorer - 8201-8300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem8p5e13 8201 8 + 5 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
 8  +  5 ; 1 3
 
Theorem8p6e14 8202 8 + 6 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
 8  +  6 ; 1 4
 
Theorem8p7e15 8203 8 + 7 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
 8  +  7 ; 1 5
 
Theorem8p8e16 8204 8 + 8 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
 8  +  8 ; 1 6
 
Theorem9p2e11 8205 9 + 2 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
 9  +  2 ; 1 1
 
Theorem9p3e12 8206 9 + 3 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 9  +  3 ; 1 2
 
Theorem9p4e13 8207 9 + 4 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
 9  +  4 ; 1 3
 
Theorem9p5e14 8208 9 + 5 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
 9  +  5 ; 1 4
 
Theorem9p6e15 8209 9 + 6 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
 9  +  6 ; 1 5
 
Theorem9p7e16 8210 9 + 7 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
 9  +  7 ; 1 6
 
Theorem9p8e17 8211 9 + 8 = 17. (Contributed by Mario Carneiro, 19-Apr-2015.)
 9  +  8 ; 1 7
 
Theorem9p9e18 8212 9 + 9 = 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
 9  +  9 ; 1 8
 
Theorem10p10e20 8213 10 + 10 = 20. (Contributed by Mario Carneiro, 19-Apr-2015.)
 10  +  10 ; 2 0
 
Theorem4t3lem 8214 Lemma for 4t3e12 8215 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)
 NN0   &     NN0   &     C  +  1   &     x.  D   &     D  +  E   =>     x.  C  E
 
Theorem4t3e12 8215 4 times 3 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 4  x.  3 ; 1 2
 
Theorem4t4e16 8216 4 times 4 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
 4  x.  4 ; 1 6
 
Theorem5t3e15 8217 5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
 5  x.  3 ; 1 5
 
Theorem5t4e20 8218 5 times 4 equals 20. (Contributed by Mario Carneiro, 19-Apr-2015.)
 5  x.  4 ; 2 0
 
Theorem5t5e25 8219 5 times 5 equals 25. (Contributed by Mario Carneiro, 19-Apr-2015.)
 5  x.  5 ; 2 5
 
Theorem6t2e12 8220 6 times 2 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 6  x.  2 ; 1 2
 
Theorem6t3e18 8221 6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
 6  x.  3 ; 1 8
 
Theorem6t4e24 8222 6 times 4 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
 6  x.  4 ; 2 4
 
Theorem6t5e30 8223 6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.)
 6  x.  5 ; 3 0
 
Theorem6t6e36 8224 6 times 6 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
 6  x.  6 ; 3 6
 
Theorem7t2e14 8225 7 times 2 equals 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
 7  x.  2 ; 1 4
 
Theorem7t3e21 8226 7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.)
 7  x.  3 ; 2 1
 
Theorem7t4e28 8227 7 times 4 equals 28. (Contributed by Mario Carneiro, 19-Apr-2015.)
 7  x.  4 ; 2 8
 
Theorem7t5e35 8228 7 times 5 equals 35. (Contributed by Mario Carneiro, 19-Apr-2015.)
 7  x.  5 ; 3 5
 
Theorem7t6e42 8229 7 times 6 equals 42. (Contributed by Mario Carneiro, 19-Apr-2015.)
 7  x.  6 ; 4 2
 
Theorem7t7e49 8230 7 times 7 equals 49. (Contributed by Mario Carneiro, 19-Apr-2015.)
 7  x.  7 ; 4 9
 
Theorem8t2e16 8231 8 times 2 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
 8  x.  2 ; 1 6
 
Theorem8t3e24 8232 8 times 3 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
 8  x.  3 ; 2 4
 
Theorem8t4e32 8233 8 times 4 equals 32. (Contributed by Mario Carneiro, 19-Apr-2015.)
 8  x.  4 ; 3 2
 
Theorem8t5e40 8234 8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.)
 8  x.  5 ; 4 0
 
Theorem8t6e48 8235 8 times 6 equals 48. (Contributed by Mario Carneiro, 19-Apr-2015.)
 8  x.  6 ; 4 8
 
Theorem8t7e56 8236 8 times 7 equals 56. (Contributed by Mario Carneiro, 19-Apr-2015.)
 8  x.  7 ; 5 6
 
Theorem8t8e64 8237 8 times 8 equals 64. (Contributed by Mario Carneiro, 19-Apr-2015.)
 8  x.  8 ; 6 4
 
Theorem9t2e18 8238 9 times 2 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
 9  x.  2 ; 1 8
 
Theorem9t3e27 8239 9 times 3 equals 27. (Contributed by Mario Carneiro, 19-Apr-2015.)
 9  x.  3 ; 2 7
 
Theorem9t4e36 8240 9 times 4 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
 9  x.  4 ; 3 6
 
Theorem9t5e45 8241 9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.)
 9  x.  5 ; 4 5
 
Theorem9t6e54 8242 9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.)
 9  x.  6 ; 5 4
 
Theorem9t7e63 8243 9 times 7 equals 63. (Contributed by Mario Carneiro, 19-Apr-2015.)
 9  x.  7 ; 6 3
 
Theorem9t8e72 8244 9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.)
 9  x.  8 ; 7 2
 
Theorem9t9e81 8245 9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.)
 9  x.  9 ; 8 1
 
Theoremdecbin0 8246 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
 NN0   =>     4  x.  2  x.  2  x.
 
Theoremdecbin2 8247 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
 NN0   =>     4  x.  +  2  2  x.  2  x.  +  1
 
Theoremdecbin3 8248 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
 NN0   =>     4  x.  +  3 
 2  x.  2  x.  +  1  +  1
 
3.4.10  Upper sets of integers
 
Syntaxcuz 8249 Extend class notation with the upper integer function. Read " ZZ>= `  M " as "the set of integers greater than or equal to  M."
 ZZ>=
 
Definitiondf-uz 8250* 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 " ZZ>= `  M " as "the set of integers greater than or equal to  M." See uzval 8251 for its value, uzssz 8268 for its relationship to  ZZ, nnuz 8284 and nn0uz 8283 for its relationships to  NN and  NN0, and eluz1 8253 and eluz2 8255 for its membership relations. (Contributed by NM, 5-Sep-2005.)

 ZZ>=  j 
 ZZ  |->  { k  ZZ  |  j  <_  k }
 
Theoremuzval 8251* The value of the upper integers function. (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
 N  ZZ  ZZ>= `  N  { k  ZZ  |  N  <_  k }
 
Theoremuzf 8252 The domain and range of the upper integers function. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 3-Nov-2013.)

 ZZ>= : ZZ --> ~P ZZ
 
Theoremeluz1 8253 Membership in the upper set of integers starting at  M. (Contributed by NM, 5-Sep-2005.)
 M  ZZ  N  ZZ>=
 `  M  N  ZZ  M  <_  N
 
Theoremeluzel2 8254 Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
 N  ZZ>=
 `  M 
 M  ZZ
 
Theoremeluz2 8255 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  ZZ. (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
 N  ZZ>=
 `  M  M  ZZ  N  ZZ  M  <_  N
 
Theoremeluz1i 8256 Membership in an upper set of integers. (Contributed by NM, 5-Sep-2005.)
 M  ZZ   =>     N  ZZ>=
 `  M  N  ZZ  M  <_  N
 
Theoremeluzuzle 8257 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.)
 ZZ  <_  C  ZZ>= `  C  ZZ>= `
 
Theoremeluzelz 8258 A member of an upper set of integers is an integer. (Contributed by NM, 6-Sep-2005.)
 N  ZZ>=
 `  M 
 N  ZZ
 
Theoremeluzelre 8259 A member of an upper set of integers is a real. (Contributed by Mario Carneiro, 31-Aug-2013.)
 N  ZZ>=
 `  M 
 N  RR
 
Theoremeluzelcn 8260 A member of an upper set of integers is a complex number. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 N  ZZ>=
 `  M 
 N  CC
 
Theoremeluzle 8261 Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.)
 N  ZZ>=
 `  M 
 M  <_  N
 
Theoremeluz 8262 Membership in an upper set of integers. (Contributed by NM, 2-Oct-2005.)
 M  ZZ  N  ZZ  N  ZZ>= `  M  M  <_  N
 
Theoremuzid 8263 Membership of the least member in an upper set of integers. (Contributed by NM, 2-Sep-2005.)
 M  ZZ  M  ZZ>= `  M
 
Theoremuzn0 8264 The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.)
 M  ran  ZZ>=  M  =/=  (/)
 
Theoremuztrn 8265 Transitive law for sets of upper integers. (Contributed by NM, 20-Sep-2005.)
 M  ZZ>= `  K  K  ZZ>= `  N  M  ZZ>= `  N
 
Theoremuztrn2 8266 Transitive law for sets of upper integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
 Z  ZZ>= `  K   =>     N  Z  M  ZZ>=
 `  N  M  Z
 
Theoremuzneg 8267 Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.)
 N  ZZ>=
 `  M  -u M  ZZ>=
 `  -u N
 
Theoremuzssz 8268 An upper set of integers is a subset of all integers. (Contributed by NM, 2-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
 ZZ>= `  M  C_ 
 ZZ
 
Theoremuzss 8269 Subset relationship for two sets of upper integers. (Contributed by NM, 5-Sep-2005.)
 N  ZZ>=
 `  M  ZZ>= `  N  C_  ZZ>= `  M
 
Theoremuztric 8270 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  ZZ  N  ZZ  N  ZZ>= `  M  M  ZZ>= `  N
 
Theoremuz11 8271 The upper integers function is one-to-one. (Contributed by NM, 12-Dec-2005.)
 M  ZZ  ZZ>= `  M  ZZ>= `  N  M  N
 
Theoremeluzp1m1 8272 Membership in the next upper set of integers. (Contributed by NM, 12-Sep-2005.)
 M  ZZ  N  ZZ>=
 `  M  +  1  N  -  1  ZZ>= `  M
 
Theoremeluzp1l 8273 Strict ordering implied by membership in the next upper set of integers. (Contributed by NM, 12-Sep-2005.)
 M  ZZ  N  ZZ>=
 `  M  +  1 
 M  <  N
 
Theoremeluzp1p1 8274 Membership in the next upper set of integers. (Contributed by NM, 5-Oct-2005.)
 N  ZZ>=
 `  M  N  +  1  ZZ>= `  M  +  1
 
Theoremeluzaddi 8275 Membership in a later upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.)
 M  ZZ   &     K  ZZ   =>     N  ZZ>=
 `  M  N  +  K  ZZ>= `  M  +  K
 
Theoremeluzsubi 8276 Membership in an earlier upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.)
 M  ZZ   &     K  ZZ   =>     N  ZZ>=
 `  M  +  K  N  -  K  ZZ>= `  M
 
Theoremeluzadd 8277 Membership in a later upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
 N  ZZ>= `  M  K  ZZ  N  +  K  ZZ>= `  M  +  K
 
Theoremeluzsub 8278 Membership in an earlier upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
 M  ZZ  K  ZZ  N  ZZ>= `  M  +  K  N  -  K  ZZ>= `  M
 
Theoremuzm1 8279 Choices for an element of an upper interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
 N  ZZ>=
 `  M  N  M  N  -  1  ZZ>= `  M
 
Theoremuznn0sub 8280 The nonnegative difference of integers is a nonnegative integer. (Contributed by NM, 4-Sep-2005.)
 N  ZZ>=
 `  M  N  -  M  NN0
 
Theoremuzin 8281 Intersection of two upper intervals of integers. (Contributed by Mario Carneiro, 24-Dec-2013.)
 M  ZZ  N  ZZ  ZZ>= `  M  i^i  ZZ>= `  N 
 ZZ>= `  if M 
 <_  N ,  N ,  M
 
Theoremuzp1 8282 Choices for an element of an upper interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
 N  ZZ>=
 `  M  N  M  N  ZZ>= `  M  +  1
 
Theoremnn0uz 8283 Nonnegative integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)

 NN0  ZZ>= `  0
 
Theoremnnuz 8284 Positive integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)

 NN  ZZ>= `  1
 
Theoremelnnuz 8285 A positive integer expressed as a member of an upper set of integers. (Contributed by NM, 6-Jun-2006.)
 N  NN  N  ZZ>= `  1
 
Theoremelnn0uz 8286 A nonnegative integer expressed as a member an upper set of integers. (Contributed by NM, 6-Jun-2006.)
 N  NN0  N  ZZ>= `  0
 
Theoremeluz2nn 8287 An integer is greater than or equal to 2 is a positive integer. (Contributed by AV, 3-Nov-2018.)
 ZZ>=
 `  2  NN
 
Theoremeluzge2nn0 8288 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  ZZ>=
 `  2 
 N  NN0
 
Theoremuzuzle23 8289 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.)
 ZZ>=
 `  3  ZZ>= `  2
 
Theoremeluzge3nn 8290 If an integer is greater than 3, then it is a positive integer. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
 N  ZZ>=
 `  3 
 N  NN
 
Theoremuz3m2nn 8291 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  ZZ>=
 `  3  N  -  2  NN
 
Theorem1eluzge0 8292 1 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
 1  ZZ>= `  0
 
Theorem2eluzge0 8293 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  ZZ>= `  0
 
Theorem2eluzge0OLD 8294 2 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.) Obsolete version of 2eluzge0 8293 as of 25-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 2  ZZ>= `  0
 
Theorem2eluzge1 8295 2 is an integer greater than or equal to 1. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
 2  ZZ>= `  1
 
Theoremuznnssnn 8296 The upper integers starting from a natural are a subset of the naturals. (Contributed by Scott Fenton, 29-Jun-2013.)
 N  NN  ZZ>= `  N  C_ 
 NN
 
Theoremraluz 8297* Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
 M  ZZ  n  ZZ>= `  M  n  ZZ  M  <_  n
 
Theoremraluz2 8298* Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
 n  ZZ>= `  M  M  ZZ  n  ZZ  M  <_  n
 
Theoremrexuz 8299* Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
 M  ZZ  n  ZZ>= `  M  n  ZZ  M  <_  n
 
Theoremrexuz2 8300* Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
 n  ZZ>= `  M  M  ZZ  n  ZZ  M  <_  n
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