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Theorem List for Metamath Proof Explorer - 10201-10300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem8p8e16 10201 8 + 8 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  +  8 )  = ; 1 6
 
Theorem9p2e11 10202 9 + 2 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  2 )  = ; 1 1
 
Theorem9p3e12 10203 9 + 3 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  3 )  = ; 1 2
 
Theorem9p4e13 10204 9 + 4 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  4 )  = ; 1 3
 
Theorem9p5e14 10205 9 + 5 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  5 )  = ; 1 4
 
Theorem9p6e15 10206 9 + 6 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  6 )  = ; 1 5
 
Theorem9p7e16 10207 9 + 7 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  7 )  = ; 1 6
 
Theorem9p8e17 10208 9 + 8 = 17. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  8 )  = ; 1 7
 
Theorem9p9e18 10209 9 + 9 = 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  9 )  = ; 1 8
 
Theorem10p10e20 10210 10 + 10 = 20. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 10  +  10 )  = ; 2 0
 
Theorem4t3lem 10211 Lemma for 4t3e12 10212 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  C  =  ( B  +  1 )   &    |-  ( A  x.  B )  =  D   &    |-  ( D  +  A )  =  E   =>    |-  ( A  x.  C )  =  E
 
Theorem4t3e12 10212 4 times 3 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 4  x.  3
 )  = ; 1 2
 
Theorem4t4e16 10213 4 times 4 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 4  x.  4
 )  = ; 1 6
 
Theorem5t3e15 10214 5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 5  x.  3
 )  = ; 1 5
 
Theorem5t4e20 10215 5 times 4 equals 20. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 5  x.  4
 )  = ; 2 0
 
Theorem5t5e25 10216 5 times 5 equals 25. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 5  x.  5
 )  = ; 2 5
 
Theorem6t2e12 10217 6 times 2 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 6  x.  2
 )  = ; 1 2
 
Theorem6t3e18 10218 6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 6  x.  3
 )  = ; 1 8
 
Theorem6t4e24 10219 6 times 4 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 6  x.  4
 )  = ; 2 4
 
Theorem6t5e30 10220 6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 6  x.  5
 )  = ; 3 0
 
Theorem6t6e36 10221 6 times 6 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 6  x.  6
 )  = ; 3 6
 
Theorem7t2e14 10222 7 times 2 equals 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  x.  2
 )  = ; 1 4
 
Theorem7t3e21 10223 7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  x.  3
 )  = ; 2 1
 
Theorem7t4e28 10224 7 times 4 equals 28. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  x.  4
 )  = ; 2 8
 
Theorem7t5e35 10225 7 times 5 equals 35. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  x.  5
 )  = ; 3 5
 
Theorem7t6e42 10226 7 times 6 equals 42. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  x.  6
 )  = ; 4 2
 
Theorem7t7e49 10227 7 times 7 equals 49. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  x.  7
 )  = ; 4 9
 
Theorem8t2e16 10228 8 times 2 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  x.  2
 )  = ; 1 6
 
Theorem8t3e24 10229 8 times 3 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  x.  3
 )  = ; 2 4
 
Theorem8t4e32 10230 8 times 4 equals 32. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  x.  4
 )  = ; 3 2
 
Theorem8t5e40 10231 8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  x.  5
 )  = ; 4 0
 
Theorem8t6e48 10232 8 times 6 equals 48. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  x.  6
 )  = ; 4 8
 
Theorem8t7e56 10233 8 times 7 equals 56. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  x.  7
 )  = ; 5 6
 
Theorem8t8e64 10234 8 times 8 equals 64. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  x.  8
 )  = ; 6 4
 
Theorem9t2e18 10235 9 times 2 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  x.  2
 )  = ; 1 8
 
Theorem9t3e27 10236 9 times 3 equals 27. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  x.  3
 )  = ; 2 7
 
Theorem9t4e36 10237 9 times 4 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  x.  4
 )  = ; 3 6
 
Theorem9t5e45 10238 9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  x.  5
 )  = ; 4 5
 
Theorem9t6e54 10239 9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  x.  6
 )  = ; 5 4
 
Theorem9t7e63 10240 9 times 7 equals 63. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  x.  7
 )  = ; 6 3
 
Theorem9t8e72 10241 9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  x.  8
 )  = ; 7 2
 
Theorem9t9e81 10242 9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  x.  9
 )  = ; 8 1
 
Theoremdecbin0 10243 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  A  e.  NN0   =>    |-  ( 4  x.  A )  =  ( 2  x.  ( 2  x.  A ) )
 
Theoremdecbin2 10244 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  A  e.  NN0   =>    |-  ( ( 4  x.  A )  +  2 )  =  ( 2  x.  ( ( 2  x.  A )  +  1 ) )
 
Theoremdecbin3 10245 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  A  e.  NN0   =>    |-  ( ( 4  x.  A )  +  3 )  =  ( ( 2  x.  ( ( 2  x.  A )  +  1 ) )  +  1 )
 
5.4.9  Upper partititions of integers
 
Syntaxcuz 10246 Extend class notation with the upper integer function. Read " ZZ>= `  M " as "the set of integers greater than or equal to  M."
 class  ZZ>=
 
Definitiondf-uz 10247* 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 10248 for its value, uzssz 10263 for its relationship to  ZZ, nnuz 10279 and nn0uz 10278 for its relationships to  NN and  NN0, and eluz1 10250 and eluz2 10252 for its membership relations. (Contributed by NM, 5-Sep-2005.)
 |- 
 ZZ>=  =  ( j  e. 
 ZZ  |->  { k  e.  ZZ  |  j  <_  k }
 )
 
Theoremuzval 10248* The value of the upper integers function. (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  ( N  e.  ZZ  ->  ( ZZ>= `  N )  =  { k  e.  ZZ  |  N  <_  k }
 )
 
Theoremuzf 10249 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 10250 Membership in the set of upper integers starting at  M. (Contributed by NM, 5-Sep-2005.)
 |-  ( M  e.  ZZ  ->  ( N  e.  ( ZZ>=
 `  M )  <->  ( N  e.  ZZ  /\  M  <_  N ) ) )
 
Theoremeluzel2 10251 Implication of membership in a set of upper integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  M  e.  ZZ )
 
Theoremeluz2 10252 Membership in a set of upper integers. We use the fact that a function's value (under our function value definition) is empty outside of its domain to show  M  e.  ZZ. (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  ( N  e.  ( ZZ>=
 `  M )  <->  ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  <_  N )
 )
 
Theoremeluz1i 10253 Membership in a set of upper integers. (Contributed by NM, 5-Sep-2005.)
 |-  M  e.  ZZ   =>    |-  ( N  e.  ( ZZ>= `  M )  <->  ( N  e.  ZZ  /\  M  <_  N ) )
 
Theoremeluzelz 10254 A member of a set of upper integers is an integer. (Contributed by NM, 6-Sep-2005.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  N  e.  ZZ )
 
Theoremeluzelre 10255 A member of a set of upper integers is a real. (Contributed by Mario Carneiro, 31-Aug-2013.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  N  e.  RR )
 
Theoremeluzle 10256 Implication of membership in a set of upper integers. (Contributed by NM, 6-Sep-2005.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  M  <_  N )
 
Theoremeluz 10257 Membership in a set of upper integers. (Contributed by NM, 2-Oct-2005.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  e.  ( ZZ>= `  M )  <->  M 
 <_  N ) )
 
Theoremuzid 10258 Membership of the least member in a set of upper integers. (Contributed by NM, 2-Sep-2005.)
 |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M ) )
 
Theoremuzn0 10259 The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.)
 |-  ( M  e.  ran  ZZ>=  ->  M  =/=  (/) )
 
Theoremuztrn 10260 Transitive law for sets of upper integers. (Contributed by NM, 20-Sep-2005.)
 |-  ( ( M  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  N ) )  ->  M  e.  ( ZZ>= `  N ) )
 
Theoremuztrn2 10261 Transitive law for sets of upper integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
 |-  Z  =  ( ZZ>= `  K )   =>    |-  ( ( N  e.  Z  /\  M  e.  ( ZZ>=
 `  N ) ) 
 ->  M  e.  Z )
 
Theoremuzneg 10262 Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  -u M  e.  ( ZZ>= `  -u N ) )
 
Theoremuzssz 10263 A set of upper 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 10264 Subset relationship for two sets of upper integers. (Contributed by NM, 5-Sep-2005.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( ZZ>= `  N )  C_  ( ZZ>= `  M )
 )
 
Theoremuztric 10265 Totality 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  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  e.  ( ZZ>= `  M )  \/  M  e.  ( ZZ>= `  N ) ) )
 
Theoremuz11 10266 The upper integers function is one-to-one. (Contributed by NM, 12-Dec-2005.)
 |-  ( M  e.  ZZ  ->  ( ( ZZ>= `  M )  =  ( ZZ>= `  N )  <->  M  =  N ) )
 
Theoremeluzp1m1 10267 Membership in the next set of upper integers. (Contributed by NM, 12-Sep-2005.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ( ZZ>=
 `  ( M  +  1 ) ) ) 
 ->  ( N  -  1
 )  e.  ( ZZ>= `  M ) )
 
Theoremeluzp1l 10268 Strict ordering implied by membership in the next set of upper integers. (Contributed by NM, 12-Sep-2005.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ( ZZ>=
 `  ( M  +  1 ) ) ) 
 ->  M  <  N )
 
Theoremeluzp1p1 10269 Membership in the next set of upper integers. (Contributed by NM, 5-Oct-2005.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( N  +  1
 )  e.  ( ZZ>= `  ( M  +  1
 ) ) )
 
Theoremeluzaddi 10270 Membership in a later set of upper integers. (Contributed by Paul Chapman, 22-Nov-2007.)
 |-  M  e.  ZZ   &    |-  K  e.  ZZ   =>    |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( N  +  K )  e.  ( ZZ>= `  ( M  +  K ) ) )
 
Theoremeluzsubi 10271 Membership in an earlier set of upper integers. (Contributed by Paul Chapman, 22-Nov-2007.)
 |-  M  e.  ZZ   &    |-  K  e.  ZZ   =>    |-  ( N  e.  ( ZZ>=
 `  ( M  +  K ) )  ->  ( N  -  K )  e.  ( ZZ>= `  M ) )
 
Theoremeluzadd 10272 Membership in a later set of upper integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( N  e.  ( ZZ>= `  M )  /\  K  e.  ZZ )  ->  ( N  +  K )  e.  ( ZZ>= `  ( M  +  K ) ) )
 
Theoremeluzsub 10273 Membership in an earlier set of upper integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( M  e.  ZZ  /\  K  e.  ZZ  /\  N  e.  ( ZZ>= `  ( M  +  K ) ) )  ->  ( N  -  K )  e.  ( ZZ>= `  M ) )
 
Theoremuzm1 10274 Choices for an element of an upper interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( N  =  M  \/  ( N  -  1
 )  e.  ( ZZ>= `  M ) ) )
 
Theoremuznn0sub 10275 The nonnegative difference of integers is a nonnegative integer. (Contributed by NM, 4-Sep-2005.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( N  -  M )  e.  NN0 )
 
Theoremuzin 10276 Intersection of two upper intervals of integers. (Contributed by Mario Carneiro, 24-Dec-2013.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ZZ>= `  M )  i^i  ( ZZ>= `  N ) )  =  ( ZZ>= `  if ( M  <_  N ,  N ,  M ) ) )
 
Theoremuzp1 10277 Choices for an element of an upper interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( N  =  M  \/  N  e.  ( ZZ>= `  ( M  +  1
 ) ) ) )
 
Theoremnn0uz 10278 Nonnegative integers expressed as a set of upper integers. (Contributed by NM, 2-Sep-2005.)
 |- 
 NN0  =  ( ZZ>= `  0 )
 
Theoremnnuz 10279 Natural numbers expressed as a set of upper integers. (Contributed by NM, 2-Sep-2005.)
 |- 
 NN  =  ( ZZ>= `  1 )
 
Theoremelnnuz 10280 A natural number expressed as a member of a set of upper integers. (Contributed by NM, 6-Jun-2006.)
 |-  ( N  e.  NN  <->  N  e.  ( ZZ>= `  1 )
 )
 
Theoremelnn0uz 10281 A nonnegative integer expressed as a member a set of upper integers. (Contributed by NM, 6-Jun-2006.)
 |-  ( N  e.  NN0  <->  N  e.  ( ZZ>= `  0 )
 )
 
Theoremuznnssnn 10282 The upper integers starting from a natural are a subset of the naturals. (Contributed by Scott Fenton, 29-Jun-2013.)
 |-  ( N  e.  NN  ->  ( ZZ>= `  N )  C_ 
 NN )
 
Theoremraluz 10283* Restricted universal quantification in a set of upper integers. (Contributed by NM, 9-Sep-2005.)
 |-  ( M  e.  ZZ  ->  ( A. n  e.  ( ZZ>= `  M ) ph 
 <-> 
 A. n  e.  ZZ  ( M  <_  n  ->  ph ) ) )
 
Theoremraluz2 10284* Restricted universal quantification in a set of upper integers. (Contributed by NM, 9-Sep-2005.)
 |-  ( A. n  e.  ( ZZ>= `  M ) ph 
 <->  ( M  e.  ZZ  ->  A. n  e.  ZZ  ( M  <_  n  ->  ph ) ) )
 
Theoremrexuz 10285* Restricted existential quantification in a set of upper integers. (Contributed by NM, 9-Sep-2005.)
 |-  ( M  e.  ZZ  ->  ( E. n  e.  ( ZZ>= `  M ) ph 
 <-> 
 E. n  e.  ZZ  ( M  <_  n  /\  ph ) ) )
 
Theoremrexuz2 10286* Restricted existential quantification in a set of upper integers. (Contributed by NM, 9-Sep-2005.)
 |-  ( E. n  e.  ( ZZ>= `  M ) ph 
 <->  ( M  e.  ZZ  /\ 
 E. n  e.  ZZ  ( M  <_  n  /\  ph ) ) )
 
Theorem2rexuz 10287* Double existential quantification in a set of upper integers. (Contributed by NM, 3-Nov-2005.)
 |-  ( E. m E. n  e.  ( ZZ>= `  m ) ph  <->  E. m  e.  ZZ  E. n  e.  ZZ  ( m  <_  n  /\  ph )
 )
 
Theorempeano2uz 10288 Second Peano postulate for a set of upper integers. (Contributed by NM, 7-Sep-2005.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( N  +  1
 )  e.  ( ZZ>= `  M ) )
 
Theorempeano2uzs 10289 Second Peano postulate for a set of upper integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( N  e.  Z  ->  ( N  +  1 )  e.  Z )
 
Theorempeano2uzr 10290 Reversed second Peano axiom for upper integers. (Contributed by NM, 2-Jan-2006.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ( ZZ>=
 `  ( M  +  1 ) ) ) 
 ->  N  e.  ( ZZ>= `  M ) )
 
Theoremuzaddcl 10291 Addition closure law for a set of upper integers. (Contributed by NM, 4-Jun-2006.)
 |-  ( ( N  e.  ( ZZ>= `  M )  /\  K  e.  NN0 )  ->  ( N  +  K )  e.  ( ZZ>= `  M ) )
 
Theoremuzind4 10292* Induction on the set of upper 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 hypothesis. (Contributed by NM, 7-Sep-2005.)
 |-  ( j  =  M  ->  ( ph  <->  ps ) )   &    |-  (
 j  =  k  ->  ( ph  <->  ch ) )   &    |-  (
 j  =  ( k  +  1 )  ->  ( ph  <->  th ) )   &    |-  (
 j  =  N  ->  (
 ph 
 <->  ta ) )   &    |-  ( M  e.  ZZ  ->  ps )   &    |-  ( k  e.  ( ZZ>= `  M )  ->  ( ch  ->  th )
 )   =>    |-  ( N  e.  ( ZZ>=
 `  M )  ->  ta )
 
Theoremuzind4ALT 10293* Induction on the set of upper 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 hypothesis. Either uzind4 10292 or uzind4ALT 10293 may be used; see comment for nnind 9780. (Contributed by NM, 7-Sep-2005.)
 |-  ( M  e.  ZZ  ->  ps )   &    |-  ( k  e.  ( ZZ>= `  M )  ->  ( ch  ->  th )
 )   &    |-  ( j  =  M  ->  ( ph  <->  ps ) )   &    |-  (
 j  =  k  ->  ( ph  <->  ch ) )   &    |-  (
 j  =  ( k  +  1 )  ->  ( ph  <->  th ) )   &    |-  (
 j  =  N  ->  (
 ph 
 <->  ta ) )   =>    |-  ( N  e.  ( ZZ>= `  M )  ->  ta )
 
Theoremuzind4s 10294* Induction on the set of upper integers that starts at an integer  M, using explicit substitution. The hypotheses are the basis and the induction hypothesis. (Contributed by NM, 4-Nov-2005.)
 |-  ( M  e.  ZZ  -> 
 [. M  /  k ]. ph )   &    |-  ( k  e.  ( ZZ>= `  M )  ->  ( ph  ->  [. (
 k  +  1 ) 
 /  k ]. ph )
 )   =>    |-  ( N  e.  ( ZZ>=
 `  M )  ->  [. N  /  k ]. ph )
 
Theoremuzind4s2 10295* Induction on the set of upper integers that starts at an integer  M, using explicit substitution. The hypotheses are the basis and the induction hypothesis. Use this instead of uzind4s 10294 when  j and  k must be distinct in  [. ( k  +  1 )  / 
j ]. ph. (Contributed by NM, 16-Nov-2005.)
 |-  ( M  e.  ZZ  -> 
 [. M  /  j ]. ph )   &    |-  ( k  e.  ( ZZ>= `  M )  ->  ( [. k  /  j ]. ph  ->  [. (
 k  +  1 ) 
 /  j ]. ph )
 )   =>    |-  ( N  e.  ( ZZ>=
 `  M )  ->  [. N  /  j ]. ph )
 
Theoremuzind4i 10296* 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 hypothesis. (Contributed by NM, 4-Sep-2005.)
 |-  M  e.  ZZ   &    |-  (
 j  =  M  ->  (
 ph 
 <->  ps ) )   &    |-  (
 j  =  k  ->  ( ph  <->  ch ) )   &    |-  (
 j  =  ( k  +  1 )  ->  ( ph  <->  th ) )   &    |-  (
 j  =  N  ->  (
 ph 
 <->  ta ) )   &    |-  ps   &    |-  (
 k  e.  ( ZZ>= `  M )  ->  ( ch 
 ->  th ) )   =>    |-  ( N  e.  ( ZZ>= `  M )  ->  ta )
 
Theoremuzwo 10297* Well-ordering principle: any non-empty subset of a set of upper integers has the least element. (Contributed by NM, 8-Oct-2005.)
 |-  ( ( S  C_  ( ZZ>= `  M )  /\  S  =/=  (/) )  ->  E. j  e.  S  A. k  e.  S  j 
 <_  k )
 
TheoremuzwoOLD 10298* Well-ordering principle: any non-empty subset of the upper integers has the least element. (Contributed by NM, 8-Oct-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( S  C_  ( ZZ>= `  M )  /\  -.  S  =  (/) )  ->  E. j  e.  S  A. k  e.  S  j 
 <_  k )
 
Theoremuzwo2 10299* Well-ordering principle: any non-empty subset of upper integers has a unique least element. (Contributed by NM, 8-Oct-2005.)
 |-  ( ( S  C_  ( ZZ>= `  M )  /\  S  =/=  (/) )  ->  E! j  e.  S  A. k  e.  S  j 
 <_  k )
 
Theoremnnwo 10300* Well-ordering principle: any non-empty set of natural numbers has a least element. Theorem I.37 (well-ordering principle) of [Apostol] p. 34. (Contributed by NM, 17-Aug-2001.)
 |-  ( ( A  C_  NN  /\  A  =/=  (/) )  ->  E. x  e.  A  A. y  e.  A  x  <_  y )
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