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Theorem List for Metamath Proof Explorer - 10201-10300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem6t6e36 10201 6 times 6 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 6  x.  6
 )  = ; 3 6
 
Theorem7t2e14 10202 7 times 2 equals 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  x.  2
 )  = ; 1 4
 
Theorem7t3e21 10203 7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  x.  3
 )  = ; 2 1
 
Theorem7t4e28 10204 7 times 4 equals 28. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  x.  4
 )  = ; 2 8
 
Theorem7t5e35 10205 7 times 5 equals 35. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  x.  5
 )  = ; 3 5
 
Theorem7t6e42 10206 7 times 6 equals 42. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  x.  6
 )  = ; 4 2
 
Theorem7t7e49 10207 7 times 7 equals 49. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  x.  7
 )  = ; 4 9
 
Theorem8t2e16 10208 8 times 2 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  x.  2
 )  = ; 1 6
 
Theorem8t3e24 10209 8 times 3 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  x.  3
 )  = ; 2 4
 
Theorem8t4e32 10210 8 times 4 equals 32. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  x.  4
 )  = ; 3 2
 
Theorem8t5e40 10211 8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  x.  5
 )  = ; 4 0
 
Theorem8t6e48 10212 8 times 6 equals 48. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  x.  6
 )  = ; 4 8
 
Theorem8t7e56 10213 8 times 7 equals 56. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  x.  7
 )  = ; 5 6
 
Theorem8t8e64 10214 8 times 8 equals 64. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  x.  8
 )  = ; 6 4
 
Theorem9t2e18 10215 9 times 2 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  x.  2
 )  = ; 1 8
 
Theorem9t3e27 10216 9 times 3 equals 27. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  x.  3
 )  = ; 2 7
 
Theorem9t4e36 10217 9 times 4 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  x.  4
 )  = ; 3 6
 
Theorem9t5e45 10218 9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  x.  5
 )  = ; 4 5
 
Theorem9t6e54 10219 9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  x.  6
 )  = ; 5 4
 
Theorem9t7e63 10220 9 times 7 equals 63. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  x.  7
 )  = ; 6 3
 
Theorem9t8e72 10221 9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  x.  8
 )  = ; 7 2
 
Theorem9t9e81 10222 9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  x.  9
 )  = ; 8 1
 
Theoremdecbin0 10223 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 10224 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 10225 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 10226 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 10227* 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 10228 for its value, uzssz 10243 for its relationship to  ZZ, nnuz 10259 and nn0uz 10258 for its relationships to  NN and  NN0, and eluz1 10230 and eluz2 10232 for its membership relations. (Contributed by NM, 5-Sep-2005.)
 |- 
 ZZ>=  =  ( j  e. 
 ZZ  |->  { k  e.  ZZ  |  j  <_  k }
 )
 
Theoremuzval 10228* 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 10229 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 10230 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 10231 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 10232 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 10233 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 10234 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 10235 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 10236 Implication of membership in a set of upper integers. (Contributed by NM, 6-Sep-2005.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  M  <_  N )
 
Theoremeluz 10237 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 10238 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 10239 The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.)
 |-  ( M  e.  ran  ZZ>=  ->  M  =/=  (/) )
 
Theoremuztrn 10240 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 10241 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 10242 Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  -u M  e.  ( ZZ>= `  -u N ) )
 
Theoremuzssz 10243 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 10244 Subset relationship for two sets of upper integers. (Contributed by NM, 5-Sep-2005.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( ZZ>= `  N )  C_  ( ZZ>= `  M )
 )
 
Theoremuztric 10245 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 10246 The upper integers function is one-to-one. (Contributed by NM, 12-Dec-2005.)
 |-  ( M  e.  ZZ  ->  ( ( ZZ>= `  M )  =  ( ZZ>= `  N )  <->  M  =  N ) )
 
Theoremeluzp1m1 10247 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 10248 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 10249 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 10250 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 10251 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 10252 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 10253 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 10254 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 10255 The nonnegative difference of integers is a nonnegative integer. (Contributed by NM, 4-Sep-2005.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( N  -  M )  e.  NN0 )
 
Theoremuzin 10256 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 10257 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 10258 Nonnegative integers expressed as a set of upper integers. (Contributed by NM, 2-Sep-2005.)
 |- 
 NN0  =  ( ZZ>= `  0 )
 
Theoremnnuz 10259 Natural numbers expressed as a set of upper integers. (Contributed by NM, 2-Sep-2005.)
 |- 
 NN  =  ( ZZ>= `  1 )
 
Theoremelnnuz 10260 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 10261 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 10262 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 10263* 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 10264* 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 10265* 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 10266* 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 10267* 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 10268 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 10269 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 10270 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 10271 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 10272* 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 10273* Alternate version of uzind4 10272 with different hypothesis order for easier use with the Metamath Proof Assistant, since "assign last" will assign the substitutions first. (This may or may not be kept permanenently, or it may replace uzind4 10272- I haven't decided yet. --NM) (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 10274* 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 10275* 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 10274 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 10276* 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 10277* Well-ordering principle: any non-empty subset of a set of upper integers has a least element. (Contributed by NM, 8-Oct-2005.)
 |-  ( ( S  C_  ( ZZ>= `  M )  /\  S  =/=  (/) )  ->  E. j  e.  S  A. k  e.  S  j 
 <_  k )
 
TheoremuzwoOLD 10278* Well-ordering principle: any non-empty subset of the upper integers has a 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 10279* 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 10280* 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 )
 
Theoremnnwof 10281* Well-ordering principle: any non-empty set of natural numbers has a least element. This version allows  x and  y to be present in  A as long as they are effectively not free. (Contributed by NM, 17-Aug-2001.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x A   &    |-  F/_ y A   =>    |-  ( ( A 
 C_  NN  /\  A  =/=  (/) )  ->  E. x  e.  A  A. y  e.  A  x  <_  y
 )
 
Theoremnnwos 10282* Well-ordering principle: any non-empty set of natural numbers has a least element (schema form). (Contributed by NM, 17-Aug-2001.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( E. x  e.  NN  ph  ->  E. x  e.  NN  ( ph  /\  A. y  e.  NN  ( ps  ->  x  <_  y
 ) ) )
 
Theoremindstr 10283* Strong Mathematical Induction for positive integers (inference schema). (Contributed by NM, 17-Aug-2001.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  ( x  e.  NN  ->  (
 A. y  e.  NN  ( y  <  x  ->  ps )  ->  ph )
 )   =>    |-  ( x  e.  NN  -> 
 ph )
 
Theoremeluznn0 10284 Membership in a nonegative set of upper integers implies membership in  NN0. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N ) )  ->  M  e.  NN0 )
 
Theoremeluz2b1 10285 Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.)
 |-  ( N  e.  ( ZZ>=
 `  2 )  <->  ( N  e.  ZZ  /\  1  <  N ) )
 
Theoremeluz2b2 10286 Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.)
 |-  ( N  e.  ( ZZ>=
 `  2 )  <->  ( N  e.  NN  /\  1  <  N ) )
 
Theoremeluz2b3 10287 Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.)
 |-  ( N  e.  ( ZZ>=
 `  2 )  <->  ( N  e.  NN  /\  N  =/=  1
 ) )
 
Theoremuz2m1nn 10288 One less than an integer greater than or equal to 2 is a positive integer. (Contributed by Paul Chapman, 17-Nov-2012.)
 |-  ( N  e.  ( ZZ>=
 `  2 )  ->  ( N  -  1
 )  e.  NN )
 
Theorem1nuz2 10289 1 is not in  ( ZZ>= `  2
). (Contributed by Paul Chapman, 21-Nov-2012.)
 |- 
 -.  1  e.  ( ZZ>=
 `  2 )
 
Theoremelnn1uz2 10290 A positive integer is either 1 or greater than or equal to 2. (Contributed by Paul Chapman, 17-Nov-2012.)
 |-  ( N  e.  NN  <->  ( N  =  1  \/  N  e.  ( ZZ>= `  2 ) ) )
 
Theoremuz2mulcl 10291 Closure of multiplication of integers greater than or equal to 2. (Contributed by Paul Chapman, 26-Oct-2012.)
 |-  ( ( M  e.  ( ZZ>= `  2 )  /\  N  e.  ( ZZ>= `  2 ) )  ->  ( M  x.  N )  e.  ( ZZ>= `  2 ) )
 
Theoremindstr2 10292* 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 hypothesis. (Contributed by Paul Chapman, 21-Nov-2012.)
 |-  ( x  =  1 
 ->  ( ph  <->  ch ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  ch   &    |-  ( x  e.  ( ZZ>= `  2 )  ->  ( A. y  e.  NN  (
 y  <  x  ->  ps )  ->  ph ) )   =>    |-  ( x  e.  NN  -> 
 ph )
 
Theoremuzinfmi 10293 Extract the lower bound of a set of upper integers as its infimum. Note that the " `'  < " argument turns supremum into infimum (for which we do not currently have a separate notation). (Contributed by NM, 7-Oct-2005.)
 |-  M  e.  ZZ   =>    |-  sup ( (
 ZZ>= `  M ) ,  RR ,  `'  <  )  =  M
 
Theoremnninfm 10294 The infimum of the set of natural numbers is one. Note that " `'  < " turns sup into inf. (Contributed by NM, 16-Jun-2005.)
 |- 
 sup ( NN ,  RR ,  `'  <  )  =  1
 
Theoremnn0infm 10295 The infimum of the set of nonnegative integers is zero. Note that " `'  < " turns sup into inf. (Contributed by NM, 16-Jun-2005.)
 |- 
 sup ( NN0 ,  RR ,  `'  <  )  =  0
 
Theoreminfmssuzle 10296 The infimum of a subset of a set of upper integers is less than or equal to all members of the subset. Note that the " `'  < " argument turns supremum into infimum (for which we do not currently have a separate notation). (Contributed by NM, 11-Oct-2005.)
 |-  ( ( S  C_  ( ZZ>= `  M )  /\  A  e.  S ) 
 ->  sup ( S ,  RR ,  `'  <  )  <_  A )
 
Theoreminfmssuzcl 10297 The infimum of a subset of a set of upper integers belongs to the subset. (Contributed by NM, 11-Oct-2005.)
 |-  ( ( S  C_  ( ZZ>= `  M )  /\  S  =/=  (/) )  ->  sup ( S ,  RR ,  `'  <  )  e.  S )
 
Theoremublbneg 10298* The image under negation of a bounded-above set of reals is bounded below. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( E. x  e. 
 RR  A. y  e.  A  y  <_  x  ->  E. x  e.  RR  A. y  e. 
 { z  e.  RR  |  -u z  e.  A } x  <_  y )
 
Theoremeqreznegel 10299* Two ways to express the image under negation of a set of integers. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( A  C_  ZZ  ->  { z  e.  RR  |  -u z  e.  A }  =  { z  e.  ZZ  |  -u z  e.  A } )
 
Theoremnegn0 10300* The image under negation of a nonempty set of reals is nonempty. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( A  C_  RR  /\  A  =/=  (/) )  ->  { z  e.  RR  |  -u z  e.  A }  =/=  (/) )
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