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Theorem List for Metamath Proof Explorer - 10101-10200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdecaddci 10101 Add two numerals  M and  N (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  N  e.  NN0   &    |-  M  = ; A B   &    |-  ( A  +  1 )  =  D   &    |-  C  e.  NN0   &    |-  ( B  +  N )  = ; 1 C   =>    |-  ( M  +  N )  = ; D C
 
Theoremdecaddci2 10102 Add two numerals  M and  N (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  N  e.  NN0   &    |-  M  = ; A B   &    |-  ( A  +  1 )  =  D   &    |-  ( B  +  N )  =  10   =>    |-  ( M  +  N )  = ; D 0
 
Theoremdecmul1c 10103 The product of a numeral with a number. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  P  e.  NN0   &    |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  N  = ; A B   &    |-  D  e.  NN0   &    |-  E  e.  NN0   &    |-  ( ( A  x.  P )  +  E )  =  C   &    |-  ( B  x.  P )  = ; E D   =>    |-  ( N  x.  P )  = ; C D
 
Theoremdecmul2c 10104 The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  P  e.  NN0   &    |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  N  = ; A B   &    |-  D  e.  NN0   &    |-  E  e.  NN0   &    |-  ( ( P  x.  A )  +  E )  =  C   &    |-  ( P  x.  B )  = ; E D   =>    |-  ( P  x.  N )  = ; C D
 
Theorem6p5lem 10105 Lemma for 6p5e11 10106 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  A  e.  NN0   &    |-  D  e.  NN0   &    |-  E  e.  NN0   &    |-  B  =  ( D  +  1 )   &    |-  C  =  ( E  +  1 )   &    |-  ( A  +  D )  = ; 1 E   =>    |-  ( A  +  B )  = ; 1 C
 
Theorem6p5e11 10106 6 + 5 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 6  +  5 )  = ; 1 1
 
Theorem6p6e12 10107 6 + 6 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 6  +  6 )  = ; 1 2
 
Theorem7p4e11 10108 7 + 4 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  +  4 )  = ; 1 1
 
Theorem7p5e12 10109 7 + 5 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  +  5 )  = ; 1 2
 
Theorem7p6e13 10110 7 + 6 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  +  6 )  = ; 1 3
 
Theorem7p7e14 10111 7 + 7 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  +  7 )  = ; 1 4
 
Theorem8p3e11 10112 8 + 3 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  +  3 )  = ; 1 1
 
Theorem8p4e12 10113 8 + 4 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  +  4 )  = ; 1 2
 
Theorem8p5e13 10114 8 + 5 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  +  5 )  = ; 1 3
 
Theorem8p6e14 10115 8 + 6 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  +  6 )  = ; 1 4
 
Theorem8p7e15 10116 8 + 7 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  +  7 )  = ; 1 5
 
Theorem8p8e16 10117 8 + 8 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  +  8 )  = ; 1 6
 
Theorem9p2e11 10118 9 + 2 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  2 )  = ; 1 1
 
Theorem9p3e12 10119 9 + 3 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  3 )  = ; 1 2
 
Theorem9p4e13 10120 9 + 4 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  4 )  = ; 1 3
 
Theorem9p5e14 10121 9 + 5 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  5 )  = ; 1 4
 
Theorem9p6e15 10122 9 + 6 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  6 )  = ; 1 5
 
Theorem9p7e16 10123 9 + 7 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  7 )  = ; 1 6
 
Theorem9p8e17 10124 9 + 8 = 17. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  8 )  = ; 1 7
 
Theorem9p9e18 10125 9 + 9 = 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  9 )  = ; 1 8
 
Theorem10p10e20 10126 10 + 10 = 20. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 10  +  10 )  = ; 2 0
 
Theorem4t3lem 10127 Lemma for 4t3e12 10128 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 10128 4 times 3 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 4  x.  3
 )  = ; 1 2
 
Theorem4t4e16 10129 4 times 4 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 4  x.  4
 )  = ; 1 6
 
Theorem5t3e15 10130 5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 5  x.  3
 )  = ; 1 5
 
Theorem5t4e20 10131 5 times 4 equals 20. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 5  x.  4
 )  = ; 2 0
 
Theorem5t5e25 10132 5 times 5 equals 25. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 5  x.  5
 )  = ; 2 5
 
Theorem6t2e12 10133 6 times 2 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 6  x.  2
 )  = ; 1 2
 
Theorem6t3e18 10134 6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 6  x.  3
 )  = ; 1 8
 
Theorem6t4e24 10135 6 times 4 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 6  x.  4
 )  = ; 2 4
 
Theorem6t5e30 10136 6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 6  x.  5
 )  = ; 3 0
 
Theorem6t6e36 10137 6 times 6 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 6  x.  6
 )  = ; 3 6
 
Theorem7t2e14 10138 7 times 2 equals 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  x.  2
 )  = ; 1 4
 
Theorem7t3e21 10139 7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  x.  3
 )  = ; 2 1
 
Theorem7t4e28 10140 7 times 4 equals 28. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  x.  4
 )  = ; 2 8
 
Theorem7t5e35 10141 7 times 5 equals 35. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  x.  5
 )  = ; 3 5
 
Theorem7t6e42 10142 7 times 6 equals 42. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  x.  6
 )  = ; 4 2
 
Theorem7t7e49 10143 7 times 7 equals 49. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  x.  7
 )  = ; 4 9
 
Theorem8t2e16 10144 8 times 2 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  x.  2
 )  = ; 1 6
 
Theorem8t3e24 10145 8 times 3 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  x.  3
 )  = ; 2 4
 
Theorem8t4e32 10146 8 times 4 equals 32. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  x.  4
 )  = ; 3 2
 
Theorem8t5e40 10147 8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  x.  5
 )  = ; 4 0
 
Theorem8t6e48 10148 8 times 6 equals 48. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  x.  6
 )  = ; 4 8
 
Theorem8t7e56 10149 8 times 7 equals 56. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  x.  7
 )  = ; 5 6
 
Theorem8t8e64 10150 8 times 8 equals 64. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  x.  8
 )  = ; 6 4
 
Theorem9t2e18 10151 9 times 2 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  x.  2
 )  = ; 1 8
 
Theorem9t3e27 10152 9 times 3 equals 27. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  x.  3
 )  = ; 2 7
 
Theorem9t4e36 10153 9 times 4 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  x.  4
 )  = ; 3 6
 
Theorem9t5e45 10154 9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  x.  5
 )  = ; 4 5
 
Theorem9t6e54 10155 9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  x.  6
 )  = ; 5 4
 
Theorem9t7e63 10156 9 times 7 equals 63. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  x.  7
 )  = ; 6 3
 
Theorem9t8e72 10157 9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  x.  8
 )  = ; 7 2
 
Theorem9t9e81 10158 9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  x.  9
 )  = ; 8 1
 
Theoremdecbin0 10159 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 10160 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 10161 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 10162 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 10163* 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 10164 for its value, uzssz 10179 for its relationship to  ZZ, nnuz 10195 and nn0uz 10194 for its relationships to  NN and  NN0, and eluz1 10166 and eluz2 10168 for its membership relations. (Contributed by NM, 5-Sep-2005.)
 |- 
 ZZ>=  =  ( j  e. 
 ZZ  |->  { k  e.  ZZ  |  j  <_  k }
 )
 
Theoremuzval 10164* 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 10165 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 10166 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 10167 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 10168 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 10169 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 10170 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 10171 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 10172 Implication of membership in a set of upper integers. (Contributed by NM, 6-Sep-2005.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  M  <_  N )
 
Theoremeluz 10173 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 10174 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 10175 The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.)
 |-  ( M  e.  ran  ZZ>=  ->  M  =/=  (/) )
 
Theoremuztrn 10176 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 10177 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 10178 Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  -u M  e.  ( ZZ>= `  -u N ) )
 
Theoremuzssz 10179 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 10180 Subset relationship for two sets of upper integers. (Contributed by NM, 5-Sep-2005.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( ZZ>= `  N )  C_  ( ZZ>= `  M )
 )
 
Theoremuztric 10181 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 10182 The upper integers function is one-to-one. (Contributed by NM, 12-Dec-2005.)
 |-  ( M  e.  ZZ  ->  ( ( ZZ>= `  M )  =  ( ZZ>= `  N )  <->  M  =  N ) )
 
Theoremeluzp1m1 10183 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 10184 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 10185 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 10186 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 10187 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 10188 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 10189 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 10190 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 10191 The nonnegative difference of integers is a nonnegative integer. (Contributed by NM, 4-Sep-2005.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( N  -  M )  e.  NN0 )
 
Theoremuzin 10192 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 10193 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 10194 Nonnegative integers expressed as a set of upper integers. (Contributed by NM, 2-Sep-2005.)
 |- 
 NN0  =  ( ZZ>= `  0 )
 
Theoremnnuz 10195 Natural numbers expressed as a set of upper integers. (Contributed by NM, 2-Sep-2005.)
 |- 
 NN  =  ( ZZ>= `  1 )
 
Theoremelnnuz 10196 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 10197 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 10198 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 10199* 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 10200* 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 ) ) )
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