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Theorem List for Intuitionistic Logic Explorer - 9101-9200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnn0le2is012 9101 A nonnegative integer which is less than or equal to 2 is either 0 or 1 or 2. (Contributed by AV, 16-Mar-2019.)
 |-  ( ( N  e.  NN0  /\  N  <_  2 )  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) )
 
Theoremnn0lem1lt 9102 Nonnegative integer ordering relation. (Contributed by NM, 21-Jun-2005.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M  <_  N  <->  ( M  -  1 )  <  N ) )
 
Theoremnnlem1lt 9103 Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
 |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  <_  N  <-> 
 ( M  -  1
 )  <  N )
 )
 
Theoremnnltlem1 9104 Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
 |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  <  N  <->  M  <_  ( N  -  1 ) ) )
 
Theoremnnm1ge0 9105 A positive integer decreased by 1 is greater than or equal to 0. (Contributed by AV, 30-Oct-2018.)
 |-  ( N  e.  NN  ->  0  <_  ( N  -  1 ) )
 
Theoremnn0ge0div 9106 Division of a nonnegative integer by a positive number is not negative. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
 |-  ( ( K  e.  NN0  /\  L  e.  NN )  ->  0  <_  ( K  /  L ) )
 
Theoremzdiv 9107* Two ways to express " M divides  N. (Contributed by NM, 3-Oct-2008.)
 |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( E. k  e.  ZZ  ( M  x.  k )  =  N  <->  ( N  /  M )  e.  ZZ ) )
 
Theoremzdivadd 9108 Property of divisibility: if  D divides  A and  B then it divides  A  +  B. (Contributed by NM, 3-Oct-2008.)
 |-  ( ( ( D  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  /  D )  e.  ZZ  /\  ( B  /  D )  e. 
 ZZ ) )  ->  ( ( A  +  B )  /  D )  e.  ZZ )
 
Theoremzdivmul 9109 Property of divisibility: if  D divides  A then it divides  B  x.  A. (Contributed by NM, 3-Oct-2008.)
 |-  ( ( ( D  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A 
 /  D )  e. 
 ZZ )  ->  (
 ( B  x.  A )  /  D )  e. 
 ZZ )
 
Theoremzextle 9110* An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\ 
 A. k  e.  ZZ  ( k  <_  M  <->  k  <_  N ) )  ->  M  =  N )
 
Theoremzextlt 9111* An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\ 
 A. k  e.  ZZ  ( k  <  M  <->  k  <  N ) )  ->  M  =  N )
 
Theoremrecnz 9112 The reciprocal of a number greater than 1 is not an integer. (Contributed by NM, 3-May-2005.)
 |-  ( ( A  e.  RR  /\  1  <  A )  ->  -.  ( 1  /  A )  e.  ZZ )
 
Theorembtwnnz 9113 A number between an integer and its successor is not an integer. (Contributed by NM, 3-May-2005.)
 |-  ( ( A  e.  ZZ  /\  A  <  B  /\  B  <  ( A  +  1 ) ) 
 ->  -.  B  e.  ZZ )
 
Theoremgtndiv 9114 A larger number does not divide a smaller positive integer. (Contributed by NM, 3-May-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A ) 
 ->  -.  ( B  /  A )  e.  ZZ )
 
Theoremhalfnz 9115 One-half is not an integer. (Contributed by NM, 31-Jul-2004.)
 |- 
 -.  ( 1  / 
 2 )  e.  ZZ
 
Theorem3halfnz 9116 Three halves is not an integer. (Contributed by AV, 2-Jun-2020.)
 |- 
 -.  ( 3  / 
 2 )  e.  ZZ
 
Theoremsuprzclex 9117* The supremum of a set of integers is an element of the set. (Contributed by Jim Kingdon, 20-Dec-2021.)
 |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  < 
 y  /\  A. y  e. 
 RR  ( y  < 
 x  ->  E. z  e.  A  y  <  z
 ) ) )   &    |-  ( ph  ->  A  C_  ZZ )   =>    |-  ( ph  ->  sup ( A ,  RR ,  <  )  e.  A )
 
Theoremprime 9118* Two ways to express " A is a prime number (or 1)." (Contributed by NM, 4-May-2005.)
 |-  ( A  e.  NN  ->  ( A. x  e. 
 NN  ( ( A 
 /  x )  e. 
 NN  ->  ( x  =  1  \/  x  =  A ) )  <->  A. x  e.  NN  ( ( 1  < 
 x  /\  x  <_  A 
 /\  ( A  /  x )  e.  NN )  ->  x  =  A ) ) )
 
Theoremmsqznn 9119 The square of a nonzero integer is a positive integer. (Contributed by NM, 2-Aug-2004.)
 |-  ( ( A  e.  ZZ  /\  A  =/=  0
 )  ->  ( A  x.  A )  e.  NN )
 
Theoremzneo 9120 No even integer equals an odd integer (i.e. no integer can be both even and odd). Exercise 10(a) of [Apostol] p. 28. (Contributed by NM, 31-Jul-2004.) (Proof shortened by Mario Carneiro, 18-May-2014.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( 2  x.  A )  =/=  (
 ( 2  x.  B )  +  1 )
 )
 
Theoremnneoor 9121 A positive integer is even or odd. (Contributed by Jim Kingdon, 15-Mar-2020.)
 |-  ( N  e.  NN  ->  ( ( N  / 
 2 )  e.  NN  \/  ( ( N  +  1 )  /  2
 )  e.  NN )
 )
 
Theoremnneo 9122 A positive integer is even or odd but not both. (Contributed by NM, 1-Jan-2006.) (Proof shortened by Mario Carneiro, 18-May-2014.)
 |-  ( N  e.  NN  ->  ( ( N  / 
 2 )  e.  NN  <->  -.  ( ( N  +  1 )  /  2
 )  e.  NN )
 )
 
Theoremnneoi 9123 A positive integer is even or odd but not both. (Contributed by NM, 20-Aug-2001.)
 |-  N  e.  NN   =>    |-  ( ( N 
 /  2 )  e. 
 NN 
 <->  -.  ( ( N  +  1 )  / 
 2 )  e.  NN )
 
Theoremzeo 9124 An integer is even or odd. (Contributed by NM, 1-Jan-2006.)
 |-  ( N  e.  ZZ  ->  ( ( N  / 
 2 )  e.  ZZ  \/  ( ( N  +  1 )  /  2
 )  e.  ZZ )
 )
 
Theoremzeo2 9125 An integer is even or odd but not both. (Contributed by Mario Carneiro, 12-Sep-2015.)
 |-  ( N  e.  ZZ  ->  ( ( N  / 
 2 )  e.  ZZ  <->  -.  ( ( N  +  1 )  /  2
 )  e.  ZZ )
 )
 
Theorempeano2uz2 9126* Second Peano postulate for upper integers. (Contributed by NM, 3-Oct-2004.)
 |-  ( ( A  e.  ZZ  /\  B  e.  { x  e.  ZZ  |  A  <_  x } )  ->  ( B  +  1
 )  e.  { x  e.  ZZ  |  A  <_  x } )
 
Theorempeano5uzti 9127* Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 25-Jul-2013.)
 |-  ( N  e.  ZZ  ->  ( ( N  e.  A  /\  A. x  e.  A  ( x  +  1 )  e.  A )  ->  { k  e. 
 ZZ  |  N  <_  k }  C_  A )
 )
 
Theorempeano5uzi 9128* Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 3-May-2014.)
 |-  N  e.  ZZ   =>    |-  ( ( N  e.  A  /\  A. x  e.  A  ( x  +  1 )  e.  A )  ->  { k  e.  ZZ  |  N  <_  k }  C_  A )
 
Theoremdfuzi 9129* An expression for the upper integers that start at  N that is analogous to dfnn2 8690 for positive integers. (Contributed by NM, 6-Jul-2005.) (Proof shortened by Mario Carneiro, 3-May-2014.)
 |-  N  e.  ZZ   =>    |-  { z  e. 
 ZZ  |  N  <_  z }  =  |^| { x  |  ( N  e.  x  /\  A. y  e.  x  ( y  +  1
 )  e.  x ) }
 
Theoremuzind 9130* Induction on the upper integers that start at  M. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by NM, 5-Jul-2005.)
 |-  ( j  =  M  ->  ( ph  <->  ps ) )   &    |-  (
 j  =  k  ->  ( ph  <->  ch ) )   &    |-  (
 j  =  ( k  +  1 )  ->  ( ph  <->  th ) )   &    |-  (
 j  =  N  ->  (
 ph 
 <->  ta ) )   &    |-  ( M  e.  ZZ  ->  ps )   &    |-  ( ( M  e.  ZZ  /\  k  e.  ZZ  /\  M  <_  k )  ->  ( ch  ->  th ) )   =>    |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  <_  N )  ->  ta )
 
Theoremuzind2 9131* Induction on the upper integers that start after an integer  M. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by NM, 25-Jul-2005.)
 |-  ( j  =  ( M  +  1 ) 
 ->  ( ph  <->  ps ) )   &    |-  (
 j  =  k  ->  ( ph  <->  ch ) )   &    |-  (
 j  =  ( k  +  1 )  ->  ( ph  <->  th ) )   &    |-  (
 j  =  N  ->  (
 ph 
 <->  ta ) )   &    |-  ( M  e.  ZZ  ->  ps )   &    |-  ( ( M  e.  ZZ  /\  k  e.  ZZ  /\  M  <  k )  ->  ( ch  ->  th ) )   =>    |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  <  N )  ->  ta )
 
Theoremuzind3 9132* Induction on the upper integers that start 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, 26-Jul-2005.)
 |-  ( j  =  M  ->  ( ph  <->  ps ) )   &    |-  (
 j  =  m  ->  ( ph  <->  ch ) )   &    |-  (
 j  =  ( m  +  1 )  ->  ( ph  <->  th ) )   &    |-  (
 j  =  N  ->  (
 ph 
 <->  ta ) )   &    |-  ( M  e.  ZZ  ->  ps )   &    |-  ( ( M  e.  ZZ  /\  m  e.  { k  e.  ZZ  |  M  <_  k }
 )  ->  ( ch  ->  th ) )   =>    |-  ( ( M  e.  ZZ  /\  N  e.  { k  e.  ZZ  |  M  <_  k }
 )  ->  ta )
 
Theoremnn0ind 9133* Principle of Mathematical Induction (inference schema) on nonnegative integers. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by NM, 13-May-2004.)
 |-  ( x  =  0 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  +  1 )  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ps   &    |-  (
 y  e.  NN0  ->  ( ch  ->  th )
 )   =>    |-  ( A  e.  NN0  ->  ta )
 
Theoremfzind 9134* Induction on the integers from  M to  N inclusive . The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  ( x  =  M  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  +  1 )  ->  ( ph  <->  th ) )   &    |-  ( x  =  K  ->  (
 ph 
 <->  ta ) )   &    |-  (
 ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  <_  N )  ->  ps )   &    |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( y  e.  ZZ  /\  M  <_  y  /\  y  <  N ) ) 
 ->  ( ch  ->  th )
 )   =>    |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  M  <_  K  /\  K  <_  N ) )  ->  ta )
 
Theoremfnn0ind 9135* Induction on the integers from  0 to  N inclusive . The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  ( x  =  0 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  +  1 )  ->  ( ph  <->  th ) )   &    |-  ( x  =  K  ->  (
 ph 
 <->  ta ) )   &    |-  ( N  e.  NN0  ->  ps )   &    |-  (
 ( N  e.  NN0  /\  y  e.  NN0  /\  y  <  N )  ->  ( ch  ->  th ) )   =>    |-  ( ( N  e.  NN0  /\  K  e.  NN0  /\  K  <_  N )  ->  ta )
 
Theoremnn0ind-raph 9136* Principle of Mathematical Induction (inference schema) on nonnegative integers. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. Raph Levien remarks: "This seems a bit painful. I wonder if an explicit substitution version would be easier." (Contributed by Raph Levien, 10-Apr-2004.)
 |-  ( x  =  0 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  +  1 )  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ps   &    |-  (
 y  e.  NN0  ->  ( ch  ->  th )
 )   =>    |-  ( A  e.  NN0  ->  ta )
 
Theoremzindd 9137* Principle of Mathematical Induction on all integers, deduction version. The first five hypotheses give the substitutions; the last three are the basis, the induction, and the extension to negative numbers. (Contributed by Paul Chapman, 17-Apr-2009.) (Proof shortened by Mario Carneiro, 4-Jan-2017.)
 |-  ( x  =  0 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  +  1 )  ->  ( ph  <->  ta ) )   &    |-  ( x  =  -u y  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  et ) )   &    |-  ( ze  ->  ps )   &    |-  ( ze  ->  ( y  e.  NN0  ->  ( ch  ->  ta )
 ) )   &    |-  ( ze  ->  ( y  e.  NN  ->  ( ch  ->  th )
 ) )   =>    |-  ( ze  ->  ( A  e.  ZZ  ->  et ) )
 
Theorembtwnz 9138* Any real number can be sandwiched between two integers. Exercise 2 of [Apostol] p. 28. (Contributed by NM, 10-Nov-2004.)
 |-  ( A  e.  RR  ->  ( E. x  e. 
 ZZ  x  <  A  /\  E. y  e.  ZZ  A  <  y ) )
 
Theoremnn0zd 9139 A positive integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  NN0 )   =>    |-  ( ph  ->  A  e.  ZZ )
 
Theoremnnzd 9140 A nonnegative integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  e.  ZZ )
 
Theoremzred 9141 An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  ZZ )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremzcnd 9142 An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  ZZ )   =>    |-  ( ph  ->  A  e.  CC )
 
Theoremznegcld 9143 Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  ZZ )   =>    |-  ( ph  ->  -u A  e.  ZZ )
 
Theorempeano2zd 9144 Deduction from second Peano postulate generalized to integers. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  ZZ )   =>    |-  ( ph  ->  ( A  +  1 )  e.  ZZ )
 
Theoremzaddcld 9145 Closure of addition of integers. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   =>    |-  ( ph  ->  ( A  +  B )  e.  ZZ )
 
Theoremzsubcld 9146 Closure of subtraction of integers. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   =>    |-  ( ph  ->  ( A  -  B )  e.  ZZ )
 
Theoremzmulcld 9147 Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   =>    |-  ( ph  ->  ( A  x.  B )  e.  ZZ )
 
Theoremzadd2cl 9148 Increasing an integer by 2 results in an integer. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
 |-  ( N  e.  ZZ  ->  ( N  +  2 )  e.  ZZ )
 
Theorembtwnapz 9149 A number between an integer and its successor is apart from any integer. (Contributed by Jim Kingdon, 6-Jan-2023.)
 |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  B  <  ( A  +  1 ) )   =>    |-  ( ph  ->  B #  C )
 
4.4.10  Decimal arithmetic
 
Syntaxcdc 9150 Constant used for decimal constructor.
 class ; A B
 
Definitiondf-dec 9151 Define the "decimal constructor", which is used to build up "decimal integers" or "numeric terms" in base  1
0. For example,  (;;; 1 0 0 0  + ;;; 2 0 0 0 )  = ;;; 3 0 0 0 1kp2ke3k 12863. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 1-Aug-2021.)
 |- ; A B  =  ( ( ( 9  +  1 )  x.  A )  +  B )
 
Theorem9p1e10 9152 9 + 1 = 10. (Contributed by Mario Carneiro, 18-Apr-2015.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 1-Aug-2021.)
 |-  ( 9  +  1 )  = ; 1 0
 
Theoremdfdec10 9153 Version of the definition of the "decimal constructor" using ; 1 0 instead of the symbol 10. Of course, this statement cannot be used as definition, because it uses the "decimal constructor". (Contributed by AV, 1-Aug-2021.)
 |- ; A B  =  ( (; 1 0  x.  A )  +  B )
 
Theoremdeceq1 9154 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  ( A  =  B  -> ; A C  = ; B C )
 
Theoremdeceq2 9155 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  ( A  =  B  -> ; C A  = ; C B )
 
Theoremdeceq1i 9156 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  =  B   =>    |- ; A C  = ; B C
 
Theoremdeceq2i 9157 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  =  B   =>    |- ; C A  = ; C B
 
Theoremdeceq12i 9158 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  =  B   &    |-  C  =  D   =>    |- ; A C  = ; B D
 
Theoremnumnncl 9159 Closure for a numeral (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  T  e.  NN0   &    |-  A  e.  NN0   &    |-  B  e.  NN   =>    |-  ( ( T  x.  A )  +  B )  e.  NN
 
Theoremnum0u 9160 Add a zero in the units place. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  T  e.  NN0   &    |-  A  e.  NN0   =>    |-  ( T  x.  A )  =  ( ( T  x.  A )  +  0 )
 
Theoremnum0h 9161 Add a zero in the higher places. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  T  e.  NN0   &    |-  A  e.  NN0   =>    |-  A  =  ( ( T  x.  0 )  +  A )
 
Theoremnumcl 9162 Closure for a decimal integer (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  T  e.  NN0   &    |-  A  e.  NN0   &    |-  B  e.  NN0   =>    |-  ( ( T  x.  A )  +  B )  e.  NN0
 
Theoremnumsuc 9163 The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  T  e.  NN0   &    |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  ( B  +  1 )  =  C   &    |-  N  =  ( ( T  x.  A )  +  B )   =>    |-  ( N  +  1 )  =  ( ( T  x.  A )  +  C )
 
Theoremdeccl 9164 Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   =>    |- ; A B  e.  NN0
 
Theorem10nn 9165 10 is a positive integer. (Contributed by NM, 8-Nov-2012.) (Revised by AV, 6-Sep-2021.)
 |- ; 1
 0  e.  NN
 
Theorem10pos 9166 The number 10 is positive. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
 |-  0  < ; 1 0
 
Theorem10nn0 9167 10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |- ; 1
 0  e.  NN0
 
Theorem10re 9168 The number 10 is real. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
 |- ; 1
 0  e.  RR
 
Theoremdecnncl 9169 Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  A  e.  NN0   &    |-  B  e.  NN   =>    |- ; A B  e.  NN
 
Theoremdec0u 9170 Add a zero in the units place. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  A  e.  NN0   =>    |-  (; 1 0  x.  A )  = ; A 0
 
Theoremdec0h 9171 Add a zero in the higher places. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  A  e.  NN0   =>    |-  A  = ; 0 A
 
Theoremnumnncl2 9172 Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  T  e.  NN   &    |-  A  e.  NN   =>    |-  ( ( T  x.  A )  +  0
 )  e.  NN
 
Theoremdecnncl2 9173 Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  A  e.  NN   =>    |- ; A 0  e.  NN
 
Theoremnumlt 9174 Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  T  e.  NN   &    |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  C  e.  NN   &    |-  B  <  C   =>    |-  ( ( T  x.  A )  +  B )  <  ( ( T  x.  A )  +  C )
 
Theoremnumltc 9175 Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  T  e.  NN   &    |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  C  e.  NN0   &    |-  D  e.  NN0   &    |-  C  <  T   &    |-  A  <  B   =>    |-  ( ( T  x.  A )  +  C )  <  ( ( T  x.  B )  +  D )
 
Theoremle9lt10 9176 A "decimal digit" (i.e. a nonnegative integer less than or equal to 9) is less then 10. (Contributed by AV, 8-Sep-2021.)
 |-  A  e.  NN0   &    |-  A  <_  9   =>    |-  A  < ; 1 0
 
Theoremdeclt 9177 Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  C  e.  NN   &    |-  B  <  C   =>    |- ; A B  < ; A C
 
Theoremdecltc 9178 Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  C  e.  NN0   &    |-  D  e.  NN0   &    |-  C  < ; 1 0   &    |-  A  <  B   =>    |- ; A C  < ; B D
 
Theoremdeclth 9179 Comparing two decimal integers (unequal higher places). (Contributed by AV, 8-Sep-2021.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  C  e.  NN0   &    |-  D  e.  NN0   &    |-  C  <_  9   &    |-  A  <  B   =>    |- ; A C  < ; B D
 
Theoremdecsuc 9180 The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  ( B  +  1 )  =  C   &    |-  N  = ; A B   =>    |-  ( N  +  1 )  = ; A C
 
Theorem3declth 9181 Comparing two decimal integers with three "digits" (unequal higher places). (Contributed by AV, 8-Sep-2021.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  C  e.  NN0   &    |-  D  e.  NN0   &    |-  E  e.  NN0   &    |-  F  e.  NN0   &    |-  A  <  B   &    |-  C  <_  9   &    |-  E  <_  9   =>    |- ;; A C E  < ;; B D F
 
Theorem3decltc 9182 Comparing two decimal integers with three "digits" (unequal higher places). (Contributed by AV, 15-Jun-2021.) (Revised by AV, 6-Sep-2021.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  C  e.  NN0   &    |-  D  e.  NN0   &    |-  E  e.  NN0   &    |-  F  e.  NN0   &    |-  A  <  B   &    |-  C  < ; 1
 0   &    |-  E  < ; 1 0   =>    |- ;; A C E  < ;; B D F
 
Theoremdecle 9183 Comparing two decimal integers (equal higher places). (Contributed by AV, 17-Aug-2021.) (Revised by AV, 8-Sep-2021.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  C  e.  NN0   &    |-  B  <_  C   =>    |- ; A B  <_ ; A C
 
Theoremdecleh 9184 Comparing two decimal integers (unequal higher places). (Contributed by AV, 17-Aug-2021.) (Revised by AV, 8-Sep-2021.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  C  e.  NN0   &    |-  D  e.  NN0   &    |-  C  <_  9   &    |-  A  <  B   =>    |- ; A C  <_ ; B D
 
Theoremdeclei 9185 Comparing a digit to a decimal integer. (Contributed by AV, 17-Aug-2021.)
 |-  A  e.  NN   &    |-  B  e.  NN0   &    |-  C  e.  NN0   &    |-  C  <_  9   =>    |-  C  <_ ; A B
 
Theoremnumlti 9186 Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  T  e.  NN   &    |-  A  e.  NN   &    |-  B  e.  NN0   &    |-  C  e.  NN0   &    |-  C  <  T   =>    |-  C  <  (
 ( T  x.  A )  +  B )
 
Theoremdeclti 9187 Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
 |-  A  e.  NN   &    |-  B  e.  NN0   &    |-  C  e.  NN0   &    |-  C  < ; 1 0   =>    |-  C  < ; A B
 
Theoremdecltdi 9188 Comparing a digit to a decimal integer. (Contributed by AV, 8-Sep-2021.)
 |-  A  e.  NN   &    |-  B  e.  NN0   &    |-  C  e.  NN0   &    |-  C  <_  9   =>    |-  C  < ; A B
 
Theoremnumsucc 9189 The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  Y  e.  NN0   &    |-  T  =  ( Y  +  1 )   &    |-  A  e.  NN0   &    |-  ( A  +  1 )  =  B   &    |-  N  =  ( ( T  x.  A )  +  Y )   =>    |-  ( N  +  1 )  =  ( ( T  x.  B )  +  0 )
 
Theoremdecsucc 9190 The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
 |-  A  e.  NN0   &    |-  ( A  +  1 )  =  B   &    |-  N  = ; A 9   =>    |-  ( N  +  1 )  = ; B 0
 
Theorem1e0p1 9191 The successor of zero. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  1  =  ( 0  +  1 )
 
Theoremdec10p 9192 Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  (; 1 0  +  A )  = ; 1 A
 
Theoremnumma 9193 Perform a multiply-add of two decimal integers  M and 
N against a fixed multiplicand  P (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  T  e.  NN0   &    |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  C  e.  NN0   &    |-  D  e.  NN0   &    |-  M  =  ( ( T  x.  A )  +  B )   &    |-  N  =  ( ( T  x.  C )  +  D )   &    |-  P  e.  NN0   &    |-  ( ( A  x.  P )  +  C )  =  E   &    |-  ( ( B  x.  P )  +  D )  =  F   =>    |-  (
 ( M  x.  P )  +  N )  =  ( ( T  x.  E )  +  F )
 
Theoremnummac 9194 Perform a multiply-add of two decimal integers  M and 
N against a fixed multiplicand  P (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  T  e.  NN0   &    |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  C  e.  NN0   &    |-  D  e.  NN0   &    |-  M  =  ( ( T  x.  A )  +  B )   &    |-  N  =  ( ( T  x.  C )  +  D )   &    |-  P  e.  NN0   &    |-  F  e.  NN0   &    |-  G  e.  NN0   &    |-  ( ( A  x.  P )  +  ( C  +  G )
 )  =  E   &    |-  (
 ( B  x.  P )  +  D )  =  ( ( T  x.  G )  +  F )   =>    |-  ( ( M  x.  P )  +  N )  =  ( ( T  x.  E )  +  F )
 
Theoremnumma2c 9195 Perform a multiply-add of two decimal integers  M and 
N against a fixed multiplicand  P (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  T  e.  NN0   &    |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  C  e.  NN0   &    |-  D  e.  NN0   &    |-  M  =  ( ( T  x.  A )  +  B )   &    |-  N  =  ( ( T  x.  C )  +  D )   &    |-  P  e.  NN0   &    |-  F  e.  NN0   &    |-  G  e.  NN0   &    |-  ( ( P  x.  A )  +  ( C  +  G )
 )  =  E   &    |-  (
 ( P  x.  B )  +  D )  =  ( ( T  x.  G )  +  F )   =>    |-  ( ( P  x.  M )  +  N )  =  ( ( T  x.  E )  +  F )
 
Theoremnumadd 9196 Add two decimal integers  M and  N (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  T  e.  NN0   &    |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  C  e.  NN0   &    |-  D  e.  NN0   &    |-  M  =  ( ( T  x.  A )  +  B )   &    |-  N  =  ( ( T  x.  C )  +  D )   &    |-  ( A  +  C )  =  E   &    |-  ( B  +  D )  =  F   =>    |-  ( M  +  N )  =  ( ( T  x.  E )  +  F )
 
Theoremnumaddc 9197 Add two decimal integers  M and  N (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  T  e.  NN0   &    |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  C  e.  NN0   &    |-  D  e.  NN0   &    |-  M  =  ( ( T  x.  A )  +  B )   &    |-  N  =  ( ( T  x.  C )  +  D )   &    |-  F  e.  NN0   &    |-  ( ( A  +  C )  +  1
 )  =  E   &    |-  ( B  +  D )  =  ( ( T  x.  1 )  +  F )   =>    |-  ( M  +  N )  =  ( ( T  x.  E )  +  F )
 
Theoremnummul1c 9198 The product of a decimal integer with a number. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  T  e.  NN0   &    |-  P  e.  NN0   &    |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  N  =  ( ( T  x.  A )  +  B )   &    |-  D  e.  NN0   &    |-  E  e.  NN0   &    |-  ( ( A  x.  P )  +  E )  =  C   &    |-  ( B  x.  P )  =  (
 ( T  x.  E )  +  D )   =>    |-  ( N  x.  P )  =  ( ( T  x.  C )  +  D )
 
Theoremnummul2c 9199 The product of a decimal integer with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  T  e.  NN0   &    |-  P  e.  NN0   &    |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  N  =  ( ( T  x.  A )  +  B )   &    |-  D  e.  NN0   &    |-  E  e.  NN0   &    |-  ( ( P  x.  A )  +  E )  =  C   &    |-  ( P  x.  B )  =  (
 ( T  x.  E )  +  D )   =>    |-  ( P  x.  N )  =  ( ( T  x.  C )  +  D )
 
Theoremdecma 9200 Perform a multiply-add of two numerals  M and  N against a fixed multiplicand  P (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  C  e.  NN0   &    |-  D  e.  NN0   &    |-  M  = ; A B   &    |-  N  = ; C D   &    |-  P  e.  NN0   &    |-  (
 ( A  x.  P )  +  C )  =  E   &    |-  ( ( B  x.  P )  +  D )  =  F   =>    |-  (
 ( M  x.  P )  +  N )  = ; E F
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