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Theorem List for Intuitionistic Logic Explorer - 8401-8500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremzeo 8401 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 8402 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 8403* 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 8404* 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 8405* 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 8406* An expression for the upper integers that start at  N that is analogous to dfnn2 7991 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 8407* 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 8408* 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 8409* 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 8410* 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 8411* 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 8412* 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 8413* 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 8414* 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 8415* 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 8416 A positive integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  NN0 )   =>    |-  ( ph  ->  A  e.  ZZ )
 
Theoremnnzd 8417 A nonnegative integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  e.  ZZ )
 
Theoremzred 8418 An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  ZZ )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremzcnd 8419 An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  ZZ )   =>    |-  ( ph  ->  A  e.  CC )
 
Theoremznegcld 8420 Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  ZZ )   =>    |-  ( ph  ->  -u A  e.  ZZ )
 
Theorempeano2zd 8421 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 8422 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 8423 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 8424 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 8425 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 )
 
3.4.9  Decimal arithmetic
 
Syntaxcdc 8426 Constant used for decimal constructor.
 class ; A B
 
Definitiondf-dec 8427 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 10250. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 1-Aug-2021.)
 |- ; A B  =  ( ( ( 9  +  1 )  x.  A )  +  B )
 
Theorem9p1e10 8428 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 8429 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 8430 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 8431 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 8432 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  =  B   =>    |- ; A C  = ; B C
 
Theoremdeceq2i 8433 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  =  B   =>    |- ; C A  = ; C B
 
Theoremdeceq12i 8434 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  =  B   &    |-  C  =  D   =>    |- ; A C  = ; B D
 
Theoremnumnncl 8435 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 8436 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 8437 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 8438 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 8439 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 8440 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 8441 10 is a positive integer. (Contributed by NM, 8-Nov-2012.) (Revised by AV, 6-Sep-2021.)
 |- ; 1
 0  e.  NN
 
Theorem10pos 8442 The number 10 is positive. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
 |-  0  < ; 1 0
 
Theorem10nn0 8443 10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |- ; 1
 0  e.  NN0
 
Theorem10re 8444 The number 10 is real. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
 |- ; 1
 0  e.  RR
 
Theoremdecnncl 8445 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 8446 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 8447 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 8448 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 8449 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 8450 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 8451 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 8452 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 8453 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 8454 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 8455 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 8456 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 8457 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 8458 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 8459 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 8460 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 8461 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 8462 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 8463 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 8464 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 8465 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 8466 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 8467 The successor of zero. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  1  =  ( 0  +  1 )
 
Theoremdec10p 8468 Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  (; 1 0  +  A )  = ; 1 A
 
Theoremnumma 8469 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 8470 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 8471 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 8472 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 8473 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 8474 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 8475 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 8476 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
 
Theoremdecmac 8477 Perform a multiply-add of two numerals  M and  N against a fixed multiplicand  P (with 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   &    |-  F  e.  NN0   &    |-  G  e.  NN0   &    |-  ( ( A  x.  P )  +  ( C  +  G )
 )  =  E   &    |-  (
 ( B  x.  P )  +  D )  = ; G F   =>    |-  ( ( M  x.  P )  +  N )  = ; E F
 
Theoremdecma2c 8478 Perform a multiply-add of two numerals  M and  N against a fixed multiplier  P (with 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   &    |-  F  e.  NN0   &    |-  G  e.  NN0   &    |-  ( ( P  x.  A )  +  ( C  +  G )
 )  =  E   &    |-  (
 ( P  x.  B )  +  D )  = ; G F   =>    |-  ( ( P  x.  M )  +  N )  = ; E F
 
Theoremdecadd 8479 Add two numerals  M and  N (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   &    |-  ( A  +  C )  =  E   &    |-  ( B  +  D )  =  F   =>    |-  ( M  +  N )  = ; E F
 
Theoremdecaddc 8480 Add two numerals  M and  N (with 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   &    |-  ( ( A  +  C )  +  1 )  =  E   &    |-  F  e.  NN0   &    |-  ( B  +  D )  = ; 1 F   =>    |-  ( M  +  N )  = ; E F
 
Theoremdecaddc2 8481 Add two numerals  M and  N (with 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   &    |-  ( ( A  +  C )  +  1 )  =  E   &    |-  ( B  +  D )  = ; 1 0   =>    |-  ( M  +  N )  = ; E 0
 
Theoremdecrmanc 8482 Perform a multiply-add of two numerals  M and  N against a fixed multiplicand  P (no carry). (Contributed by AV, 16-Sep-2021.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  N  e.  NN0   &    |-  M  = ; A B   &    |-  P  e.  NN0   &    |-  ( A  x.  P )  =  E   &    |-  ( ( B  x.  P )  +  N )  =  F   =>    |-  (
 ( M  x.  P )  +  N )  = ; E F
 
Theoremdecrmac 8483 Perform a multiply-add of two numerals  M and  N against a fixed multiplicand  P (with carry). (Contributed by AV, 16-Sep-2021.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  N  e.  NN0   &    |-  M  = ; A B   &    |-  P  e.  NN0   &    |-  F  e.  NN0   &    |-  G  e.  NN0   &    |-  ( ( A  x.  P )  +  G )  =  E   &    |-  ( ( B  x.  P )  +  N )  = ; G F   =>    |-  ( ( M  x.  P )  +  N )  = ; E F
 
Theoremdecaddm10 8484 The sum of two multiples of 10 is a multiple of 10. (Contributed by AV, 30-Jul-2021.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   =>    |-  (; A 0  + ; B 0 )  = ; ( A  +  B )
 0
 
Theoremdecaddi 8485 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   &    |-  ( B  +  N )  =  C   =>    |-  ( M  +  N )  = ; A C
 
Theoremdecaddci 8486 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 8487 Add two numerals  M and  N (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  N  e.  NN0   &    |-  M  = ; A B   &    |-  ( A  +  1 )  =  D   &    |-  ( B  +  N )  = ; 1 0   =>    |-  ( M  +  N )  = ; D 0
 
Theoremdecsubi 8488 Difference between a numeral  M and a nonnegative integer  N (no underflow). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  N  e.  NN0   &    |-  M  = ; A B   &    |-  ( A  +  1 )  =  D   &    |-  ( B  -  N )  =  C   =>    |-  ( M  -  N )  = ; A C
 
Theoremdecmul1 8489 The product of a numeral with a number (no carry). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)
 |-  P  e.  NN0   &    |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  N  = ; A B   &    |-  D  e.  NN0   &    |-  ( A  x.  P )  =  C   &    |-  ( B  x.  P )  =  D   =>    |-  ( N  x.  P )  = ; C D
 
Theoremdecmul1c 8490 The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
 |-  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 8491 The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
 |-  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
 
Theoremdecmulnc 8492 The product of a numeral with a number (no carry). (Contributed by AV, 15-Jun-2021.)
 |-  N  e.  NN0   &    |-  A  e.  NN0   &    |-  B  e.  NN0   =>    |-  ( N  x. ; A B )  = ; ( N  x.  A ) ( N  x.  B )
 
Theorem11multnc 8493 The product of 11 (as numeral) with a number (no carry). (Contributed by AV, 15-Jun-2021.)
 |-  N  e.  NN0   =>    |-  ( N  x. ; 1 1 )  = ; N N
 
Theoremdecmul10add 8494 A multiplication of a number and a numeral expressed as addition with first summand as multiple of 10. (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  M  e.  NN0   &    |-  E  =  ( M  x.  A )   &    |-  F  =  ( M  x.  B )   =>    |-  ( M  x. ; A B )  =  (; E 0  +  F )
 
Theorem6p5lem 8495 Lemma for 6p5e11 8498 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
 
Theorem5p5e10 8496 5 + 5 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
 |-  ( 5  +  5 )  = ; 1 0
 
Theorem6p4e10 8497 6 + 4 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
 |-  ( 6  +  4 )  = ; 1 0
 
Theorem6p5e11 8498 6 + 5 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  ( 6  +  5 )  = ; 1 1
 
Theorem6p6e12 8499 6 + 6 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 6  +  6 )  = ; 1 2
 
Theorem7p3e10 8500 7 + 3 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
 |-  ( 7  +  3 )  = ; 1 0
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