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Theorem List for Intuitionistic Logic Explorer - 9401-9500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnn0ind-raph 9401* 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 9402* 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 9403* 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 9404 A positive integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  NN0 )   =>    |-  ( ph  ->  A  e.  ZZ )
 
Theoremnnzd 9405 A nonnegative integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  e.  ZZ )
 
Theoremzred 9406 An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  ZZ )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremzcnd 9407 An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  ZZ )   =>    |-  ( ph  ->  A  e.  CC )
 
Theoremznegcld 9408 Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  ZZ )   =>    |-  ( ph  ->  -u A  e.  ZZ )
 
Theorempeano2zd 9409 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 9410 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 9411 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 9412 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 9413 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 9414 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 9415 Constant used for decimal constructor.
 class ; A B
 
Definitiondf-dec 9416 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 14954. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 1-Aug-2021.)
 |- ; A B  =  ( ( ( 9  +  1 )  x.  A )  +  B )
 
Theorem9p1e10 9417 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 9418 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 9419 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 9420 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 9421 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  =  B   =>    |- ; A C  = ; B C
 
Theoremdeceq2i 9422 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  =  B   =>    |- ; C A  = ; C B
 
Theoremdeceq12i 9423 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  =  B   &    |-  C  =  D   =>    |- ; A C  = ; B D
 
Theoremnumnncl 9424 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 9425 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 9426 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 9427 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 9428 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 9429 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 9430 10 is a positive integer. (Contributed by NM, 8-Nov-2012.) (Revised by AV, 6-Sep-2021.)
 |- ; 1
 0  e.  NN
 
Theorem10pos 9431 The number 10 is positive. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
 |-  0  < ; 1 0
 
Theorem10nn0 9432 10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |- ; 1
 0  e.  NN0
 
Theorem10re 9433 The number 10 is real. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
 |- ; 1
 0  e.  RR
 
Theoremdecnncl 9434 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 9435 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 9436 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 9437 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 9438 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 9439 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 9440 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 9441 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 9442 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 9443 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 9444 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 9445 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 9446 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 9447 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 9448 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 9449 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 9450 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 9451 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 9452 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 9453 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 9454 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 9455 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 9456 The successor of zero. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  1  =  ( 0  +  1 )
 
Theoremdec10p 9457 Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  (; 1 0  +  A )  = ; 1 A
 
Theoremnumma 9458 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 9459 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 9460 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 9461 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 9462 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 9463 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 9464 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 9465 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 9466 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 9467 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 9468 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 9469 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 9470 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 9471 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 9472 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 9473 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 9474 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 9475 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 9476 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 9477 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 9478 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 9479 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 9480 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 9481 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 9482 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 9483 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 9484 Lemma for 6p5e11 9487 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 9485 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 9486 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 9487 6 + 5 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  ( 6  +  5 )  = ; 1 1
 
Theorem6p6e12 9488 6 + 6 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 6  +  6 )  = ; 1 2
 
Theorem7p3e10 9489 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
 
Theorem7p4e11 9490 7 + 4 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  ( 7  +  4 )  = ; 1 1
 
Theorem7p5e12 9491 7 + 5 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  +  5 )  = ; 1 2
 
Theorem7p6e13 9492 7 + 6 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  +  6 )  = ; 1 3
 
Theorem7p7e14 9493 7 + 7 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  +  7 )  = ; 1 4
 
Theorem8p2e10 9494 8 + 2 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
 |-  ( 8  +  2 )  = ; 1 0
 
Theorem8p3e11 9495 8 + 3 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  ( 8  +  3 )  = ; 1 1
 
Theorem8p4e12 9496 8 + 4 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  +  4 )  = ; 1 2
 
Theorem8p5e13 9497 8 + 5 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  +  5 )  = ; 1 3
 
Theorem8p6e14 9498 8 + 6 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  +  6 )  = ; 1 4
 
Theorem8p7e15 9499 8 + 7 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  +  7 )  = ; 1 5
 
Theorem8p8e16 9500 8 + 8 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  +  8 )  = ; 1 6
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