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Theorem List for Metamath Proof Explorer - 10101-10200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremuzind3 10101* 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 hypothesis. (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 )
 
TheoremuzindOLD 10102* Induction on the upper integers that start at an integer  B. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis.

Warning: The HTML proof page is 3/4 megabyte in size. An attempt to shorten it is on my to-do list. Anyone is welcome to try. (Contributed by NM, 11-May-2004.) (New usage is discouraged.)

 |-  ( x  =  B  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  +  1 )  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ps   &    |-  (
 ( ( y  e. 
 ZZ  /\  B  e.  ZZ )  /\  B  <_  y )  ->  ( ch  ->  th ) )   =>    |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  B  <_  A )  ->  ta )
 
Theoremuzind3OLD 10103* Induction on the set of upper integers that starts at  B. The first four hypotheses give us the substitution instances we need, and the last two are the basis and the induction hypothesis. (Contributed by NM, 9-Nov-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( x  =  B  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  +  1 )  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ps   &    |-  (
 ( B  e.  ZZ  /\  y  e.  { z  e.  ZZ  |  B  <_  z } )  ->  ( ch  ->  th ) )   =>    |-  ( ( B  e.  ZZ  /\  A  e.  { z  e.  ZZ  |  B  <_  z }
 )  ->  ta )
 
Theoremnn0ind 10104* 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 hypothesis. (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 10105* 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 hypothesis. (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 10106* 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 hypothesis. (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 )
 
Theoremnn0indALT 10107* 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 hypothesis. (Contributed by NM, 28-Nov-2005.)
 |-  ( y  e.  NN0  ->  ( ch  ->  th )
 )   &    |- 
 ps   &    |-  ( x  =  0 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  +  1 )  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   =>    |-  ( A  e.  NN0 
 ->  ta )
 
Theoremnn0ind-raph 10108* 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 hypothesis. 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 10109* 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 10110* 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 10111 A natural number is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  NN0 )   =>    |-  ( ph  ->  A  e.  ZZ )
 
Theoremnnzd 10112 A nonnegative integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  e.  ZZ )
 
Theoremzred 10113 An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  ZZ )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremzcnd 10114 An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  ZZ )   =>    |-  ( ph  ->  A  e.  CC )
 
Theoremznegcld 10115 Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  ZZ )   =>    |-  ( ph  ->  -u A  e.  ZZ )
 
Theorempeano2zd 10116 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 10117 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 10118 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 10119 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 )
 
5.4.8  Decimal arithmetic
 
Syntaxcdc 10120 Constant used for decimal constructor.
 class ; A B
 
Definitiondf-dec 10121 Define the "decimal constructor", which is used to build up "decimal integers" or "numeric terms" in base 10. For example,  (;;; 1 0 0 0  + ;;; 2 0 0 0 )  = ;;; 3 0 0 0 1kp2ke3k 20810. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |- ; A B  =  ( ( 10  x.  A )  +  B )
 
Theoremdecex 10122 A decimal number is a set. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |- ; A B  e.  _V
 
Theoremdeceq1 10123 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  ( A  =  B  -> ; A C  = ; B C )
 
Theoremdeceq2 10124 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  ( A  =  B  -> ; C A  = ; C B )
 
Theoremdeceq1i 10125 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  =  B   =>    |- ; A C  = ; B C
 
Theoremdeceq2i 10126 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  =  B   =>    |- ; C A  = ; C B
 
Theoremdeceq12i 10127 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  =  B   &    |-  C  =  D   =>    |- ; A C  = ; B D
 
Theoremnumnncl 10128 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 10129 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 10130 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 10131 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 10132 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 )
 
Theoremdecnncl 10133 Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  e.  NN0   &    |-  B  e.  NN   =>    |- ; A B  e.  NN
 
Theoremdeccl 10134 Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   =>    |- ; A B  e.  NN0
 
Theoremdec0u 10135 Add a zero in the units place. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  e.  NN0   =>    |-  ( 10  x.  A )  = ; A 0
 
Theoremdec0h 10136 Add a zero in the higher places. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  e.  NN0   =>    |-  A  = ; 0 A
 
Theoremnumnncl2 10137 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 10138 Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  e.  NN   =>    |- ; A 0  e.  NN
 
Theoremnumlt 10139 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 10140 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 )
 
Theoremdeclt 10141 Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  C  e.  NN   &    |-  B  <  C   =>    |- ; A B  < ; A C
 
Theoremdecltc 10142 Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  C  e.  NN0   &    |-  D  e.  NN0   &    |-  C  <  10   &    |-  A  <  B   =>    |- ; A C  < ; B D
 
Theoremdecsuc 10143 The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  ( B  +  1 )  =  C   &    |-  N  = ; A B   =>    |-  ( N  +  1 )  = ; A C
 
Theoremnumlti 10144 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 10145 Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  A  e.  NN   &    |-  B  e.  NN0   &    |-  C  e.  NN0   &    |-  C  <  10   =>    |-  C  < ; A B
 
Theoremnumsucc 10146 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 10147 The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  A  e.  NN0   &    |-  ( A  +  1 )  =  B   &    |-  N  = ; A 9   =>    |-  ( N  +  1 )  = ; B 0
 
Theorem1e0p1 10148 The successor of zero. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  1  =  ( 0  +  1 )
 
Theoremdec10p 10149 Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 10  +  A )  = ; 1 A
 
Theoremdec10 10150 The decimal form of 10. NB: In our presentations of large numbers later on, we will use our symbol for 10 at the highest digits when advantageous, because we can use this theorem to convert back to "long form" (where each digit is in the range 0-9) with no extra effort. However, we cannot do this for lower digits while maintaining the ease of use of the decimal system, since it requires nontrivial number knowledge (more than just equality theorems) to convert back. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |- 
 10  = ; 1 0
 
Theoremnumma 10151 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 10152 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 10153 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 10154 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 10155 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 10156 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 10157 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 10158 Perform a multiply-add of two numerals  M and  N against a fixed multiplicand  P (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  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 10159 Perform a multiply-add of two numerals  M and  N against a fixed multiplicand  P (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  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 10160 Perform a multiply-add of two numerals  M and  N against a fixed multiplicand  P (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  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 10161 Add two numerals  M and  N (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  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 10162 Add two numerals  M and  N (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  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 10163 Add two numerals  M and  N (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  C  e.  NN0   &    |-  D  e.  NN0   &    |-  M  = ; A B   &    |-  N  = ; C D   &    |-  ( ( A  +  C )  +  1 )  =  E   &    |-  ( B  +  D )  =  10   =>    |-  ( M  +  N )  = ; E 0
 
Theoremdecaddi 10164 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 10165 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 10166 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 10167 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 10168 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 10169 Lemma for 6p5e11 10170 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 10170 6 + 5 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 6  +  5 )  = ; 1 1
 
Theorem6p6e12 10171 6 + 6 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 6  +  6 )  = ; 1 2
 
Theorem7p4e11 10172 7 + 4 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  +  4 )  = ; 1 1
 
Theorem7p5e12 10173 7 + 5 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  +  5 )  = ; 1 2
 
Theorem7p6e13 10174 7 + 6 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  +  6 )  = ; 1 3
 
Theorem7p7e14 10175 7 + 7 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  +  7 )  = ; 1 4
 
Theorem8p3e11 10176 8 + 3 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  +  3 )  = ; 1 1
 
Theorem8p4e12 10177 8 + 4 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  +  4 )  = ; 1 2
 
Theorem8p5e13 10178 8 + 5 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  +  5 )  = ; 1 3
 
Theorem8p6e14 10179 8 + 6 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  +  6 )  = ; 1 4
 
Theorem8p7e15 10180 8 + 7 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  +  7 )  = ; 1 5
 
Theorem8p8e16 10181 8 + 8 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  +  8 )  = ; 1 6
 
Theorem9p2e11 10182 9 + 2 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  2 )  = ; 1 1
 
Theorem9p3e12 10183 9 + 3 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  3 )  = ; 1 2
 
Theorem9p4e13 10184 9 + 4 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  4 )  = ; 1 3
 
Theorem9p5e14 10185 9 + 5 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  5 )  = ; 1 4
 
Theorem9p6e15 10186 9 + 6 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  6 )  = ; 1 5
 
Theorem9p7e16 10187 9 + 7 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  7 )  = ; 1 6
 
Theorem9p8e17 10188 9 + 8 = 17. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  8 )  = ; 1 7
 
Theorem9p9e18 10189 9 + 9 = 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  9 )  = ; 1 8
 
Theorem10p10e20 10190 10 + 10 = 20. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 10  +  10 )  = ; 2 0
 
Theorem4t3lem 10191 Lemma for 4t3e12 10192 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 10192 4 times 3 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 4  x.  3
 )  = ; 1 2
 
Theorem4t4e16 10193 4 times 4 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 4  x.  4
 )  = ; 1 6
 
Theorem5t3e15 10194 5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 5  x.  3
 )  = ; 1 5
 
Theorem5t4e20 10195 5 times 4 equals 20. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 5  x.  4
 )  = ; 2 0
 
Theorem5t5e25 10196 5 times 5 equals 25. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 5  x.  5
 )  = ; 2 5
 
Theorem6t2e12 10197 6 times 2 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 6  x.  2
 )  = ; 1 2
 
Theorem6t3e18 10198 6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 6  x.  3
 )  = ; 1 8
 
Theorem6t4e24 10199 6 times 4 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 6  x.  4
 )  = ; 2 4
 
Theorem6t5e30 10200 6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 6  x.  5
 )  = ; 3 0
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