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Theorem List for Intuitionistic Logic Explorer - 9201-9300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdeceq2i 9201 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  =  B   =>    |- ; C A  = ; C B
 
Theoremdeceq12i 9202 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  =  B   &    |-  C  =  D   =>    |- ; A C  = ; B D
 
Theoremnumnncl 9203 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 9204 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 9205 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 9206 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 9207 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 9208 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 9209 10 is a positive integer. (Contributed by NM, 8-Nov-2012.) (Revised by AV, 6-Sep-2021.)
 |- ; 1
 0  e.  NN
 
Theorem10pos 9210 The number 10 is positive. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
 |-  0  < ; 1 0
 
Theorem10nn0 9211 10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |- ; 1
 0  e.  NN0
 
Theorem10re 9212 The number 10 is real. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
 |- ; 1
 0  e.  RR
 
Theoremdecnncl 9213 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 9214 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 9215 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 9216 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 9217 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 9218 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 9219 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 9220 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 9221 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 9222 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 9223 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 9224 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 9225 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 9226 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 9227 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 9228 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 9229 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 9230 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 9231 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 9232 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 9233 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 9234 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 9235 The successor of zero. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  1  =  ( 0  +  1 )
 
Theoremdec10p 9236 Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  (; 1 0  +  A )  = ; 1 A
 
Theoremnumma 9237 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 9238 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 9239 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 9240 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 9241 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 9242 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 9243 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 9244 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 9245 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 9246 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 9247 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 9248 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 9249 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 9250 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 9251 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 9252 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 9253 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 9254 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 9255 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 9256 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 9257 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 9258 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 9259 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 9260 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 9261 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 9262 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 9263 Lemma for 6p5e11 9266 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 9264 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 9265 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 9266 6 + 5 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  ( 6  +  5 )  = ; 1 1
 
Theorem6p6e12 9267 6 + 6 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 6  +  6 )  = ; 1 2
 
Theorem7p3e10 9268 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 9269 7 + 4 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  ( 7  +  4 )  = ; 1 1
 
Theorem7p5e12 9270 7 + 5 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  +  5 )  = ; 1 2
 
Theorem7p6e13 9271 7 + 6 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  +  6 )  = ; 1 3
 
Theorem7p7e14 9272 7 + 7 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  +  7 )  = ; 1 4
 
Theorem8p2e10 9273 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 9274 8 + 3 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  ( 8  +  3 )  = ; 1 1
 
Theorem8p4e12 9275 8 + 4 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  +  4 )  = ; 1 2
 
Theorem8p5e13 9276 8 + 5 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  +  5 )  = ; 1 3
 
Theorem8p6e14 9277 8 + 6 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  +  6 )  = ; 1 4
 
Theorem8p7e15 9278 8 + 7 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  +  7 )  = ; 1 5
 
Theorem8p8e16 9279 8 + 8 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  +  8 )  = ; 1 6
 
Theorem9p2e11 9280 9 + 2 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  ( 9  +  2 )  = ; 1 1
 
Theorem9p3e12 9281 9 + 3 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  3 )  = ; 1 2
 
Theorem9p4e13 9282 9 + 4 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  4 )  = ; 1 3
 
Theorem9p5e14 9283 9 + 5 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  5 )  = ; 1 4
 
Theorem9p6e15 9284 9 + 6 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  6 )  = ; 1 5
 
Theorem9p7e16 9285 9 + 7 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  7 )  = ; 1 6
 
Theorem9p8e17 9286 9 + 8 = 17. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  8 )  = ; 1 7
 
Theorem9p9e18 9287 9 + 9 = 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  9 )  = ; 1 8
 
Theorem10p10e20 9288 10 + 10 = 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  (; 1 0  + ; 1 0 )  = ; 2
 0
 
Theorem10m1e9 9289 10 - 1 = 9. (Contributed by AV, 6-Sep-2021.)
 |-  (; 1 0  -  1
 )  =  9
 
Theorem4t3lem 9290 Lemma for 4t3e12 9291 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 9291 4 times 3 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 4  x.  3
 )  = ; 1 2
 
Theorem4t4e16 9292 4 times 4 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 4  x.  4
 )  = ; 1 6
 
Theorem5t2e10 9293 5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 4-Sep-2021.)
 |-  ( 5  x.  2
 )  = ; 1 0
 
Theorem5t3e15 9294 5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  ( 5  x.  3
 )  = ; 1 5
 
Theorem5t4e20 9295 5 times 4 equals 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  ( 5  x.  4
 )  = ; 2 0
 
Theorem5t5e25 9296 5 times 5 equals 25. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  ( 5  x.  5
 )  = ; 2 5
 
Theorem6t2e12 9297 6 times 2 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 6  x.  2
 )  = ; 1 2
 
Theorem6t3e18 9298 6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 6  x.  3
 )  = ; 1 8
 
Theorem6t4e24 9299 6 times 4 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 6  x.  4
 )  = ; 2 4
 
Theorem6t5e30 9300 6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  ( 6  x.  5
 )  = ; 3 0
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