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Theorem List for Intuitionistic Logic Explorer - 8101-8200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremzsubcld 8101 Closure of subtraction of integers. (Contributed by Mario Carneiro, 28-May-2016.)
 ZZ   &     ZZ   =>     - 
 ZZ
 
Theoremzmulcld 8102 Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016.)
 ZZ   &     ZZ   =>     x. 
 ZZ
 
Theoremzadd2cl 8103 Increasing an integer by 2 results in an integer. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
 N  ZZ  N  +  2  ZZ
 
3.4.9  Decimal arithmetic
 
Syntaxcdc 8104 Constant used for decimal constructor.
;
 
Definitiondf-dec 8105 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. (Contributed by Mario Carneiro, 17-Apr-2015.)
;  10  x.  +
 
Theoremdeceq1 8106 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; C ; C
 
Theoremdeceq2 8107 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; C ; C
 
Theoremdeceq1i 8108 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
   =>    ; C ; C
 
Theoremdeceq2i 8109 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
   =>    ; C ; C
 
Theoremdeceq12i 8110 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
   &     C  D   =>    ; C ; D
 
Theoremnumnncl 8111 Closure for a numeral (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)
 T  NN0   &     NN0   &     NN   =>     T  x.  +  NN
 
Theoremnum0u 8112 Add a zero in the units place. (Contributed by Mario Carneiro, 18-Feb-2014.)
 T  NN0   &     NN0   =>     T  x.  T  x.  +  0
 
Theoremnum0h 8113 Add a zero in the higher places. (Contributed by Mario Carneiro, 18-Feb-2014.)
 T  NN0   &     NN0   =>     T  x.  0  +
 
Theoremnumcl 8114 Closure for a decimal integer (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)
 T  NN0   &     NN0   &     NN0   =>     T  x.  +  NN0
 
Theoremnumsuc 8115 The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 T  NN0   &     NN0   &     NN0   &     +  1  C   &     N  T  x.  +    =>     N  +  1  T  x.  +  C
 
Theoremdecnncl 8116 Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.)
 NN0   &     NN   =>    ;  NN
 
Theoremdeccl 8117 Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.)
 NN0   &     NN0   =>    ;  NN0
 
Theoremdec0u 8118 Add a zero in the units place. (Contributed by Mario Carneiro, 17-Apr-2015.)
 NN0   =>     10  x. ; 0
 
Theoremdec0h 8119 Add a zero in the higher places. (Contributed by Mario Carneiro, 17-Apr-2015.)
 NN0   =>    ; 0
 
Theoremnumnncl2 8120 Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 9-Mar-2015.)
 T  NN   &     NN   =>     T  x.  +  0  NN
 
Theoremdecnncl2 8121 Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 17-Apr-2015.)
 NN   =>    ; 0  NN
 
Theoremnumlt 8122 Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)
 T  NN   &     NN0   &     NN0   &     C  NN   &     <  C   =>     T  x.  +  <  T  x.  +  C
 
Theoremnumltc 8123 Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)
 T  NN   &     NN0   &     NN0   &     C  NN0   &     D  NN0   &     C  <  T   &     <    =>     T  x.  +  C  <  T  x.  +  D
 
Theoremdeclt 8124 Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 17-Apr-2015.)
 NN0   &     NN0   &     C  NN   &     <  C   =>    ;  < ; C
 
Theoremdecltc 8125 Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)
 NN0   &     NN0   &     C  NN0   &     D  NN0   &     C  <  10   &     <    =>    ; C  < ; D
 
Theoremdecsuc 8126 The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 17-Apr-2015.)
 NN0   &     NN0   &     +  1  C   &     N ;   =>     N  +  1 ; C
 
Theoremnumlti 8127 Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 T  NN   &     NN   &     NN0   &     C  NN0   &     C  <  T   =>     C  <  T  x.  +
 
Theoremdeclti 8128 Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 NN   &     NN0   &     C  NN0   &     C  <  10   =>     C  < ;
 
Theoremnumsucc 8129 The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 Y  NN0   &     T  Y  +  1   &     NN0   &     +  1    &     N  T  x.  +  Y   =>     N  +  1  T  x.  +  0
 
Theoremdecsucc 8130 The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 NN0   &     +  1    &     N ; 9   =>     N  +  1 ; 0
 
Theorem1e0p1 8131 The successor of zero. (Contributed by Mario Carneiro, 18-Feb-2014.)
 1  0  +  1
 
Theoremdec10p 8132 Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 10  + ; 1
 
Theoremdec10 8133 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 8134 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  NN0   &     NN0   &     NN0   &     C  NN0   &     D  NN0   &     M  T  x.  +    &     N  T  x.  C  +  D   &     P  NN0   &     x.  P  +  C  E   &     x.  P  +  D  F   =>     M  x.  P  +  N  T  x.  E  +  F
 
Theoremnummac 8135 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  NN0   &     NN0   &     NN0   &     C  NN0   &     D  NN0   &     M  T  x.  +    &     N  T  x.  C  +  D   &     P  NN0   &     F  NN0   &     G  NN0   &     x.  P  +  C  +  G  E   &     x.  P  +  D  T  x.  G  +  F   =>     M  x.  P  +  N  T  x.  E  +  F
 
Theoremnumma2c 8136 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  NN0   &     NN0   &     NN0   &     C  NN0   &     D  NN0   &     M  T  x.  +    &     N  T  x.  C  +  D   &     P  NN0   &     F  NN0   &     G  NN0   &     P  x.  +  C  +  G  E   &     P  x.  +  D  T  x.  G  +  F   =>     P  x.  M  +  N  T  x.  E  +  F
 
Theoremnumadd 8137 Add two decimal integers  M and  N (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 T  NN0   &     NN0   &     NN0   &     C  NN0   &     D  NN0   &     M  T  x.  +    &     N  T  x.  C  +  D   &     +  C  E   &     +  D  F   =>     M  +  N  T  x.  E  +  F
 
Theoremnumaddc 8138 Add two decimal integers  M and  N (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 T  NN0   &     NN0   &     NN0   &     C  NN0   &     D  NN0   &     M  T  x.  +    &     N  T  x.  C  +  D   &     F  NN0   &     +  C  +  1  E   &     +  D  T  x.  1  +  F   =>     M  +  N  T  x.  E  +  F
 
Theoremnummul1c 8139 The product of a decimal integer with a number. (Contributed by Mario Carneiro, 18-Feb-2014.)
 T  NN0   &     P  NN0   &     NN0   &     NN0   &     N  T  x.  +    &     D  NN0   &     E  NN0   &     x.  P  +  E  C   &     x.  P  T  x.  E  +  D   =>     N  x.  P  T  x.  C  +  D
 
Theoremnummul2c 8140 The product of a decimal integer with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 T  NN0   &     P  NN0   &     NN0   &     NN0   &     N  T  x.  +    &     D  NN0   &     E  NN0   &     P  x.  +  E  C   &     P  x.  T  x.  E  +  D   =>     P  x.  N  T  x.  C  +  D
 
Theoremdecma 8141 Perform a multiply-add of two numerals  M and  N against a fixed multiplicand  P (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 NN0   &     NN0   &     C  NN0   &     D  NN0   &     M ;   &     N ; C D   &     P  NN0   &     x.  P  +  C  E   &     x.  P  +  D  F   =>     M  x.  P  +  N ; E F
 
Theoremdecmac 8142 Perform a multiply-add of two numerals  M and  N against a fixed multiplicand  P (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 NN0   &     NN0   &     C  NN0   &     D  NN0   &     M ;   &     N ; C D   &     P  NN0   &     F  NN0   &     G  NN0   &     x.  P  +  C  +  G  E   &     x.  P  +  D ; G F   =>     M  x.  P  +  N ; E F
 
Theoremdecma2c 8143 Perform a multiply-add of two numerals  M and  N against a fixed multiplicand  P (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 NN0   &     NN0   &     C  NN0   &     D  NN0   &     M ;   &     N ; C D   &     P  NN0   &     F  NN0   &     G  NN0   &     P  x.  +  C  +  G  E   &     P  x.  +  D ; G F   =>     P  x.  M  +  N ; E F
 
Theoremdecadd 8144 Add two numerals  M and  N (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 NN0   &     NN0   &     C  NN0   &     D  NN0   &     M ;   &     N ; C D   &     +  C  E   &     +  D  F   =>     M  +  N ; E F
 
Theoremdecaddc 8145 Add two numerals  M and  N (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 NN0   &     NN0   &     C  NN0   &     D  NN0   &     M ;   &     N ; C D   &     +  C  +  1  E   &     F  NN0   &     +  D ; 1 F   =>     M  +  N ; E F
 
Theoremdecaddc2 8146 Add two numerals  M and  N (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 NN0   &     NN0   &     C  NN0   &     D  NN0   &     M ;   &     N ; C D   &     +  C  +  1  E   &     +  D  10   =>     M  +  N ; E 0
 
Theoremdecaddi 8147 Add two numerals  M and  N (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 NN0   &     NN0   &     N  NN0   &     M ;   &     +  N  C   =>     M  +  N ; C
 
Theoremdecaddci 8148 Add two numerals  M and  N (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 NN0   &     NN0   &     N  NN0   &     M ;   &     +  1  D   &     C  NN0   &     +  N ; 1 C   =>     M  +  N ; D C
 
Theoremdecaddci2 8149 Add two numerals  M and  N (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 NN0   &     NN0   &     N  NN0   &     M ;   &     +  1  D   &     +  N  10   =>     M  +  N ; D 0
 
Theoremdecmul1c 8150 The product of a numeral with a number. (Contributed by Mario Carneiro, 18-Feb-2014.)
 P  NN0   &     NN0   &     NN0   &     N ;   &     D  NN0   &     E  NN0   &     x.  P  +  E  C   &     x.  P ; E D   =>     N  x.  P ; C D
 
Theoremdecmul2c 8151 The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 P  NN0   &     NN0   &     NN0   &     N ;   &     D  NN0   &     E  NN0   &     P  x.  +  E  C   &     P  x. ; E D   =>     P  x.  N ; C D
 
Theorem6p5lem 8152 Lemma for 6p5e11 8153 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)
 NN0   &     D  NN0   &     E  NN0   &     D  +  1   &     C  E  +  1   &     +  D ; 1 E   =>     + ; 1 C
 
Theorem6p5e11 8153 6 + 5 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
 6  +  5 ; 1 1
 
Theorem6p6e12 8154 6 + 6 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 6  +  6 ; 1 2
 
Theorem7p4e11 8155 7 + 4 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
 7  +  4 ; 1 1
 
Theorem7p5e12 8156 7 + 5 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 7  +  5 ; 1 2
 
Theorem7p6e13 8157 7 + 6 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
 7  +  6 ; 1 3
 
Theorem7p7e14 8158 7 + 7 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
 7  +  7 ; 1 4
 
Theorem8p3e11 8159 8 + 3 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
 8  +  3 ; 1 1
 
Theorem8p4e12 8160 8 + 4 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 8  +  4 ; 1 2
 
Theorem8p5e13 8161 8 + 5 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
 8  +  5 ; 1 3
 
Theorem8p6e14 8162 8 + 6 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
 8  +  6 ; 1 4
 
Theorem8p7e15 8163 8 + 7 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
 8  +  7 ; 1 5
 
Theorem8p8e16 8164 8 + 8 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
 8  +  8 ; 1 6
 
Theorem9p2e11 8165 9 + 2 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
 9  +  2 ; 1 1
 
Theorem9p3e12 8166 9 + 3 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 9  +  3 ; 1 2
 
Theorem9p4e13 8167 9 + 4 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
 9  +  4 ; 1 3
 
Theorem9p5e14 8168 9 + 5 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
 9  +  5 ; 1 4
 
Theorem9p6e15 8169 9 + 6 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
 9  +  6 ; 1 5
 
Theorem9p7e16 8170 9 + 7 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
 9  +  7 ; 1 6
 
Theorem9p8e17 8171 9 + 8 = 17. (Contributed by Mario Carneiro, 19-Apr-2015.)
 9  +  8 ; 1 7
 
Theorem9p9e18 8172 9 + 9 = 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
 9  +  9 ; 1 8
 
Theorem10p10e20 8173 10 + 10 = 20. (Contributed by Mario Carneiro, 19-Apr-2015.)
 10  +  10 ; 2 0
 
Theorem4t3lem 8174 Lemma for 4t3e12 8175 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)
 NN0   &     NN0   &     C  +  1   &     x.  D   &     D  +  E   =>     x.  C  E
 
Theorem4t3e12 8175 4 times 3 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 4  x.  3 ; 1 2
 
Theorem4t4e16 8176 4 times 4 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
 4  x.  4 ; 1 6
 
Theorem5t3e15 8177 5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
 5  x.  3 ; 1 5
 
Theorem5t4e20 8178 5 times 4 equals 20. (Contributed by Mario Carneiro, 19-Apr-2015.)
 5  x.  4 ; 2 0
 
Theorem5t5e25 8179 5 times 5 equals 25. (Contributed by Mario Carneiro, 19-Apr-2015.)
 5  x.  5 ; 2 5
 
Theorem6t2e12 8180 6 times 2 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 6  x.  2 ; 1 2
 
Theorem6t3e18 8181 6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
 6  x.  3 ; 1 8
 
Theorem6t4e24 8182 6 times 4 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
 6  x.  4 ; 2 4
 
Theorem6t5e30 8183 6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.)
 6  x.  5 ; 3 0
 
Theorem6t6e36 8184 6 times 6 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
 6  x.  6 ; 3 6
 
Theorem7t2e14 8185 7 times 2 equals 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
 7  x.  2 ; 1 4
 
Theorem7t3e21 8186 7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.)
 7  x.  3 ; 2 1
 
Theorem7t4e28 8187 7 times 4 equals 28. (Contributed by Mario Carneiro, 19-Apr-2015.)
 7  x.  4 ; 2 8
 
Theorem7t5e35 8188 7 times 5 equals 35. (Contributed by Mario Carneiro, 19-Apr-2015.)
 7  x.  5 ; 3 5
 
Theorem7t6e42 8189 7 times 6 equals 42. (Contributed by Mario Carneiro, 19-Apr-2015.)
 7  x.  6 ; 4 2
 
Theorem7t7e49 8190 7 times 7 equals 49. (Contributed by Mario Carneiro, 19-Apr-2015.)
 7  x.  7 ; 4 9
 
Theorem8t2e16 8191 8 times 2 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
 8  x.  2 ; 1 6
 
Theorem8t3e24 8192 8 times 3 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
 8  x.  3 ; 2 4
 
Theorem8t4e32 8193 8 times 4 equals 32. (Contributed by Mario Carneiro, 19-Apr-2015.)
 8  x.  4 ; 3 2
 
Theorem8t5e40 8194 8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.)
 8  x.  5 ; 4 0
 
Theorem8t6e48 8195 8 times 6 equals 48. (Contributed by Mario Carneiro, 19-Apr-2015.)
 8  x.  6 ; 4 8
 
Theorem8t7e56 8196 8 times 7 equals 56. (Contributed by Mario Carneiro, 19-Apr-2015.)
 8  x.  7 ; 5 6
 
Theorem8t8e64 8197 8 times 8 equals 64. (Contributed by Mario Carneiro, 19-Apr-2015.)
 8  x.  8 ; 6 4
 
Theorem9t2e18 8198 9 times 2 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
 9  x.  2 ; 1 8
 
Theorem9t3e27 8199 9 times 3 equals 27. (Contributed by Mario Carneiro, 19-Apr-2015.)
 9  x.  3 ; 2 7
 
Theorem9t4e36 8200 9 times 4 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
 9  x.  4 ; 3 6
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