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Theorem List for Metamath Proof Explorer - 10001-10100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremzsubcld 10001 Closure of subtraction of integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzmulcld 10002 Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016.)

5.4.8  Decimal arithmetic

Syntaxcdc 10003 Constant used for decimal constructor.
;

Definitiondf-dec 10004 Define the "decimal constructor", which is used to build up "decimal integers" or "numeric terms" in base 10. For example, ;;; ;;; ;;; 1kp2ke3k 20646. (Contributed by Mario Carneiro, 17-Apr-2015.)
;

Theoremdecex 10005 A decimal number is a set. (Contributed by Mario Carneiro, 17-Apr-2015.)
;

Theoremdeceq1 10006 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theoremdeceq2 10007 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theoremdeceq1i 10008 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theoremdeceq2i 10009 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theoremdeceq12i 10010 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theoremnumnncl 10011 Closure for a numeral (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnum0u 10012 Add a zero in the units place. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnum0h 10013 Add a zero in the higher places. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumcl 10014 Closure for a decimal integer (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumsuc 10015 The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdecnncl 10016 Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.)
;

Theoremdeccl 10017 Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.)
;

Theoremdec0u 10018 Add a zero in the units place. (Contributed by Mario Carneiro, 17-Apr-2015.)
;

Theoremdec0h 10019 Add a zero in the higher places. (Contributed by Mario Carneiro, 17-Apr-2015.)
;

Theoremnumnncl2 10020 Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 9-Mar-2015.)

Theoremdecnncl2 10021 Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 17-Apr-2015.)
;

Theoremnumlt 10022 Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumltc 10023 Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdeclt 10024 Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theoremdecltc 10025 Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)
; ;

Theoremdecsuc 10026 The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 17-Apr-2015.)
;       ;

Theoremnumlti 10027 Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdeclti 10028 Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
;

Theoremnumsucc 10029 The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdecsucc 10030 The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;       ;

Theorem1e0p1 10031 The successor of zero. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdec10p 10032 Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theoremdec10 10033 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.)
;

Theoremnumma 10034 Perform a multiply-add of two decimal integers and against a fixed multiplicand (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnummac 10035 Perform a multiply-add of two decimal integers and against a fixed multiplicand (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumma2c 10036 Perform a multiply-add of two decimal integers and against a fixed multiplicand (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumadd 10037 Add two decimal integers and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumaddc 10038 Add two decimal integers and (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnummul1c 10039 The product of a decimal integer with a number. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnummul2c 10040 The product of a decimal integer with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdecma 10041 Perform a multiply-add of two numerals and against a fixed multiplicand (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;       ;                            ;

Theoremdecmac 10042 Perform a multiply-add of two numerals and against a fixed multiplicand (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;       ;                                   ;       ;

Theoremdecma2c 10043 Perform a multiply-add of two numerals and against a fixed multiplicand (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;       ;                                   ;       ;

Theoremdecadd 10044 Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;       ;                     ;

Theoremdecaddc 10045 Add two numerals and (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;       ;                     ;       ;

Theoremdecaddc2 10046 Add two numerals and (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;       ;                     ;

Theoremdecaddi 10047 Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;              ;

Theoremdecaddci 10048 Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;                     ;       ;

Theoremdecaddci2 10049 Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;                     ;

Theoremdecmul1c 10050 The product of a numeral with a number. (Contributed by Mario Carneiro, 18-Feb-2014.)
;                            ;       ;

Theoremdecmul2c 10051 The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;                            ;       ;

Theorem6p5lem 10052 Lemma for 6p5e11 10053 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)
;       ;

Theorem6p5e11 10053 6 + 5 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem6p6e12 10054 6 + 6 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem7p4e11 10055 7 + 4 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem7p5e12 10056 7 + 5 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem7p6e13 10057 7 + 6 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem7p7e14 10058 7 + 7 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8p3e11 10059 8 + 3 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8p4e12 10060 8 + 4 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8p5e13 10061 8 + 5 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8p6e14 10062 8 + 6 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8p7e15 10063 8 + 7 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8p8e16 10064 8 + 8 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9p2e11 10065 9 + 2 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9p3e12 10066 9 + 3 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9p4e13 10067 9 + 4 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9p5e14 10068 9 + 5 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9p6e15 10069 9 + 6 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9p7e16 10070 9 + 7 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9p8e17 10071 9 + 8 = 17. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9p9e18 10072 9 + 9 = 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem10p10e20 10073 10 + 10 = 20. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem4t3lem 10074 Lemma for 4t3e12 10075 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)

Theorem4t3e12 10075 4 times 3 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem4t4e16 10076 4 times 4 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem5t3e15 10077 5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem5t4e20 10078 5 times 4 equals 20. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem5t5e25 10079 5 times 5 equals 25. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem6t2e12 10080 6 times 2 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem6t3e18 10081 6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem6t4e24 10082 6 times 4 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem6t5e30 10083 6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem6t6e36 10084 6 times 6 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem7t2e14 10085 7 times 2 equals 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem7t3e21 10086 7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem7t4e28 10087 7 times 4 equals 28. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem7t5e35 10088 7 times 5 equals 35. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem7t6e42 10089 7 times 6 equals 42. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem7t7e49 10090 7 times 7 equals 49. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8t2e16 10091 8 times 2 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8t3e24 10092 8 times 3 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8t4e32 10093 8 times 4 equals 32. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8t5e40 10094 8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8t6e48 10095 8 times 6 equals 48. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8t7e56 10096 8 times 7 equals 56. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8t8e64 10097 8 times 8 equals 64. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9t2e18 10098 9 times 2 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9t3e27 10099 9 times 3 equals 27. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9t4e36 10100 9 times 4 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

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