P. 96 of Theorem List - Intuitionistic Logic Explorer

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**Theorem List for Intuitionistic Logic Explorer -** 9501-9600 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	declt 9501	Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	decltc 9502	Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
; ; ;

Theorem	declth 9503	Comparing two decimal integers (unequal higher places). (Contributed by AV, 8-Sep-2021.)
; ;

Theorem	decsuc 9504	The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	3declth 9505	Comparing two decimal integers with three "digits" (unequal higher places). (Contributed by AV, 8-Sep-2021.)
;; ;;

Theorem	3decltc 9506	Comparing two decimal integers with three "digits" (unequal higher places). (Contributed by AV, 15-Jun-2021.) (Revised by AV, 6-Sep-2021.)
; ; ;; ;;

Theorem	decle 9507	Comparing two decimal integers (equal higher places). (Contributed by AV, 17-Aug-2021.) (Revised by AV, 8-Sep-2021.)
; ;

Theorem	decleh 9508	Comparing two decimal integers (unequal higher places). (Contributed by AV, 17-Aug-2021.) (Revised by AV, 8-Sep-2021.)
; ;

Theorem	declei 9509	Comparing a digit to a decimal integer. (Contributed by AV, 17-Aug-2021.)
;

Theorem	numlti 9510	Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	declti 9511	Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	decltdi 9512	Comparing a digit to a decimal integer. (Contributed by AV, 8-Sep-2021.)
;

Theorem	numsucc 9513	The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	decsucc 9514	The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	1e0p1 9515	The successor of zero. (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	dec10p 9516	Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	numma 9517	Perform a multiply-add of two decimal integers and against a fixed multiplicand (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	nummac 9518	Perform a multiply-add of two decimal integers and against a fixed multiplicand (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	numma2c 9519	Perform a multiply-add of two decimal integers and against a fixed multiplicand (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	numadd 9520	Add two decimal integers and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	numaddc 9521	Add two decimal integers and (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	nummul1c 9522	The product of a decimal integer with a number. (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	nummul2c 9523	The product of a decimal integer with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	decma 9524	Perform a multiply-add of two numerals and against a fixed multiplicand (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
; ; ;

Theorem	decmac 9525	Perform a multiply-add of two numerals and against a fixed multiplicand (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
; ; ; ;

Theorem	decma2c 9526	Perform a multiply-add of two numerals and against a fixed multiplier (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
; ; ; ;

Theorem	decadd 9527	Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
; ; ;

Theorem	decaddc 9528	Add two numerals and (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
; ; ; ;

Theorem	decaddc2 9529	Add two numerals and (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
; ; ; ;

Theorem	decrmanc 9530	Perform a multiply-add of two numerals and against a fixed multiplicand (no carry). (Contributed by AV, 16-Sep-2021.)
; ;

Theorem	decrmac 9531	Perform a multiply-add of two numerals and against a fixed multiplicand (with carry). (Contributed by AV, 16-Sep-2021.)
; ; ;

Theorem	decaddm10 9532	The sum of two multiples of 10 is a multiple of 10. (Contributed by AV, 30-Jul-2021.)
; ; ;

Theorem	decaddi 9533	Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
; ;

Theorem	decaddci 9534	Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
; ; ;

Theorem	decaddci2 9535	Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
; ; ;

Theorem	decsubi 9536	Difference between a numeral and a nonnegative integer (no underflow). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	decmul1 9537	The product of a numeral with a number (no carry). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	decmul1c 9538	The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
; ; ;

Theorem	decmul2c 9539	The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
; ; ;

Theorem	decmulnc 9540	The product of a numeral with a number (no carry). (Contributed by AV, 15-Jun-2021.)
; ;

Theorem	11multnc 9541	The product of 11 (as numeral) with a number (no carry). (Contributed by AV, 15-Jun-2021.)
; ;

Theorem	decmul10add 9542	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.)
; ;

Theorem	6p5lem 9543	Lemma for 6p5e11 9546 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)
; ;

Theorem	5p5e10 9544	5 + 5 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
;

Theorem	6p4e10 9545	6 + 4 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
;

Theorem	6p5e11 9546	6 + 5 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	6p6e12 9547	6 + 6 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7p3e10 9548	7 + 3 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
;

Theorem	7p4e11 9549	7 + 4 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	7p5e12 9550	7 + 5 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7p6e13 9551	7 + 6 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7p7e14 9552	7 + 7 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8p2e10 9553	8 + 2 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
;

Theorem	8p3e11 9554	8 + 3 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	8p4e12 9555	8 + 4 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8p5e13 9556	8 + 5 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8p6e14 9557	8 + 6 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8p7e15 9558	8 + 7 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8p8e16 9559	8 + 8 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p2e11 9560	9 + 2 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	9p3e12 9561	9 + 3 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p4e13 9562	9 + 4 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p5e14 9563	9 + 5 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p6e15 9564	9 + 6 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p7e16 9565	9 + 7 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p8e17 9566	9 + 8 = 17. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p9e18 9567	9 + 9 = 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	10p10e20 9568	10 + 10 = 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ; ;

Theorem	10m1e9 9569	10 - 1 = 9. (Contributed by AV, 6-Sep-2021.)
;

Theorem	4t3lem 9570	Lemma for 4t3e12 9571 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)


Theorem	4t3e12 9571	4 times 3 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	4t4e16 9572	4 times 4 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	5t2e10 9573	5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 4-Sep-2021.)
;

Theorem	5t3e15 9574	5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	5t4e20 9575	5 times 4 equals 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	5t5e25 9576	5 times 5 equals 25. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	6t2e12 9577	6 times 2 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	6t3e18 9578	6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	6t4e24 9579	6 times 4 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	6t5e30 9580	6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	6t6e36 9581	6 times 6 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	7t2e14 9582	7 times 2 equals 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t3e21 9583	7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t4e28 9584	7 times 4 equals 28. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t5e35 9585	7 times 5 equals 35. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t6e42 9586	7 times 6 equals 42. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t7e49 9587	7 times 7 equals 49. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t2e16 9588	8 times 2 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t3e24 9589	8 times 3 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t4e32 9590	8 times 4 equals 32. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t5e40 9591	8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	8t6e48 9592	8 times 6 equals 48. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	8t7e56 9593	8 times 7 equals 56. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t8e64 9594	8 times 8 equals 64. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t2e18 9595	9 times 2 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t3e27 9596	9 times 3 equals 27. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t4e36 9597	9 times 4 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t5e45 9598	9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t6e54 9599	9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t7e63 9600	9 times 7 equals 63. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

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