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	10nn 9501	10 is a positive integer. (Contributed by NM, 8-Nov-2012.) (Revised by AV, 6-Sep-2021.)
;

Theorem	10pos 9502	The number 10 is positive. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
;

Theorem	10nn0 9503	10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	10re 9504	The number 10 is real. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
;

Theorem	decnncl 9505	Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	dec0u 9506	Add a zero in the units place. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	dec0h 9507	Add a zero in the higher places. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	numnncl2 9508	Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 9-Mar-2015.)


Theorem	decnncl2 9509	Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	numlt 9510	Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	numltc 9511	Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	le9lt10 9512	A "decimal digit" (i.e. a nonnegative integer less than or equal to 9) is less then 10. (Contributed by AV, 8-Sep-2021.)
;

Theorem	declt 9513	Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	decltc 9514	Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
; ; ;

Theorem	declth 9515	Comparing two decimal integers (unequal higher places). (Contributed by AV, 8-Sep-2021.)
; ;

Theorem	decsuc 9516	The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	3declth 9517	Comparing two decimal integers with three "digits" (unequal higher places). (Contributed by AV, 8-Sep-2021.)
;; ;;

Theorem	3decltc 9518	Comparing two decimal integers with three "digits" (unequal higher places). (Contributed by AV, 15-Jun-2021.) (Revised by AV, 6-Sep-2021.)
; ; ;; ;;

Theorem	decle 9519	Comparing two decimal integers (equal higher places). (Contributed by AV, 17-Aug-2021.) (Revised by AV, 8-Sep-2021.)
; ;

Theorem	decleh 9520	Comparing two decimal integers (unequal higher places). (Contributed by AV, 17-Aug-2021.) (Revised by AV, 8-Sep-2021.)
; ;

Theorem	declei 9521	Comparing a digit to a decimal integer. (Contributed by AV, 17-Aug-2021.)
;

Theorem	numlti 9522	Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	declti 9523	Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	decltdi 9524	Comparing a digit to a decimal integer. (Contributed by AV, 8-Sep-2021.)
;

Theorem	numsucc 9525	The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	decsucc 9526	The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	1e0p1 9527	The successor of zero. (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	dec10p 9528	Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	numma 9529	Perform a multiply-add of two decimal integers and against a fixed multiplicand (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	nummac 9530	Perform a multiply-add of two decimal integers and against a fixed multiplicand (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	numma2c 9531	Perform a multiply-add of two decimal integers and against a fixed multiplicand (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	numadd 9532	Add two decimal integers and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	numaddc 9533	Add two decimal integers and (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	nummul1c 9534	The product of a decimal integer with a number. (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	nummul2c 9535	The product of a decimal integer with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	decma 9536	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 9537	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 9538	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 9539	Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
; ; ;

Theorem	decaddc 9540	Add two numerals and (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
; ; ; ;

Theorem	decaddc2 9541	Add two numerals and (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
; ; ; ;

Theorem	decrmanc 9542	Perform a multiply-add of two numerals and against a fixed multiplicand (no carry). (Contributed by AV, 16-Sep-2021.)
; ;

Theorem	decrmac 9543	Perform a multiply-add of two numerals and against a fixed multiplicand (with carry). (Contributed by AV, 16-Sep-2021.)
; ; ;

Theorem	decaddm10 9544	The sum of two multiples of 10 is a multiple of 10. (Contributed by AV, 30-Jul-2021.)
; ; ;

Theorem	decaddi 9545	Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
; ;

Theorem	decaddci 9546	Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
; ; ;

Theorem	decaddci2 9547	Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
; ; ;

Theorem	decsubi 9548	Difference between a numeral and a nonnegative integer (no underflow). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	decmul1 9549	The product of a numeral with a number (no carry). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	decmul1c 9550	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 9551	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 9552	The product of a numeral with a number (no carry). (Contributed by AV, 15-Jun-2021.)
; ;

Theorem	11multnc 9553	The product of 11 (as numeral) with a number (no carry). (Contributed by AV, 15-Jun-2021.)
; ;

Theorem	decmul10add 9554	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 9555	Lemma for 6p5e11 9558 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)
; ;

Theorem	5p5e10 9556	5 + 5 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
;

Theorem	6p4e10 9557	6 + 4 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
;

Theorem	6p5e11 9558	6 + 5 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	6p6e12 9559	6 + 6 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7p3e10 9560	7 + 3 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
;

Theorem	7p4e11 9561	7 + 4 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	7p5e12 9562	7 + 5 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7p6e13 9563	7 + 6 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7p7e14 9564	7 + 7 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8p2e10 9565	8 + 2 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
;

Theorem	8p3e11 9566	8 + 3 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	8p4e12 9567	8 + 4 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8p5e13 9568	8 + 5 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8p6e14 9569	8 + 6 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8p7e15 9570	8 + 7 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8p8e16 9571	8 + 8 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p2e11 9572	9 + 2 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	9p3e12 9573	9 + 3 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p4e13 9574	9 + 4 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p5e14 9575	9 + 5 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p6e15 9576	9 + 6 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p7e16 9577	9 + 7 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p8e17 9578	9 + 8 = 17. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p9e18 9579	9 + 9 = 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	10p10e20 9580	10 + 10 = 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ; ;

Theorem	10m1e9 9581	10 - 1 = 9. (Contributed by AV, 6-Sep-2021.)
;

Theorem	4t3lem 9582	Lemma for 4t3e12 9583 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)


Theorem	4t3e12 9583	4 times 3 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	4t4e16 9584	4 times 4 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	5t2e10 9585	5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 4-Sep-2021.)
;

Theorem	5t3e15 9586	5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	5t4e20 9587	5 times 4 equals 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	5t5e25 9588	5 times 5 equals 25. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	6t2e12 9589	6 times 2 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	6t3e18 9590	6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	6t4e24 9591	6 times 4 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	6t5e30 9592	6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	6t6e36 9593	6 times 6 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	7t2e14 9594	7 times 2 equals 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t3e21 9595	7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t4e28 9596	7 times 4 equals 28. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t5e35 9597	7 times 5 equals 35. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t6e42 9598	7 times 6 equals 42. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t7e49 9599	7 times 7 equals 49. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t2e16 9600	8 times 2 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

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