P. 86 of Theorem List - Intuitionistic Logic Explorer

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**Theorem List for Intuitionistic Logic Explorer -** 8501-8600 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	numlt 8501	Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	numltc 8502	Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	le9lt10 8503	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 8504	Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	decltc 8505	Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
; ; ;

Theorem	declth 8506	Comparing two decimal integers (unequal higher places). (Contributed by AV, 8-Sep-2021.)
; ;

Theorem	decsuc 8507	The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	3declth 8508	Comparing two decimal integers with three "digits" (unequal higher places). (Contributed by AV, 8-Sep-2021.)
;; ;;

Theorem	3decltc 8509	Comparing two decimal integers with three "digits" (unequal higher places). (Contributed by AV, 15-Jun-2021.) (Revised by AV, 6-Sep-2021.)
; ; ;; ;;

Theorem	decle 8510	Comparing two decimal integers (equal higher places). (Contributed by AV, 17-Aug-2021.) (Revised by AV, 8-Sep-2021.)
; ;

Theorem	decleh 8511	Comparing two decimal integers (unequal higher places). (Contributed by AV, 17-Aug-2021.) (Revised by AV, 8-Sep-2021.)
; ;

Theorem	declei 8512	Comparing a digit to a decimal integer. (Contributed by AV, 17-Aug-2021.)
;

Theorem	numlti 8513	Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	declti 8514	Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	decltdi 8515	Comparing a digit to a decimal integer. (Contributed by AV, 8-Sep-2021.)
;

Theorem	numsucc 8516	The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	decsucc 8517	The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	1e0p1 8518	The successor of zero. (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	dec10p 8519	Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	numma 8520	Perform a multiply-add of two decimal integers and against a fixed multiplicand (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	nummac 8521	Perform a multiply-add of two decimal integers and against a fixed multiplicand (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	numma2c 8522	Perform a multiply-add of two decimal integers and against a fixed multiplicand (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	numadd 8523	Add two decimal integers and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	numaddc 8524	Add two decimal integers and (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	nummul1c 8525	The product of a decimal integer with a number. (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	nummul2c 8526	The product of a decimal integer with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	decma 8527	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 8528	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 8529	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 8530	Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
; ; ;

Theorem	decaddc 8531	Add two numerals and (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
; ; ; ;

Theorem	decaddc2 8532	Add two numerals and (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
; ; ; ;

Theorem	decrmanc 8533	Perform a multiply-add of two numerals and against a fixed multiplicand (no carry). (Contributed by AV, 16-Sep-2021.)
; ;

Theorem	decrmac 8534	Perform a multiply-add of two numerals and against a fixed multiplicand (with carry). (Contributed by AV, 16-Sep-2021.)
; ; ;

Theorem	decaddm10 8535	The sum of two multiples of 10 is a multiple of 10. (Contributed by AV, 30-Jul-2021.)
; ; ;

Theorem	decaddi 8536	Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
; ;

Theorem	decaddci 8537	Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
; ; ;

Theorem	decaddci2 8538	Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
; ; ;

Theorem	decsubi 8539	Difference between a numeral and a nonnegative integer (no underflow). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	decmul1 8540	The product of a numeral with a number (no carry). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	decmul1c 8541	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 8542	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 8543	The product of a numeral with a number (no carry). (Contributed by AV, 15-Jun-2021.)
; ;

Theorem	11multnc 8544	The product of 11 (as numeral) with a number (no carry). (Contributed by AV, 15-Jun-2021.)
; ;

Theorem	decmul10add 8545	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 8546	Lemma for 6p5e11 8549 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)
; ;

Theorem	5p5e10 8547	5 + 5 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
;

Theorem	6p4e10 8548	6 + 4 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
;

Theorem	6p5e11 8549	6 + 5 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	6p6e12 8550	6 + 6 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7p3e10 8551	7 + 3 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
;

Theorem	7p4e11 8552	7 + 4 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	7p5e12 8553	7 + 5 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7p6e13 8554	7 + 6 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7p7e14 8555	7 + 7 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8p2e10 8556	8 + 2 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
;

Theorem	8p3e11 8557	8 + 3 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	8p4e12 8558	8 + 4 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8p5e13 8559	8 + 5 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8p6e14 8560	8 + 6 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8p7e15 8561	8 + 7 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8p8e16 8562	8 + 8 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p2e11 8563	9 + 2 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	9p3e12 8564	9 + 3 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p4e13 8565	9 + 4 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p5e14 8566	9 + 5 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p6e15 8567	9 + 6 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p7e16 8568	9 + 7 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p8e17 8569	9 + 8 = 17. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p9e18 8570	9 + 9 = 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	10p10e20 8571	10 + 10 = 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ; ;

Theorem	10m1e9 8572	10 - 1 = 9. (Contributed by AV, 6-Sep-2021.)
;

Theorem	4t3lem 8573	Lemma for 4t3e12 8574 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)


Theorem	4t3e12 8574	4 times 3 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	4t4e16 8575	4 times 4 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	5t2e10 8576	5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 4-Sep-2021.)
;

Theorem	5t3e15 8577	5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	5t4e20 8578	5 times 4 equals 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	5t5e25 8579	5 times 5 equals 25. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	6t2e12 8580	6 times 2 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	6t3e18 8581	6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	6t4e24 8582	6 times 4 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	6t5e30 8583	6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	6t6e36 8584	6 times 6 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	7t2e14 8585	7 times 2 equals 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t3e21 8586	7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t4e28 8587	7 times 4 equals 28. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t5e35 8588	7 times 5 equals 35. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t6e42 8589	7 times 6 equals 42. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t7e49 8590	7 times 7 equals 49. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t2e16 8591	8 times 2 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t3e24 8592	8 times 3 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t4e32 8593	8 times 4 equals 32. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t5e40 8594	8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	8t6e48 8595	8 times 6 equals 48. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	8t7e56 8596	8 times 7 equals 56. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t8e64 8597	8 times 8 equals 64. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t2e18 8598	9 times 2 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t3e27 8599	9 times 3 equals 27. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t4e36 8600	9 times 4 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

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