P. 95 of Theorem List - Intuitionistic Logic Explorer

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**Theorem List for Intuitionistic Logic Explorer -** 9401-9500 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	decaddm10 9401	The sum of two multiples of 10 is a multiple of 10. (Contributed by AV, 30-Jul-2021.)
; ; ;

Theorem	decaddi 9402	Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
; ;

Theorem	decaddci 9403	Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
; ; ;

Theorem	decaddci2 9404	Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
; ; ;

Theorem	decsubi 9405	Difference between a numeral and a nonnegative integer (no underflow). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	decmul1 9406	The product of a numeral with a number (no carry). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	decmul1c 9407	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 9408	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 9409	The product of a numeral with a number (no carry). (Contributed by AV, 15-Jun-2021.)
; ;

Theorem	11multnc 9410	The product of 11 (as numeral) with a number (no carry). (Contributed by AV, 15-Jun-2021.)
; ;

Theorem	decmul10add 9411	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 9412	Lemma for 6p5e11 9415 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)
; ;

Theorem	5p5e10 9413	5 + 5 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
;

Theorem	6p4e10 9414	6 + 4 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
;

Theorem	6p5e11 9415	6 + 5 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	6p6e12 9416	6 + 6 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7p3e10 9417	7 + 3 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
;

Theorem	7p4e11 9418	7 + 4 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	7p5e12 9419	7 + 5 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7p6e13 9420	7 + 6 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7p7e14 9421	7 + 7 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8p2e10 9422	8 + 2 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
;

Theorem	8p3e11 9423	8 + 3 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	8p4e12 9424	8 + 4 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8p5e13 9425	8 + 5 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8p6e14 9426	8 + 6 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8p7e15 9427	8 + 7 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8p8e16 9428	8 + 8 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p2e11 9429	9 + 2 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	9p3e12 9430	9 + 3 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p4e13 9431	9 + 4 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p5e14 9432	9 + 5 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p6e15 9433	9 + 6 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p7e16 9434	9 + 7 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p8e17 9435	9 + 8 = 17. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p9e18 9436	9 + 9 = 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	10p10e20 9437	10 + 10 = 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ; ;

Theorem	10m1e9 9438	10 - 1 = 9. (Contributed by AV, 6-Sep-2021.)
;

Theorem	4t3lem 9439	Lemma for 4t3e12 9440 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)


Theorem	4t3e12 9440	4 times 3 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	4t4e16 9441	4 times 4 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	5t2e10 9442	5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 4-Sep-2021.)
;

Theorem	5t3e15 9443	5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	5t4e20 9444	5 times 4 equals 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	5t5e25 9445	5 times 5 equals 25. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	6t2e12 9446	6 times 2 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	6t3e18 9447	6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	6t4e24 9448	6 times 4 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	6t5e30 9449	6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	6t6e36 9450	6 times 6 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	7t2e14 9451	7 times 2 equals 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t3e21 9452	7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t4e28 9453	7 times 4 equals 28. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t5e35 9454	7 times 5 equals 35. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t6e42 9455	7 times 6 equals 42. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t7e49 9456	7 times 7 equals 49. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t2e16 9457	8 times 2 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t3e24 9458	8 times 3 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t4e32 9459	8 times 4 equals 32. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t5e40 9460	8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	8t6e48 9461	8 times 6 equals 48. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	8t7e56 9462	8 times 7 equals 56. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t8e64 9463	8 times 8 equals 64. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t2e18 9464	9 times 2 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t3e27 9465	9 times 3 equals 27. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t4e36 9466	9 times 4 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t5e45 9467	9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t6e54 9468	9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t7e63 9469	9 times 7 equals 63. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t8e72 9470	9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t9e81 9471	9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t11e99 9472	9 times 11 equals 99. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	9lt10 9473	9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.)
;

Theorem	8lt10 9474	8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.)
;

Theorem	7lt10 9475	7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
;

Theorem	6lt10 9476	6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
;

Theorem	5lt10 9477	5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
;

Theorem	4lt10 9478	4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
;

Theorem	3lt10 9479	3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
;

Theorem	2lt10 9480	2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
;

Theorem	1lt10 9481	1 is less than 10. (Contributed by NM, 7-Nov-2012.) (Revised by Mario Carneiro, 9-Mar-2015.) (Revised by AV, 8-Sep-2021.)
;

Theorem	decbin0 9482	Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	decbin2 9483	Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	decbin3 9484	Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	halfthird 9485	Half minus a third. (Contributed by Scott Fenton, 8-Jul-2015.)


Theorem	5recm6rec 9486	One fifth minus one sixth. (Contributed by Scott Fenton, 9-Jan-2017.)
;

4.4.11 Upper sets of integers

Syntax	cuz 9487	Extend class notation with the upper integer function. Read " " as "the set of integers greater than or equal to ".


Definition	df-uz 9488*	Define a function whose value at is the semi-infinite set of contiguous integers starting at , which we will also call the upper integers starting at . Read " " as "the set of integers greater than or equal to ". See uzval 9489 for its value, uzssz 9506 for its relationship to , nnuz 9522 and nn0uz 9521 for its relationships to and , and eluz1 9491 and eluz2 9493 for its membership relations. (Contributed by NM, 5-Sep-2005.)


Theorem	uzval 9489*	The value of the upper integers function. (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)


Theorem	uzf 9490	The domain and range of the upper integers function. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 3-Nov-2013.)


Theorem	eluz1 9491	Membership in the upper set of integers starting at . (Contributed by NM, 5-Sep-2005.)
$/\$

Theorem	eluzel2 9492	Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)


Theorem	eluz2 9493	Membership in an upper set of integers. We use the fact that a function's value (under our function value definition) is empty outside of its domain to show . (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
$/\$ $/\$

Theorem	eluz1i 9494	Membership in an upper set of integers. (Contributed by NM, 5-Sep-2005.)
$/\$

Theorem	eluzuzle 9495	An integer in an upper set of integers is an element of an upper set of integers with a smaller bound. (Contributed by Alexander van der Vekens, 17-Jun-2018.)
$/\$

Theorem	eluzelz 9496	A member of an upper set of integers is an integer. (Contributed by NM, 6-Sep-2005.)


Theorem	eluzelre 9497	A member of an upper set of integers is a real. (Contributed by Mario Carneiro, 31-Aug-2013.)


Theorem	eluzelcn 9498	A member of an upper set of integers is a complex number. (Contributed by Glauco Siliprandi, 29-Jun-2017.)


Theorem	eluzle 9499	Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.)


Theorem	eluz 9500	Membership in an upper set of integers. (Contributed by NM, 2-Oct-2005.)
$/\$

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