P. 85 of Theorem List - Intuitionistic Logic Explorer

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**Theorem List for Intuitionistic Logic Explorer -** 8401-8500 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	numadd 8401	Add two decimal integers and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	numaddc 8402	Add two decimal integers and (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	nummul1c 8403	The product of a decimal integer with a number. (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	nummul2c 8404	The product of a decimal integer with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	decma 8405	Perform a multiply-add of two numerals and against a fixed multiplicand (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
; ; ;

Theorem	decmac 8406	Perform a multiply-add of two numerals and against a fixed multiplicand (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
; ; ; ;

Theorem	decma2c 8407	Perform a multiply-add of two numerals and against a fixed multiplicand (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
; ; ; ;

Theorem	decadd 8408	Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
; ; ;

Theorem	decaddc 8409	Add two numerals and (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
; ; ; ;

Theorem	decaddc2 8410	Add two numerals and (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
; ; ;

Theorem	decaddi 8411	Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
; ;

Theorem	decaddci 8412	Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
; ; ;

Theorem	decaddci2 8413	Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
; ;

Theorem	decmul1c 8414	The product of a numeral with a number. (Contributed by Mario Carneiro, 18-Feb-2014.)
; ; ;

Theorem	decmul2c 8415	The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
; ; ;

Theorem	6p5lem 8416	Lemma for 6p5e11 8417 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)
; ;

Theorem	6p5e11 8417	6 + 5 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	6p6e12 8418	6 + 6 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7p4e11 8419	7 + 4 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7p5e12 8420	7 + 5 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7p6e13 8421	7 + 6 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7p7e14 8422	7 + 7 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8p3e11 8423	8 + 3 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8p4e12 8424	8 + 4 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8p5e13 8425	8 + 5 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8p6e14 8426	8 + 6 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8p7e15 8427	8 + 7 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8p8e16 8428	8 + 8 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p2e11 8429	9 + 2 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p3e12 8430	9 + 3 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p4e13 8431	9 + 4 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p5e14 8432	9 + 5 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p6e15 8433	9 + 6 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p7e16 8434	9 + 7 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p8e17 8435	9 + 8 = 17. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p9e18 8436	9 + 9 = 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	10p10e20 8437	10 + 10 = 20. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	4t3lem 8438	Lemma for 4t3e12 8439 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)


Theorem	4t3e12 8439	4 times 3 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	4t4e16 8440	4 times 4 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	5t3e15 8441	5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	5t4e20 8442	5 times 4 equals 20. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	5t5e25 8443	5 times 5 equals 25. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	6t2e12 8444	6 times 2 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	6t3e18 8445	6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	6t4e24 8446	6 times 4 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	6t5e30 8447	6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	6t6e36 8448	6 times 6 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t2e14 8449	7 times 2 equals 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t3e21 8450	7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t4e28 8451	7 times 4 equals 28. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t5e35 8452	7 times 5 equals 35. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t6e42 8453	7 times 6 equals 42. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t7e49 8454	7 times 7 equals 49. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t2e16 8455	8 times 2 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t3e24 8456	8 times 3 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t4e32 8457	8 times 4 equals 32. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t5e40 8458	8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t6e48 8459	8 times 6 equals 48. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t7e56 8460	8 times 7 equals 56. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t8e64 8461	8 times 8 equals 64. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t2e18 8462	9 times 2 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t3e27 8463	9 times 3 equals 27. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t4e36 8464	9 times 4 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t5e45 8465	9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t6e54 8466	9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t7e63 8467	9 times 7 equals 63. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t8e72 8468	9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t9e81 8469	9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	decbin0 8470	Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	decbin2 8471	Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	decbin3 8472	Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)


3.4.10 Upper sets of integers

Syntax	cuz 8473	Extend class notation with the upper integer function. Read " " as "the set of integers greater than or equal to ."


Definition	df-uz 8474*	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 8475 for its value, uzssz 8492 for its relationship to , nnuz 8508 and nn0uz 8507 for its relationships to and , and eluz1 8477 and eluz2 8479 for its membership relations. (Contributed by NM, 5-Sep-2005.)


Theorem	uzval 8475*	The value of the upper integers function. (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)


Theorem	uzf 8476	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 8477	Membership in the upper set of integers starting at . (Contributed by NM, 5-Sep-2005.)
$/\$

Theorem	eluzel2 8478	Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)


Theorem	eluz2 8479	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 8480	Membership in an upper set of integers. (Contributed by NM, 5-Sep-2005.)
$/\$

Theorem	eluzuzle 8481	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 8482	A member of an upper set of integers is an integer. (Contributed by NM, 6-Sep-2005.)


Theorem	eluzelre 8483	A member of an upper set of integers is a real. (Contributed by Mario Carneiro, 31-Aug-2013.)


Theorem	eluzelcn 8484	A member of an upper set of integers is a complex number. (Contributed by Glauco Siliprandi, 29-Jun-2017.)


Theorem	eluzle 8485	Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.)


Theorem	eluz 8486	Membership in an upper set of integers. (Contributed by NM, 2-Oct-2005.)
$/\$

Theorem	uzid 8487	Membership of the least member in an upper set of integers. (Contributed by NM, 2-Sep-2005.)


Theorem	uzn0 8488	The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.)


Theorem	uztrn 8489	Transitive law for sets of upper integers. (Contributed by NM, 20-Sep-2005.)
$/\$

Theorem	uztrn2 8490	Transitive law for sets of upper integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
$/\$

Theorem	uzneg 8491	Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.)


Theorem	uzssz 8492	An upper set of integers is a subset of all integers. (Contributed by NM, 2-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)


Theorem	uzss 8493	Subset relationship for two sets of upper integers. (Contributed by NM, 5-Sep-2005.)


Theorem	uztric 8494	Trichotomy of the ordering relation on integers, stated in terms of upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 25-Jun-2013.)
$/\$ $\/$

Theorem	uz11 8495	The upper integers function is one-to-one. (Contributed by NM, 12-Dec-2005.)


Theorem	eluzp1m1 8496	Membership in the next upper set of integers. (Contributed by NM, 12-Sep-2005.)
$/\$

Theorem	eluzp1l 8497	Strict ordering implied by membership in the next upper set of integers. (Contributed by NM, 12-Sep-2005.)
$/\$

Theorem	eluzp1p1 8498	Membership in the next upper set of integers. (Contributed by NM, 5-Oct-2005.)


Theorem	eluzaddi 8499	Membership in a later upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.)


Theorem	eluzsubi 8500	Membership in an earlier upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.)

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