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**Theorem List for Metamath Proof Explorer -** 10401-10500 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	9p3e12 10401	9 + 3 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p4e13 10402	9 + 4 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p5e14 10403	9 + 5 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p6e15 10404	9 + 6 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p7e16 10405	9 + 7 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p8e17 10406	9 + 8 = 17. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p9e18 10407	9 + 9 = 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	10p10e20 10408	10 + 10 = 20. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	4t3lem 10409	Lemma for 4t3e12 10410 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)


Theorem	4t3e12 10410	4 times 3 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	4t4e16 10411	4 times 4 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	5t3e15 10412	5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	5t4e20 10413	5 times 4 equals 20. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	5t5e25 10414	5 times 5 equals 25. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	6t2e12 10415	6 times 2 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	6t3e18 10416	6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	6t4e24 10417	6 times 4 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	6t5e30 10418	6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	6t6e36 10419	6 times 6 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t2e14 10420	7 times 2 equals 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t3e21 10421	7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t4e28 10422	7 times 4 equals 28. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t5e35 10423	7 times 5 equals 35. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t6e42 10424	7 times 6 equals 42. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t7e49 10425	7 times 7 equals 49. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t2e16 10426	8 times 2 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t3e24 10427	8 times 3 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t4e32 10428	8 times 4 equals 32. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t5e40 10429	8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t6e48 10430	8 times 6 equals 48. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t7e56 10431	8 times 7 equals 56. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t8e64 10432	8 times 8 equals 64. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t2e18 10433	9 times 2 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t3e27 10434	9 times 3 equals 27. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t4e36 10435	9 times 4 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t5e45 10436	9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t6e54 10437	9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t7e63 10438	9 times 7 equals 63. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t8e72 10439	9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t9e81 10440	9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	decbin0 10441	Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	decbin2 10442	Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	decbin3 10443	Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)


5.4.9 Upper partititions of integers

Syntax	cuz 10444	Extend class notation with the upper integer function. Read " " as "the set of integers greater than or equal to ."


Definition	df-uz 10445*	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 10446 for its value, uzssz 10461 for its relationship to , nnuz 10477 and nn0uz 10476 for its relationships to and , and eluz1 10448 and eluz2 10450 for its membership relations. (Contributed by NM, 5-Sep-2005.)


Theorem	uzval 10446*	The value of the upper integers function. (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)


Theorem	uzf 10447	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 10448	Membership in the set of upper integers starting at . (Contributed by NM, 5-Sep-2005.)
$/\$

Theorem	eluzel2 10449	Implication of membership in a set of upper integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)


Theorem	eluz2 10450	Membership in a set of upper 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 10451	Membership in a set of upper integers. (Contributed by NM, 5-Sep-2005.)
$/\$

Theorem	eluzelz 10452	A member of a set of upper integers is an integer. (Contributed by NM, 6-Sep-2005.)


Theorem	eluzelre 10453	A member of a set of upper integers is a real. (Contributed by Mario Carneiro, 31-Aug-2013.)


Theorem	eluzle 10454	Implication of membership in a set of upper integers. (Contributed by NM, 6-Sep-2005.)


Theorem	eluz 10455	Membership in a set of upper integers. (Contributed by NM, 2-Oct-2005.)
$/\$

Theorem	uzid 10456	Membership of the least member in a set of upper integers. (Contributed by NM, 2-Sep-2005.)


Theorem	uzn0 10457	The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.)


Theorem	uztrn 10458	Transitive law for sets of upper integers. (Contributed by NM, 20-Sep-2005.)
$/\$

Theorem	uztrn2 10459	Transitive law for sets of upper integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
$/\$

Theorem	uzneg 10460	Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.)


Theorem	uzssz 10461	A set of upper integers is a subset of all integers. (Contributed by NM, 2-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)


Theorem	uzss 10462	Subset relationship for two sets of upper integers. (Contributed by NM, 5-Sep-2005.)


Theorem	uztric 10463	Totality 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 10464	The upper integers function is one-to-one. (Contributed by NM, 12-Dec-2005.)


Theorem	eluzp1m1 10465	Membership in the next set of upper integers. (Contributed by NM, 12-Sep-2005.)
$/\$

Theorem	eluzp1l 10466	Strict ordering implied by membership in the next set of upper integers. (Contributed by NM, 12-Sep-2005.)
$/\$

Theorem	eluzp1p1 10467	Membership in the next set of upper integers. (Contributed by NM, 5-Oct-2005.)


Theorem	eluzaddi 10468	Membership in a later set of upper integers. (Contributed by Paul Chapman, 22-Nov-2007.)


Theorem	eluzsubi 10469	Membership in an earlier set of upper integers. (Contributed by Paul Chapman, 22-Nov-2007.)


Theorem	eluzadd 10470	Membership in a later set of upper integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
$/\$

Theorem	eluzsub 10471	Membership in an earlier set of upper integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
$/\$ $/\$

Theorem	uzm1 10472	Choices for an element of an upper interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
$\/$

Theorem	uznn0sub 10473	The nonnegative difference of integers is a nonnegative integer. (Contributed by NM, 4-Sep-2005.)


Theorem	uzin 10474	Intersection of two upper intervals of integers. (Contributed by Mario Carneiro, 24-Dec-2013.)
$/\$

Theorem	uzp1 10475	Choices for an element of an upper interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
$\/$

Theorem	nn0uz 10476	Nonnegative integers expressed as a set of upper integers. (Contributed by NM, 2-Sep-2005.)


Theorem	nnuz 10477	Natural numbers expressed as a set of upper integers. (Contributed by NM, 2-Sep-2005.)


Theorem	elnnuz 10478	A natural number expressed as a member of a set of upper integers. (Contributed by NM, 6-Jun-2006.)


Theorem	elnn0uz 10479	A nonnegative integer expressed as a member a set of upper integers. (Contributed by NM, 6-Jun-2006.)


Theorem	uznnssnn 10480	The upper integers starting from a natural are a subset of the naturals. (Contributed by Scott Fenton, 29-Jun-2013.)


Theorem	raluz 10481*	Restricted universal quantification in a set of upper integers. (Contributed by NM, 9-Sep-2005.)


Theorem	raluz2 10482*	Restricted universal quantification in a set of upper integers. (Contributed by NM, 9-Sep-2005.)


Theorem	rexuz 10483*	Restricted existential quantification in a set of upper integers. (Contributed by NM, 9-Sep-2005.)
$/\$

Theorem	rexuz2 10484*	Restricted existential quantification in a set of upper integers. (Contributed by NM, 9-Sep-2005.)
$/\$ $/\$

Theorem	2rexuz 10485*	Double existential quantification in a set of upper integers. (Contributed by NM, 3-Nov-2005.)
$/\$

Theorem	peano2uz 10486	Second Peano postulate for a set of upper integers. (Contributed by NM, 7-Sep-2005.)


Theorem	peano2uzs 10487	Second Peano postulate for a set of upper integers. (Contributed by Mario Carneiro, 26-Dec-2013.)


Theorem	peano2uzr 10488	Reversed second Peano axiom for upper integers. (Contributed by NM, 2-Jan-2006.)
$/\$

Theorem	uzaddcl 10489	Addition closure law for a set of upper integers. (Contributed by NM, 4-Jun-2006.)
$/\$

Theorem	uzind4 10490*	Induction on the set of upper integers that starts at an integer . The first four hypotheses give us the substitution instances we need, and the last two are the basis and the induction hypothesis. (Contributed by NM, 7-Sep-2005.)


Theorem	uzind4ALT 10491*	Induction on the set of upper integers that starts at an integer . The last four hypotheses give us the substitution instances we need; the first two are the basis and the induction hypothesis. Either uzind4 10490 or uzind4ALT 10491 may be used; see comment for nnind 9974. (Contributed by NM, 7-Sep-2005.)


Theorem	uzind4s 10492*	Induction on the set of upper integers that starts at an integer , using explicit substitution. The hypotheses are the basis and the induction hypothesis. (Contributed by NM, 4-Nov-2005.)


Theorem	uzind4s2 10493*	Induction on the set of upper integers that starts at an integer , using explicit substitution. The hypotheses are the basis and the induction hypothesis. Use this instead of uzind4s 10492 when and must be distinct in . (Contributed by NM, 16-Nov-2005.)


Theorem	uzind4i 10494*	Induction on the upper integers that start at . The first hypothesis specifies the lower bound, the next four give us the substitution instances we need, and the last two are the basis and the induction hypothesis. (Contributed by NM, 4-Sep-2005.)


Theorem	uzwo 10495*	Well-ordering principle: any non-empty subset of a set of upper integers has the least element. (Contributed by NM, 8-Oct-2005.)
$/\$

Theorem	uzwoOLD 10496*	Well-ordering principle: any non-empty subset of the upper integers has the least element. (Contributed by NM, 8-Oct-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
$/\$

Theorem	uzwo2 10497*	Well-ordering principle: any non-empty subset of upper integers has a unique least element. (Contributed by NM, 8-Oct-2005.)
$/\$

Theorem	nnwo 10498*	Well-ordering principle: any non-empty set of natural numbers has a least element. Theorem I.37 (well-ordering principle) of [Apostol] p. 34. (Contributed by NM, 17-Aug-2001.)
$/\$

Theorem	nnwof 10499*	Well-ordering principle: any non-empty set of natural numbers has a least element. This version allows and to be present in as long as they are effectively not free. (Contributed by NM, 17-Aug-2001.) (Revised by Mario Carneiro, 15-Oct-2016.)
$/\$

Theorem	nnwos 10500*	Well-ordering principle: any non-empty set of natural numbers has a least element (schema form). (Contributed by NM, 17-Aug-2001.)
$/\$

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