mmtheorems72 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 7101-7200 - Page 72 of 175

Type	Label	Description
Statement
Theorem	maxle 7101	Two ways of saying the maximum of two numbers is less than or equal to a third.
|- ((A e. RR /\ B e. RR /\ C e. RR) -> (if(A <_ B, B, A) <_ C <-> (A <_ C /\ B <_ C)))
Theorem	min1 7102	The minimum of two numbers is less than or equal to the first.
|- ((A e. RR /\ B e. RR) -> if(A <_ B, A, B) <_ A)
Theorem	min2 7103	The minimum of two numbers is less than or equal to the second.
|- ((A e. RR /\ B e. RR) -> if(A <_ B, A, B) <_ B)
Theorem	lemin 7104	Two ways of saying a number is less than or equal to the minimum of two others.
|- ((A e. RR /\ B e. RR /\ C e. RR) -> (A <_ if(B <_ C, B, C) <-> (A <_ B /\ A <_ C)))
Theorem	maxlt 7105	Two ways of saying the maximum of two numbers is less than a third.
|- ((A e. RR /\ B e. RR /\ C e. RR) -> (if(A <_ B, B, A) < C <-> (A < C /\ B < C)))
Theorem	ltmin 7106	Two ways of saying a number is less than the minimum of two others.
|- ((A e. RR /\ B e. RR /\ C e. RR) -> (A < if(B <_ C, B, C) <-> (A < B /\ A < C)))
Theorem	squeeze0 7107	If a nonnegative number is less than any positive number, it is zero.
|- ((A e. RR /\ 0 <_ A /\ A.x e. RR (0 < x -> A < x)) -> A = 0)
Natural numbers (as a subset of complex numbers)
Definition	df-n 7108	The natural numbers of analysis start at one (unlike the ordinal natural numbers, i.e. the members of the set om, df-om 3950, which start at zero). This is the convention used by most analysis books, and it is often convenient in proofs because we don't have to worry about division by zero. See nnind 7120 for the principle of mathematical induction. See dfnn2 7119 for a slight variant. See df-n0 7309 for the set of nonnegative integers NN0 starting at zero. See dfn2 7321 for NN defined in terms of NN0.
|- NN = |^|{x | (1 e. x /\ A.y e. x (y + 1) e. x)}
Theorem	peano5nni 7109	Peano's inductive postulate. Theorem I.36 (principle of mathematical induction) of [Apostol] p. 34.
|- A e. _V   =>   |- ((1 e. A /\ A.x e. A (x + 1) e. A) -> NN C_ A)
Theorem	nnssre 7110	The natural numbers are a subset of the reals.
|- NN C_ RR
Theorem	nnsscn 7111	The natural numbers are a subset of the complex numbers.
|- NN C_ CC
Theorem	nnre 7112	A natural number is a real number.
|- (A e. NN -> A e. RR)
Theorem	nncn 7113	A natural number is a complex number.
|- (A e. NN -> A e. CC)
Theorem	nnrei 7114	A natural number is a real number.
|- A e. NN   =>   |- A e. RR
Theorem	nncni 7115	A natural number is a complex number.
|- A e. NN   =>   |- A e. CC
Theorem	nnex 7116	The set of natural numbers exists.
|- NN e. _V
Theorem	1nn 7117	Peano postulate: 1 is a natural number.
|- 1 e. NN
Theorem	peano2nn 7118	Peano postulate: a successor of a natural number is a natural number.
|- (A e. NN -> (A + 1) e. NN)
Theorem	dfnn2 7119	Alternate definition of the set of natural numbers. Definition of positive integers in [Apostol] p. 22.
|- NN = |^|{x | (x C_ RR /\ 1 e. x /\ A.y e. x (y + 1) e. x)}
Principle of mathematical induction
Theorem	nnind 7120	Principle of Mathematical Induction (inference schema). The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis. See nnaddcl 7123 for an example of its use. See nn0ind 7424 for induction on nonnegative integers and uzind 7417, uzind4 7619 for induction on an arbitrary set of upper integers. See indstr 7630 for strong induction.
|- (x = 1 -> (ph <-> ps))   &   |- (x = y -> (ph <-> ch))   &   |- (x = (y + 1) -> (ph <-> th))   &   |- (x = A -> (ph <-> ta))   &   |- ps   &   |- (y e. NN -> (ch -> th))   =>   |- (A e. NN -> ta)
Theorem	nnindALT 7121	Principle of Mathematical Induction (inference schema). The last four hypotheses give us the substitution instances we need; the first two are the induction hypothesis and the basis. (This ALT version of nnind 7120 is easier to use with the Proof Assistant since 'assign last' will be applied to the substitution instances first. We may switch to it as the official version.)
|- (y e. NN -> (ch -> th))   &   |- ps   &   |- (x = 1 -> (ph <-> ps))   &   |- (x = y -> (ph <-> ch))   &   |- (x = (y + 1) -> (ph <-> th))   &   |- (x = A -> (ph <-> ta))   =>   |- (A e. NN -> ta)
Natural numbers (cont.)
Theorem	nn1suc 7122	If a statement holds for 1 and also holds for a successor, it holds for all natural numbers. The first three hypotheses give us the substitution instances we need; the last two show that it holds for 1 and for a successor.
|- (x = 1 -> (ph <-> ps))   &   |- (x = (y + 1) -> (ph <-> ch))   &   |- (x = A -> (ph <-> th))   &   |- ps   &   |- (y e. NN -> ch)   =>   |- (A e. NN -> th)
Theorem	nnaddcl 7123	Closure of addition of natural numbers, proved by induction on the second addend.
|- ((A e. NN /\ B e. NN) -> (A + B) e. NN)
Theorem	nnmulcl 7124	Closure of multiplication of natural numbers.
|- ((A e. NN /\ B e. NN) -> (A x. B) e. NN)
Theorem	nn2ge 7125	There exists a natural number greater than or equal to any two others.
|- ((A e. NN /\ B e. NN) -> E.x e. NN (A <_ x /\ B <_ x))
Theorem	nnge1 7126	A natural number is one or greater.
|- (A e. NN -> 1 <_ A)
Theorem	nngt1ne1 7127	A natural number is greater than one iff it is not equal to one.
|- (A e. NN -> (1 < A <-> A =/= 1))
Theorem	nnle1eq1 7128	A natural number is less than or equal to one iff it is equal to one.
|- (A e. NN -> (A <_ 1 <-> A = 1))
Theorem	nngt0 7129	A natural number is positive.
|- (A e. NN -> 0 < A)
Theorem	lt1nnn 7130	A number less than one is not a natural number.
|- ((A e. RR /\ A < 1) -> -. A e. NN)
Theorem	0nnn 7131	Zero is not a natural number.
|- -. 0 e. NN
Theorem	nnne0 7132	A natural number is nonzero.
|- (A e. NN -> A =/= 0)
Theorem	nngt0i 7133	A natural number is positive (inference version).
|- A e. NN   =>   |- 0 < A
Theorem	nnne0i 7134	A natural number is nonzero (inference version).
|- A e. NN   =>   |- A =/= 0
Theorem	nndivre 7135	The quotient of a real and a natural number is real.
|- ((A e. RR /\ N e. NN) -> (A / N) e. RR)
Theorem	nnrecre 7136	The reciprocal of a natural number is real.
|- (N e. NN -> (1 / N) e. RR)
Theorem	nnrecgt0 7137	The reciprocal of a natural number is positive.
|- (A e. NN -> 0 < (1 / A))
Theorem	nnleltp1 7138	Natural number ordering relation.
|- ((A e. NN /\ B e. NN) -> (A <_ B <-> A < (B + 1)))
Theorem	nnltp1le 7139	Natural number ordering relation.
|- ((A e. NN /\ B e. NN) -> (A < B <-> (A + 1) <_ B))
Theorem	nnsubi 7140	Subtraction of natural numbers.
|- A e. NN   &   |- B e. NN   =>   |- (A < B <-> (B - A) e. NN)
Theorem	nnsub 7141	Subtraction of natural numbers.
|- ((A e. NN /\ B e. NN) -> (A < B <-> (B - A) e. NN))
Theorem	nnaddm1cl 7142	Closure of addition of natural numbers minus one.
|- ((A e. NN /\ B e. NN) -> ((A + B) - 1) e. NN)
Theorem	nndiv 7143	Two ways to express "A divides B " for natural numbers.
|- ((A e. NN /\ B e. NN) -> (E.x e. NN (A x. x) = B <-> (B / A) e. NN))
Theorem	nndivtr 7144	Transitive property of divisibility: if A divides B and B divides C, then A divides C. Typically C would be an integer, although the theorem holds for complex C.
|- (((A e. NN /\ B e. NN /\ C e. CC) /\ ((B / A) e. NN /\ (C / B) e. NN)) -> (C / A) e. NN)
Decimal representation of numbers
Syntax	c2 7145	Extend class notation to include the number 2.
class 2
Syntax	c3 7146	Extend class notation to include the number 3.
class 3
Syntax	c4 7147	Extend class notation to include the number 4.
class 4
Syntax	c5 7148	Extend class notation to include the number 5.
class 5
Syntax	c6 7149	Extend class notation to include the number 6.
class 6
Syntax	c7 7150	Extend class notation to include the number 7.
class 7
Syntax	c8 7151	Extend class notation to include the number 8.
class 8
Syntax	c9 7152	Extend class notation to include the number 9.
class 9
Syntax	c10 7153	Extend class notation to include the number 10.
class 10
Definition	df-2 7154	Define the number 2. Note that the numbers 0 and 1 are constants defined as primitives of the complex number axiom system (see df-0 6393 and df-1 6394). Note: Only the digits 0 through 9 (df-0 6393 through df-9 7161) and the number 10 (df-10 7162) are explicitly defined. Integers can be exhibited as sums of powers of 10 or as some other expression built from operations on the numbers 0 through 10. For example, the prime number 823541 can be expressed as (7^7) - 2. Decimals can be expressed as ratios of integers, as in cos2bnd 8741. (Fortunately, most abstract math rarely requires numbers larger than 4. Even in Wiles' proof of Fermat's Last Theorem, the largest number used appears to be 12.) A decimal representation of numbers may be added at some point in the future if it is deemed useful. Ideas for a clean, eliminable definition are welcome. (An awkward earlier definition was deleted from the database on 18-Sep-1999.)
|- 2 = (1 + 1)
Definition	df-3 7155	Define the number 3.
|- 3 = (2 + 1)
Definition	df-4 7156	Define the number 4.
|- 4 = (3 + 1)
Definition	df-5 7157	Define the number 5.
|- 5 = (4 + 1)
Definition	df-6 7158	Define the number 6.
|- 6 = (5 + 1)
Definition	df-7 7159	Define the number 7.
|- 7 = (6 + 1)
Definition	df-8 7160	Define the number 8.
|- 8 = (7 + 1)
Definition	df-9 7161	Define the number 9.
|- 9 = (8 + 1)
Definition	df-10 7162	Define the number 10. See remarks under df-2 7154.
|- 10 = (9 + 1)
Theorem	2re 7163	The number 2 is real.
|- 2 e. RR
Theorem	2cn 7164	The number 2 is a complex number.
|- 2 e. CC
Theorem	3re 7165	The number 3 is real.
|- 3 e. RR
Theorem	4re 7166	The number 4 is real.
|- 4 e. RR
Theorem	5re 7167	The number 5 is real.
|- 5 e. RR
Theorem	6re 7168	The number 6 is real.
|- 6 e. RR
Theorem	7re 7169	The number 7 is real.
|- 7 e. RR
Theorem	8re 7170	The number 8 is real.
|- 8 e. RR
Theorem	9re 7171	The number 9 is real.
|- 9 e. RR
Theorem	10re 7172	The number 10 is real.
|- 10 e. RR
Theorem	2pos 7173	The number 2 is positive.
|- 0 < 2
Theorem	2ne0 7174	The number 2 is nonzero.
|- 2 =/= 0
Theorem	3pos 7175	The number 3 is positive.
|- 0 < 3
Theorem	4pos 7176	The number 4 is positive.
|- 0 < 4
Theorem	5pos 7177	The number 5 is positive.
|- 0 < 5
Theorem	6pos 7178	The number 6 is positive.
|- 0 < 6
Theorem	7pos 7179	The number 7 is positive.
|- 0 < 7
Theorem	8pos 7180	The number 8 is positive.
|- 0 < 8
Theorem	9pos 7181	The number 9 is positive.
|- 0 < 9
Theorem	10pos 7182	The number 10 is positive.
|- 0 < 10
Theorem	2nn 7183	2 is a natural number.
|- 2 e. NN
Theorem	3nn 7184	3 is a natural number.
|- 3 e. NN
Some properties of specific numbers
Theorem	2p2e4 7185	Two plus two equals four. For more information, see "2+2=4 Trivia" on the Metamath Proof Explorer Home Page: http://us.metamath.org/mpegif/mmset.html#trivia.
|- (2 + 2) = 4
Theorem	4nn 7186	4 is a natural number.
|- 4 e. NN
Theorem	2timesi 7187	Two times a number.
|- A e. CC   =>   |- (2 x. A) = (A + A)
Theorem	2times 7188	Two times a number.
|- (A e. CC -> (2 x. A) = (A + A))
Theorem	times2 7189	A number times 2.
|- (A e. CC -> (A x. 2) = (A + A))
Theorem	times2i 7190	A number times 2.
|- A e. CC   =>   |- (A x. 2) = (A + A)
Theorem	3p2e5 7191	3 + 2 = 5.
|- (3 + 2) = 5
Theorem	3p3e6 7192	3 + 3 = 6.
|- (3 + 3) = 6
Theorem	4p2e6 7193	4 + 2 = 6.
|- (4 + 2) = 6
Theorem	4p3e7 7194	4 + 3 = 7.
|- (4 + 3) = 7
Theorem	4p4e8 7195	4 + 4 = 8.
|- (4 + 4) = 8
Theorem	5p2e7 7196	5 + 2 = 7.
|- (5 + 2) = 7
Theorem	5p3e8 7197	5 + 3 = 8.
|- (5 + 3) = 8
Theorem	5p4e9 7198	5 + 4 = 9.
|- (5 + 4) = 9
Theorem	5p5e10 7199	5 + 5 = 10.
|- (5 + 5) = 10
Theorem	6p2e8 7200	6 + 2 = 8.
|- (6 + 2) = 8