mmtheorems61 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 6001-6100 - Page 61 of 108

Type	Label	Description
Statement
Theorem	10pos 6001	The number 10 is positive.
|- 0 < 10
Theorem	2nn 6002	2 is a natural number.
|- 2 e. NN
Theorem	3nn 6003	3 is a natural number.
|- 3 e. NN
Some properties of specific numbers
Theorem	2p2e4 6004	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 6005	4 is a natural number.
|- 4 e. NN
Theorem	2times 6006	Two times a number.
|- A e. CC   =>   |- (2 x. A) = (A + A)
Theorem	2timest 6007	Two times a number.
|- (A e. CC -> (2 x. A) = (A + A))
Theorem	times2t 6008	A number times 2.
|- (A e. CC -> (A x. 2) = (A + A))
Theorem	times2 6009	A number times 2.
|- A e. CC   =>   |- (A x. 2) = (A + A)
Theorem	3p2e5 6010	3 + 2 = 5.
|- (3 + 2) = 5
Theorem	3p3e6 6011	3 + 3 = 6.
|- (3 + 3) = 6
Theorem	4p2e6 6012	4 + 2 = 6.
|- (4 + 2) = 6
Theorem	4p3e7 6013	4 + 3 = 7.
|- (4 + 3) = 7
Theorem	4p4e8 6014	4 + 4 = 8.
|- (4 + 4) = 8
Theorem	5p2e7 6015	5 + 2 = 7.
|- (5 + 2) = 7
Theorem	5p3e8 6016	5 + 3 = 8.
|- (5 + 3) = 8
Theorem	5p4e9 6017	5 + 4 = 9.
|- (5 + 4) = 9
Theorem	5p5e10 6018	5 + 5 = 10.
|- (5 + 5) = 10
Theorem	6p2e8 6019	6 + 2 = 8.
|- (6 + 2) = 8
Theorem	6p3e9 6020	6 + 3 = 9.
|- (6 + 3) = 9
Theorem	6p4e10 6021	6 + 4 = 10.
|- (6 + 4) = 10
Theorem	7p2e9 6022	7 + 2 = 9.
|- (7 + 2) = 9
Theorem	7p3e10 6023	7 + 3 = 10.
|- (7 + 3) = 10
Theorem	8p2e10 6024	8 + 2 = 10.
|- (8 + 2) = 10
Theorem	2t2e4 6025	2 times 2 equals 4.
|- (2 x. 2) = 4
Theorem	3t2e6 6026	3 times 2 equals 6.
|- (3 x. 2) = 6
Theorem	3t3e9 6027	3 times 3 equals 9.
|- (3 x. 3) = 9
Theorem	4t2e8 6028	4 times 2 equals 8.
|- (4 x. 2) = 8
Theorem	5t2e10 6029	5 times 2 equals 10.
|- (5 x. 2) = 10
Theorem	4d2e2 6030	One half of four is two.
|- (4 / 2) = 2
Theorem	1lt2 6031	1 is less than 2.
|- 1 < 2
Theorem	halfgt0 6032	One-half is greater than zero.
|- 0 < (1 / 2)
Theorem	halflt1 6033	One-half is less than one.
|- (1 / 2) < 1
Theorem	8th4div3 6034	An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)
|- ((1 / 8) x. (4 / 3)) = (1 / 6)
Theorem	halfpm6th 6035	One half plus or minus one sixth. (Contributed by Paul Chapman, 17-Jan-2008.)
|- (((1 / 2) - (1 / 6)) = (1 / 3) /\ ((1 / 2) + (1 / 6)) = (2 / 3))
Theorem	halfclt 6036	Closure of half of a number (frequently used special case).
|- (A e. CC -> (A / 2) e. CC)
Theorem	rehalfclt 6037	Real closure of half.
|- (A e. RR -> (A / 2) e. RR)
Theorem	half0t 6038	Half of a number is zero iff the number is zero.
|- (A e. CC -> ((A / 2) = 0 <-> A = 0))
Theorem	halfpost 6039	A positive number is greater than its half.
|- (A e. RR -> (0 < A <-> (A / 2) < A))
Theorem	halfpos2t 6040	A number is positive iff its half is positive.
|- (A e. RR -> (0 < A <-> 0 < (A / 2)))
Theorem	halfnneg2t 6041	A number is nonnegative iff its half is nonnegative.
|- (A e. RR -> (0 <_ A <-> 0 <_ (A / 2)))
Theorem	2halvest 6042	Two halves make a whole.
|- (A e. CC -> ((A / 2) + (A / 2)) = A)
Theorem	halfaddsubcl 6043	Closure of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)
|- ((A e. CC /\ B e. CC) -> (((A + B) / 2) e. CC /\ ((A - B) / 2) e. CC))
Theorem	halfaddsubt 6044	Sum and difference of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)
|- ((A e. CC /\ B e. CC) -> ((((A + B) / 2) + ((A - B) / 2)) = A /\ (((A + B) / 2) - ((A - B) / 2)) = B))
Theorem	lt2halvest 6045	A sum is less than the whole if each term is less than half.
|- ((A e. RR /\ B e. RR /\ C e. RR) -> ((A < (C / 2) /\ B < (C / 2)) -> (A + B) < C))
Theorem	nominpos 6046	There is no smallest positive real number.
|- -. E.x e. RR (0 < x /\ -. E.y e. RR (0 < y /\ y < x))
Theorem	avglet 6047	The average of two numbers is less than or equal to at least one of them.
|- ((A e. RR /\ B e. RR) -> (((A + B) / 2) <_ A \/ ((A + B) / 2) <_ B))
Completeness Axiom and Suprema
Theorem	lbreu 6048	If a set of reals contains a lower bound, it contains a unique lower bound.
|- ((S (_ RR /\ E.x e. S A.y e. S x <_ y) -> E!x e. S A.y e. S x <_ y)
Theorem	lbcl 6049	If a set of reals contains a lower bound, it contains a unique lower bound that belongs to the set.
|- ((S (_ RR /\ E.x e. S A.y e. S x <_ y) -> U.{x e. S | A.y e. S x <_ y} e. S)
Theorem	lble 6050	If a set of reals contains a lower bound, the lower bound is less than or equal to all members of the set.
|- ((S (_ RR /\ E.x e. S A.y e. S x <_ y /\ A e. S) -> U.{x e. S | A.y e. S x <_ y} <_ A)
Theorem	lbinfm 6051	If a set of reals contains a lower bound, the lower bound is its infimum.
|- ((S (_ RR /\ E.x e. S A.y e. S x <_ y) -> sup(S, RR, `' < ) = U.{x e. S | A.y e. S x <_ y})
Theorem	lbinfmcl 6052	If a set of reals contains a lower bound, it contains its infimum.
|- ((S (_ RR /\ E.x e. S A.y e. S x <_ y) -> sup(S, RR, `' < ) e. S)
Theorem	lbinfmle 6053	If a set of reals contains a lower bound, its infmimum is less than or equal to all members of the set.
|- ((S (_ RR /\ E.x e. S A.y e. S x <_ y /\ A e. S) -> sup(S, RR, `' < ) <_ A)
Theorem	sup2 6054	A non-empty, bounded-above set of reals has a supremum. Stronger version of completeness axiom (it has a slightly weaker antecedent).
|- ((A (_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A (y < x \/ y = x)) -> E.x e. RR (A.y e. A -. x < y /\ A.y e. RR (y < x -> E.z e. A y < z)))
Theorem	sup3 6055	A version of the completeness axiom for reals.
|- ((A (_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A y <_ x) -> E.x e. RR (A.y e. A -. x < y /\ A.y e. RR (y < x -> E.z e. A y < z)))
Theorem	infm3lem 6056	Lemma for infm3 6057.
Theorem	infm3 6057	The completeness axiom for reals in terms of infimum: a non-empty, bounded-below set of reals has a infimum. (This theorem is the dual of sup3 6055.)
|- ((A (_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A x <_ y) -> E.x e. RR (A.y e. A -. y < x /\ A.y e. RR (x < y -> E.z e. A z < y)))
Theorem	suprcl 6058	Closure of supremum of a non-empty bounded set of reals.
|- ((A (_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A y <_ x) -> sup(A, RR, < ) e. RR)
Theorem	suprub 6059	A member of a non-empty bounded set of reals is less than or equal to the set's upper bound.
|- (((A (_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A y <_ x) /\ B e. A) -> B <_ sup(A, RR, < ))
Theorem	suprlub 6060	The supremum of a non-empty bounded set of reals is the least upper bound.
|- (((A (_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A y <_ x) /\ (B e. RR /\ B < sup(A, RR, < ))) -> E.z e. A B < z)
Theorem	suprnub 6061	An upper bound is not less than the supremum of a non-empty bounded set of reals.
|- (((A (_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A y <_ x) /\ (B e. RR /\ A.z e. A -. B < z)) -> -. B < sup(A, RR, < ))
Theorem	suprleub 6062	The supremum of a non-empty bounded set of reals is less than or equal to an upper bound.
|- (((A (_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A y <_ x) /\ (B e. RR /\ A.z e. A z <_ B)) -> sup(A, RR, < ) <_ B)
Theorem	sup3i 6063	A version of the completeness axiom for reals.
|- (A (_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A y <_ x)   =>   |- E.x e. RR (A.y e. A -. x < y /\ A.y e. RR (y < x -> E.z e. A y < z))
Theorem	suprcli 6064	Closure of supremum of a non-empty bounded set of reals.
|- (A (_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A y <_ x)   =>   |- sup(A, RR, < ) e. RR
Theorem	suprubi 6065	A member of a non-empty bounded set of reals is less than or equal to the set's upper bound.
|- (A (_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A y <_ x)   =>   |- (B e. A -> B <_ sup(A, RR, < ))
Theorem	suprlubi 6066	The supremum of a non-empty bounded set of reals is the least upper bound.
|- (A (_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A y <_ x)   =>   |- ((B e. RR /\ B < sup(A, RR, < )) -> E.z e. A B < z)
Theorem	suprnubi 6067	An upper bound is not less than the supremum of a non-empty bounded set of reals.
|- (A (_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A y <_ x)   =>   |- ((B e. RR /\ A.z e. A -. B < z) -> -. B < sup(A, RR, < ))
Theorem	suprleubi 6068	The supremum of a non-empty bounded set of reals is less than or equal to an upper bound.
|- (