mmtheorems61 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 6001-6100 - Page 61 of 107

Type	Label	Description
Statement
Theorem	avglet 6001	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 6002	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 6003	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 6004	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 6005	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 6006	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 6007	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 6008	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 6009	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 6010	Lemma for infm3 6011.
Theorem	infm3 6011	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 6009.)
|- ((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 6012	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 6013	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 6014	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 6015	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 6016	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 6017	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 6018	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 6019	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 6020	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 6021	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 6022	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	reuunineg 6023	The negative of the unique real such that ph.
|- (x = -uy -> (ph <-> ps))   =>   |- (E!x e. RR ph -> U.{x e. RR | ph} = -uU.{y e. RR | ps})
Theorem	dfinfmr 6024	The infimum (expressed as supremum with converse 'less-than') of a set of reals A.
|- (A (_ RR -> sup(A, RR, `' < ) = U.{x e. RR | (A.y e. A x <_ y /\ A.y e. RR (x < y -> E.z e. A z < y))})
Theorem	infmsup 6025	The infimum (expressed as supremum with converse 'less-than') of a set of reals A is the negative of the supremum of the negatives of its elements. The antecedent ensures that A is nonempty and has a lower bound.
|- ((A (_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A x <_ y) -> sup(A, RR, `' < ) = -usup({z e. RR | -uz e. A}, RR, < ))
Theorem	infmrcl 6026	Closure of infimum of a non-empty bounded set of reals.
|- ((A (_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A x <_ y) -> sup(A, RR, `' < ) e. RR)
Theorem	nnunb 6027	The set of natural numbers is unbounded above. Theorem I.28 of [Apostol] p. 26.
|- -. E.x e. RR A.y e. NN (y < x \/ y = x)
Theorem	arch 6028	Archimedean property of real numbers. For any real number, there is an integer greater than it. Theorem I.29 of [Apostol] p. 26.
|- (A e. RR -> E.n e. NN A < n)
Theorem	nnreclt 6029	There exists a natural number whose reciprocal is less than a given positive real. Exercise 3 of [Apostol] p. 28.
|- ((A e. RR /\ 0 < A) -> E.n e. NN (1 / n) < A)
Theorem	bndndx 6030	A bounded real sequence A(k) is less than or equal to at least one of its indices.
|- (E.x e. RR A.k e. NN (A e. RR /\ A <_ x) -> E.k e. NN A <_ k)
Supremum on the extended reals
Theorem	xrsupexmnf 6031	Adding minus infinity to a set does not affect the existence of its supremum.
|- (E.x e. RR* (A.y e. A -. x < y /\ A.y e. RR* (y < x -> E.z e. A y < z)) -> E.x e. RR* (A.y e. (A u. { -oo}) -. x < y /\ A.y e. RR* (y < x -> E.z e. (A u. { -oo})y < z)))
Theorem	xrinfmexpnf 6032	Adding plus infinity to a set does not affect the existence of its infimum.
|- (E.x e. RR* (A.y e. A -. y < x /\ A.y e. RR* (x < y -> E.z e. A z < y)) -> E.x e. RR* (A.y e. (A u. { +oo}) -. y < x /\ A.y e. RR* (x < y -> E.z e. (A u. { +oo})z < y)))
Theorem	xrsupsslem 6033	Lemma for xrsupss 6035.
Theorem	xrinfmsslem 6034	Lemma for xrinfmss 6036.
Theorem	xrsupss 6035	Any subset of extended reals has a supremum.
|- (A (_ RR* -> E.x e. RR* (A.y e. A -. x < y /\ A.y e. RR* (y < x -> E.z e. A y < z)))
Theorem	xrinfmss 6036	Any subset of extended reals has an infimum.
|- (A (_ RR* -> E.x e. RR* (A.y e. A -. y < x /\ A.y e. RR* (x < y -> E.z e. A z < y)))
Theorem	xrub 6037	By quantifying only over reals, we can specify any extended real upper bound for any set of extended reals.
|- ((A (_ RR* /\ B e. RR*) -> (A.x e. RR (x < B -> E.y e. A x < y) <-> A.x e. RR* (x < B -> E.y e. A x < y)))
Theorem	supxr 6038	The supremum of a set of extended reals.
|- (((A (_ RR* /\ B e. RR*) /\ (A.x e. A -. B < x /\ A.x e. RR (x < B -> E.y e. A x < y))) -> sup(A, RR*, < ) = B)
Theorem	supxr2 6039	The supremum of a set of extended reals.
|- (((A (_ RR* /\ B e. RR*) /\ (A.x e. A x <_ B /\ A.x e. RR (x < B -> E.y e. A x < y))) -> sup(A, RR*, < ) = B)
Theorem	supxrre 6040	The real and extended real suprema match when the real supremum exists.
|- ((A (_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A y <_ x) -> sup(A, RR*, < ) = sup(A, RR, < ))
Theorem	supxrcl 6041	The supremum of an arbitrary set of extended reals is an extended real.
|- (A (_ RR* -> sup(A,