Nuprl Lemma : upper-bounds-closed
∀[A:Set(ℝ)]. closed-rset(upper-bounds(A))
Proof
Definitions occuring in Statement :
upper-bounds: upper-bounds(A),
closed-rset: closed-rset(A),
rset: Set(ℝ),
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
upper-bounds: upper-bounds(A),
closed-rset: closed-rset(A),
upper-bound: A ≤ b,
member-closure: y ∈ closure(A),
rset-member: x ∈ A,
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
and: P ∧ Q,
rset: Set(ℝ),
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T}
Lemmas referenced :
subtype_rel_self,
istype-nat,
converges-to_wf,
rleq_wf,
real_wf,
le_witness_for_triv,
rset_wf,
constant-limit,
req_weakening,
rleq-limit
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
universeIsType,
applyEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
instantiate,
extract_by_obid,
isectElimination,
universeEquality,
inhabitedIsType,
productIsType,
functionIsType,
lambdaEquality_alt,
because_Cache,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
functionIsTypeImplies,
independent_functionElimination
Latex:
\mforall{}[A:Set(\mBbbR{})]. closed-rset(upper-bounds(A))
Date html generated:
2019_10_29-AM-10_41_07
Last ObjectModification:
2019_04_19-PM-06_28_39
Theory : reals
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