Nuprl Lemma : range_sup-const
∀I:{I:Interval| icompact(I)} . ∀[c:ℝ]. (sup{c | x ∈ I} = c)
Proof
Definitions occuring in Statement :
range_sup: sup{f[x] | x ∈ I},
icompact: icompact(I),
interval: Interval,
req: x = y,
real: ℝ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
rfun: I ⟶ℝ,
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
top: Top,
istype: istype(T),
ifun: ifun(f;I),
real-fun: real-fun(f;a;b),
implies: P ⇒ Q,
sup: sup(A) = b,
and: P ∧ Q,
icompact: icompact(I),
sq_stable: SqStable(P),
i-nonvoid: i-nonvoid(I),
exists: ∃x:A. B[x],
squash: ↓T,
upper-bound: A ≤ b,
rset-member: x ∈ A,
rrange: f[x](x∈I),
cand: A c∧ B
Lemmas referenced :
range_sup-bound,
istype-top,
subtype_rel_dep_function,
top_wf,
real_wf,
i-member_wf,
istype-void,
req_weakening,
req_wf,
rccint_wf,
left-endpoint_wf,
right-endpoint_wf,
ifun_wf,
rleq_weakening_equal,
range_sup-property,
rleq_antisymmetry,
req_witness,
range_sup_wf,
interval_wf,
icompact_wf,
sq_stable__rleq
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isect_memberFormation_alt,
sqequalRule,
setElimination,
rename,
dependent_set_memberEquality_alt,
lambdaEquality_alt,
applyEquality,
isectElimination,
setEquality,
setIsType,
universeIsType,
independent_isectElimination,
isect_memberEquality_alt,
voidElimination,
because_Cache,
inhabitedIsType,
productElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_pairFormation_alt,
independent_pairFormation,
productIsType
Latex:
\mforall{}I:\{I:Interval| icompact(I)\} . \mforall{}[c:\mBbbR{}]. (sup\{c | x \mmember{} I\} = c)
Date html generated:
2019_10_30-AM-07_44_43
Last ObjectModification:
2019_04_29-PM-10_55_09
Theory : reals
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