Nuprl Lemma : ifun-iff-continuous
∀I:Interval. (icompact(I) ⇒ (∀f:I ⟶ℝ. (ifun(λx.f[x];I) ⇐⇒ f[x] continuous for x ∈ I)))
Proof
Definitions occuring in Statement :
ifun: ifun(f;I),
continuous: f[x] continuous for x ∈ I,
icompact: icompact(I),
rfun: I ⟶ℝ,
interval: Interval,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
lambda: λx.A[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
prop: ℙ,
so_apply: x[s],
rfun: I ⟶ℝ,
rev_implies: P ⇐ Q,
label: ...$L... t,
ifun: ifun(f;I),
top: Top,
icompact: icompact(I)
Lemmas referenced :
ifun-continuous,
ifun_wf,
real_wf,
i-member_wf,
continuous_wf,
rfun_wf,
icompact_wf,
interval_wf,
icompact-is-rccint,
left_endpoint_rccint_lemma,
istype-void,
right_endpoint_rccint_lemma,
real-fun-iff-continuous,
left-endpoint_wf,
right-endpoint_wf,
rccint_wf,
icompact-endpoints-rleq,
real-cont-iff-continuous
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
sqequalRule,
dependent_set_memberEquality_alt,
because_Cache,
universeIsType,
isectElimination,
independent_isectElimination,
lambdaEquality_alt,
applyEquality,
setIsType,
isect_memberEquality_alt,
voidElimination,
productElimination
Latex:
\mforall{}I:Interval. (icompact(I) {}\mRightarrow{} (\mforall{}f:I {}\mrightarrow{}\mBbbR{}. (ifun(\mlambda{}x.f[x];I) \mLeftarrow{}{}\mRightarrow{} f[x] continuous for x \mmember{} I)))
Date html generated:
2019_10_30-AM-07_16_51
Last ObjectModification:
2019_10_09-PM-06_43_53
Theory : reals
Home
Index