Nuprl Lemma : ifun-continuous
∀I:Interval. (icompact(I) ⇒ (∀f:{f:I ⟶ℝ| ifun(f;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],
implies: P ⇒ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
icompact: icompact(I),
and: P ∧ Q,
i-nonvoid: i-nonvoid(I),
exists: ∃x:A. B[x],
top: Top,
guard: {T},
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
so_apply: x[s],
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
sq_stable: SqStable(P),
squash: ↓T,
ifun: ifun(f;I),
real-fun: real-fun(f;a;b)
Lemmas referenced :
icompact-is-rccint,
rfun_wf,
ifun_wf,
icompact_wf,
interval_wf,
member_rccint_lemma,
istype-void,
rleq_transitivity,
left-endpoint_wf,
right-endpoint_wf,
real-cont-iff-continuous,
subtype_rel_self,
real_wf,
i-member_wf,
rccint_wf,
real-fun-iff-continuous,
sq_stable__ifun
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
setIsType,
universeIsType,
sqequalRule,
productElimination,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
because_Cache,
independent_functionElimination,
lambdaEquality_alt,
applyEquality,
setElimination,
rename,
setEquality,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}I:Interval. (icompact(I) {}\mRightarrow{} (\mforall{}f:\{f:I {}\mrightarrow{}\mBbbR{}| ifun(f;I)\} . f[x] continuous for x \mmember{} I))
Date html generated:
2019_10_30-AM-07_16_06
Last ObjectModification:
2019_10_09-PM-06_39_22
Theory : reals
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