Nuprl Lemma : sup-range
∀I:{I:Interval| icompact(I)} . ∀f:I ⟶ℝ. (f[x] continuous for x ∈ I ⇒ (∃y:ℝ. sup(f(x)(x∈I)) = y))
Proof
Definitions occuring in Statement :
continuous: f[x] continuous for x ∈ I,
rrange: f[x](x∈I),
icompact: icompact(I),
r-ap: f(x),
rfun: I ⟶ℝ,
interval: Interval,
sup: sup(A) = b,
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
sq_stable: SqStable(P),
squash: ↓T,
so_apply: x[s],
r-ap: f(x),
rfun: I ⟶ℝ,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x.t[x],
label: ...$L... t
Lemmas referenced :
rrange_wf,
totally-bounded-sup,
icompact_wf,
interval_wf,
rfun_wf,
continuous_wf,
i-member_wf,
real_wf,
sq_stable__i-member,
r-ap_wf,
sq_stable__icompact,
continuous-compact-range-totally-bounded
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
independent_functionElimination,
introduction,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
because_Cache,
lambdaEquality,
isectElimination,
independent_isectElimination,
setEquality,
applyEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}I:\{I:Interval| icompact(I)\} . \mforall{}f:I {}\mrightarrow{}\mBbbR{}. (f[x] continuous for x \mmember{} I {}\mRightarrow{} (\mexists{}y:\mBbbR{}. sup(f(x)(x\mmember{}I)) = y))
Date html generated:
2016_05_18-AM-09_15_30
Last ObjectModification:
2016_01_17-AM-02_37_53
Theory : reals
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