Nuprl Lemma : Inorm-bound
∀[I:{I:Interval| icompact(I)} ]. ∀[f:I ⟶ℝ]. ∀[mc:f[x] continuous for x ∈ I]. ∀[x:{r:ℝ| r ∈ I} ]. (|f[x]| ≤ ||f[x]||_I)
Proof
Definitions occuring in Statement :
Inorm: ||f[x]||_I,
continuous: f[x] continuous for x ∈ I,
icompact: icompact(I),
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
rleq: x ≤ y,
rabs: |x|,
real: ℝ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
so_lambda: λ2x.t[x],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s],
prop: ℙ,
subtype_rel: A ⊆r B,
Inorm: ||f[x]||_I,
sup: sup(A) = b,
upper-bound: A ≤ b,
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
sq_stable__i-member,
rset-member-rrange,
continuous-abs-subtype,
range-sup-property,
icompact_wf,
interval_wf,
rfun_wf,
continuous_wf,
set_wf,
nat_plus_wf,
rabs_wf,
i-member_wf,
real_wf,
Inorm_wf,
rsub_wf,
less_than'_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairEquality,
because_Cache,
lemma_by_obid,
isectElimination,
applyEquality,
setEquality,
hypothesis,
setElimination,
rename,
minusEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}[I:\{I:Interval| icompact(I)\} ]. \mforall{}[f:I {}\mrightarrow{}\mBbbR{}]. \mforall{}[mc:f[x] continuous for x \mmember{} I]. \mforall{}[x:\{r:\mBbbR{}| r \mmember{} I\} ].
(|f[x]| \mleq{} ||f[x]||\_I)
Date html generated:
2016_05_18-AM-09_17_20
Last ObjectModification:
2016_01_17-AM-02_40_10
Theory : reals
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