Nuprl Lemma : strictly-increasing-on-closed-interval
∀a,b:ℝ. ∀f:[a, b] ⟶ℝ.
(f[x] strictly-increasing for x ∈ (a, b) ⇒ f[x] increasing for x ∈ [a, b] ⇒ f[x] strictly-increasing for x ∈ [a, b])
Proof
Definitions occuring in Statement :
strictly-increasing-on-interval: f[x] strictly-increasing for x ∈ I,
increasing-on-interval: f[x] increasing for x ∈ I,
rfun: I ⟶ℝ,
rooint: (l, u),
rccint: [l, u],
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
subtype_rel: A ⊆r B,
so_apply: x[s],
rfun: I ⟶ℝ,
label: ...$L... t,
so_lambda: λ2x.t[x],
prop: ℙ,
uimplies: b supposing a,
guard: {T},
uall: ∀[x:A]. B[x],
cand: A c∧ B,
and: P ∧ Q,
implies: P ⇒ Q,
top: Top,
member: t ∈ T,
subinterval: I ⊆ J ,
all: ∀x:A. B[x],
strictly-increasing-on-interval: f[x] strictly-increasing for x ∈ I,
exists: ∃x:A. B[x],
squash: ↓T,
sq_stable: SqStable(P),
increasing-on-interval: f[x] increasing for x ∈ I
Lemmas referenced :
rfun_wf,
subtype_rel_sets,
rooint_wf,
strictly-increasing-on-interval_wf,
real_wf,
i-member_wf,
rccint_wf,
increasing-on-interval_wf,
rless_wf,
rleq_weakening_rless,
member_rccint_lemma,
member_rooint_lemma,
set_wf,
ravg_wf,
ravg-between,
exists_wf,
sq_stable__rless,
rleq_wf,
rless_transitivity1,
rless_transitivity2,
sq_stable__rleq
Rules used in proof :
independent_functionElimination,
setEquality,
dependent_set_memberEquality,
rename,
setElimination,
applyEquality,
lambdaEquality,
because_Cache,
productEquality,
independent_pairFormation,
independent_isectElimination,
hypothesisEquality,
isectElimination,
productElimination,
hypothesis,
voidEquality,
voidElimination,
isect_memberEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
sqequalRule,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
dependent_pairFormation,
imageElimination,
baseClosed,
imageMemberEquality
Latex:
\mforall{}a,b:\mBbbR{}. \mforall{}f:[a, b] {}\mrightarrow{}\mBbbR{}.
(f[x] strictly-increasing for x \mmember{} (a, b)
{}\mRightarrow{} f[x] increasing for x \mmember{} [a, b]
{}\mRightarrow{} f[x] strictly-increasing for x \mmember{} [a, b])
Date html generated:
2017_10_03-PM-00_29_19
Last ObjectModification:
2017_07_30-PM-08_50_55
Theory : reals
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