Nuprl Lemma : strict-monotonic-is-locally-non-constant
∀I:Interval. ∀f:I ⟶ℝ.
(((∀x,y:{x:ℝ| x ∈ I} . ((x < y) ⇒ ((f x) < (f y)))) ∨ (∀x,y:{x:ℝ| x ∈ I} . ((x < y) ⇒ ((f y) < (f x)))))
⇒ (∀a,b:{x:ℝ| x ∈ I} . ∀x:ℝ. locally-non-constant(f;a;b;x)))
Proof
Definitions occuring in Statement :
locally-non-constant: locally-non-constant(f;a;b;c),
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
rless: x < y,
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
set: {x:A| B[x]} ,
apply: f a
Definitions unfolded in proof :
rfun: I ⟶ℝ,
so_apply: x[s],
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
or: P ∨ Q,
uimplies: b supposing a,
guard: {T},
rneq: x ≠ y,
locally-non-constant: locally-non-constant(f;a;b;c),
squash: ↓T,
sq_stable: SqStable(P),
and: P ∧ Q,
exists: ∃x:A. B[x],
r-ap: f(x)
Lemmas referenced :
interval_wf,
rfun_wf,
rless_wf,
all_wf,
or_wf,
i-member_wf,
set_wf,
real_wf,
rless_transitivity2,
rleq_weakening_rless,
rless_transitivity1,
i-member-between,
rleq_wf,
rneq-cases,
rleq_transitivity,
sq_stable__i-member,
r-ap_wf,
rneq_wf,
rleq_weakening_equal
Rules used in proof :
dependent_set_memberEquality,
applyEquality,
functionEquality,
because_Cache,
rename,
setElimination,
setEquality,
hypothesisEquality,
lambdaEquality,
sqequalRule,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
inrFormation,
inlFormation,
unionElimination,
independent_isectElimination,
independent_functionElimination,
dependent_functionElimination,
imageElimination,
baseClosed,
imageMemberEquality,
productEquality,
independent_pairFormation,
dependent_pairFormation
Latex:
\mforall{}I:Interval. \mforall{}f:I {}\mrightarrow{}\mBbbR{}.
(((\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} I\} . ((x < y) {}\mRightarrow{} ((f x) < (f y))))
\mvee{} (\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} I\} . ((x < y) {}\mRightarrow{} ((f y) < (f x)))))
{}\mRightarrow{} (\mforall{}a,b:\{x:\mBbbR{}| x \mmember{} I\} . \mforall{}x:\mBbbR{}. locally-non-constant(f;a;b;x)))
Date html generated:
2017_10_03-AM-10_31_43
Last ObjectModification:
2017_07_29-PM-00_45_02
Theory : reals
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