Nuprl Lemma : concave-positive-nonzero-on
∀I:Interval. ∀f:I ⟶ℝ.
((∀x,y:ℝ. ((x ∈ I) ⇒ (y ∈ I) ⇒ (x = y) ⇒ (f[x] = f[y])))
⇒ (∀x:ℝ. ((x ∈ I) ⇒ (r0 < f[x])))
⇒ concave-on(I;x.f[x])
⇒ f[x]≠r0 for x ∈ I)
Proof
Definitions occuring in Statement :
concave-on: concave-on(I;x.f[x]),
nonzero-on: f[x]≠r0 for x ∈ I,
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
rless: x < y,
req: x = y,
int-to-real: r(n),
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
nonzero-on: f[x]≠r0 for x ∈ I,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
label: ...$L... t,
rfun: I ⟶ℝ,
nat_plus: ℕ+,
uimplies: b supposing a,
sq_stable: SqStable(P),
and: P ∧ Q,
squash: ↓T,
sq_exists: ∃x:A [B[x]],
cand: A c∧ B,
guard: {T},
subinterval: I ⊆ J ,
iff: P ⇐⇒ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
stable: Stable{P},
not: ¬A,
or: P ∨ Q,
false: False,
exists: ∃x:A. B[x],
concave-on: concave-on(I;x.f[x]),
i-member: r ∈ I,
rccint: [l, u],
rge: x ≥ y,
rbetween: x≤y≤z
Lemmas referenced :
set_wf,
nat_plus_wf,
icompact_wf,
i-approx_wf,
concave-on_wf,
i-member_wf,
real_wf,
all_wf,
rless_wf,
int-to-real_wf,
req_wf,
rfun_wf,
interval_wf,
i-approx-is-subinterval,
less_than_wf,
sq_stable__i-member,
left-endpoint_wf,
i-approx-finite,
icompact-endpoints,
right-endpoint_wf,
rmin_wf,
rleq_wf,
rabs_wf,
rmin_strict_ub,
rleq_weakening_rless,
rleq_functionality,
req_weakening,
rabs-of-nonneg,
stable__rleq,
false_wf,
or_wf,
not_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
i-member-compact,
sq_stable__icompact,
rbetween-convex,
i-member-convex,
radd_wf,
rmul_wf,
rsub_wf,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rmin-lb-convex,
equal_wf,
not-rless,
rleq_antisymmetry,
icompact-endpoints-rleq,
rmin_functionality,
rleq_transitivity,
rmin-rleq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
sqequalRule,
lambdaEquality,
hypothesisEquality,
applyEquality,
setElimination,
rename,
dependent_set_memberEquality,
setEquality,
functionEquality,
natural_numberEquality,
because_Cache,
dependent_functionElimination,
independent_isectElimination,
independent_functionElimination,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberFormation,
independent_pairFormation,
productEquality,
unionElimination,
voidElimination,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}I:Interval. \mforall{}f:I {}\mrightarrow{}\mBbbR{}.
((\mforall{}x,y:\mBbbR{}. ((x \mmember{} I) {}\mRightarrow{} (y \mmember{} I) {}\mRightarrow{} (x = y) {}\mRightarrow{} (f[x] = f[y])))
{}\mRightarrow{} (\mforall{}x:\mBbbR{}. ((x \mmember{} I) {}\mRightarrow{} (r0 < f[x])))
{}\mRightarrow{} concave-on(I;x.f[x])
{}\mRightarrow{} f[x]\mneq{}r0 for x \mmember{} I)
Date html generated:
2018_05_22-PM-02_20_25
Last ObjectModification:
2017_10_20-PM-05_17_44
Theory : reals
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