Nuprl Lemma : nonzero-on-implies
∀I:Interval. ∀f:I ⟶ℝ. (f[x]≠r0 for x ∈ I ⇒ (∀x:ℝ. ((x ∈ I) ⇒ f[x] ≠ r0)))
Proof
Definitions occuring in Statement :
nonzero-on: f[x]≠r0 for x ∈ I,
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
rneq: x ≠ y,
int-to-real: r(n),
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
exists: ∃x:A. B[x],
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s],
nonzero-on: f[x]≠r0 for x ∈ I,
icompact: icompact(I),
and: P ∧ Q,
cand: A c∧ B,
i-nonvoid: i-nonvoid(I),
sq_exists: ∃x:{A| B[x]},
sq_stable: SqStable(P),
guard: {T},
uimplies: b supposing a,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
rabs-positive-iff,
rless_transitivity1,
rabs_wf,
int-to-real_wf,
sq_stable__rless,
i-approx-finite,
i-approx-closed,
i-approx_wf,
icompact_wf,
interval_wf,
rfun_wf,
nonzero-on_wf,
real_wf,
i-member_wf,
i-member-witness
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
productElimination,
isectElimination,
sqequalRule,
lambdaEquality,
applyEquality,
setEquality,
dependent_set_memberEquality,
independent_pairFormation,
because_Cache,
dependent_pairFormation,
setElimination,
rename,
natural_numberEquality,
introduction,
independent_isectElimination,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}I:Interval. \mforall{}f:I {}\mrightarrow{}\mBbbR{}. (f[x]\mneq{}r0 for x \mmember{} I {}\mRightarrow{} (\mforall{}x:\mBbbR{}. ((x \mmember{} I) {}\mRightarrow{} f[x] \mneq{} r0)))
Date html generated:
2016_05_18-AM-09_19_14
Last ObjectModification:
2016_01_17-AM-02_40_45
Theory : reals
Home
Index