Nuprl Lemma : function-values-near-same-sign
∀I:Interval. ∀f:{x:ℝ| x ∈ I} ⟶ ℝ.
(icompact(I)
⇒ (∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f[x] = f[y])))
⇒ (∀x:{x:ℝ| x ∈ I}
((r0 < |f[x]|)
⇒ (∃d:{d:ℝ| r0 < d}
∀y:{x:ℝ| x ∈ I} . ((|x - y| ≤ d) ⇒ ((r0 < f[x] ⇐⇒ r0 < f[y]) ∧ (f[x] < r0 ⇐⇒ f[y] < r0)))))))
Proof
Definitions occuring in Statement :
icompact: icompact(I),
i-member: r ∈ I,
interval: Interval,
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rsub: x - y,
req: x = y,
int-to-real: r(n),
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
rfun: I ⟶ℝ,
continuous: f[x] continuous for x ∈ I,
nat_plus: ℕ+,
rless: x < y,
sq_exists: ∃x:A [B[x]],
uall: ∀[x:A]. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
false: False,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
so_apply: x[s],
cand: A c∧ B,
sq_stable: SqStable(P),
rneq: x ≠ y,
uiff: uiff(P;Q),
rge: x ≥ y,
req_int_terms: t1 ≡ t2,
real: ℝ,
rdiv: (x/y),
less_than: a < b,
less_than': less_than'(a;b)
Lemmas referenced :
function-is-continuous,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
icompact_wf,
squash_wf,
true_wf,
i-approx-of-compact,
subtype_rel_self,
iff_weakening_equal,
i-approx_wf,
small-reciprocal-real,
rabs_wf,
i-member_wf,
rless_wf,
int-to-real_wf,
sq_stable__rless,
sq_stable__i-member,
rabs-difference-bound-rleq,
rdiv_wf,
rless-int,
intformand_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_term_value_var_lemma,
rleq_wf,
rsub_wf,
real_wf,
req_wf,
interval_wf,
rleq_weakening_rless,
radd_wf,
rless-implies-rless,
itermSubtract_wf,
itermAdd_wf,
req-iff-rsub-is-0,
rless_functionality,
req_weakening,
rabs-of-nonneg,
rless_functionality_wrt_implies,
rleq_weakening_equal,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rleq-int-fractions2,
sq_stable__less_than,
decidable__le,
intformle_wf,
itermMultiply_wf,
int_formula_prop_le_lemma,
int_term_value_mul_lemma,
rabs-strict-ub,
rless-int-fractions2,
rless_transitivity2,
radd-preserves-rleq,
rminus_wf,
rmul_wf,
rinv_wf2,
itermMinus_wf,
rleq_functionality,
req_transitivity,
rinv-as-rdiv,
real_term_value_minus_lemma,
real_term_value_mul_lemma,
radd-preserves-rless,
rless_irreflexivity,
rabs-of-nonpos,
rmul_reverses_rless,
rminus_functionality,
rmul_preserves_rless,
rmul-rinv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
sqequalRule,
independent_functionElimination,
hypothesis,
dependent_set_memberEquality_alt,
natural_numberEquality,
setElimination,
rename,
isectElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
universeIsType,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
productElimination,
independent_pairFormation,
closedConclusion,
inrFormation_alt,
int_eqEquality,
functionIsType,
productIsType,
setIsType,
addEquality,
inhabitedIsType,
multiplyEquality,
minusEquality
Latex:
\mforall{}I:Interval. \mforall{}f:\{x:\mBbbR{}| x \mmember{} I\} {}\mrightarrow{} \mBbbR{}.
(icompact(I)
{}\mRightarrow{} (\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} I\} . ((x = y) {}\mRightarrow{} (f[x] = f[y])))
{}\mRightarrow{} (\mforall{}x:\{x:\mBbbR{}| x \mmember{} I\}
((r0 < |f[x]|)
{}\mRightarrow{} (\mexists{}d:\{d:\mBbbR{}| r0 < d\}
\mforall{}y:\{x:\mBbbR{}| x \mmember{} I\}
((|x - y| \mleq{} d) {}\mRightarrow{} ((r0 < f[x] \mLeftarrow{}{}\mRightarrow{} r0 < f[y]) \mwedge{} (f[x] < r0 \mLeftarrow{}{}\mRightarrow{} f[y] < r0)))))))
Date html generated:
2019_10_30-AM-07_47_43
Last ObjectModification:
2019_01_13-PM-07_20_40
Theory : reals
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