Nuprl Lemma : function-diff-small-or-interval-proper
∀I:Interval. ∀f:I ⟶ℝ. ∀x,y:{x:ℝ| x ∈ I} . ∀e:{e:ℝ| r0 < e} .
(f[x] continuous for x ∈ I ⇒ ((|f[x] - f[y]| ≤ e) ∨ (r0 < |x - y|)))
Proof
Definitions occuring in Statement :
continuous: f[x] continuous for x ∈ I,
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
sq_stable: SqStable(P),
squash: ↓T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
so_apply: x[s],
prop: ℙ,
continuous: f[x] continuous for x ∈ I,
i-approx: i-approx(I;n),
rccint: [l, u],
nat_plus: ℕ+,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
iff: P ⇐⇒ Q,
top: Top,
exists: ∃x:A. B[x],
sq_exists: ∃x:{A| B[x]},
subinterval: I ⊆ J ,
cand: A c∧ B,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
rev_implies: P ⇐ Q,
rless: x < y,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
subtype_rel: A ⊆r B,
label: ...$L... t
Lemmas referenced :
rmin-rmax-subinterval,
sq_stable__i-member,
continuous_functionality_wrt_subinterval,
i-member_wf,
real_wf,
rccint_wf,
rmin_wf,
rmax_wf,
less_than_wf,
rccint-icompact,
rmin-rleq-rmax,
icompact_wf,
member_rccint_lemma,
small-reciprocal-real,
sq_stable__and,
rless_wf,
int-to-real_wf,
all_wf,
rleq_wf,
rabs_wf,
rsub_wf,
rdiv_wf,
rless-int,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
sq_stable__rless,
sq_stable__all,
sq_stable__rleq,
less_than'_wf,
nat_plus_wf,
squash_wf,
continuous_wf,
set_wf,
rfun_wf,
interval_wf,
rless-cases,
rmin-rleq,
rleq-rmax,
rleq_weakening_rless,
rless_transitivity2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
setElimination,
rename,
hypothesis,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
because_Cache,
isectElimination,
lambdaEquality,
applyEquality,
dependent_set_memberEquality,
setEquality,
natural_numberEquality,
independent_pairFormation,
productElimination,
isect_memberEquality,
voidElimination,
voidEquality,
functionEquality,
productEquality,
independent_isectElimination,
inrFormation,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
independent_pairEquality,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
inlFormation
Latex:
\mforall{}I:Interval. \mforall{}f:I {}\mrightarrow{}\mBbbR{}. \mforall{}x,y:\{x:\mBbbR{}| x \mmember{} I\} . \mforall{}e:\{e:\mBbbR{}| r0 < e\} .
(f[x] continuous for x \mmember{} I {}\mRightarrow{} ((|f[x] - f[y]| \mleq{} e) \mvee{} (r0 < |x - y|)))
Date html generated:
2016_10_26-AM-09_55_54
Last ObjectModification:
2016_08_29-PM-10_09_17
Theory : reals
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