Nuprl Lemma : dense-in-reals-implies
∀X:ℝ ⟶ ℙ
(dense-in-interval((-∞, ∞);X)
⇒ (∀x:ℝ. ∀y:{y:ℝ| y = x} . ((X x) ⇒ (X y)))
⇒ (∀x:ℝ. ∃x':ℝ. ((x' = x) ∧ (∀k:ℕ+. ∃y:ℝ. ((y = x' ∈ (ℕ+k ⟶ ℤ)) ∧ (X y))))))
Proof
Definitions occuring in Statement :
dense-in-interval: dense-in-interval(I;X),
riiint: (-∞, ∞),
req: x = y,
real: ℝ,
int_seg: {i..j-},
nat_plus: ℕ+,
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
real: ℝ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
cand: A c∧ B,
le: A ≤ B,
false: False,
not: ¬A,
top: Top
Lemmas referenced :
dense-in-reals-implies-accel,
accelerate_wf,
less_than_wf,
subtype_rel_sets,
nat_plus_wf,
regular-int-seq_wf,
real-regular,
accelerate-req,
req_wf,
all_wf,
exists_wf,
real_wf,
equal_wf,
int_seg_wf,
subtype_rel_dep_function,
int_seg_subtype_nat_plus,
false_wf,
subtype_rel_self,
dense-in-interval_wf,
riiint_wf,
i-member_wf,
member_riiint_lemma,
set_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
dependent_pairFormation,
isectElimination,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
baseClosed,
applyEquality,
functionEquality,
intEquality,
because_Cache,
lambdaEquality,
functionExtensionality,
independent_isectElimination,
setElimination,
rename,
setEquality,
productElimination,
productEquality,
universeEquality,
instantiate,
cumulativity,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}X:\mBbbR{} {}\mrightarrow{} \mBbbP{}
(dense-in-interval((-\minfty{}, \minfty{});X)
{}\mRightarrow{} (\mforall{}x:\mBbbR{}. \mforall{}y:\{y:\mBbbR{}| y = x\} . ((X x) {}\mRightarrow{} (X y)))
{}\mRightarrow{} (\mforall{}x:\mBbbR{}. \mexists{}x':\mBbbR{}. ((x' = x) \mwedge{} (\mforall{}k:\mBbbN{}\msupplus{}. \mexists{}y:\mBbbR{}. ((y = x') \mwedge{} (X y))))))
Date html generated:
2017_10_03-AM-10_10_38
Last ObjectModification:
2017_09_19-PM-01_03_15
Theory : reals
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