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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