Nuprl Lemma : better-continuity-for-reals
∀x:ℝ. ∃x':{x':ℝ| x' = x} . ∀g:ℕ ⟶ ℝ. (lim n→∞.g n = x ⇒ (∀P:ℝ ⟶ 𝔹. ∃z:{z:ℝ| P z = P x'} . (∃n:{ℕ| (z = (g n))})))
Proof
Definitions occuring in Statement :
converges-to: lim n→∞.x[n] = y,
req: x = y,
real: ℝ,
nat: ℕ,
bool: 𝔹,
all: ∀x:A. B[x],
sq_exists: ∃x:{A| B[x]},
exists: ∃x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
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,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
so_apply: x[s],
sq_exists: ∃x:{A| B[x]}
Lemmas referenced :
connectedness-main-lemma,
accelerate-req,
less_than_wf,
req_wf,
all_wf,
nat_wf,
real_wf,
converges-to_wf,
bool_wf,
exists_wf,
equal_wf,
sq_exists_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
dependent_pairFormation,
isectElimination,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
baseClosed,
productElimination,
because_Cache,
functionEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
setEquality,
setElimination,
rename
Latex:
\mforall{}x:\mBbbR{}
\mexists{}x':\{x':\mBbbR{}| x' = x\}
\mforall{}g:\mBbbN{} {}\mrightarrow{} \mBbbR{}. (lim n\mrightarrow{}\minfty{}.g n = x {}\mRightarrow{} (\mforall{}P:\mBbbR{} {}\mrightarrow{} \mBbbB{}. \mexists{}z:\{z:\mBbbR{}| P z = P x'\} . (\mexists{}n:\{\mBbbN{}| (z = (g n))\})))
Date html generated:
2017_10_03-AM-10_09_45
Last ObjectModification:
2017_09_13-PM-03_26_02
Theory : reals
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