Nuprl Lemma : cantor-to-interval_wf1
∀a,b:ℝ. ∀f:ℕ ⟶ 𝔹. (cantor-to-interval(a;b;f) ∈ {x:ℝ| lim n→∞.fst(cantor-interval(a;b;f;n)) = x} ) supposing a ≤ b
Proof
Definitions occuring in Statement :
cantor-to-interval: cantor-to-interval(a;b;f),
cantor-interval: cantor-interval(a;b;f;n),
converges-to: lim n→∞.x[n] = y,
rleq: x ≤ y,
real: ℝ,
nat: ℕ,
bool: 𝔹,
uimplies: b supposing a,
pi1: fst(t),
all: ∀x:A. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
cantor-to-interval: cantor-to-interval(a;b;f),
prop: ℙ,
uall: ∀[x:A]. B[x],
pi1: fst(t),
cantor-interval-converges-ext,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
top: Top,
so_apply: x[s],
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
converges: x[n]↓ as n→∞,
exists: ∃x:A. B[x]
Lemmas referenced :
nat_wf,
bool_wf,
rleq_wf,
real_wf,
cantor-interval-converges-ext,
all_wf,
isect_wf,
converges_wf,
cantor-interval_wf,
pi1_wf_top,
equal_wf,
uimplies_subtype,
subtype_rel_dep_function,
int_seg_wf,
int_seg_subtype_nat,
false_wf,
subtype_rel_self,
converges-to_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
functionEquality,
extract_by_obid,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
isectElimination,
applyEquality,
instantiate,
productEquality,
productElimination,
independent_pairEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_functionElimination,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
independent_pairFormation,
functionExtensionality,
dependent_set_memberEquality
Latex:
\mforall{}a,b:\mBbbR{}. \mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbB{}. (cantor-to-interval(a;b;f) \mmember{} \{x:\mBbbR{}| lim n\mrightarrow{}\minfty{}.fst(cantor-interval(a;b;f;n)) = x\} ) \000Csupposing a \mleq{} b
Date html generated:
2017_10_03-AM-09_54_08
Last ObjectModification:
2017_07_28-AM-08_03_31
Theory : reals
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