Nuprl Lemma : inhabited-covers-real-implies-ext
∀[A,B:ℝ ⟶ ℙ].
((∃a:ℝ. A[a])
⇒ (∃b:ℝ. B[b])
⇒ (∀r:ℝ. (A[r] ∨ B[r]))
⇒ (∃f,g:ℕ ⟶ ℝ. ∃x:ℝ. ((∀n:ℕ. A[f n]) ∧ (∀n:ℕ. B[g n]) ∧ lim n→∞.f n = x ∧ lim n→∞.g n = x)))
Proof
Definitions occuring in Statement :
converges-to: lim n→∞.x[n] = y,
real: ℝ,
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
member: t ∈ T,
so_apply: x[s],
so_lambda: λ2x.t[x],
ifthenelse: if b then t else f fi ,
subtract: n - m,
spreadn: spread3,
pi1: fst(t),
top: Top,
inhabited-covers-real-implies,
common-limit-midpoints,
cover-seq-property-ext,
canonical-bound-property,
converges-iff-cauchy,
sq-all-large-and,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a
Lemmas referenced :
inhabited-covers-real-implies,
istype-void,
lifting-strict-callbyvalue,
strict4-spread,
cover-seq-0,
common-limit-midpoints,
cover-seq-property-ext,
canonical-bound-property,
converges-iff-cauchy,
sq-all-large-and
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality_alt,
voidElimination,
isectElimination,
baseClosed,
independent_isectElimination
Latex:
\mforall{}[A,B:\mBbbR{} {}\mrightarrow{} \mBbbP{}].
((\mexists{}a:\mBbbR{}. A[a])
{}\mRightarrow{} (\mexists{}b:\mBbbR{}. B[b])
{}\mRightarrow{} (\mforall{}r:\mBbbR{}. (A[r] \mvee{} B[r]))
{}\mRightarrow{} (\mexists{}f,g:\mBbbN{} {}\mrightarrow{} \mBbbR{}. \mexists{}x:\mBbbR{}. ((\mforall{}n:\mBbbN{}. A[f n]) \mwedge{} (\mforall{}n:\mBbbN{}. B[g n]) \mwedge{} lim n\mrightarrow{}\minfty{}.f n = x \mwedge{} lim n\mrightarrow{}\minfty{}.g n = x)))
Date html generated:
2019_10_30-AM-07_19_19
Last ObjectModification:
2019_02_12-AM-11_16_40
Theory : reals
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