Nuprl Lemma : i-member-proper-iff
∀I:Interval. (iproper(I) ⇒ (∀r:ℝ. (r ∈ I ⇐⇒ ∃n:ℕ+. (iproper(i-approx(I;n)) ∧ (r ∈ i-approx(I;n))))))
Proof
Definitions occuring in Statement :
i-approx: i-approx(I;n),
i-member: r ∈ I,
iproper: iproper(I),
interval: Interval,
real: ℝ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
int_upper: {i...},
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
false: False,
nat_plus: ℕ+,
guard: {T},
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
interval: Interval,
i-approx: i-approx(I;n),
i-member: r ∈ I,
iff: P ⇐⇒ Q,
rless: x < y,
sq_exists: ∃x:A [B[x]],
cand: A c∧ B,
rccint: [l, u],
rev_implies: P ⇐ Q,
iproper: iproper(I),
left-endpoint: left-endpoint(I),
pi1: fst(t),
endpoints: endpoints(I),
outl: outl(x),
right-endpoint: right-endpoint(I),
pi2: snd(t),
i-finite: i-finite(I),
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
rneq: x ≠ y,
so_lambda: λ2x.t[x],
so_apply: x[s],
rge: x ≥ y,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
bool: 𝔹,
unit: Unit,
it: ⋅,
rev_uimplies: rev_uimplies(P;Q),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
real: ℝ,
sq_stable: SqStable(P),
squash: ↓T,
rdiv: (x/y),
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
rgt: x > y
Latex:
\mforall{}I:Interval
(iproper(I) {}\mRightarrow{} (\mforall{}r:\mBbbR{}. (r \mmember{} I \mLeftarrow{}{}\mRightarrow{} \mexists{}n:\mBbbN{}\msupplus{}. (iproper(i-approx(I;n)) \mwedge{} (r \mmember{} i-approx(I;n))))))
Date html generated:
2020_05_20-AM-11_32_26
Last ObjectModification:
2020_01_06-PM-00_58_45
Theory : reals
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