Nuprl Lemma : sup-iff-closure
∀[A:Set(ℝ)]. ∀x:ℝ. (sup(A) = x ⇐⇒ A ≤ x ∧ x ∈ closure(A))
Proof
Definitions occuring in Statement :
member-closure: y ∈ closure(A),
sup: sup(A) = b,
upper-bound: A ≤ b,
rset: Set(ℝ),
real: ℝ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
subtype_rel: A ⊆r B,
rset: Set(ℝ),
so_apply: x[s],
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
prop: ℙ,
uimplies: b supposing a,
le: A ≤ B,
rnonneg: rnonneg(x),
rleq: x ≤ y,
upper-bound: A ≤ b,
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
sup: sup(A) = b,
member-closure: y ∈ closure(A),
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
decidable: Dec(P),
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
nat_plus: ℕ+,
pi1: fst(t),
rset-member: x ∈ A,
cand: A c∧ B,
ge: i ≥ j ,
nat: ℕ,
sq_exists: ∃x:A [B[x]],
converges-to: lim n→∞.x[n] = y,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
rless: x < y,
req_int_terms: t1 ≡ t2,
rdiv: (x/y),
rge: x ≥ y,
squash: ↓T,
sq_stable: SqStable(P),
real: ℝ
Latex:
\mforall{}[A:Set(\mBbbR{})]. \mforall{}x:\mBbbR{}. (sup(A) = x \mLeftarrow{}{}\mRightarrow{} A \mleq{} x \mwedge{} x \mmember{} closure(A))
Date html generated:
2020_05_20-AM-11_28_35
Last ObjectModification:
2020_03_19-PM-01_18_44
Theory : reals
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