Nuprl Lemma : m-open-cover-iff
∀[X:Type]. ∀[d:metric(X)]. ∀[I:Type]. ∀[A:I ⟶ X ⟶ ℙ].
(m-open-cover(X;d;I;i,x.A[i;x])
⇐⇒ (∀i:I. m-open(X;d;x.A[i;x])) ∧ (∃b:X ⟶ ℕ+. ∃c:X ⟶ I. ∀x,y:X. ((mdist(d;x;y) ≤ (r1/r(b x))) ⇒ A[c x;y])))
Proof
Definitions occuring in Statement :
m-open-cover: m-open-cover(X;d;I;i,x.A[i; x]),
m-open: m-open(X;d;x.A[x]),
mdist: mdist(d;x;y),
metric: metric(X),
rdiv: (x/y),
rleq: x ≤ y,
int-to-real: r(n),
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
m-open-cover: m-open-cover(X;d;I;i,x.A[i; x]),
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
prop: ℙ,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nat_plus: ℕ+,
guard: {T},
m-open: m-open(X;d;x.A[x]),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
pi1: fst(t),
rneq: x ≠ y,
or: P ∨ Q,
decidable: Dec(P),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[I:Type]. \mforall{}[A:I {}\mrightarrow{} X {}\mrightarrow{} \mBbbP{}].
(m-open-cover(X;d;I;i,x.A[i;x])
\mLeftarrow{}{}\mRightarrow{} (\mforall{}i:I. m-open(X;d;x.A[i;x]))
\mwedge{} (\mexists{}b:X {}\mrightarrow{} \mBbbN{}\msupplus{}. \mexists{}c:X {}\mrightarrow{} I. \mforall{}x,y:X. ((mdist(d;x;y) \mleq{} (r1/r(b x))) {}\mRightarrow{} A[c x;y])))
Date html generated:
2020_05_20-AM-11_56_12
Last ObjectModification:
2020_01_12-PM-01_13_54
Theory : reals
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