Nuprl Lemma : continuous-monotone-isect
∀[A:Type]. ∀[F:A ⟶ Type ⟶ Type]. ContinuousMonotone(T.⋂a:A. F[a;T]) supposing ∀a:A. ContinuousMonotone(T.F[a;T])
Proof
Definitions occuring in Statement :
continuous-monotone: ContinuousMonotone(T.F[T]),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
isect: ⋂x:A. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
so_apply: x[s1;s2],
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
continuous-monotone: ContinuousMonotone(T.F[T]),
and: P ∧ Q,
type-monotone: Monotone(T.F[T]),
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
type-continuous: Continuous(T.F[T]),
prop: ℙ,
strong-type-continuous: Continuous+(T.F[T]),
ext-eq: A ≡ B,
guard: {T}
Lemmas referenced :
subtype_rel_isect,
equal_wf,
subtype_rel_wf,
nat_wf,
subtype_rel_weakening,
all_wf,
continuous-monotone_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
isectEquality,
cumulativity,
hypothesisEquality,
applyEquality,
functionExtensionality,
lambdaEquality,
productElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
hypothesis,
lambdaFormation,
dependent_functionElimination,
independent_functionElimination,
because_Cache,
axiomEquality,
isect_memberEquality,
universeEquality,
independent_pairEquality,
functionEquality,
instantiate
Latex:
\mforall{}[A:Type]. \mforall{}[F:A {}\mrightarrow{} Type {}\mrightarrow{} Type].
ContinuousMonotone(T.\mcap{}a:A. F[a;T]) supposing \mforall{}a:A. ContinuousMonotone(T.F[a;T])
Date html generated:
2017_04_14-AM-07_36_34
Last ObjectModification:
2017_02_27-PM-03_08_46
Theory : subtype_1
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