Nuprl Lemma : strong-continuous-dep-isect
∀A:Type. ∀G:T:Type ⟶ A ⟶ Type. ((∀a:A. Continuous+(T.G[T;a])) ⇒ Continuous+(T.x:A ⋂ G[T;x]))
Proof
Definitions occuring in Statement :
dep-isect: x:A ⋂ B[x],
strong-type-continuous: Continuous+(T.F[T]),
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
strong-type-continuous: Continuous+(T.F[T]),
uall: ∀[x:A]. B[x],
member: t ∈ T,
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A
Lemmas referenced :
nat_wf,
dep-isect_wf,
all_wf,
strong-type-continuous_wf,
false_wf,
le_wf,
dep-isect-subtype,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaEquality,
isectEquality,
cumulativity,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
applyEquality,
productElimination,
independent_pairEquality,
axiomEquality,
functionEquality,
universeEquality,
instantiate,
isectElimination,
dependent_set_memberEquality,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
functionExtensionality,
dependentIntersection_memberEquality,
isect_memberEquality,
independent_functionElimination,
dependentIntersectionElimination
Latex:
\mforall{}A:Type. \mforall{}G:T:Type {}\mrightarrow{} A {}\mrightarrow{} Type. ((\mforall{}a:A. Continuous+(T.G[T;a])) {}\mRightarrow{} Continuous+(T.x:A \mcap{} G[T;x]))
Date html generated:
2018_05_21-PM-06_21_27
Last ObjectModification:
2018_05_19-PM-05_32_20
Theory : dependent!intersection
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