Nuprl Lemma : continuous-monotone-product
∀[F,G:Type ⟶ Type].
(ContinuousMonotone(T.F[T] × G[T])) supposing (ContinuousMonotone(T.G[T]) and ContinuousMonotone(T.F[T]))
Proof
Definitions occuring in Statement :
continuous-monotone: ContinuousMonotone(T.F[T]),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
so_apply: x[s],
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]),
type-continuous: Continuous(T.F[T]),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
strong-type-continuous: Continuous+(T.F[T]),
ext-eq: A ≡ B,
guard: {T},
prop: ℙ
Lemmas referenced :
subtype_rel_simple_product,
subtype_rel_wf,
strong-continuous-product,
nat_wf,
subtype_rel_weakening,
continuous-monotone_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
applyEquality,
hypothesisEquality,
independent_isectElimination,
hypothesis,
productElimination,
axiomEquality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
universeEquality,
lambdaEquality,
isectEquality,
independent_pairEquality,
functionEquality,
cumulativity,
productEquality
Latex:
\mforall{}[F,G:Type {}\mrightarrow{} Type].
(ContinuousMonotone(T.F[T] \mtimes{} G[T])) supposing
(ContinuousMonotone(T.G[T]) and
ContinuousMonotone(T.F[T]))
Date html generated:
2016_05_13-PM-04_09_52
Last ObjectModification:
2015_12_26-AM-11_22_32
Theory : subtype_1
Home
Index