Nuprl Lemma : strong-continuous-depproduct
∀[A:Type]. ∀[G:T:Type ⟶ A ⟶ Type]. Continuous+(T.x:A × G[T;x]) supposing ∀a:A. Continuous+(T.G[T;a])
Proof
Definitions occuring in Statement :
strong-type-continuous: Continuous+(T.F[T]),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
product: 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,
strong-type-continuous: Continuous+(T.F[T]),
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
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,
implies: P ⇒ Q,
pi1: fst(t),
pi2: snd(t),
istype: istype(T)
Lemmas referenced :
nat_wf,
strong-type-continuous_wf,
false_wf,
le_wf,
member_wf,
pi2_wf,
subtype_rel_product
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
Error :lambdaEquality_alt,
Error :isectIsType,
Error :universeIsType,
because_Cache,
Error :productIsType,
hypothesisEquality,
applyEquality,
Error :isect_memberEquality_alt,
productElimination,
thin,
Error :dependent_pairEquality_alt,
functionExtensionality,
universeEquality,
extract_by_obid,
hypothesis,
isectEquality,
sqequalHypSubstitution,
independent_pairEquality,
axiomEquality,
Error :functionIsType,
Error :inhabitedIsType,
isectElimination,
equalityTransitivity,
equalitySymmetry,
rename,
Error :dependent_set_memberEquality_alt,
natural_numberEquality,
Error :lambdaFormation_alt,
Error :equalityIsType1,
dependent_functionElimination,
independent_functionElimination,
cumulativity,
applyLambdaEquality,
hyp_replacement,
dependent_pairEquality,
lambdaEquality,
independent_isectElimination
Latex:
\mforall{}[A:Type]. \mforall{}[G:T:Type {}\mrightarrow{} A {}\mrightarrow{} Type].
Continuous+(T.x:A \mtimes{} G[T;x]) supposing \mforall{}a:A. Continuous+(T.G[T;a])
Date html generated:
2019_06_20-PM-00_27_45
Last ObjectModification:
2018_09_30-PM-00_43_44
Theory : subtype_1
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