Nuprl Lemma : strong-continuous-product
∀[F,G:Type ⟶ Type]. (Continuous+(T.F[T] × G[T])) supposing (Continuous+(T.G[T]) and Continuous+(T.F[T]))
Proof
Definitions occuring in Statement :
strong-type-continuous: Continuous+(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,
strong-type-continuous: Continuous+(T.F[T]),
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
all: ∀x:A. B[x],
pi1: fst(t),
pi2: snd(t)
Lemmas referenced :
nat_wf,
strong-type-continuous_wf,
pi1_wf,
pi2_wf,
false_wf,
le_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaEquality,
isectEquality,
extract_by_obid,
hypothesis,
productEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
universeEquality,
thin,
isect_memberEquality,
productElimination,
independent_pairEquality,
sqequalHypSubstitution,
isectElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
axiomEquality,
functionEquality,
cumulativity,
dependent_set_memberEquality,
natural_numberEquality,
lambdaFormation,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[F,G:Type {}\mrightarrow{} Type].
(Continuous+(T.F[T] \mtimes{} G[T])) supposing (Continuous+(T.G[T]) and Continuous+(T.F[T]))
Date html generated:
2017_04_14-AM-07_36_21
Last ObjectModification:
2017_02_27-PM-03_08_39
Theory : subtype_1
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