Nuprl Lemma : union-continuous-type-monotone
∀[F:Type ⟶ Type]. (Monotone(T.F[T])) supposing (union-continuous{i:l}(T.F[T]) and (∀A,B:Type. (A ≡ B
⇒ F[A] ≡ F[B])))
Proof
Definitions occuring in Statement :
union-continuous: union-continuous{i:l}(T.F[T])
,
type-monotone: Monotone(T.F[T])
,
ext-eq: A ≡ B
,
uimplies: b supposing a
,
uall: ∀[x:A]. B[x]
,
so_apply: x[s]
,
all: ∀x:A. B[x]
,
implies: P
⇒ Q
,
function: x:A ⟶ B[x]
,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x]
,
member: t ∈ T
,
uimplies: b supposing a
,
type-monotone: Monotone(T.F[T])
,
subtype_rel: A ⊆r B
,
so_apply: x[s]
,
prop: ℙ
,
so_lambda: λ2x.t[x]
,
implies: P
⇒ Q
,
union-continuous: union-continuous{i:l}(T.F[T])
,
tunion: ⋃x:A.B[x]
,
ifthenelse: if b then t else f fi
,
btrue: tt
,
pi2: snd(t)
,
all: ∀x:A. B[x]
,
ext-eq: A ≡ B
,
and: P ∧ Q
,
bool: 𝔹
,
bfalse: ff
,
guard: {T}
Lemmas referenced :
subtype_rel_transitivity,
bfalse_wf,
tunion_wf,
btrue_wf,
ifthenelse_wf,
bool_wf,
ext-eq_wf,
all_wf,
union-continuous_wf,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaEquality,
sqequalHypSubstitution,
applyEquality,
hypothesisEquality,
sqequalRule,
axiomEquality,
hypothesis,
lemma_by_obid,
isectElimination,
thin,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
universeEquality,
instantiate,
cumulativity,
functionEquality,
imageMemberEquality,
dependent_pairEquality,
baseClosed,
dependent_functionElimination,
independent_functionElimination,
productElimination,
independent_pairFormation,
imageElimination,
unionElimination,
independent_isectElimination
Latex:
\mforall{}[F:Type {}\mrightarrow{} Type]
(Monotone(T.F[T])) supposing
(union-continuous\{i:l\}(T.F[T]) and
(\mforall{}A,B:Type. (A \mequiv{} B {}\mRightarrow{} F[A] \mequiv{} F[B])))
Date html generated:
2016_05_13-PM-04_10_24
Last ObjectModification:
2016_01_14-PM-07_29_50
Theory : subtype_1
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