Nuprl Lemma : isect-mono
∀A:Type. ∀B:A ⟶ Type. ((∀a:A. mono(B[a])) ⇒ mono(⋂a:A. B[a]))
Proof
Definitions occuring in Statement :
mono: mono(T),
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
isect: ⋂x:A. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
mono: mono(T),
member: t ∈ T,
subtype_rel: A ⊆r B,
so_apply: x[s],
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
uimplies: b supposing a
Lemmas referenced :
is-above_wf,
base_wf,
all_wf,
mono_wf,
is-above-subtype
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
isect_memberEquality,
sqequalHypSubstitution,
sqequalRule,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
applyEquality,
lambdaEquality,
isectElimination,
because_Cache,
equalityTransitivity,
equalitySymmetry,
isectEquality,
independent_functionElimination,
lemma_by_obid,
functionEquality,
cumulativity,
universeEquality,
independent_isectElimination
Latex:
\mforall{}A:Type. \mforall{}B:A {}\mrightarrow{} Type. ((\mforall{}a:A. mono(B[a])) {}\mRightarrow{} mono(\mcap{}a:A. B[a]))
Date html generated:
2016_05_13-PM-04_13_52
Last ObjectModification:
2015_12_26-AM-11_10_24
Theory : subtype_1
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