Nuprl Lemma : list-prod-set-type
∀[A,T:Type]. ∀[L:(A × T) List]. ∀[P:T ⟶ ℙ]. L ∈ (A × {x:T| P[x]} ) List supposing (∀p∈L.P[snd(p)])
Proof
Definitions occuring in Statement :
l_all: (∀x∈L.P[x]),
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
pi2: snd(t),
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
pi2: snd(t)
Lemmas referenced :
list-set-type2,
pi2_wf,
subtype_rel_list,
l_all_wf,
l_member_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
productEquality,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesis,
independent_isectElimination,
setEquality,
universeEquality,
because_Cache,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
isect_memberEquality,
functionEquality,
cumulativity,
productElimination,
independent_pairEquality,
dependent_set_memberEquality
Latex:
\mforall{}[A,T:Type]. \mforall{}[L:(A \mtimes{} T) List]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. L \mmember{} (A \mtimes{} \{x:T| P[x]\} ) List supposing (\mforall{}p\mmember{}L.P[snd(p)])
Date html generated:
2016_05_14-AM-07_49_02
Last ObjectModification:
2015_12_26-PM-04_45_19
Theory : list_1
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