Nuprl Lemma : quotient-isect-base2
∀[T:Type]. ∀[E:T ⟶ T ⟶ ℙ]. x,y:T//E[x;y] ⋂ Base ⊆r T ⋂ Base supposing EquivRel(T;x,y.E[x;y])
Proof
Definitions occuring in Statement :
equiv_rel: EquivRel(T;x,y.E[x; y]),
isect2: T1 ⋂ T2,
quotient: x,y:A//B[x; y],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
function: x:A ⟶ B[x],
base: Base,
universe: Type
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
guard: {T},
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
subtype_rel: A ⊆r B,
prop: ℙ
Lemmas referenced :
isect2_wf,
quotient_wf,
base_wf,
subtype_rel_self,
subtype_rel_functionality_wrt_iff,
quotient-isect-base,
ext-eq_weakening,
equiv_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
independent_isectElimination,
hypothesis,
because_Cache,
isect_memberFormation,
introduction,
productElimination,
axiomEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[E:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. x,y:T//E[x;y] \mcap{} Base \msubseteq{}r T \mcap{} Base supposing EquivRel(T;x,y.E[x;y])
Date html generated:
2016_05_14-AM-06_08_09
Last ObjectModification:
2015_12_26-AM-11_48_39
Theory : quot_1
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