Nuprl Lemma : isect2_quotient
∀[T:Type]. ∀[E1,E2:T ⟶ T ⟶ ℙ].
(x,y:T//E1[x;y] ⋂ x,y:T//E2[x;y] ≡ x,y:T//(E1[x;y] ∧ E2[x;y])) supposing
(EquivRel(T;x,y.E1[x;y]) and
EquivRel(T;x,y.E2[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],
ext-eq: A ≡ B,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
and: 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,
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
prop: ℙ,
implies: P ⇒ Q,
all: ∀x:A. B[x],
cand: A c∧ B,
guard: {T},
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
isect2: T1 ⋂ T2,
quotient: x,y:A//B[x; y]
Lemmas referenced :
equiv_rel_wf,
istype-universe,
subtype_by_equality,
isect2_wf,
quotient_wf,
equiv_rel_and,
istype-base,
isect2_decomp,
isect2_subtype_rel2,
equal_functionality_wrt_subtype_rel2,
isect2_subtype_rel,
base_wf,
quotient-isect-base2,
quotient-member-eq,
subtype_rel_self,
bool_wf,
equal-wf-base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
axiomEquality,
hypothesis,
Error :universeIsType,
extract_by_obid,
isectElimination,
hypothesisEquality,
Error :lambdaEquality_alt,
applyEquality,
Error :inhabitedIsType,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
Error :functionIsType,
because_Cache,
universeEquality,
instantiate,
independent_isectElimination,
productEquality,
independent_functionElimination,
Error :lambdaFormation_alt,
Error :equalityIstype,
sqequalBase,
equalitySymmetry,
equalityTransitivity,
functionExtensionality,
cumulativity,
equalityElimination,
unionElimination,
isect_memberEquality,
pertypeElimination,
promote_hyp,
dependent_functionElimination,
Error :productIsType,
lambdaEquality,
pointwiseFunctionalityForEquality,
lemma_by_obid
Latex:
\mforall{}[T:Type]. \mforall{}[E1,E2:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(x,y:T//E1[x;y] \mcap{} x,y:T//E2[x;y] \mequiv{} x,y:T//(E1[x;y] \mwedge{} E2[x;y])) supposing
(EquivRel(T;x,y.E1[x;y]) and
EquivRel(T;x,y.E2[x;y]))
Date html generated:
2019_06_20-PM-00_32_28
Last ObjectModification:
2018_11_25-PM-01_55_03
Theory : quot_1
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