Nuprl Lemma : quotient-isect-base
∀[T:Type]. ∀[E:T ⟶ T ⟶ ℙ]. x,y:T//E[x;y] ⋂ Base ≡ 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],
ext-eq: A ≡ B,
uimplies: b supposing a,
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 :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
isect2: T1 ⋂ T2,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
cand: A c∧ B,
bfalse: ff,
or: P ∨ Q,
prop: ℙ,
quotient: x,y:A//B[x; y]
Lemmas referenced :
isect2_decomp,
quotient_wf,
istype-universe,
base_wf,
isect2_subtype_rel2,
bool_wf,
isect2_wf,
isect2_subtype_rel3,
subtype_quotient,
subtype_rel_wf,
equiv_rel_wf,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
Error :lambdaEquality_alt,
isect_memberEquality,
sqequalHypSubstitution,
unionElimination,
thin,
equalityElimination,
sqequalRule,
extract_by_obid,
isectElimination,
because_Cache,
applyEquality,
hypothesisEquality,
Error :inhabitedIsType,
hypothesis,
independent_isectElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
Error :universeIsType,
Error :inlFormation_alt,
independent_pairEquality,
axiomEquality,
Error :isect_memberEquality_alt,
Error :functionIsType,
universeEquality,
pertypeElimination,
Error :productIsType,
Error :equalityIsType4,
instantiate
Latex:
\mforall{}[T:Type]. \mforall{}[E:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. x,y:T//E[x;y] \mcap{} Base \mequiv{} T \mcap{} Base supposing EquivRel(T;x,y.E[x;y])
Date html generated:
2019_06_20-PM-00_32_20
Last ObjectModification:
2018_10_06-PM-03_56_25
Theory : quot_1
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