Nuprl Lemma : subtype_rel_quotient
∀[A,B:Type]. ∀[E:B ⟶ B ⟶ ℙ]. ((x,y:A//E[x;y]) ⊆r (x,y:B//E[x;y])) supposing (EquivRel(B;x,y.E[x;y]) and (A ⊆r B))
Proof
Definitions occuring in Statement :
equiv_rel: EquivRel(T;x,y.E[x; y]),
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],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
prop: ℙ,
implies: P ⇒ Q,
quotient: x,y:A//B[x; y],
and: P ∧ Q,
all: ∀x:A. B[x],
guard: {T}
Lemmas referenced :
quotient_wf,
equiv_rel_subtype,
equiv_rel_wf,
subtype_rel_wf,
quotient-member-eq,
equal-wf-base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaEquality,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
sqequalRule,
applyEquality,
hypothesis,
because_Cache,
independent_isectElimination,
universeEquality,
independent_functionElimination,
axiomEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
pointwiseFunctionalityForEquality,
pertypeElimination,
productElimination,
dependent_functionElimination,
productEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[E:B {}\mrightarrow{} B {}\mrightarrow{} \mBbbP{}].
((x,y:A//E[x;y]) \msubseteq{}r (x,y:B//E[x;y])) supposing (EquivRel(B;x,y.E[x;y]) and (A \msubseteq{}r B))
Date html generated:
2016_05_14-AM-06_08_05
Last ObjectModification:
2015_12_26-AM-11_48_32
Theory : quot_1
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