Nuprl Lemma : quotient_subtype_quotient
∀[A,B:Type]. ∀[EA:A ⟶ A ⟶ ℙ]. ∀[EB:B ⟶ B ⟶ ℙ].
((x,y:A//EA[x;y]) ⊆r (x,y:B//EB[x;y])) supposing
((∀x,y:A. (EA[x;y] ⇒ EB[x;y])) and
EquivRel(A;x,y.EA[x;y]) and
EquivRel(B;x,y.EB[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],
all: ∀x:A. B[x],
implies: 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,
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
quotient: x,y:A//B[x; y],
and: P ∧ Q,
all: ∀x:A. B[x],
guard: {T},
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
quotient_wf,
quotient-member-eq,
equal-wf-base,
subtype_rel_self,
all_wf,
equiv_rel_wf,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaEquality,
pointwiseFunctionalityForEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
applyEquality,
independent_isectElimination,
hypothesis,
pertypeElimination,
productElimination,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
productEquality,
because_Cache,
instantiate,
axiomEquality,
functionEquality,
isect_memberEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[EA:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}]. \mforall{}[EB:B {}\mrightarrow{} B {}\mrightarrow{} \mBbbP{}].
((x,y:A//EA[x;y]) \msubseteq{}r (x,y:B//EB[x;y])) supposing
((\mforall{}x,y:A. (EA[x;y] {}\mRightarrow{} EB[x;y])) and
EquivRel(A;x,y.EA[x;y]) and
EquivRel(B;x,y.EB[x;y]) and
(A \msubseteq{}r B))
Date html generated:
2018_05_21-PM-00_04_41
Last ObjectModification:
2018_05_11-AM-10_50_13
Theory : quot_1
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