Nuprl Lemma : quotient_wf
∀[T:Type]. ∀[E:T ⟶ T ⟶ ℙ]. x,y:T//E[x;y] ∈ Type supposing EquivRel(T;x,y.E[x;y])
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,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
and: P ∧ Q,
subtype_rel: A ⊆r B,
cand: A c∧ B,
guard: {T},
equiv_rel: EquivRel(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
all: ∀x:A. B[x],
implies: P ⇒ Q,
quotient: x,y:A//B[x; y],
sym: Sym(T;x,y.E[x; y])
Lemmas referenced :
equiv_rel_wf,
equal-wf-base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
lemma_by_obid,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality,
applyEquality,
isect_memberEquality,
because_Cache,
functionEquality,
cumulativity,
universeEquality,
productEquality,
productElimination,
independent_pairFormation,
dependent_functionElimination,
independent_functionElimination,
pertypeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[E:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. x,y:T//E[x;y] \mmember{} Type supposing EquivRel(T;x,y.E[x;y])
Date html generated:
2016_05_14-AM-06_07_42
Last ObjectModification:
2015_12_26-AM-11_48_38
Theory : quot_1
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