Nuprl Lemma : quotient-equality
∀[T:Type]. ∀[E1,E2:T ⟶ T ⟶ ℙ].
((x,y:T//E1[x;y]) = (x,y:T//E2[x;y]) ∈ Type) supposing (EquivRel(T;x,y.E1[x;y]) and (∀x,y:T. (E2[x;y] ⇐⇒ E1[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],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
quotient: x,y:A//B[x; y],
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
cand: A c∧ B,
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,
guard: {T},
so_lambda: λ2x y.t[x; y],
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sym: Sym(T;x,y.E[x; y])
Lemmas referenced :
equal-wf-base,
equiv_rel_wf,
all_wf,
iff_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
pertypeEquality,
productEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
hypothesis,
applyEquality,
functionExtensionality,
cumulativity,
lambdaEquality,
universeEquality,
productElimination,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
dependent_functionElimination,
independent_functionElimination,
functionEquality,
isect_memberFormation,
isect_memberEquality,
axiomEquality
Latex:
\mforall{}[T:Type]. \mforall{}[E1,E2:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
((x,y:T//E1[x;y]) = (x,y:T//E2[x;y])) supposing
(EquivRel(T;x,y.E1[x;y]) and
(\mforall{}x,y:T. (E2[x;y] \mLeftarrow{}{}\mRightarrow{} E1[x;y])))
Date html generated:
2016_10_21-AM-09_43_41
Last ObjectModification:
2016_08_09-PM-00_18_17
Theory : quot_1
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