Nuprl Lemma : quotient-squash
∀[T:Type]. ∀[E:T ⟶ T ⟶ ℙ]. x,y:T//E[x;y] ≡ x,y:T//(↓E[x;y]) 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],
ext-eq: A ≡ B,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
squash: ↓T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
implies: P ⇒ Q,
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
quotient: x,y:A//B[x; y],
all: ∀x:A. B[x],
squash: ↓T,
prop: ℙ
Lemmas referenced :
equiv_rel_wf,
equal-wf-base,
quotient-member-eq,
squash_wf,
quotient_wf,
equiv_rel_squash
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
independent_functionElimination,
hypothesis,
independent_pairFormation,
pointwiseFunctionalityForEquality,
independent_isectElimination,
pertypeElimination,
productElimination,
because_Cache,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
productEquality,
universeEquality,
imageElimination,
independent_pairEquality,
axiomEquality,
isect_memberEquality,
functionEquality,
cumulativity
Latex:
\mforall{}[T:Type]. \mforall{}[E:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. x,y:T//E[x;y] \mequiv{} x,y:T//(\mdownarrow{}E[x;y]) supposing EquivRel(T;x,y.E[x;y])
Date html generated:
2016_05_14-AM-06_08_07
Last ObjectModification:
2016_01_14-PM-07_33_19
Theory : quot_1
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