Nuprl Lemma : quotient-implies-squash
∀[P,Q:ℙ]. ((P ⇒ Q) ⇒ {⇃(P) ⇒ (↓Q)})
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
squash: ↓T,
implies: P ⇒ Q,
true: True
Definitions unfolded in proof :
guard: {T},
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
prop: ℙ,
quotient: x,y:A//B[x; y],
and: P ∧ Q,
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a
Lemmas referenced :
equiv_rel_true,
quotient_wf,
true_wf,
equal-wf-base,
squash_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
rename,
pointwiseFunctionalityForEquality,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
pertypeElimination,
productElimination,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
productEquality,
because_Cache,
imageElimination,
cumulativity,
lambdaEquality,
independent_isectElimination,
functionEquality,
dependent_functionElimination,
universeEquality,
isect_memberEquality
Latex:
\mforall{}[P,Q:\mBbbP{}]. ((P {}\mRightarrow{} Q) {}\mRightarrow{} \{\00D9(P) {}\mRightarrow{} (\mdownarrow{}Q)\})
Date html generated:
2016_05_19-AM-11_56_56
Last ObjectModification:
2016_05_18-AM-08_21_36
Theory : quot_1
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