Nuprl Lemma : quotient-mono
∀[T:Type]. (mono(T) ⇒ (∀E:T ⟶ T ⟶ ℙ. (EquivRel(T;x,y.E[x;y]) ⇒ mono(x,y:T//E[x;y]))))
Proof
Definitions occuring in Statement :
equiv_rel: EquivRel(T;x,y.E[x; y]),
mono: mono(T),
quotient: x,y:A//B[x; y],
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,
implies: P ⇒ Q,
all: ∀x:A. B[x],
mono: mono(T),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
prop: ℙ,
quotient: x,y:A//B[x; y],
and: P ∧ Q,
is-above: is-above(T;a;z),
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
cand: A c∧ B,
trans: Trans(T;x,y.E[x; y]),
sym: Sym(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y]),
equiv_rel: EquivRel(T;x,y.E[x; y])
Lemmas referenced :
is-above_wf,
quotient_wf,
istype-base,
equiv_rel_wf,
mono_wf,
istype-universe,
subtype_rel_self,
sqle_wf_base,
quotient-member-eq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
Error :lambdaFormation_alt,
hypothesis,
Error :universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
Error :lambdaEquality_alt,
applyEquality,
because_Cache,
independent_isectElimination,
Error :inhabitedIsType,
Error :functionIsType,
universeEquality,
dependent_functionElimination,
axiomEquality,
Error :functionIsTypeImplies,
instantiate,
pointwiseFunctionalityForEquality,
pertypeElimination,
productElimination,
Error :productIsType,
Error :equalityIstype,
sqequalBase,
equalitySymmetry,
equalityTransitivity,
Error :equalityIsType2,
independent_pairFormation,
Error :dependent_pairFormation_alt,
independent_functionElimination,
applyLambdaEquality,
hyp_replacement
Latex:
\mforall{}[T:Type]. (mono(T) {}\mRightarrow{} (\mforall{}E:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}. (EquivRel(T;x,y.E[x;y]) {}\mRightarrow{} mono(x,y:T//E[x;y]))))
Date html generated:
2019_06_20-PM-00_32_22
Last ObjectModification:
2018_11_24-AM-09_34_59
Theory : quot_1
Home
Index