Nuprl Lemma : mfun-class-strong-subtype
∀[X,Y:Type]. ∀[d:metric(X)]. ∀[d':metric(Y)]. ∀[A:Type].
strong-subtype(mfun-class(X;d;A;d');mfun-class(X;d;Y;d')) supposing metric-subspace(Y;d';A)
Proof
Definitions occuring in Statement :
mfun-class: mfun-class(X;dx;Y;dy),
metric-subspace: metric-subspace(X;d;A),
metric: metric(X),
strong-subtype: strong-subtype(A;B),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
strong-subtype: strong-subtype(A;B),
member: t ∈ T,
metric-subspace: metric-subspace(X;d;A),
and: P ∧ Q,
subtype_rel: A ⊆r B,
cand: A c∧ B,
sq_stable: SqStable(P),
implies: P ⇒ Q,
mfun-class: mfun-class(X;dx;Y;dy),
quotient: x,y:A//B[x; y],
so_lambda: λ2x y.t[x; y],
mfun: FUN(X ⟶ Y),
so_apply: x[s1;s2],
all: ∀x:A. B[x],
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
squash: ↓T,
exists: ∃x:A. B[x],
is-mfun: f:FUN(X;Y),
meqfun: meqfun(d;A;f;g),
equiv_rel: EquivRel(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
sym: Sym(T;x,y.E[x; y])
Lemmas referenced :
sq_stable__subtype_rel,
mfun-class_wf,
metric-on-subtype,
quotient-member-eq,
mfun_wf,
meqfun_wf,
meqfun-equiv-rel-mfun,
mfun-subtype,
subtype_rel_set,
is-mfun_wf,
subtype_rel_dep_function,
equal_wf,
metric-subspace_wf,
metric_wf,
istype-universe,
meqfun-equiv-rel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
applyEquality,
independent_isectElimination,
hypothesis,
sqequalRule,
independent_functionElimination,
lambdaEquality_alt,
pointwiseFunctionalityForEquality,
pertypeElimination,
promote_hyp,
setElimination,
rename,
inhabitedIsType,
universeIsType,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
productIsType,
equalityIstype,
sqequalBase,
functionEquality,
functionIsType,
lambdaFormation_alt,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
setEquality,
productEquality,
setIsType,
instantiate,
universeEquality,
functionExtensionality,
applyLambdaEquality,
dependent_set_memberEquality_alt
Latex:
\mforall{}[X,Y:Type]. \mforall{}[d:metric(X)]. \mforall{}[d':metric(Y)]. \mforall{}[A:Type].
strong-subtype(mfun-class(X;d;A;d');mfun-class(X;d;Y;d')) supposing metric-subspace(Y;d';A)
Date html generated:
2019_10_30-AM-06_33_29
Last ObjectModification:
2019_10_02-AM-10_07_11
Theory : reals
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