Nuprl Lemma : mfun-strong-subtype
∀[X,Y:Type]. ∀[d:metric(X)]. ∀[d':metric(Y)]. ∀[A:Type].
strong-subtype(FUN(X ⟶ A);FUN(X ⟶ Y)) supposing metric-subspace(Y;d';A)
Proof
Definitions occuring in Statement :
metric-subspace: metric-subspace(X;d;A),
mfun: FUN(X ⟶ Y),
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,
squash: ↓T,
exists: ∃x:A. B[x],
prop: ℙ,
mfun: FUN(X ⟶ Y),
is-mfun: f:FUN(X;Y),
all: ∀x:A. B[x],
guard: {T}
Lemmas referenced :
sq_stable__subtype_rel,
mfun_wf,
metric-on-subtype,
mfun-subtype,
equal_wf,
is-mfun_wf,
metric-subspace_wf,
metric_wf,
istype-universe,
meq_weakening
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,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
setEquality,
productEquality,
lambdaEquality_alt,
setElimination,
rename,
applyLambdaEquality,
dependent_set_memberEquality_alt,
universeIsType,
because_Cache,
setIsType,
productIsType,
equalityIstype,
inhabitedIsType,
instantiate,
universeEquality,
functionExtensionality,
dependent_functionElimination,
equalitySymmetry,
equalityTransitivity,
lambdaFormation_alt
Latex:
\mforall{}[X,Y:Type]. \mforall{}[d:metric(X)]. \mforall{}[d':metric(Y)]. \mforall{}[A:Type].
strong-subtype(FUN(X {}\mrightarrow{} A);FUN(X {}\mrightarrow{} Y)) supposing metric-subspace(Y;d';A)
Date html generated:
2019_10_30-AM-06_32_28
Last ObjectModification:
2019_10_02-AM-10_06_18
Theory : reals
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