Nuprl Lemma : metric-on-subtype
∀[X,Y:Type]. metric(X) ⊆r metric(Y) supposing Y ⊆r X
Proof
Definitions occuring in Statement :
metric: metric(X),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
member: t ∈ T,
metric: metric(X),
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
istype: istype(T),
cand: A c∧ B,
prop: ℙ,
guard: {T}
Lemmas referenced :
subtype_rel_dep_function,
real_wf,
rleq_wf,
int-to-real_wf,
req_wf,
radd_wf,
metric_wf,
subtype_rel_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaEquality_alt,
sqequalHypSubstitution,
setElimination,
thin,
rename,
cut,
dependent_set_memberEquality_alt,
productElimination,
functionExtensionality,
applyEquality,
hypothesisEquality,
hypothesis,
sqequalRule,
introduction,
extract_by_obid,
isectElimination,
universeIsType,
inhabitedIsType,
independent_isectElimination,
lambdaFormation_alt,
independent_pairFormation,
because_Cache,
productIsType,
functionIsType,
natural_numberEquality,
instantiate,
universeEquality,
dependent_functionElimination
Latex:
\mforall{}[X,Y:Type]. metric(X) \msubseteq{}r metric(Y) supposing Y \msubseteq{}r X
Date html generated:
2019_10_29-AM-10_53_16
Last ObjectModification:
2019_10_02-AM-09_34_54
Theory : reals
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