Nuprl Lemma : mdist-m-cont
∀[X:Type]. ∀d:metric(X). ∀a:X. m-cont-real-fun(X;d;x.mdist(d;a;x))
Proof
Definitions occuring in Statement :
m-cont-real-fun: m-cont-real-fun(X;d;x.f[x]),
mdist: mdist(d;x;y),
metric: metric(X),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
m-cont-real-fun: m-cont-real-fun(X;d;x.f[x]),
exists: ∃x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
prop: ℙ,
uimplies: b supposing a,
uiff: uiff(P;Q),
rge: x ≥ y,
guard: {T},
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top
Lemmas referenced :
rabs-difference-bound-iff,
mdist_wf,
rless_wf,
rabs_wf,
rsub_wf,
real_wf,
int-to-real_wf,
metric_wf,
istype-universe,
mdist-triangle-inequality,
radd_wf,
rless-implies-rless,
itermSubtract_wf,
itermVar_wf,
itermAdd_wf,
req-iff-rsub-is-0,
rless_functionality_wrt_implies,
rsub_functionality_wrt_rleq,
rleq_weakening_equal,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_const_lemma,
rless_functionality,
radd_functionality,
mdist-symm,
req_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
dependent_pairFormation_alt,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesis,
setElimination,
rename,
because_Cache,
productElimination,
independent_functionElimination,
independent_pairFormation,
universeIsType,
sqequalRule,
functionIsType,
setIsType,
natural_numberEquality,
instantiate,
universeEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination
Latex:
\mforall{}[X:Type]. \mforall{}d:metric(X). \mforall{}a:X. m-cont-real-fun(X;d;x.mdist(d;a;x))
Date html generated:
2019_10_30-AM-06_27_49
Last ObjectModification:
2019_10_02-AM-10_03_01
Theory : reals
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