Nuprl Lemma : mtb-cantor-map-onto
∀[X:Type]. ∀[d:metric(X)]. ∀[cmplt:mcomplete(X with d)]. ∀[mtb:m-TB(X;d)]. ∀[x:X].
mtb-cantor-map(d;cmplt;mtb;mtb-point-cantor(mtb;x)) ≡ x
Proof
Definitions occuring in Statement :
mtb-cantor-map: mtb-cantor-map(d;cmplt;mtb;p),
mtb-point-cantor: mtb-point-cantor(mtb;p),
m-TB: m-TB(X;d),
mcomplete: mcomplete(M),
mk-metric-space: X with d,
meq: x ≡ y,
metric: metric(X),
uall: ∀[x:A]. B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
implies: P ⇒ Q,
false: False,
mtb-cantor-map: mtb-cantor-map(d;cmplt;mtb;p),
guard: {T},
squash: ↓T,
prop: ℙ,
metric: metric(X),
so_lambda: λ2x.t[x],
true: True,
so_apply: x[s],
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
meq: x ≡ y
Lemmas referenced :
mtb-point-cantor-seq-regular,
m-regularize-of-regular,
mtb-seq_wf,
mtb-point-cantor_wf,
m-k-regular-monotone,
istype-void,
istype-le,
istype-false,
m-regularize-mcauchy,
cauchy-mlimit-unique,
nat_wf,
mconverges-to_wf,
squash_wf,
true_wf,
istype-nat,
real_wf,
subtype_rel_self,
iff_weakening_equal,
mtb-seq-mtb-point-cantor-mconverges-to,
req_witness,
mtb-cantor-map_wf,
int-to-real_wf,
m-TB_wf,
mcomplete_wf,
mk-metric-space_wf,
metric_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
because_Cache,
hypothesis,
independent_isectElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
sqequalRule,
lambdaFormation_alt,
voidElimination,
equalityTransitivity,
equalitySymmetry,
functionExtensionality,
applyEquality,
lambdaEquality_alt,
imageElimination,
universeIsType,
functionIsType,
setElimination,
rename,
imageMemberEquality,
baseClosed,
instantiate,
productElimination,
independent_functionElimination,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
universeEquality
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[cmplt:mcomplete(X with d)]. \mforall{}[mtb:m-TB(X;d)]. \mforall{}[x:X].
mtb-cantor-map(d;cmplt;mtb;mtb-point-cantor(mtb;x)) \mequiv{} x
Date html generated:
2019_10_30-AM-07_10_12
Last ObjectModification:
2019_10_09-AM-09_40_57
Theory : reals
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