Nuprl Lemma : m-cont-real-fun-is-mfun
∀[X:Type]. ∀[d:metric(X)]. ∀[f:X ⟶ ℝ]. (m-cont-real-fun(X;d;x.f[x]) ⇒ λx.f[x]:FUN(X;ℝ))
Proof
Definitions occuring in Statement :
m-cont-real-fun: m-cont-real-fun(X;d;x.f[x]),
is-mfun: f:FUN(X;Y),
rmetric: rmetric(),
metric: metric(X),
real: ℝ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
implies: P ⇒ Q,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
so_apply: x[s],
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
is-mfun: f:FUN(X;Y),
all: ∀x:A. B[x],
rmetric: rmetric(),
meq: x ≡ y,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
m-cont-real-fun: m-cont-real-fun(X;d;x.f[x]),
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
rneq: x ≠ y,
guard: {T},
rless: x < y,
sq_exists: ∃x:A [B[x]],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
metric: metric(X),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
absval: |i|,
req_int_terms: t1 ≡ t2,
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
req-iff-rabs-rleq,
rless-int-fractions2,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
intformand_wf,
int_formula_prop_and_lemma,
rless_wf,
rleq_weakening_rless,
rabs_wf,
rsub_wf,
nat_plus_wf,
meq_wf,
m-cont-real-fun_wf,
req_witness,
rmetric_wf,
real_wf,
metric_wf,
istype-universe,
itermSubtract_wf,
req-int,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
req_functionality,
rabs_functionality,
rsub_functionality,
req_weakening,
req_transitivity,
rabs-int,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
sq_stable__rless,
mdist_wf,
rless_functionality,
mdist-same,
mdist_functionality,
meq_weakening
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
applyEquality,
hypothesisEquality,
hypothesis,
productElimination,
independent_functionElimination,
natural_numberEquality,
isectElimination,
setElimination,
rename,
multiplyEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
universeIsType,
dependent_set_memberEquality_alt,
closedConclusion,
because_Cache,
inrFormation_alt,
independent_pairFormation,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
functionIsType,
isectIsTypeImplies,
instantiate,
universeEquality,
minusEquality,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[f:X {}\mrightarrow{} \mBbbR{}]. (m-cont-real-fun(X;d;x.f[x]) {}\mRightarrow{} \mlambda{}x.f[x]:FUN(X;\mBbbR{}))
Date html generated:
2019_10_30-AM-06_27_29
Last ObjectModification:
2019_10_02-AM-10_02_43
Theory : reals
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