Nuprl Lemma : continuous-composition-from-maps-compact
∀I,J:Interval. ∀f:I ⟶ℝ. ∀g:J ⟶ℝ.
(maps-compact(I;J;x.f[x]) ⇒ f[x] continuous for x ∈ I ⇒ g[x] continuous for x ∈ J ⇒ g[f[x]] continuous for x ∈ I)
Proof
Definitions occuring in Statement :
maps-compact: maps-compact(I;J;x.f[x]),
continuous: f[x] continuous for x ∈ I,
rfun: I ⟶ℝ,
interval: Interval,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
continuous: f[x] continuous for x ∈ I,
member: t ∈ T,
maps-compact: maps-compact(I;J;x.f[x]),
exists: ∃x:A. B[x],
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s],
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rless: x < y,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top
Lemmas referenced :
small-reciprocal-real,
rless_wf,
int-to-real_wf,
nat_plus_wf,
icompact_wf,
i-approx_wf,
continuous_wf,
real_wf,
i-member_wf,
maps-compact_wf,
rfun_wf,
interval_wf,
istype-less_than,
rleq_wf,
rabs_wf,
rsub_wf,
rdiv_wf,
rless-int,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
i-member-approx,
rless_transitivity2,
rleq_weakening_rless
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
rename,
cut,
hypothesis,
setElimination,
introduction,
extract_by_obid,
dependent_set_memberEquality_alt,
universeIsType,
isectElimination,
natural_numberEquality,
setIsType,
sqequalRule,
lambdaEquality_alt,
applyEquality,
inhabitedIsType,
productIsType,
functionIsType,
because_Cache,
closedConclusion,
independent_isectElimination,
inrFormation_alt,
independent_functionElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
promote_hyp
Latex:
\mforall{}I,J:Interval. \mforall{}f:I {}\mrightarrow{}\mBbbR{}. \mforall{}g:J {}\mrightarrow{}\mBbbR{}.
(maps-compact(I;J;x.f[x])
{}\mRightarrow{} f[x] continuous for x \mmember{} I
{}\mRightarrow{} g[x] continuous for x \mmember{} J
{}\mRightarrow{} g[f[x]] continuous for x \mmember{} I)
Date html generated:
2019_10_30-AM-07_47_14
Last ObjectModification:
2018_11_12-AM-10_54_20
Theory : reals
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