Nuprl Lemma : continuous-composition
∀I:Interval
(iproper(I)
⇒ (∀J:Interval. ∀f:{x:ℝ| x ∈ I} ⟶ {y:ℝ| y ∈ J} . ∀g:J ⟶ℝ.
(f[x] continuous for x ∈ I ⇒ g[x] continuous for x ∈ J ⇒ g[f[x]] continuous for x ∈ I)))
Proof
Definitions occuring in Statement :
continuous: f[x] continuous for x ∈ I,
rfun: I ⟶ℝ,
i-member: r ∈ I,
iproper: iproper(I),
interval: Interval,
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
subtype_rel: A ⊆r B,
rfun: I ⟶ℝ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
uimplies: b supposing a,
label: ...$L... t,
guard: {T}
Lemmas referenced :
continuous-implies-functional,
subtype_rel_dep_function,
real_wf,
i-member_wf,
set_wf,
function-is-continuous,
req_wf,
continuous_wf,
rfun_wf,
interval_wf,
iproper_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
applyEquality,
sqequalRule,
isectElimination,
because_Cache,
lambdaEquality,
setEquality,
hypothesis,
independent_isectElimination,
setElimination,
rename,
independent_functionElimination,
functionExtensionality,
dependent_set_memberEquality,
functionEquality
Latex:
\mforall{}I:Interval
(iproper(I)
{}\mRightarrow{} (\mforall{}J:Interval. \mforall{}f:\{x:\mBbbR{}| x \mmember{} I\} {}\mrightarrow{} \{y:\mBbbR{}| y \mmember{} J\} . \mforall{}g:J {}\mrightarrow{}\mBbbR{}.
(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:
2016_10_26-AM-10_00_33
Last ObjectModification:
2016_09_12-PM-01_38_06
Theory : reals
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