Nuprl Lemma : total-function-limit
∀f:ℝ ⟶ ℝ. ∀y:ℝ. ∀x:ℕ ⟶ ℝ. ((∀x,y:ℝ. ((x = y) ⇒ (f[x] = f[y]))) ⇒ lim n→∞.x[n] = y ⇒ lim n→∞.f[x[n]] = f[y])
Proof
Definitions occuring in Statement :
converges-to: lim n→∞.x[n] = y,
req: x = y,
real: ℝ,
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
rfun: I ⟶ℝ,
top: Top,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
implies: P ⇒ Q,
r-ap: f(x),
true: True
Lemmas referenced :
continuous-limit,
riiint_wf,
member_riiint_lemma,
subtype_rel_dep_function,
real_wf,
true_wf,
subtype_rel_self,
set_wf,
all_wf,
req_wf,
nat_wf,
function-is-continuous,
iproper-riiint,
i-member_wf,
converges-to_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_functionElimination,
thin,
hypothesis,
lambdaFormation,
hypothesisEquality,
applyEquality,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
isectElimination,
lambdaEquality,
setEquality,
independent_isectElimination,
setElimination,
rename,
because_Cache,
independent_functionElimination,
functionEquality,
functionExtensionality,
natural_numberEquality
Latex:
\mforall{}f:\mBbbR{} {}\mrightarrow{} \mBbbR{}. \mforall{}y:\mBbbR{}. \mforall{}x:\mBbbN{} {}\mrightarrow{} \mBbbR{}.
((\mforall{}x,y:\mBbbR{}. ((x = y) {}\mRightarrow{} (f[x] = f[y]))) {}\mRightarrow{} lim n\mrightarrow{}\minfty{}.x[n] = y {}\mRightarrow{} lim n\mrightarrow{}\minfty{}.f[x[n]] = f[y])
Date html generated:
2016_10_26-AM-09_52_10
Last ObjectModification:
2016_09_05-AM-08_42_16
Theory : reals
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