Nuprl Lemma : series-diverges-rmul
∀[x:ℕ ⟶ ℝ]. (Σn.x[n]↑ ⇒ (∀c:ℝ. (c ≠ r0 ⇒ Σn.c * x[n]↑)))
Proof
Definitions occuring in Statement :
series-diverges: Σn.x[n]↑,
rneq: x ≠ y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
series-diverges: Σn.x[n]↑,
diverges: n.x[n]↑,
exists: ∃x:A. B[x],
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
prop: ℙ,
so_lambda: λ2x.t[x],
nat: ℕ,
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rsub: x - y,
rleq: x ≤ y,
rnonneg: rnonneg(x),
real: ℝ
Lemmas referenced :
rmul_wf,
rabs_wf,
nat_wf,
rless_wf,
int-to-real_wf,
all_wf,
exists_wf,
le_wf,
rleq_wf,
rsub_wf,
rsum_wf,
int_seg_subtype_nat,
false_wf,
int_seg_wf,
rneq_wf,
real_wf,
series-diverges_wf,
rmul-is-positive,
rabs-neq-zero,
rleq_functionality,
req_weakening,
rabs_functionality,
rsub_functionality,
rsum_linearity2,
equal_wf,
radd_wf,
rminus_wf,
req_functionality,
req_transitivity,
rmul-distrib,
radd_functionality,
rmul_over_rminus,
rabs-rmul,
rmul_preserves_rleq2,
zero-rleq-rabs,
less_than'_wf,
nat_plus_wf,
rmul_comm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
independent_pairFormation,
productEquality,
natural_numberEquality,
sqequalRule,
lambdaEquality,
because_Cache,
setElimination,
rename,
applyEquality,
functionExtensionality,
addEquality,
independent_isectElimination,
functionEquality,
dependent_functionElimination,
independent_functionElimination,
inlFormation,
promote_hyp,
equalityTransitivity,
equalitySymmetry,
addLevel,
impliesFunctionality,
independent_pairEquality,
voidElimination,
minusEquality,
axiomEquality
Latex:
\mforall{}[x:\mBbbN{} {}\mrightarrow{} \mBbbR{}]. (\mSigma{}n.x[n]\muparrow{} {}\mRightarrow{} (\mforall{}c:\mBbbR{}. (c \mneq{} r0 {}\mRightarrow{} \mSigma{}n.c * x[n]\muparrow{})))
Date html generated:
2017_10_03-AM-09_18_52
Last ObjectModification:
2017_07_28-AM-07_44_07
Theory : reals
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