Nuprl Lemma : series-sum-linear3
∀x:ℕ ⟶ ℝ. ∀a,c:ℝ. (Σn.x[n] = a ⇒ Σn.x[n] * c = a * c)
Proof
Definitions occuring in Statement :
series-sum: Σn.x[n] = a,
rmul: a * b,
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 :
series-sum: Σn.x[n] = a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
converges-to: lim n→∞.x[n] = y,
member: t ∈ T,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
nat_plus: ℕ+,
sq_type: SQType(T),
guard: {T},
nat: ℕ,
subtype_rel: A ⊆r B,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
sq_exists: ∃x:{A| B[x]},
rneq: x ≠ y,
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
rleq: x ≤ y,
rnonneg: rnonneg(x),
rsub: x - y,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y
Lemmas referenced :
integer-bound,
equal_wf,
set-value-type,
less_than_wf,
int-value-type,
subtype_base_sq,
nat_plus_wf,
set_subtype_base,
int_subtype_base,
converges-to_wf,
rsum_wf,
nat_wf,
int_seg_subtype_nat,
false_wf,
int_seg_wf,
real_wf,
mul_nat_plus,
le_wf,
all_wf,
rleq_wf,
rabs_wf,
rsub_wf,
rmul_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_properties,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rless_wf,
rmul_preserves_rleq2,
mul_bounds_1b,
zero-rleq-rabs,
less_than'_wf,
radd_wf,
rminus_wf,
rleq_functionality,
req_inversion,
rabs-rmul,
req_weakening,
rabs_functionality,
rmul-distrib2,
radd_functionality,
rsum_linearity3,
rmul_over_rminus,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rleq-int-fractions2,
decidable__le,
intformle_wf,
itermMultiply_wf,
int_formula_prop_le_lemma,
int_term_value_mul_lemma,
uiff_transitivity,
rmul-int-rdiv,
rmul_comm,
rleq-int-fractions
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
because_Cache,
cutEval,
dependent_set_memberEquality,
isectElimination,
equalityTransitivity,
hypothesis,
equalitySymmetry,
lambdaEquality,
independent_isectElimination,
intEquality,
natural_numberEquality,
setElimination,
rename,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
applyEquality,
functionExtensionality,
addEquality,
independent_pairFormation,
functionEquality,
inrFormation,
unionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
isect_memberFormation,
multiplyEquality,
independent_pairEquality,
minusEquality,
axiomEquality
Latex:
\mforall{}x:\mBbbN{} {}\mrightarrow{} \mBbbR{}. \mforall{}a,c:\mBbbR{}. (\mSigma{}n.x[n] = a {}\mRightarrow{} \mSigma{}n.x[n] * c = a * c)
Date html generated:
2017_10_03-AM-09_18_01
Last ObjectModification:
2017_07_28-AM-07_43_19
Theory : reals
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