Nuprl Lemma : rsum_linearity3
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. ∀[y:ℝ]. (Σ{x[k] * y | n≤k≤m} = (Σ{x[k] | n≤k≤m} * y))
Proof
Definitions occuring in Statement :
rsum: Σ{x[k] | n≤k≤m},
req: x = y,
rmul: a * b,
real: ℝ,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rsum: Σ{x[k] | n≤k≤m},
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
prop: ℙ,
all: ∀x:A. B[x],
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a)
Lemmas referenced :
req_witness,
rsum_wf,
rmul_wf,
int_seg_wf,
real_wf,
value-type-has-value,
int-value-type,
from-upto_wf,
list_wf,
le_wf,
less_than_wf,
valueall-type-has-valueall,
list-valueall-type,
real-valueall-type,
map_wf,
evalall-reduce,
valueall-type-real-list,
radd-list-linearity3,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
addEquality,
natural_numberEquality,
hypothesis,
independent_functionElimination,
isect_memberEquality,
because_Cache,
functionEquality,
intEquality,
independent_isectElimination,
setEquality,
productEquality,
lambdaFormation,
callbyvalueReduce,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[x:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}]. \mforall{}[y:\mBbbR{}]. (\mSigma{}\{x[k] * y | n\mleq{}k\mleq{}m\} = (\mSigma{}\{x[k] | n\mleq{}k\mleq{}m\} * y))
Date html generated:
2017_10_03-AM-08_59_21
Last ObjectModification:
2017_07_28-AM-07_38_51
Theory : reals
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