Nuprl Lemma : rsum-constant
∀[n,m:ℤ]. ∀[a:ℝ]. (Σ{a | n≤k≤m} = (a * Σ{r1 | n≤k≤m}))
Proof
Definitions occuring in Statement :
rsum: Σ{x[k] | n≤k≤m},
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
all: ∀x:A. B[x],
prop: ℙ
Lemmas referenced :
req_witness,
rsum_wf,
int_seg_wf,
rmul_wf,
int-to-real_wf,
real_wf,
le_wf,
req_weakening,
req_functionality,
req_inversion,
rsum_linearity2,
rsum_functionality2,
rmul-one-both
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
addEquality,
natural_numberEquality,
hypothesis,
independent_functionElimination,
isect_memberEquality,
because_Cache,
intEquality,
independent_isectElimination,
productElimination,
lambdaFormation
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[a:\mBbbR{}]. (\mSigma{}\{a | n\mleq{}k\mleq{}m\} = (a * \mSigma{}\{r1 | n\mleq{}k\mleq{}m\}))
Date html generated:
2016_05_18-AM-07_47_21
Last ObjectModification:
2015_12_28-AM-01_03_17
Theory : reals
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