Nuprl Lemma : sum_unroll_hi_q
∀[i,j:ℤ]. ∀[E:{i..j-} ⟶ ℚ]. (Σi ≤ k < j. E[k] = (Σi ≤ k < j - 1. E[k] + E[j - 1]) ∈ ℚ) supposing i < j
Proof
Definitions occuring in Statement :
qsum: Σa ≤ j < b. E[j],
qadd: r + s,
rationals: ℚ,
int_seg: {i..j-},
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
qrng: <ℚ+*>,
rng_car: |r|,
pi1: fst(t),
rng_plus: +r,
pi2: snd(t),
infix_ap: x f y,
qsum: Σa ≤ j < b. E[j]
Lemmas referenced :
rng_sum_unroll_hi,
qrng_wf,
crng_subtype_rng
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
hypothesis,
applyEquality,
sqequalRule
Latex:
\mforall{}[i,j:\mBbbZ{}]. \mforall{}[E:\{i..j\msupminus{}\} {}\mrightarrow{} \mBbbQ{}]. (\mSigma{}i \mleq{} k < j. E[k] = (\mSigma{}i \mleq{} k < j - 1. E[k] + E[j - 1])) supposing i < j
Date html generated:
2020_05_20-AM-09_24_54
Last ObjectModification:
2020_01_26-AM-11_24_35
Theory : rationals
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