Nuprl Lemma : sum_split_q
∀[a,b,c:ℤ].
(∀[E:{a..c-} ⟶ ℚ]. (Σa ≤ j < c. E[j] = (Σa ≤ j < b. E[j] + Σb ≤ j < c. E[j]) ∈ ℚ)) supposing ((b ≤ c) and (a ≤ b))
Proof
Definitions occuring in Statement :
qsum: Σa ≤ j < b. E[j],
qadd: r + s,
rationals: ℚ,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
function: x:A ⟶ B[x],
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_split,
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{}[a,b,c:\mBbbZ{}].
(\mforall{}[E:\{a..c\msupminus{}\} {}\mrightarrow{} \mBbbQ{}]. (\mSigma{}a \mleq{} j < c. E[j] = (\mSigma{}a \mleq{} j < b. E[j] + \mSigma{}b \mleq{} j < c. E[j]))) supposing
((b \mleq{} c) and
(a \mleq{} b))
Date html generated:
2020_05_20-AM-09_25_22
Last ObjectModification:
2020_02_03-PM-02_17_25
Theory : rationals
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