Nuprl Lemma : prod_sum_l_q
∀[a,b:ℤ]. ∀[E:{a..b-} ⟶ ℚ]. ∀[u:ℚ]. ((u * Σa ≤ j < b. E[j]) = Σa ≤ j < b. u * E[j] ∈ ℚ) supposing a ≤ b
Proof
Definitions occuring in Statement :
qsum: Σa ≤ j < b. E[j],
qmul: 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_times: *,
pi2: snd(t),
infix_ap: x f y,
qsum: Σa ≤ j < b. E[j]
Lemmas referenced :
rng_times_sum_l,
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:\mBbbZ{}].
\mforall{}[E:\{a..b\msupminus{}\} {}\mrightarrow{} \mBbbQ{}]. \mforall{}[u:\mBbbQ{}]. ((u * \mSigma{}a \mleq{} j < b. E[j]) = \mSigma{}a \mleq{} j < b. u * E[j]) supposing a \mleq{} b
Date html generated:
2020_05_20-AM-09_25_36
Last ObjectModification:
2020_02_04-PM-01_58_34
Theory : rationals
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