Nuprl Lemma : sum_shift_q
∀[a,b:ℤ]. ∀[E:{a..b-} ⟶ ℚ]. ∀[k:ℤ]. (Σa ≤ j < b. E[j] = Σa + k ≤ j < b + k. E[j - k] ∈ ℚ) supposing a ≤ b
Proof
Definitions occuring in Statement :
qsum: Σa ≤ j < b. E[j],
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],
subtract: n - m,
add: n + m,
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),
qsum: Σa ≤ j < b. E[j]
Lemmas referenced :
rng_sum_shift,
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{}[k:\mBbbZ{}]. (\mSigma{}a \mleq{} j < b. E[j] = \mSigma{}a + k \mleq{} j < b + k. E[j - k]) supposing a \mleq{} b
Date html generated:
2020_05_20-AM-09_25_15
Last ObjectModification:
2020_02_03-PM-02_15_32
Theory : rationals
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