Nuprl Lemma : rng_sum_unroll_lo
∀[r:Rng]. ∀[i,j:ℤ].
∀[E:{i..j-} ⟶ |r|]. ((Σ(r) i ≤ k < j. E[k]) = (E[i] +r (Σ(r) i + 1 ≤ k < j. E[k])) ∈ |r|) supposing i < j
Proof
Definitions occuring in Statement :
rng_sum: rng_sum,
rng: Rng,
rng_plus: +r,
rng_car: |r|,
int_seg: {i..j-},
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
so_apply: x[s],
function: x:A ⟶ B[x],
add: 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,
grp: Group{i},
mon: Mon,
imon: IMonoid,
prop: ℙ,
rng_sum: rng_sum,
add_grp_of_rng: r↓+gp,
grp_car: |g|,
pi1: fst(t),
grp_op: *,
pi2: snd(t),
uimplies: b supposing a,
rng: Rng
Lemmas referenced :
mon_itop_unroll_lo,
add_grp_of_rng_wf_a,
grp_sig_wf,
monoid_p_wf,
grp_car_wf,
grp_op_wf,
grp_id_wf,
inverse_wf,
grp_inv_wf,
int_seg_wf,
rng_car_wf,
less_than_wf,
rng_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
sqequalRule,
lambdaEquality,
setElimination,
rename,
setEquality,
cumulativity,
isect_memberEquality,
axiomEquality,
functionEquality,
equalityTransitivity,
equalitySymmetry,
intEquality
Latex:
\mforall{}[r:Rng]. \mforall{}[i,j:\mBbbZ{}].
\mforall{}[E:\{i..j\msupminus{}\} {}\mrightarrow{} |r|]. ((\mSigma{}(r) i \mleq{} k < j. E[k]) = (E[i] +r (\mSigma{}(r) i + 1 \mleq{} k < j. E[k])))
supposing i < j
Date html generated:
2016_05_15-PM-00_28_09
Last ObjectModification:
2015_12_26-PM-11_58_33
Theory : rings_1
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