Nuprl Lemma : rng_sum_split
∀[r:Rng]. ∀[a,b,c:ℤ].
(∀[E:{a..c-} ⟶ |r|]. ((Σ(r) a ≤ j < c. E[j]) = ((Σ(r) a ≤ j < b. E[j]) +r (Σ(r) b ≤ j < c. E[j])) ∈ |r|)) supposing
((b ≤ c) and
(a ≤ b))
Proof
Definitions occuring in Statement :
rng_sum: rng_sum,
rng: Rng,
rng_plus: +r,
rng_car: |r|,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
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,
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_split,
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,
le_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{}[a,b,c:\mBbbZ{}].
(\mforall{}[E:\{a..c\msupminus{}\} {}\mrightarrow{} |r|]
((\mSigma{}(r) a \mleq{} j < c. E[j]) = ((\mSigma{}(r) a \mleq{} j < b. E[j]) +r (\mSigma{}(r) b \mleq{} j < c. E[j])))) supposing
((b \mleq{} c) and
(a \mleq{} b))
Date html generated:
2016_05_15-PM-00_28_14
Last ObjectModification:
2015_12_26-PM-11_58_15
Theory : rings_1
Home
Index