Nuprl Lemma : rng_sum_shift
∀[r:Rng]. ∀[a,b:ℤ].
∀[E:{a..b-} ⟶ |r|]. ∀[k:ℤ]. ((Σ(r) a ≤ j < b. E[j]) = (Σ(r) a + k ≤ j < b + k. E[j - k]) ∈ |r|) supposing a ≤ b
Proof
Definitions occuring in Statement :
rng_sum: rng_sum,
rng: Rng,
rng_car: |r|,
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,
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),
uimplies: b supposing a,
rng: Rng
Lemmas referenced :
mon_itop_shift,
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,
intEquality,
functionEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[r:Rng]. \mforall{}[a,b:\mBbbZ{}].
\mforall{}[E:\{a..b\msupminus{}\} {}\mrightarrow{} |r|]. \mforall{}[k:\mBbbZ{}]. ((\mSigma{}(r) a \mleq{} j < b. E[j]) = (\mSigma{}(r) a + k \mleq{} j < b + k. E[j - k]))
supposing a \mleq{} b
Date html generated:
2016_05_15-PM-00_28_12
Last ObjectModification:
2015_12_26-PM-11_58_29
Theory : rings_1
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