Nuprl Lemma : rng_minus_sum
∀[r:Rng]. ∀[a,b:ℤ].
∀[F:{a..b-} ⟶ |r|]. ((-r (Σ(r) a ≤ i < b. F[i])) = (Σ(r) a ≤ i < b. -r F[i]) ∈ |r|) supposing a ≤ b
Proof
Definitions occuring in Statement :
rng_sum: rng_sum,
rng: Rng,
rng_minus: -r,
rng_car: |r|,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
apply: f a,
function: x:A ⟶ B[x],
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
true: True,
squash: ↓T,
prop: ℙ,
rng: Rng,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
and: P ∧ Q,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
rng_wf,
le_wf,
int_seg_wf,
rng_car_wf,
equal_wf,
rng_one_wf,
rng_minus_wf,
rng_times_sum_l,
rng_sum_wf,
squash_wf,
true_wf,
rng_times_over_minus,
rng_times_one,
iff_weakening_equal
Rules used in proof :
intEquality,
equalityTransitivity,
axiomEquality,
isect_memberEquality,
functionEquality,
baseClosed,
imageMemberEquality,
natural_numberEquality,
imageElimination,
lambdaEquality,
sqequalRule,
equalitySymmetry,
hyp_replacement,
because_Cache,
rename,
setElimination,
applyEquality,
hypothesis,
independent_isectElimination,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
functionExtensionality,
universeEquality,
productElimination,
independent_functionElimination
Latex:
\mforall{}[r:Rng]. \mforall{}[a,b:\mBbbZ{}].
\mforall{}[F:\{a..b\msupminus{}\} {}\mrightarrow{} |r|]. ((-r (\mSigma{}(r) a \mleq{} i < b. F[i])) = (\mSigma{}(r) a \mleq{} i < b. -r F[i])) supposing a \mleq{} b
Date html generated:
2018_05_21-PM-03_15_12
Last ObjectModification:
2017_12_14-AM-10_03_39
Theory : rings_1
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