Nuprl Lemma : rng_sum_is_0
∀[r:Rng]. ∀[a,b:ℤ]. ∀[F:{a..b-} ⟶ |r|].
(Σ(r) a ≤ j < b. F[j]) = 0 ∈ |r| supposing (a ≤ b) ∧ (∀j:{a..b-}. (F[j] = 0 ∈ |r|))
Proof
Definitions occuring in Statement :
rng_sum: rng_sum,
rng: Rng,
rng_zero: 0,
rng_car: |r|,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
and: P ∧ Q,
function: x:A ⟶ B[x],
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
so_apply: x[s],
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
subtype_rel: A ⊆r B,
true: True,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
rng: Rng,
squash: ↓T,
prop: ℙ,
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rng_zero_wf,
all_wf,
le_wf,
iff_weakening_equal,
rng_wf,
int_seg_wf,
true_wf,
squash_wf,
rng_sum_wf,
rng_car_wf,
equal_wf,
rng_sum_0
Rules used in proof :
axiomEquality,
isect_memberEquality,
functionExtensionality,
productEquality,
independent_functionElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
dependent_functionElimination,
intEquality,
functionEquality,
equalityTransitivity,
rename,
setElimination,
because_Cache,
imageElimination,
lambdaEquality,
applyEquality,
sqequalRule,
equalitySymmetry,
hyp_replacement,
independent_isectElimination,
productElimination,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}[r:Rng]. \mforall{}[a,b:\mBbbZ{}]. \mforall{}[F:\{a..b\msupminus{}\} {}\mrightarrow{} |r|].
(\mSigma{}(r) a \mleq{} j < b. F[j]) = 0 supposing (a \mleq{} b) \mwedge{} (\mforall{}j:\{a..b\msupminus{}\}. (F[j] = 0))
Date html generated:
2018_05_21-PM-03_15_19
Last ObjectModification:
2018_01_02-PM-02_26_34
Theory : rings_1
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