Nuprl Lemma : rng_sum_0
∀[r:Rng]. ∀[a,b:ℤ]. (Σ(r) a ≤ j < b. 0) = 0 ∈ |r| supposing a ≤ b
Proof
Definitions occuring in Statement :
rng_sum: rng_sum,
rng: Rng,
rng_zero: 0,
rng_car: |r|,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
prop: ℙ,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
true: True,
so_apply: x[s],
rng: Rng,
so_lambda: λ2x.t[x],
squash: ↓T,
and: P ∧ Q,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Lemmas referenced :
rng_wf,
le_wf,
rng_times_sum_l,
rng_sum_wf,
rng_car_wf,
int_seg_wf,
rng_zero_wf,
equal_wf,
squash_wf,
true_wf,
rng_times_zero,
iff_weakening_equal
Rules used in proof :
intEquality,
independent_isectElimination,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut,
equalitySymmetry,
natural_numberEquality,
because_Cache,
rename,
setElimination,
lambdaEquality,
sqequalRule,
applyEquality,
imageElimination,
equalityTransitivity,
universeEquality,
productElimination,
functionEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}[r:Rng]. \mforall{}[a,b:\mBbbZ{}]. (\mSigma{}(r) a \mleq{} j < b. 0) = 0 supposing a \mleq{} b
Date html generated:
2018_05_21-PM-03_15_15
Last ObjectModification:
2017_12_11-PM-05_10_32
Theory : rings_1
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