Nuprl Lemma : empty_support
∀[n:ℕ]. ∀[f:ℕn ⟶ ℤ]. Σ(f[x] | x < n) = 0 ∈ ℤ supposing ∀x:ℕn. (f[x] = 0 ∈ ℤ)
Proof
Definitions occuring in Statement :
sum: Σ(f[x] | x < k),
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
istype-universe,
sum-is-zero,
int_seg_wf,
subtype_rel_self,
iff_weakening_equal,
istype-int,
int_subtype_base,
istype-nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
applyEquality,
thin,
Error :lambdaEquality_alt,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
Error :universeIsType,
Error :inhabitedIsType,
instantiate,
universeEquality,
intEquality,
sqequalRule,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
imageMemberEquality,
baseClosed,
because_Cache,
productElimination,
independent_functionElimination,
Error :functionIsType,
Error :equalityIsType4,
Error :isect_memberEquality_alt,
axiomEquality,
Error :isectIsTypeImplies
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} \mBbbZ{}]. \mSigma{}(f[x] | x < n) = 0 supposing \mforall{}x:\mBbbN{}n. (f[x] = 0)
Date html generated:
2019_06_20-PM-01_18_27
Last ObjectModification:
2018_10_18-PM-00_43_37
Theory : int_2
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