### Nuprl Lemma : sum-is-zero

`∀[n:ℕ]. ∀[f:ℕn ⟶ ℤ].  Σ(f[x] | x < n) = 0 ∈ ℤ supposing ∀i:ℕn. (f[i] = 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]` all: `∀x:A. B[x]` true: `True` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` implies: `P `` Q` nat: `ℕ` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` top: `Top`
Lemmas referenced :  equal_wf squash_wf true_wf istype-universe sum_functionality subtype_rel_self iff_weakening_equal int_seg_wf istype-int int_subtype_base nat_wf sum_constant nat_properties decidable__equal_int full-omega-unsat intformnot_wf intformeq_wf itermMultiply_wf itermConstant_wf itermVar_wf int_formula_prop_not_lemma istype-void int_formula_prop_eq_lemma int_term_value_mul_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf
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 because_Cache closedConclusion natural_numberEquality independent_isectElimination Error :lambdaFormation_alt,  dependent_functionElimination imageMemberEquality baseClosed productElimination independent_functionElimination Error :functionIsType,  setElimination rename Error :equalityIsType4,  Error :isect_memberEquality_alt,  axiomEquality Error :isectIsTypeImplies,  unionElimination approximateComputation Error :dependent_pairFormation_alt,  int_eqEquality voidElimination

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[f:\mBbbN{}n  {}\mrightarrow{}  \mBbbZ{}].    \mSigma{}(f[x]  |  x  <  n)  =  0  supposing  \mforall{}i:\mBbbN{}n.  (f[i]  =  0)

Date html generated: 2019_06_20-PM-01_18_04
Last ObjectModification: 2018_10_16-PM-04_30_17

Theory : int_2

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