Nuprl Lemma : rsum-zero-req

`∀[n,m:ℤ]. ∀[f:{n..m + 1-} ⟶ ℝ].  Σ{f[k] | n≤k≤m} = r0 supposing ∀k:{n..m + 1-}. (f[k] = r0)`

Proof

Definitions occuring in Statement :  rsum: `Σ{x[k] | n≤k≤m}` req: `x = y` int-to-real: `r(n)` real: `ℝ` int_seg: `{i..j-}` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` all: `∀x:A. B[x]` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` so_lambda: `λ2x.t[x]` so_apply: `x[s]` implies: `P `` Q` prop: `ℙ` pointwise-req: `x[k] = y[k] for k ∈ [n,m]` all: `∀x:A. B[x]` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top`
Lemmas referenced :  rsum-zero req_witness rsum_wf int_seg_wf int-to-real_wf all_wf req_wf real_wf rsum_functionality le_wf decidable__lt satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermVar_wf itermAdd_wf itermConstant_wf intformle_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_add_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_wf lelt_wf req_transitivity
Rules used in proof :  cut introduction extract_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation hypothesis sqequalHypSubstitution isectElimination thin hypothesisEquality sqequalRule lambdaEquality applyEquality functionExtensionality addEquality natural_numberEquality independent_functionElimination isect_memberEquality because_Cache equalityTransitivity equalitySymmetry functionEquality intEquality independent_isectElimination lambdaFormation dependent_set_memberEquality independent_pairFormation dependent_functionElimination unionElimination dependent_pairFormation int_eqEquality voidElimination voidEquality computeAll

Latex:
\mforall{}[n,m:\mBbbZ{}].  \mforall{}[f:\{n..m  +  1\msupminus{}\}  {}\mrightarrow{}  \mBbbR{}].    \mSigma{}\{f[k]  |  n\mleq{}k\mleq{}m\}  =  r0  supposing  \mforall{}k:\{n..m  +  1\msupminus{}\}.  (f[k]  =  r0)

Date html generated: 2016_10_26-AM-09_16_54
Last ObjectModification: 2016_10_10-PM-01_24_23

Theory : reals

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