### Nuprl Lemma : rsum-difference2

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

Proof

Definitions occuring in Statement :  rsum: `Σ{x[k] | n≤k≤m}` rsub: `x - y` req: `x = y` real: `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]` 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]` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` implies: `P `` Q` not: `¬A` top: `Top` prop: `ℙ` uiff: `uiff(P;Q)` rev_uimplies: `rev_uimplies(P;Q)` rsub: `x - y` itermConstant: `"const"` req_int_terms: `t1 ≡ t2` real_term_value: `real_term_value(f;t)` int_term_ind: int_term_ind itermSubtract: `left (-) right` itermAdd: `left (+) right` itermVar: `vvar` itermMinus: `"-"num`
Lemmas referenced :  rsum-split req_witness rsub_wf rsum_wf int_seg_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 decidable__le all_wf req_wf le_wf real_wf radd_wf rminus_wf req_functionality rsub_functionality req_weakening uiff_transitivity req_inversion radd-assoc radd-ac real_term_polynomial itermSubtract_wf itermMinus_wf int-to-real_wf req-iff-rsub-is-0 radd_functionality rminus_functionality rsum_functionality2
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality independent_isectElimination hypothesis sqequalRule lambdaEquality applyEquality functionExtensionality because_Cache addEquality natural_numberEquality setElimination rename dependent_set_memberEquality productElimination independent_pairFormation dependent_functionElimination unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll independent_functionElimination equalityTransitivity equalitySymmetry functionEquality lambdaFormation lemma_by_obid

Latex:
\mforall{}[k,n,m:\mBbbZ{}].  \mforall{}[x,y:\{k..m  +  1\msupminus{}\}  {}\mrightarrow{}  \mBbbR{}].
((\mSigma{}\{x[i]  |  k\mleq{}i\mleq{}m\}  -  \mSigma{}\{y[i]  |  k\mleq{}i\mleq{}n\})  =  \mSigma{}\{x[i]  |  n  +  1\mleq{}i\mleq{}m\})  supposing
((\mforall{}i:\{k..n  +  1\msupminus{}\}.  (x[i]  =  y[i]))  and
(n  \mleq{}  m)  and
(k  \mleq{}  n))

Date html generated: 2017_10_03-AM-08_58_52
Last ObjectModification: 2017_07_28-AM-07_38_30

Theory : reals

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