Nuprl Lemma : rroot-exists1

`∀i:{2...}. ∀x:{x:ℝ| (↑isEven(i)) `` (r0 ≤ x)} .`
`  ∃q:{q:ℕ ⟶ ℝ| `
`      (∀n,m:ℕ.  (((r0 ≤ (q n)) ∧ (r0 ≤ (q m))) ∨ (((q n) ≤ r0) ∧ ((q m) ≤ r0))))`
`      ∧ ((↑isEven(i)) `` (∀m:ℕ. (r0 ≤ (q m))))} `
`   lim n→∞.q n^i = x`

Proof

Definitions occuring in Statement :  converges-to: `lim n→∞.x[n] = y` rleq: `x ≤ y` rnexp: `x^k1` int-to-real: `r(n)` real: `ℝ` isEven: `isEven(n)` int_upper: `{i...}` nat: `ℕ` assert: `↑b` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` or: `P ∨ Q` and: `P ∧ Q` set: `{x:A| B[x]} ` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n`
Definitions unfolded in proof :  all: `∀x:A. B[x]` member: `t ∈ T` exists: `∃x:A. B[x]` and: `P ∧ Q` cand: `A c∧ B` or: `P ∨ Q` uall: `∀[x:A]. B[x]` so_lambda: `λ2x.t[x]` prop: `ℙ` so_apply: `x[s]` implies: `P `` Q` int_upper: `{i...}` subtype_rel: `A ⊆r B` nat: `ℕ` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A`
Lemmas referenced :  rroot-exists-part1 all_wf nat_wf or_wf rleq_wf int-to-real_wf assert_wf isEven_wf converges-to_wf rnexp_wf int_upper_subtype_nat false_wf le_wf real_wf int_upper_wf
Rules used in proof :  cut lemma_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation hypothesis sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality productElimination dependent_pairFormation sqequalRule independent_pairFormation dependent_set_memberEquality productEquality isectElimination lambdaEquality because_Cache natural_numberEquality applyEquality functionEquality setElimination rename setEquality

Latex:
\mforall{}i:\{2...\}.  \mforall{}x:\{x:\mBbbR{}|  (\muparrow{}isEven(i))  {}\mRightarrow{}  (r0  \mleq{}  x)\}  .
\mexists{}q:\{q:\mBbbN{}  {}\mrightarrow{}  \mBbbR{}|
(\mforall{}n,m:\mBbbN{}.    (((r0  \mleq{}  (q  n))  \mwedge{}  (r0  \mleq{}  (q  m)))  \mvee{}  (((q  n)  \mleq{}  r0)  \mwedge{}  ((q  m)  \mleq{}  r0))))
\mwedge{}  ((\muparrow{}isEven(i))  {}\mRightarrow{}  (\mforall{}m:\mBbbN{}.  (r0  \mleq{}  (q  m))))\}
lim  n\mrightarrow{}\minfty{}.q  n\^{}i  =  x

Date html generated: 2016_05_18-AM-09_33_08
Last ObjectModification: 2015_12_27-PM-11_18_47

Theory : reals

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