### Nuprl Lemma : rroot-exists

`∀i:{2...}. ∀x:{x:ℝ| (↑isEven(i)) `` (r0 ≤ x)} .  (∃y:{ℝ| (((↑isEven(i)) `` (r0 ≤ y)) ∧ (y^i = x))})`

Proof

Definitions occuring in Statement :  rleq: `x ≤ y` rnexp: `x^k1` req: `x = y` int-to-real: `r(n)` real: `ℝ` isEven: `isEven(n)` int_upper: `{i...}` assert: `↑b` all: `∀x:A. B[x]` sq_exists: `∃x:{A| B[x]}` implies: `P `` Q` and: `P ∧ Q` set: `{x:A| B[x]} ` natural_number: `\$n`
Definitions unfolded in proof :  all: `∀x:A. B[x]` member: `t ∈ T` exists: `∃x:A. B[x]` implies: `P `` Q` so_lambda: `λ2x.t[x]` so_apply: `x[s]` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` converges: `x[n]↓ as n→∞` sq_exists: `∃x:{A| B[x]}` cand: `A c∧ B` prop: `ℙ` uall: `∀[x:A]. B[x]` int_upper: `{i...}` subtype_rel: `A ⊆r B` nat: `ℕ` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` uimplies: `b supposing a` sq_stable: `SqStable(P)` squash: `↓T` guard: `{T}` top: `Top` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` uiff: `uiff(P;Q)` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` nequal: `a ≠ b ∈ T ` subtract: `n - m` rev_uimplies: `rev_uimplies(P;Q)`
Lemmas referenced :  rroot-exists1-ext rroot-exists-part2 converges-iff-cauchy nat_wf assert_wf isEven_wf rleq_wf int-to-real_wf req_wf rnexp_wf int_upper_subtype_nat false_wf le_wf set_wf real_wf int_upper_wf constant-rleq-limit sq_stable__rleq unique-limit rnexp_zero_lemma constant-limit req_weakening rmul-limit converges-to_wf subtract_wf nat_properties int_upper_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermSubtract_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_subtract_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf less_than_wf primrec-wf2 equal_wf converges-to_functionality rmul_wf eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int intformeq_wf int_formula_prop_eq_lemma int_subtype_base rmul_comm req_functionality rnexp_unroll rmul-one-both
Rules used in proof :  cut introduction extract_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation hypothesis sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality productElimination independent_functionElimination sqequalRule lambdaEquality applyEquality setElimination rename dependent_set_memberFormation isectElimination independent_pairFormation productEquality functionEquality because_Cache natural_numberEquality dependent_set_memberEquality independent_isectElimination functionExtensionality imageMemberEquality baseClosed imageElimination isect_memberEquality voidElimination voidEquality unionElimination dependent_pairFormation int_eqEquality intEquality computeAll equalityTransitivity equalitySymmetry equalityElimination promote_hyp instantiate cumulativity

Latex:
\mforall{}i:\{2...\}.  \mforall{}x:\{x:\mBbbR{}|  (\muparrow{}isEven(i))  {}\mRightarrow{}  (r0  \mleq{}  x)\}  .    (\mexists{}y:\{\mBbbR{}|  (((\muparrow{}isEven(i))  {}\mRightarrow{}  (r0  \mleq{}  y))  \mwedge{}  (y\^{}i  =  x))\})

Date html generated: 2017_10_03-AM-10_39_20
Last ObjectModification: 2017_07_28-AM-08_15_44

Theory : reals

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