### Nuprl Lemma : IVT-rpolynomial1

`∀n:ℕ. ∀a:ℕn + 1 ⟶ ℝ.`
`  (((Σi≤n. a_i * r0^i) < r0) `` (r0 < (Σi≤n. a_i * r1^i)) `` (∃x:{x:ℝ| x ∈ [r0, r1]} . ((Σi≤n. a_i * x^i) = r0)))`

Proof

Definitions occuring in Statement :  rccint: `[l, u]` i-member: `r ∈ I` rpolynomial: `(Σi≤n. a_i * x^i)` rless: `x < y` req: `x = y` int-to-real: `r(n)` real: `ℝ` int_seg: `{i..j-}` nat: `ℕ` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` set: `{x:A| B[x]} ` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` uall: `∀[x:A]. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` true: `True` prop: `ℙ` so_lambda: `λ2x.t[x]` rfun: `I ⟶ℝ` so_apply: `x[s]` uimplies: `b supposing a` r-ap: `f(x)` top: `Top` guard: `{T}` sq_exists: `∃x:A [B[x]]` exists: `∃x:A. B[x]` sq_stable: `SqStable(P)` nat: `ℕ` pointwise-req: `x[k] = y[k] for k ∈ [n,m]` int_seg: `{i..j-}` rless: `x < y` nat_plus: `ℕ+` ge: `i ≥ j ` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False`
Lemmas referenced :  IVT-locally-non-constant int-to-real_wf rless-int rless_wf rpolynomial_wf real_wf i-member_wf rccint_wf req_wf member_rccint_lemma istype-void rleq_weakening_rless rpolynomial-locally-non-zero rleq_wf sq_stable__req int_seg_wf istype-nat req_weakening nat_plus_properties nat_properties decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf istype-int int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf decidable__lt intformless_wf itermAdd_wf int_formula_prop_less_lemma int_term_value_add_lemma istype-le istype-less_than rpolynomial_functionality
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation_alt cut introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination thin isectElimination natural_numberEquality hypothesis productElimination independent_functionElimination sqequalRule independent_pairFormation imageMemberEquality hypothesisEquality baseClosed dependent_set_memberEquality_alt universeIsType lambdaEquality_alt setElimination rename setIsType independent_isectElimination because_Cache isect_memberEquality_alt voidElimination dependent_pairFormation_alt productIsType imageElimination functionIsType addEquality applyEquality unionElimination approximateComputation int_eqEquality

Latex:
\mforall{}n:\mBbbN{}.  \mforall{}a:\mBbbN{}n  +  1  {}\mrightarrow{}  \mBbbR{}.
(((\mSigma{}i\mleq{}n.  a\_i  *  r0\^{}i)  <  r0)
{}\mRightarrow{}  (r0  <  (\mSigma{}i\mleq{}n.  a\_i  *  r1\^{}i))
{}\mRightarrow{}  (\mexists{}x:\{x:\mBbbR{}|  x  \mmember{}  [r0,  r1]\}  .  ((\mSigma{}i\mleq{}n.  a\_i  *  x\^{}i)  =  r0)))

Date html generated: 2019_10_30-AM-09_13_13
Last ObjectModification: 2019_02_13-AM-11_20_28

Theory : reals

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