### Nuprl Lemma : IVT-rpolynomial2

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

Proof

Definitions occuring in Statement :  rccint: `[l, u]` i-member: `r ∈ I` rpolynomial: `(Σi≤n. a_i * x^i)` rleq: `x ≤ y` rless: `x < y` req: `x = y` 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` prop: `ℙ` uall: `∀[x:A]. B[x]` nat: `ℕ` exists: `∃x:A. B[x]` uimplies: `b supposing a` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` itermConstant: `"const"` req_int_terms: `t1 ≡ t2` false: `False` not: `¬A` top: `Top` uiff: `uiff(P;Q)` guard: `{T}` rpolynomial: `(Σi≤n. a_i * x^i)` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` le: `A ≤ B` less_than': `less_than'(a;b)` so_apply: `x[s]` pointwise-req: `x[k] = y[k] for k ∈ [n,m]` int_seg: `{i..j-}` lelt: `i ≤ j < k` rless: `x < y` sq_exists: `∃x:{A| B[x]}` nat_plus: `ℕ+` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` rsub: `x - y` i-member: `r ∈ I` rccint: `[l, u]` cand: `A c∧ B` rev_uimplies: `rev_uimplies(P;Q)` rge: `x ≥ y` rleq: `x ≤ y` rnonneg: `rnonneg(x)` real: `ℝ`
Lemmas referenced :  rless_wf rpolynomial_wf int_seg_wf rleq_wf real_wf nat_wf IVT-rpolynomial1 int-to-real_wf rsub_wf radd_wf rmul_wf rless_functionality req_weakening radd-preserves-rless real_term_polynomial itermSubtract_wf itermAdd_wf itermVar_wf real_term_value_const_lemma real_term_value_sub_lemma real_term_value_add_lemma real_term_value_var_lemma req-iff-rsub-is-0 itermConstant_wf rless_transitivity2 rleq_weakening rsum_functionality rnexp_wf int_seg_subtype_nat false_wf le_wf nat_plus_properties nat_properties decidable__lt satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_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 rnexp_functionality itermMultiply_wf real_term_value_mul_lemma rmul_functionality req_inversion rless_transitivity1 req_functionality radd-preserves-req member_rccint_lemma req_wf rminus_wf trivial-rleq-radd rleq_functionality_wrt_implies rleq_weakening_equal rmul_functionality_wrt_rleq2 itermMinus_wf real_term_value_minus_lemma radd_functionality_wrt_rleq rminus_functionality_wrt_rleq rmul_preserves_rleq2 rleq-implies-rleq less_than'_wf nat_plus_wf rleq_functionality req_transitivity radd_functionality rminus_functionality rpolynomial-composition1
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality functionExtensionality applyEquality natural_numberEquality addEquality setElimination rename hypothesis functionEquality lemma_by_obid dependent_functionElimination productElimination independent_functionElimination because_Cache independent_isectElimination sqequalRule computeAll lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation dependent_set_memberEquality unionElimination dependent_pairFormation inlFormation productEquality equalityTransitivity equalitySymmetry isect_memberFormation independent_pairEquality minusEquality axiomEquality

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

Date html generated: 2017_10_03-PM-00_37_27
Last ObjectModification: 2017_07_28-AM-08_44_26

Theory : reals

Home Index