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
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