Nuprl Lemma : rmul-rpolynomial
∀[n:ℕ]. ∀[a:ℕn + 1 ⟶ ℝ]. ∀[x,b:ℝ]. ((b * (Σi≤n. a_i * x^i)) = (Σi≤n. λi.(b * (a i))_i * x^i))
Proof
Definitions occuring in Statement :
rpolynomial: (Σi≤n. a_i * x^i),
req: x = y,
rmul: a * b,
real: ℝ,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rpolynomial: (Σi≤n. a_i * x^i),
nat: ℕ,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
so_apply: x[s],
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
req_witness,
rmul_wf,
rpolynomial_wf,
int_seg_wf,
real_wf,
istype-nat,
rsum_wf,
rnexp_wf,
int_seg_subtype_nat,
istype-false,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
req_weakening,
req_functionality,
rsum_functionality2,
rmul_assoc,
req_inversion,
rsum_linearity2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
lambdaEquality_alt,
applyEquality,
universeIsType,
natural_numberEquality,
addEquality,
setElimination,
rename,
independent_functionElimination,
inhabitedIsType,
isect_memberEquality_alt,
because_Cache,
isectIsTypeImplies,
functionIsType,
independent_isectElimination,
independent_pairFormation,
lambdaFormation_alt,
dependent_set_memberEquality_alt,
dependent_functionElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
voidElimination,
productIsType,
productElimination
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[a:\mBbbN{}n + 1 {}\mrightarrow{} \mBbbR{}]. \mforall{}[x,b:\mBbbR{}]. ((b * (\mSigma{}i\mleq{}n. a\_i * x\^{}i)) = (\mSigma{}i\mleq{}n. \mlambda{}i.(b * (a i))\_i * x\^{}i))
Date html generated:
2019_10_29-AM-10_13_05
Last ObjectModification:
2019_01_06-PM-01_25_54
Theory : reals
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