Nuprl Lemma : rpolynomial-rmul
∀[n:ℕ]. ∀[a:ℕn + 1 ⟶ ℝ]. ∀[x,b:ℝ]. (((Σi≤n. a_i * x^i) * b) = (Σi≤n. λi.((a i) * b)_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,
nat: ℕ,
pointwise-req: x[k] = y[k] for k ∈ [n,m],
all: ∀x:A. B[x],
so_apply: x[s],
implies: P ⇒ Q,
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
guard: {T}
Lemmas referenced :
rmul-rpolynomial,
real_wf,
int_seg_wf,
istype-nat,
rmul_comm,
rpolynomial_wf,
rmul_wf,
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,
rpolynomial_functionality,
req_inversion,
req_transitivity
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
inhabitedIsType,
universeIsType,
functionIsType,
natural_numberEquality,
addEquality,
setElimination,
rename,
lambdaEquality_alt,
applyEquality,
because_Cache,
lambdaFormation_alt,
sqequalRule,
dependent_set_memberEquality_alt,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
productIsType
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[a:\mBbbN{}n + 1 {}\mrightarrow{} \mBbbR{}]. \mforall{}[x,b:\mBbbR{}]. (((\mSigma{}i\mleq{}n. a\_i * x\^{}i) * b) = (\mSigma{}i\mleq{}n. \mlambda{}i.((a i) * b)\_i * x\^{}i))
Date html generated:
2019_10_29-AM-10_13_18
Last ObjectModification:
2019_01_06-PM-01_38_44
Theory : reals
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