Nuprl Lemma : poly-deriv-linear
∀n:ℕ. ∀a,b:ℕn + 1 ⟶ ℝ. ∀c,d:ℝ. ∀i:ℕn.
((poly-deriv(λi.((c * (a i)) + (d * (b i)))) i) = ((c * (poly-deriv(a) i)) + (d * (poly-deriv(b) i))))
Proof
Definitions occuring in Statement :
poly-deriv: poly-deriv(a),
req: x = y,
rmul: a * b,
radd: a + b,
real: ℝ,
int_seg: {i..j-},
nat: ℕ,
all: ∀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 :
less_than: a < b,
le: A ≤ B,
prop: ℙ,
subtract: n - m,
top: Top,
not: ¬A,
implies: P ⇒ Q,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
lelt: i ≤ j < k,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
poly-deriv: poly-deriv(a),
all: ∀x:A. B[x],
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
rmul-ac,
req_transitivity,
rmul_comm,
rmul_functionality,
radd_functionality,
rmul-distrib,
req_functionality,
uiff_transitivity,
int_seg_wf,
real_wf,
nat_wf,
req_wf,
rmul_wf,
int-to-real_wf,
radd_wf,
add-member-int_seg2,
nat_properties,
decidable__le,
subtract_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermSubtract_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
add-subtract-cancel,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
lelt_wf,
req_weakening
Rules used in proof :
introduction,
computeAll,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
unionElimination,
dependent_functionElimination,
independent_pairFormation,
dependent_set_memberEquality,
independent_isectElimination,
productElimination,
because_Cache,
applyEquality,
addEquality,
functionEquality,
hypothesis,
hypothesisEquality,
rename,
setElimination,
natural_numberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
lemma_by_obid,
cut,
sqequalRule,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_functionElimination
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b:\mBbbN{}n + 1 {}\mrightarrow{} \mBbbR{}. \mforall{}c,d:\mBbbR{}. \mforall{}i:\mBbbN{}n.
((poly-deriv(\mlambda{}i.((c * (a i)) + (d * (b i)))) i)
= ((c * (poly-deriv(a) i)) + (d * (poly-deriv(b) i))))
Date html generated:
2016_05_18-AM-10_08_19
Last ObjectModification:
2016_01_17-AM-00_38_41
Theory : reals
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