Nuprl Lemma : rdiv-rpolynomial
∀[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) supposing b ≠ r0
Proof
Definitions occuring in Statement :
rpolynomial: (Σi≤n. a_i * x^i),
rdiv: (x/y),
rneq: x ≠ y,
req: x = y,
int-to-real: r(n),
real: ℝ,
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
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,
uimplies: b supposing a,
rdiv: (x/y),
implies: P ⇒ Q,
nat: ℕ,
prop: ℙ
Lemmas referenced :
rpolynomial-rmul,
rinv_wf2,
req_witness,
rdiv_wf,
rpolynomial_wf,
int_seg_wf,
rneq_wf,
int-to-real_wf,
real_wf,
istype-nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
independent_isectElimination,
lambdaEquality_alt,
applyEquality,
universeIsType,
natural_numberEquality,
addEquality,
setElimination,
rename,
isect_memberEquality_alt,
because_Cache,
isectIsTypeImplies,
inhabitedIsType,
functionIsType
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) supposing b \mneq{} r0
Date html generated:
2019_10_29-AM-10_13_31
Last ObjectModification:
2019_01_06-PM-01_39_22
Theory : reals
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