Nuprl Lemma : fps-deriv-div
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[f,g:PowerSeries(X;r)]. ∀[x:X]. ∀[u:|r|].
d(f÷g)/dx = (((df/dx*g)-(f*dg/dx))÷(g*g)) ∈ PowerSeries(X;r) supposing (g[{}] * u) = 1 ∈ |r|
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-deriv: df/dx,
fps-div: (f÷g),
fps-mul: (f*g),
fps-sub: (f-g),
fps-coeff: f[b],
power-series: PowerSeries(X;r),
empty-bag: {},
deq: EqDecider(T),
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
universe: Type,
equal: s = t ∈ T,
crng: CRng,
rng_one: 1,
rng_times: *,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
infix_ap: x f y,
crng: CRng,
rng: Rng,
true: True,
squash: ↓T,
prop: ℙ,
and: P ∧ Q,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
fps-sub: (f-g)
Lemmas referenced :
rng_times_wf,
fps-coeff_wf,
empty-bag_wf,
rng_one_wf,
rng_car_wf,
power-series_wf,
crng_wf,
deq_wf,
valueall-type_wf,
istype-universe,
fps-mul-coeff0,
rng_zero_wf,
rng_plus_wf,
equal_wf,
squash_wf,
true_wf,
rng_times_over_plus,
subtype_rel_self,
rng_times_zero,
rng_times_assoc,
rng_plus_zero,
iff_weakening_equal,
crng_times_ac_1,
rng_times_one,
fps-one_wf,
fps-div_wf,
fps-deriv-mul,
fps-deriv_wf,
fps-div-property,
fps-deriv-one,
fps-mul_wf,
fps-neg_wf,
fps-div-unique,
fps-add_wf,
mul_over_plus_fps,
mul_zero_fps,
mul_comm_fps,
mul_ac_1_fps,
fps-zero_wf,
fps-mul-comm,
mul_one_fps,
abmonoid_comm_fps,
mul_assoc_fps,
abmonoid_ac_1_fps,
iabgrp_op_inv_assoc_fps,
mon_ident_fps,
fps-mul-div,
fps-sub_wf,
fps-mul-assoc,
mul_over_minus_fps
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
hypothesis,
equalityIstype,
inhabitedIsType,
hypothesisEquality,
applyEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
sqequalRule,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
universeIsType,
instantiate,
universeEquality,
Error :memTop,
because_Cache,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
lambdaEquality_alt,
imageElimination,
productElimination,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination,
hyp_replacement,
applyLambdaEquality,
lambdaEquality,
independent_pairFormation,
dependent_functionElimination
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[f,g:PowerSeries(X;r)]. \mforall{}[x:X]. \mforall{}[u:|r|].
d(f\mdiv{}g)/dx = (((df/dx*g)-(f*dg/dx))\mdiv{}(g*g)) supposing (g[\{\}] * u) = 1
supposing valueall-type(X)
Date html generated:
2020_05_20-AM-09_07_22
Last ObjectModification:
2019_12_26-PM-04_07_10
Theory : power!series
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