Nuprl Lemma : derivative-rdiv
∀I:Interval. ∀f1,f2,g1,g2:I ⟶ℝ.
((∀x,y:{t:ℝ| t ∈ I} . ((x = y) ⇒ (g1[x] = g1[y])))
⇒ (∀x,y:{t:ℝ| t ∈ I} . ((x = y) ⇒ (g2[x] = g2[y])))
⇒ f2[x]≠r0 for x ∈ I
⇒ (∀x,y:{t:ℝ| t ∈ I} . ((x = y) ⇒ (f2[x] = f2[y])))
⇒ d(f1[x])/dx = λx.g1[x] on I
⇒ d(f2[x])/dx = λx.g2[x] on I
⇒ d((f1[x]/f2[x]))/dx = λx.((f2[x] * g1[x]) - f1[x] * g2[x]/f2[x] * f2[x]) on I)
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
nonzero-on: f[x]≠r0 for x ∈ I,
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
rdiv: (x/y),
rsub: x - y,
req: x = y,
rmul: a * b,
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
uimplies: b supposing a,
rneq: x ≠ y,
sq_stable: SqStable(P),
squash: ↓T,
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
label: ...$L... t,
false: False,
not: ¬A,
rat_term_to_real: rat_term_to_real(f;t),
rtermDivide: num "/" denom,
rat_term_ind: rat_term_ind,
rtermVar: rtermVar(var),
pi1: fst(t),
true: True,
rtermMultiply: left "*" right,
rtermConstant: "const",
pi2: snd(t),
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
rtermSubtract: left "-" right,
rtermAdd: left "+" right,
rtermMinus: rtermMinus(num)
Lemmas referenced :
derivative-rinv,
nonzero-on-implies,
derivative-mul,
rdiv_wf,
int-to-real_wf,
sq_stable__i-member,
real_wf,
i-member_wf,
req_wf,
rmul-nonzero,
rdiv_functionality,
rminus_wf,
rmul_wf,
rminus_functionality,
rmul_functionality,
derivative_wf,
nonzero-on_wf,
rfun_wf,
interval_wf,
radd_wf,
rsub_wf,
assert-rat-term-eq2,
rtermMultiply_wf,
rtermVar_wf,
rtermDivide_wf,
rtermConstant_wf,
istype-int,
derivative_functionality,
rtermAdd_wf,
rtermMinus_wf,
rtermSubtract_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
because_Cache,
sqequalRule,
lambdaEquality_alt,
isectElimination,
closedConclusion,
natural_numberEquality,
applyEquality,
independent_isectElimination,
setElimination,
rename,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality_alt,
functionIsType,
setIsType,
universeIsType,
productElimination,
independent_pairFormation,
inhabitedIsType,
int_eqEquality,
approximateComputation
Latex:
\mforall{}I:Interval. \mforall{}f1,f2,g1,g2:I {}\mrightarrow{}\mBbbR{}.
((\mforall{}x,y:\{t:\mBbbR{}| t \mmember{} I\} . ((x = y) {}\mRightarrow{} (g1[x] = g1[y])))
{}\mRightarrow{} (\mforall{}x,y:\{t:\mBbbR{}| t \mmember{} I\} . ((x = y) {}\mRightarrow{} (g2[x] = g2[y])))
{}\mRightarrow{} f2[x]\mneq{}r0 for x \mmember{} I
{}\mRightarrow{} (\mforall{}x,y:\{t:\mBbbR{}| t \mmember{} I\} . ((x = y) {}\mRightarrow{} (f2[x] = f2[y])))
{}\mRightarrow{} d(f1[x])/dx = \mlambda{}x.g1[x] on I
{}\mRightarrow{} d(f2[x])/dx = \mlambda{}x.g2[x] on I
{}\mRightarrow{} d((f1[x]/f2[x]))/dx = \mlambda{}x.((f2[x] * g1[x]) - f1[x] * g2[x]/f2[x] * f2[x]) on I)
Date html generated:
2019_10_30-AM-09_04_01
Last ObjectModification:
2019_04_02-AM-09_45_53
Theory : reals
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