Nuprl Lemma : derivative-int-rdiv
∀a:ℤ-o. ∀I:Interval. ∀f,f':I ⟶ℝ. (d(f[x])/dx = λx.f'[x] on I ⇒ d((f[x])/a)/dx = λx.(f'[x])/a on I)
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
rfun: I ⟶ℝ,
interval: Interval,
int-rdiv: (a)/k1,
int_nzero: ℤ-o,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
int_nzero: ℤ-o,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
not: ¬A,
nequal: a ≠ b ∈ T ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
label: ...$L... t,
rfun: I ⟶ℝ,
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
derivative-rdiv-const,
int-to-real_wf,
rneq-int,
int_nzero_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformnot_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
set_subtype_base,
nequal_wf,
int_subtype_base,
derivative_wf,
real_wf,
i-member_wf,
rfun_wf,
interval_wf,
int_nzero_wf,
rdiv_wf,
int-rdiv_wf,
req_weakening,
derivative_functionality,
req_functionality,
int-rdiv-req
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
setElimination,
rename,
hypothesisEquality,
hypothesis,
independent_functionElimination,
because_Cache,
natural_numberEquality,
productElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
equalityIstype,
applyEquality,
intEquality,
baseClosed,
sqequalBase,
equalitySymmetry,
setIsType,
inhabitedIsType
Latex:
\mforall{}a:\mBbbZ{}\msupminus{}\msupzero{}. \mforall{}I:Interval. \mforall{}f,f':I {}\mrightarrow{}\mBbbR{}.
(d(f[x])/dx = \mlambda{}x.f'[x] on I {}\mRightarrow{} d((f[x])/a)/dx = \mlambda{}x.(f'[x])/a on I)
Date html generated:
2019_10_30-AM-09_04_32
Last ObjectModification:
2019_01_03-AM-11_25_06
Theory : reals
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