Nuprl Lemma : derivative_functionality
∀[I:Interval]. ∀[f1,f2,g1,g2:I ⟶ℝ].
(rfun-eq(I;f1;f2) ⇒ rfun-eq(I;g1;g2) ⇒ λx.g1[x] = d(f1[x])/dx on I ⇒ λx.g2[x] = d(f2[x])/dx on I)
Proof
Definitions occuring in Statement :
derivative: λz.g[z] = d(f[x])/dx on I,
rfun-eq: rfun-eq(I;f;g),
rfun: I ⟶ℝ,
interval: Interval,
uall: ∀[x:A]. B[x],
so_apply: x[s],
implies: P ⇒ Q
Definitions unfolded in proof :
top: Top,
not: ¬A,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
sq_exists: ∃x:{A| B[x]},
rless: x < y,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
uimplies: b supposing a,
nat_plus: ℕ+,
all: ∀x:A. B[x],
and: P ∧ Q,
so_apply: x[s],
rfun: I ⟶ℝ,
label: ...$L... t,
so_lambda: λ2x.t[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
derivative: λz.g[z] = d(f[x])/dx on I,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
uiff: uiff(P;Q)
Lemmas referenced :
rmul_functionality,
rsub_functionality,
rabs_functionality,
req_weakening,
rleq_functionality,
r-ap_wf,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
icompact_wf,
nat_plus_wf,
less_than_wf,
all_wf,
derivative_wf,
real_wf,
i-member_wf,
rfun-eq_wf,
rfun_wf,
interval_wf,
rleq_wf,
rabs_wf,
rsub_wf,
i-approx_wf,
i-member-approx,
rmul_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rless_wf
Rules used in proof :
computeAll,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
unionElimination,
inrFormation,
independent_isectElimination,
natural_numberEquality,
dependent_set_memberEquality,
rename,
setElimination,
independent_functionElimination,
dependent_functionElimination,
promote_hyp,
productElimination,
independent_pairFormation,
because_Cache,
hypothesis,
setEquality,
applyEquality,
lambdaEquality,
sqequalRule,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
lemma_by_obid,
cut,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
introduction,
productEquality,
functionEquality
Latex:
\mforall{}[I:Interval]. \mforall{}[f1,f2,g1,g2:I {}\mrightarrow{}\mBbbR{}].
(rfun-eq(I;f1;f2)
{}\mRightarrow{} rfun-eq(I;g1;g2)
{}\mRightarrow{} \mlambda{}x.g1[x] = d(f1[x])/dx on I
{}\mRightarrow{} \mlambda{}x.g2[x] = d(f2[x])/dx on I)
Date html generated:
2016_05_18-AM-09_59_27
Last ObjectModification:
2016_01_17-AM-00_42_25
Theory : reals
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