Nuprl Lemma : derivative_wf
∀[I:Interval]. ∀[f,g:I ⟶ℝ]. (λx.g[x] = d(f[x])/dx on I ∈ ℙ)
Proof
Definitions occuring in Statement :
derivative: λz.g[z] = d(f[x])/dx on I,
rfun: I ⟶ℝ,
interval: Interval,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
member: t ∈ T
Definitions unfolded in proof :
prop: ℙ,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
all: ∀x:A. B[x],
derivative: λz.g[z] = d(f[x])/dx on I,
so_lambda: λ2x.t[x],
and: P ∧ Q,
so_apply: x[s],
rfun: I ⟶ℝ,
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top
Lemmas referenced :
interval_wf,
rfun_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_plus_properties,
rless-int,
rdiv_wf,
rmul_wf,
rsub_wf,
rabs_wf,
rleq_wf,
int-to-real_wf,
rless_wf,
real_wf,
sq_exists_wf,
i-approx_wf,
icompact_wf,
nat_plus_wf,
all_wf,
i-member-approx,
i-member_wf
Rules used in proof :
isectElimination,
because_Cache,
dependent_set_memberEquality,
hypothesis,
independent_functionElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
sqequalHypSubstitution,
lemma_by_obid,
cut,
isect_memberFormation,
introduction,
sqequalRule,
lambdaEquality,
setEquality,
lambdaFormation,
setElimination,
rename,
productEquality,
natural_numberEquality,
functionEquality,
applyEquality,
independent_isectElimination,
inrFormation,
productElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
axiomEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[I:Interval]. \mforall{}[f,g:I {}\mrightarrow{}\mBbbR{}]. (\mlambda{}x.g[x] = d(f[x])/dx on I \mmember{} \mBbbP{})
Date html generated:
2016_05_18-AM-09_59_10
Last ObjectModification:
2016_01_17-AM-00_41_24
Theory : reals
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