Nuprl Lemma : derivative-of-integral
∀I:Interval. ∀a:{a:ℝ| a ∈ I} . ∀f:{f:I ⟶ℝ| ∀x,y:{a:ℝ| a ∈ I} . ((x = y) ⇒ ((f x) = (f y)))} .
d(a_∫-x f[t] dt)/dx = λt.f[t] on I
Proof
Definitions occuring in Statement :
integral: a_∫-b f[x] dx,
derivative: d(f[x])/dx = λz.g[z] on I,
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
req: x = y,
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a
Definitions unfolded in proof :
all: ∀x:A. B[x],
derivative: d(f[x])/dx = λz.g[z] on I,
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
implies: P ⇒ Q,
sq_stable: SqStable(P),
squash: ↓T,
prop: ℙ,
rfun: I ⟶ℝ,
ifun: ifun(f;I),
real-fun: real-fun(f;a;b),
top: Top,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
i-finite: i-finite(I),
rccint: [l, u],
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
so_apply: x[s],
cand: A c∧ B,
subinterval: I ⊆ J ,
iff: P ⇐⇒ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
guard: {T},
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
or: P ∨ Q,
icompact: icompact(I),
continuous: f[x] continuous for x ∈ I,
nat_plus: ℕ+,
less_than: a < b,
less_than': less_than'(a;b),
sq_exists: ∃x:A [B[x]],
rneq: x ≠ y,
rev_implies: P ⇐ Q,
rless: x < y,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
rge: x ≥ y,
i-length: |I|,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
rdiv: (x/y)
Lemmas referenced :
sq_stable__and,
icompact_wf,
i-approx_wf,
iproper_wf,
sq_stable__icompact,
sq_stable__iproper,
nat_plus_wf,
rfun_wf,
req_wf,
real_wf,
i-member_wf,
interval_wf,
rmin-rmax-subinterval,
sq_stable__i-member,
member_rccint_lemma,
istype-void,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
sq_stable__req,
subtype_rel_sets_simple,
rccint_wf,
left-endpoint_wf,
right-endpoint_wf,
rleq_wf,
subinterval_wf,
rmin_wf,
rmax_wf,
ifun_wf,
rccint-icompact,
rmin-rleq-rmax,
integral_wf,
rsub_wf,
radd-preserves-req,
integral-additive,
radd_wf,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
req_functionality,
req-iff-rsub-is-0,
real_polynomial_null,
int-to-real_wf,
istype-int,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rmin3-rmax3-subinterval,
rfun_subtype,
rmax_ub,
rmin_lb,
rleq_weakening_equal,
rabs_wf,
rmul_wf,
rabs_functionality,
rsub_functionality,
req_weakening,
req_inversion,
integral-const,
integral-rsub,
i-approx-is-subinterval,
ifun-continuous,
icompact-is-rccint,
i-approx-finite,
i-approx-approx,
istype-less_than,
subtype_rel_self,
sq_exists_wf,
rless_wf,
all_wf,
rdiv_wf,
rless-int,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_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,
sq_stable__rless,
I-norm_wf,
rleq_functionality_wrt_implies,
rabs-integral,
i-member-diff-bound,
rmin-rleq,
rleq-rmax,
i-length_wf,
rleq_functionality,
rmax-minus-rmin,
rabs-difference-symmetry,
I-norm-rleq,
sq_stable__rleq,
le_witness_for_triv,
rmul_preserves_rleq2,
zero-rleq-rabs,
rminus_wf,
itermMultiply_wf,
itermMinus_wf,
rinv_wf2,
rmul_functionality,
req_transitivity,
rinv-mul-as-rdiv,
real_term_value_mul_lemma,
real_term_value_minus_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
setElimination,
thin,
rename,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
productElimination,
isect_memberEquality_alt,
because_Cache,
independent_functionElimination,
dependent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
setIsType,
inhabitedIsType,
productIsType,
universeIsType,
functionIsType,
applyEquality,
voidElimination,
lambdaEquality_alt,
independent_isectElimination,
independent_pairFormation,
natural_numberEquality,
productEquality,
dependent_set_memberEquality_alt,
equalityTransitivity,
equalitySymmetry,
approximateComputation,
int_eqEquality,
inlFormation_alt,
equalityIsType1,
closedConclusion,
dependent_set_memberFormation_alt,
functionExtensionality,
setEquality,
functionEquality,
inrFormation_alt,
unionElimination,
dependent_pairFormation_alt,
isect_memberFormation_alt,
functionIsTypeImplies,
promote_hyp
Latex:
\mforall{}I:Interval. \mforall{}a:\{a:\mBbbR{}| a \mmember{} I\} . \mforall{}f:\{f:I {}\mrightarrow{}\mBbbR{}| \mforall{}x,y:\{a:\mBbbR{}| a \mmember{} I\} . ((x = y) {}\mRightarrow{} ((f x) = (f y)))\} .
d(a\_\mint{}\msupminus{}x f[t] dt)/dx = \mlambda{}t.f[t] on I
Date html generated:
2019_10_30-AM-11_39_15
Last ObjectModification:
2018_11_12-AM-10_53_35
Theory : reals_2
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