Nuprl Lemma : integral-single
∀[a:ℝ]. ∀[f:{f:[a, a] ⟶ℝ| ifun(f;[a, a])} ]. (a_∫-a f[x] dx = r0)
Proof
Definitions occuring in Statement :
integral: a_∫-b f[x] dx,
ifun: ifun(f;I),
rfun: I ⟶ℝ,
rccint: [l, u],
req: x = y,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
and: P ∧ Q,
cand: A c∧ B,
uimplies: b supposing a,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
integral: a_∫-b f[x] dx,
rfun: I ⟶ℝ,
so_apply: x[s],
label: ...$L... t,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
itermConstant: "const",
req_int_terms: t1 ≡ t2,
top: Top,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
sq_stable__rleq,
rmin_wf,
rmax_wf,
real_wf,
rleq_weakening_equal,
rleq_wf,
squash_wf,
true_wf,
rmin-idempotent-eq,
iff_weakening_equal,
rmin_ub,
rmax_lb,
req_witness,
i-member_wf,
rccint_wf,
ifun_wf,
icompact_wf,
rfun_wf,
interval_wf,
eta_conv,
rccint-icompact,
ifun_subtype_3,
rmin-rleq-rmax,
integral_wf,
int-to-real_wf,
set_wf,
rsub_wf,
rmin-rleq,
Riemann-integral_wf,
rmin-idempotent,
real_term_polynomial,
itermSubtract_wf,
itermConstant_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
req-iff-rsub-is-0,
req_functionality,
rsub_functionality,
Riemann-integral-single,
req_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
because_Cache,
independent_isectElimination,
independent_pairFormation,
applyEquality,
lambdaEquality,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
universeEquality,
productElimination,
dependent_functionElimination,
setElimination,
rename,
dependent_set_memberEquality,
setEquality,
computeAll,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}[a:\mBbbR{}]. \mforall{}[f:\{f:[a, a] {}\mrightarrow{}\mBbbR{}| ifun(f;[a, a])\} ]. (a\_\mint{}\msupminus{}a f[x] dx = r0)
Date html generated:
2017_10_04-PM-10_15_48
Last ObjectModification:
2017_07_28-AM-08_47_47
Theory : reals_2
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