Nuprl Lemma : integral-additive-lemma
∀[m,M:ℝ].
∀[f:{f:[m, M] ⟶ℝ| ifun(f;[m, M])} ]. ∀[a,b:{x:ℝ| x ∈ [m, M]} ].
(a_∫-b f[x] dx = (∫ f[x] dx on [m, b] - ∫ f[x] dx on [m, a]))
supposing m ≤ M
Proof
Definitions occuring in Statement :
integral: a_∫-b f[x] dx,
Riemann-integral: ∫ f[x] dx on [a, b],
ifun: ifun(f;I),
rfun: I ⟶ℝ,
rccint: [l, u],
i-member: r ∈ I,
rleq: x ≤ y,
rsub: x - y,
req: x = y,
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
integral: a_∫-b f[x] dx,
member: t ∈ T,
rfun: I ⟶ℝ,
so_apply: x[s],
prop: ℙ,
squash: ↓T,
label: ...$L... t,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
guard: {T},
rev_implies: P ⇐ Q,
top: Top,
sq_stable: SqStable(P),
so_lambda: λ2x.t[x],
i-member: r ∈ I,
rccint: [l, u],
cand: A c∧ B,
or: P ∨ Q,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
i-member_wf,
rccint_wf,
real_wf,
ifun_wf,
squash_wf,
icompact_wf,
rfun_wf,
interval_wf,
eta_conv,
rccint-icompact,
iff_weakening_equal,
sq_stable__req,
rsub_wf,
ifun_subtype_3,
rleq_weakening_equal,
member_rccint_lemma,
sq_stable__rleq,
Riemann-integral_wf,
subtype_rel_sets,
rleq_wf,
set_wf,
rmin_ub,
rmin_wf,
rmin-rleq,
rmin_lb,
Riemann-integral-additive,
radd_wf,
real_term_polynomial,
itermSubtract_wf,
itermVar_wf,
itermAdd_wf,
int-to-real_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
req-iff-rsub-is-0,
req_functionality,
req_weakening,
rsub_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
setElimination,
thin,
rename,
dependent_set_memberEquality,
sqequalRule,
lambdaEquality,
applyEquality,
sqequalHypSubstitution,
hypothesisEquality,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
setEquality,
imageElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
dependent_functionElimination,
productElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
universeEquality,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaFormation,
independent_pairFormation,
inlFormation,
natural_numberEquality,
computeAll,
int_eqEquality,
intEquality
Latex:
\mforall{}[m,M:\mBbbR{}].
\mforall{}[f:\{f:[m, M] {}\mrightarrow{}\mBbbR{}| ifun(f;[m, M])\} ]. \mforall{}[a,b:\{x:\mBbbR{}| x \mmember{} [m, M]\} ].
(a\_\mint{}\msupminus{}b f[x] dx = (\mint{} f[x] dx on [m, b] - \mint{} f[x] dx on [m, a]))
supposing m \mleq{} M
Date html generated:
2017_10_04-PM-10_15_51
Last ObjectModification:
2017_07_28-AM-08_47_49
Theory : reals_2
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