Nuprl Lemma : integral-additive
∀[a,b,c:ℝ]. ∀[f:{f:[rmin(a;rmin(b;c)), rmax(a;rmax(b;c))] ⟶ℝ| ifun(f;[rmin(a;rmin(b;c)), rmax(a;rmax(b;c))])} ].
(a_∫-b f[x] dx = (a_∫-c f[x] dx + c_∫-b f[x] dx))
Proof
Definitions occuring in Statement :
integral: a_∫-b f[x] dx,
ifun: ifun(f;I),
rfun: I ⟶ℝ,
rccint: [l, u],
rmin: rmin(x;y),
rmax: rmax(x;y),
req: x = y,
radd: a + b,
real: ℝ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
or: P ∨ Q,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
so_apply: x[s],
rfun: I ⟶ℝ,
squash: ↓T,
label: ...$L... t,
subtype_rel: A ⊆r B,
guard: {T},
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
top: Top,
cand: A c∧ B,
rev_uimplies: rev_uimplies(P;Q),
rsub: x - y
Lemmas referenced :
rmin_lb,
rmin_wf,
rmax_wf,
rleq-rmax,
rleq_wf,
set_wf,
rfun_wf,
rccint_wf,
ifun_wf,
rccint-icompact,
real_wf,
integral-additive-lemma,
i-member_wf,
squash_wf,
icompact_wf,
interval_wf,
eta_conv,
iff_weakening_equal,
ifun_subtype_3,
rmin_ub,
rleq_weakening_equal,
rmin-rleq,
rmin-rleq-rmax,
rmax_lb,
rmax_ub,
integral_wf,
rsub_wf,
Riemann-integral_wf,
radd_wf,
member_rccint_lemma,
req_functionality,
radd_functionality,
req_wf,
rminus_wf,
int-to-real_wf,
req_weakening,
uiff_transitivity,
req_inversion,
radd-assoc,
radd_comm,
req_transitivity,
radd-ac,
radd-rminus-both,
radd-zero-both
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_isectElimination,
inlFormation,
productElimination,
sqequalRule,
lambdaEquality,
because_Cache,
dependent_functionElimination,
independent_functionElimination,
setElimination,
rename,
dependent_set_memberEquality,
applyEquality,
setEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
universeEquality,
independent_pairFormation,
inrFormation,
isect_memberEquality,
voidElimination,
voidEquality,
productEquality,
natural_numberEquality
Latex:
\mforall{}[a,b,c:\mBbbR{}]. \mforall{}[f:\{f:[rmin(a;rmin(b;c)), rmax(a;rmax(b;c))] {}\mrightarrow{}\mBbbR{}|
ifun(f;[rmin(a;rmin(b;c)), rmax(a;rmax(b;c))])\} ].
(a\_\mint{}\msupminus{}b f[x] dx = (a\_\mint{}\msupminus{}c f[x] dx + c\_\mint{}\msupminus{}b f[x] dx))
Date html generated:
2016_10_26-PM-00_07_54
Last ObjectModification:
2016_09_12-PM-05_38_45
Theory : reals_2
Home
Index