Nuprl Lemma : integral-int-rmul
∀[a,b:ℝ]. ∀[f:{f:[rmin(a;b), rmax(a;b)] ⟶ℝ| ifun(f;[rmin(a;b), rmax(a;b)])} ]. ∀[c:ℤ].
(a_∫-b c * f[x] dx = c * a_∫-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),
int-rmul: k1 * a,
req: x = y,
real: ℝ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
set: {x:A| B[x]} ,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rfun: I ⟶ℝ,
so_apply: x[s],
prop: ℙ,
ifun: ifun(f;I),
all: ∀x:A. B[x],
top: Top,
real-fun: real-fun(f;a;b),
implies: P ⇒ Q,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
so_lambda: λ2x.t[x],
squash: ↓T,
label: ...$L... t,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
rev_implies: P ⇐ Q
Lemmas referenced :
req_functionality,
int-rmul_wf,
i-member_wf,
rccint_wf,
rmin_wf,
rmax_wf,
left_endpoint_rccint_lemma,
istype-void,
right_endpoint_rccint_lemma,
int-rmul_functionality,
req_weakening,
req_wf,
ifun_wf,
rccint-icompact,
rmin-rleq-rmax,
integral_wf,
rmul_wf,
int-to-real_wf,
rmul_functionality,
eta_conv,
real_wf,
equal_wf,
rfun_wf,
iff_weakening_equal,
integral_functionality,
int-rmul-req,
member_rccint_lemma,
rleq_wf,
integral-rmul-const,
req_witness,
squash_wf,
icompact_wf,
interval_wf,
true_wf,
istype-universe,
subtype_rel_self,
istype-int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
dependent_set_memberEquality_alt,
sqequalRule,
lambdaEquality_alt,
hypothesisEquality,
applyEquality,
hypothesis,
setIsType,
because_Cache,
universeIsType,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
lambdaFormation_alt,
independent_functionElimination,
independent_isectElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
imageElimination,
setEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_pairFormation,
productIsType,
inhabitedIsType,
instantiate,
universeEquality,
isectIsTypeImplies
Latex:
\mforall{}[a,b:\mBbbR{}]. \mforall{}[f:\{f:[rmin(a;b), rmax(a;b)] {}\mrightarrow{}\mBbbR{}| ifun(f;[rmin(a;b), rmax(a;b)])\} ]. \mforall{}[c:\mBbbZ{}].
(a\_\mint{}\msupminus{}b c * f[x] dx = c * a\_\mint{}\msupminus{}b f[x] dx)
Date html generated:
2019_10_30-AM-11_38_48
Last ObjectModification:
2019_01_01-PM-04_05_25
Theory : reals_2
Home
Index