Nuprl Lemma : integral-rmul-const
∀[a,b:ℝ]. ∀[f:{f:[rmin(a;b), rmax(a;b)] ⟶ℝ| ifun(f;[rmin(a;b), rmax(a;b)])} ]. ∀[c:ℝ].
(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),
req: x = y,
rmul: 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,
integral: a_∫-b f[x] dx,
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),
so_lambda: λ2x.t[x],
iff: P ⇐⇒ Q,
squash: ↓T,
label: ...$L... t,
subtype_rel: A ⊆r B,
guard: {T},
rev_implies: P ⇐ Q,
rsub: x - y
Lemmas referenced :
req_witness,
rmul_wf,
i-member_wf,
rccint_wf,
rmin_wf,
rmax_wf,
real_wf,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
req_functionality,
rmul_functionality,
req_weakening,
req_wf,
set_wf,
ifun_wf,
rccint-icompact,
rmin-rleq-rmax,
integral_wf,
squash_wf,
icompact_wf,
rfun_wf,
interval_wf,
eta_conv,
iff_weakening_equal,
rsub_wf,
ifun_subtype_3,
rleq_weakening_equal,
rmin-rleq,
rleq-rmax,
Riemann-integral_wf,
rleq_wf,
radd_wf,
rminus_wf,
rsub_functionality,
Riemann-integral-rmul-const,
uiff_transitivity,
req_transitivity,
rmul-distrib,
radd_functionality,
rmul_over_rminus,
rminus_functionality,
rmul_comm,
radd_comm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
dependent_set_memberEquality,
sqequalRule,
lambdaEquality,
hypothesisEquality,
applyEquality,
hypothesis,
setEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaFormation,
independent_functionElimination,
because_Cache,
independent_isectElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality
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:\mBbbR{}].
(a\_\mint{}\msupminus{}b c * f[x] dx = (c * a\_\mint{}\msupminus{}b f[x] dx))
Date html generated:
2016_10_26-PM-00_07_33
Last ObjectModification:
2016_09_12-PM-05_38_40
Theory : reals_2
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