Nuprl Lemma : Riemann-integral-rmul-const
∀[a:ℝ]. ∀[b:{b:ℝ| a ≤ b} ]. ∀[f:{f:[a, b] ⟶ℝ| ifun(f;[a, b])} ]. ∀[c:ℝ].
(∫ c * f[x] dx on [a, b] = (c * ∫ f[x] dx on [a, b]))
Proof
Definitions occuring in Statement :
Riemann-integral: ∫ f[x] dx on [a, b],
ifun: ifun(f;I),
rfun: I ⟶ℝ,
rccint: [l, u],
rleq: 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 :
so_apply: x[s],
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
label: ...$L... t,
rfun: I ⟶ℝ,
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,
subtype_rel: A ⊆r B,
sq_stable: SqStable(P),
squash: ↓T,
nat_plus: ℕ+,
nat: ℕ,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
subtract: n - m,
less_than': less_than'(a;b),
true: True,
guard: {T}
Lemmas referenced :
req_witness,
Riemann-integral_wf,
rmul_wf,
i-member_wf,
rccint_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,
subtype_rel_self,
rfun_wf,
sq_stable__rleq,
rleq_wf,
Riemann-sums-converge-to,
unique-limit,
Riemann-sum_wf,
decidable__lt,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
less_than_wf,
nat_wf,
converges-to_functionality,
Riemann-sum-rmul-constant,
rmul-limit,
constant-limit
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
dependent_set_memberEquality,
because_Cache,
hypothesis,
lambdaEquality,
applyEquality,
setEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaFormation,
independent_functionElimination,
independent_isectElimination,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
addEquality,
natural_numberEquality,
unionElimination,
independent_pairFormation,
intEquality,
minusEquality
Latex:
\mforall{}[a:\mBbbR{}]. \mforall{}[b:\{b:\mBbbR{}| a \mleq{} b\} ]. \mforall{}[f:\{f:[a, b] {}\mrightarrow{}\mBbbR{}| ifun(f;[a, b])\} ]. \mforall{}[c:\mBbbR{}].
(\mint{} c * f[x] dx on [a, b] = (c * \mint{} f[x] dx on [a, b]))
Date html generated:
2016_10_26-PM-00_02_49
Last ObjectModification:
2016_09_12-PM-05_38_03
Theory : reals_2
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