Nuprl Lemma : Riemann-integral-rminus
∀[a:ℝ]. ∀[b:{b:ℝ| a ≤ b} ]. ∀[f:{f:[a, b] ⟶ℝ| ifun(f;[a, b])} ]. (∫ -(f[x]) dx on [a, b] = -(∫ 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,
rminus: -(x),
real: ℝ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
set: {x:A| B[x]}
Definitions unfolded in proof :
squash: ↓T,
sq_stable: SqStable(P),
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
so_lambda: λ2x.t[x],
rev_uimplies: rev_uimplies(P;Q),
and: P ∧ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
implies: P ⇒ Q,
real-fun: real-fun(f;a;b),
top: Top,
all: ∀x:A. B[x],
ifun: ifun(f;I),
prop: ℙ,
rfun: I ⟶ℝ,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_apply: x[s]
Lemmas referenced :
member_rccint_lemma,
Riemann-integral_functionality,
Riemann-integral-rmul-const,
req_inversion,
rminus-as-rmul,
rmul_functionality,
int-to-real_wf,
rmul_wf,
rleq_wf,
sq_stable__rleq,
rfun_wf,
subtype_rel_self,
Riemann-integral_wf,
rccint-icompact,
ifun_wf,
set_wf,
req_wf,
req_weakening,
rminus_functionality,
req_functionality,
right_endpoint_rccint_lemma,
left_endpoint_rccint_lemma,
real_wf,
rccint_wf,
i-member_wf,
rminus_wf,
req_witness
Rules used in proof :
productEquality,
independent_pairFormation,
natural_numberEquality,
minusEquality,
imageElimination,
baseClosed,
imageMemberEquality,
equalitySymmetry,
equalityTransitivity,
productElimination,
independent_isectElimination,
because_Cache,
independent_functionElimination,
lambdaFormation,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
setEquality,
hypothesis,
hypothesisEquality,
applyEquality,
lambdaEquality,
dependent_set_memberEquality,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution
Latex:
\mforall{}[a:\mBbbR{}]. \mforall{}[b:\{b:\mBbbR{}| a \mleq{} b\} ]. \mforall{}[f:\{f:[a, b] {}\mrightarrow{}\mBbbR{}| ifun(f;[a, b])\} ].
(\mint{} -(f[x]) dx on [a, b] = -(\mint{} f[x] dx on [a, b]))
Date html generated:
2016_11_11-AM-07_14_20
Last ObjectModification:
2016_11_09-PM-00_18_35
Theory : reals_2
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