Nuprl Lemma : Riemann-integral-same-endpoints
∀[a:ℝ]. ∀[f:{f:[a, a] ⟶ℝ| ifun(f;[a, a])} ]. (∫ f[x] dx on [a, a] = r0)
Proof
Definitions occuring in Statement :
Riemann-integral: ∫ f[x] dx on [a, b],
ifun: ifun(f;I),
rfun: I ⟶ℝ,
rccint: [l, u],
req: x = y,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x.t[x],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s],
ifun: ifun(f;I),
all: ∀x:A. B[x],
top: Top,
real-fun: real-fun(f;a;b),
implies: P ⇒ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q
Lemmas referenced :
req_witness,
Riemann-integral_wf,
rleq_weakening_equal,
rleq_wf,
i-member_wf,
rccint_wf,
real_wf,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
req_functionality,
req_weakening,
req_wf,
set_wf,
ifun_wf,
rccint-icompact,
int-to-real_wf,
rfun_wf,
Riemann-integral-single
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
independent_isectElimination,
hypothesis,
dependent_set_memberEquality,
sqequalRule,
setElimination,
rename,
lambdaEquality,
applyEquality,
setEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaFormation,
independent_functionElimination,
productElimination,
natural_numberEquality
Latex:
\mforall{}[a:\mBbbR{}]. \mforall{}[f:\{f:[a, a] {}\mrightarrow{}\mBbbR{}| ifun(f;[a, a])\} ]. (\mint{} f[x] dx on [a, a] = r0)
Date html generated:
2016_10_26-PM-00_03_26
Last ObjectModification:
2016_09_12-PM-05_38_18
Theory : reals_2
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