Nuprl Lemma : Riemann-integral-single
∀[a,b:ℝ]. ∀[f:{f:[a, b] ⟶ℝ| ifun(f;[a, b])} ]. (∫ f[x] dx on [a, b] = r0) supposing 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],
req: x = y,
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
guard: {T},
prop: ℙ,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
so_lambda: λ2x.t[x],
label: ...$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,
iff: P ⇐⇒ Q,
rsub: x - y
Lemmas referenced :
rabs-Riemann-integral,
rleq_weakening,
rleq_wf,
radd-preserves-req,
rsub_wf,
req_witness,
Riemann-integral_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,
radd_wf,
rminus_wf,
req_inversion,
rabs_wf,
rmul_wf,
I-norm_wf,
icompact_wf,
uiff_transitivity,
radd_functionality,
radd_comm,
radd-rminus-assoc,
radd-zero-both,
rleq_functionality,
rmul_functionality,
rmul-zero-both,
rabs-as-rmax,
rmax_lb,
radd-preserves-rleq,
req_transitivity,
rminus-as-rmul,
rmul-identity1,
rmul-distrib2,
radd-int,
rleq_antisymmetry
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_isectElimination,
dependent_set_memberEquality,
because_Cache,
productElimination,
sqequalRule,
setElimination,
rename,
lambdaEquality,
applyEquality,
setEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaFormation,
independent_functionElimination,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
minusEquality,
addEquality
Latex:
\mforall{}[a,b:\mBbbR{}]. \mforall{}[f:\{f:[a, b] {}\mrightarrow{}\mBbbR{}| ifun(f;[a, b])\} ]. (\mint{} f[x] dx on [a, b] = r0) supposing a = b
Date html generated:
2016_10_26-PM-00_03_23
Last ObjectModification:
2016_09_12-PM-05_38_16
Theory : reals_2
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