Nuprl Lemma : Riemann-integral_functionality_endpoints
∀[a:ℝ]. ∀[b:{b:ℝ| a ≤ b} ]. ∀[f:{f:[a, b] ⟶ℝ| ifun(f;[a, b])} ].
∀a',b':ℝ. (∫ f[x] dx on [a, b] = ∫ f[x] dx on [a', b']) supposing ((a = a') and (b = 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,
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
sq_stable: SqStable(P),
guard: {T},
squash: ↓T,
prop: ℙ,
rfun: I ⟶ℝ,
so_apply: x[s],
label: ...$L... t,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
cand: A c∧ B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
sq_stable__subtype_rel,
rfun_wf,
rccint_wf,
ifun_wf,
rccint-icompact,
sq_stable__rleq,
req_inversion,
rleq_transitivity,
rleq_weakening,
ifun_subtype_3,
i-member_wf,
real_wf,
squash_wf,
icompact_wf,
interval_wf,
eta_conv,
iff_weakening_equal,
sq_stable__req,
Riemann-integral_wf,
rleq_wf,
req_wf,
set_wf,
Riemann-integral-additive,
rleq_weakening_equal,
subtype_rel_transitivity,
Riemann-integral-single,
radd_wf,
int-to-real_wf,
req_functionality,
req_weakening,
radd_functionality,
radd-zero-both,
uiff_transitivity,
radd_comm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
setEquality,
hypothesisEquality,
hypothesis,
independent_isectElimination,
dependent_functionElimination,
productElimination,
independent_functionElimination,
setElimination,
rename,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality,
lambdaEquality,
applyEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
independent_pairFormation,
natural_numberEquality
Latex:
\mforall{}[a:\mBbbR{}]. \mforall{}[b:\{b:\mBbbR{}| a \mleq{} b\} ]. \mforall{}[f:\{f:[a, b] {}\mrightarrow{}\mBbbR{}| ifun(f;[a, b])\} ].
\mforall{}a',b':\mBbbR{}. (\mint{} f[x] dx on [a, b] = \mint{} f[x] dx on [a', b']) supposing ((a = a') and (b = b'))
Date html generated:
2016_10_26-PM-00_05_51
Last ObjectModification:
2016_09_12-PM-05_38_21
Theory : reals_2
Home
Index