Nuprl Lemma : rlog-integral-non-zero
∀a,b:ℝ. ((r0 < a) ⇒ (a < b) ⇒ (¬(∫ (r1/t) dt on [a, b] = r0)))
Proof
Definitions occuring in Statement :
Riemann-integral: ∫ f[x] dx on [a, b],
rdiv: (x/y),
rless: x < y,
req: x = y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
rneq: x ≠ y,
or: P ∨ Q,
sq_stable: SqStable(P),
top: Top,
and: P ∧ Q,
squash: ↓T,
rfun: I ⟶ℝ,
ifun: ifun(f;I),
real-fun: real-fun(f;a;b),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
not: ¬A,
false: False,
req_int_terms: t1 ≡ t2,
itermConstant: "const",
cand: A c∧ B,
true: True,
less_than': less_than'(a;b),
less_than: a < b
Lemmas referenced :
rless_wf,
int-to-real_wf,
real_wf,
Riemann-integral-lower-bound,
rleq_weakening_rless,
rleq_wf,
rdiv_wf,
rless_transitivity2,
i-member_wf,
rccint_wf,
sq_stable__rless,
member_rccint_lemma,
rless_transitivity1,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
req_functionality,
rdiv_functionality,
req_weakening,
req_wf,
set_wf,
ifun_wf,
rccint-icompact,
rneq_functionality,
rmul-one-both,
rmul-rdiv-cancel,
rmul-ac,
rmul_comm,
rmul_functionality,
rmul-assoc,
req_inversion,
uiff_transitivity,
rmul-rdiv-cancel2,
rleq_functionality,
rmul_wf,
rmul_preserves_rleq,
req-iff-rsub-is-0,
real_term_value_var_lemma,
real_term_value_sub_lemma,
real_term_value_const_lemma,
itermConstant_wf,
itermVar_wf,
itermSubtract_wf,
real_term_polynomial,
rless-implies-rless,
rsub_wf,
rmul-is-positive,
rmul-zero-both,
rless_functionality,
rless-int,
rmul_preserves_rless,
sq_stable_rneq,
rless_irreflexivity,
Riemann-integral_wf,
rleq_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
hypothesis,
natural_numberEquality,
independent_isectElimination,
dependent_set_memberEquality,
sqequalRule,
dependent_functionElimination,
inrFormation,
independent_functionElimination,
setElimination,
rename,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
lambdaEquality,
setEquality,
productEquality,
intEquality,
int_eqEquality,
computeAll,
independent_pairFormation,
inlFormation,
addLevel,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}a,b:\mBbbR{}. ((r0 < a) {}\mRightarrow{} (a < b) {}\mRightarrow{} (\mneg{}(\mint{} (r1/t) dt on [a, b] = r0)))
Date html generated:
2017_10_04-PM-10_26_42
Last ObjectModification:
2017_07_28-AM-08_50_05
Theory : reals_2
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