Nuprl Lemma : Riemann-integral-nonneg
∀[a:ℝ]. ∀[b:{b:ℝ| a ≤ b} ]. ∀[f:{f:[a, b] ⟶ℝ| ifun(f;[a, b])} ].
r0 ≤ ∫ f[x] dx on [a, b] supposing ∀x:ℝ. ((x ∈ [a, b]) ⇒ (r0 ≤ f[x]))
Proof
Definitions occuring in Statement :
Riemann-integral: ∫ f[x] dx on [a, b],
ifun: ifun(f;I),
rfun: I ⟶ℝ,
rccint: [l, u],
i-member: r ∈ I,
rleq: x ≤ y,
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
rfun: I ⟶ℝ,
so_apply: x[s],
prop: ℙ,
squash: ↓T,
label: ...$L... t,
iff: P ⇐⇒ Q,
sq_stable: SqStable(P),
subtype_rel: A ⊆r B,
guard: {T},
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
req_int_terms: t1 ≡ t2,
top: Top
Lemmas referenced :
Riemann-integral-lower-bound,
int-to-real_wf,
less_than'_wf,
rsub_wf,
i-member_wf,
rccint_wf,
real_wf,
ifun_wf,
squash_wf,
icompact_wf,
rfun_wf,
interval_wf,
eta_conv,
rccint-icompact,
sq_stable__rleq,
iff_weakening_equal,
Riemann-integral_wf,
nat_plus_wf,
all_wf,
rleq_wf,
set_wf,
rmul_wf,
rleq_weakening,
itermSubtract_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
req-iff-rsub-is-0,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_const_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
natural_numberEquality,
independent_isectElimination,
sqequalRule,
lambdaEquality,
productElimination,
independent_pairEquality,
voidElimination,
applyEquality,
setElimination,
rename,
dependent_set_memberEquality,
setEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
independent_functionElimination,
imageMemberEquality,
baseClosed,
universeEquality,
minusEquality,
axiomEquality,
functionEquality,
approximateComputation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality
Latex:
\mforall{}[a:\mBbbR{}]. \mforall{}[b:\{b:\mBbbR{}| a \mleq{} b\} ]. \mforall{}[f:\{f:[a, b] {}\mrightarrow{}\mBbbR{}| ifun(f;[a, b])\} ].
r0 \mleq{} \mint{} f[x] dx on [a, b] supposing \mforall{}x:\mBbbR{}. ((x \mmember{} [a, b]) {}\mRightarrow{} (r0 \mleq{} f[x]))
Date html generated:
2018_05_22-PM-02_57_55
Last ObjectModification:
2017_10_23-PM-05_21_28
Theory : reals_2
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