Nuprl Lemma : integral_wf
∀a:ℝ. ∀b:{b:ℝ| a ≤ b} . ∀f:[a, b] ⟶ℝ. ∀mc:f[x] continuous for x ∈ [a, b]. (∫ f[x] dx on [a, b] ∈ ℝ)
Proof
Definitions occuring in Statement :
integral: ∫ f[x] dx on [a, b],
continuous: f[x] continuous for x ∈ I,
rfun: I ⟶ℝ,
rccint: [l, u],
rleq: x ≤ y,
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
member: t ∈ T,
set: {x:A| B[x]}
Definitions unfolded in proof :
label: ...$L... t,
true: True,
less_than': less_than'(a;b),
subtract: n - m,
uimplies: b supposing a,
uiff: uiff(P;Q),
false: False,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
not: ¬A,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
and: P ∧ Q,
le: A ≤ B,
nat: ℕ,
nat_plus: ℕ+,
rfun: I ⟶ℝ,
prop: ℙ,
so_lambda: λ2x.t[x],
top: Top,
exists: ∃x:A. B[x],
converges: x[n]↓ as n→∞,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
integral: ∫ f[x] dx on [a, b],
member: t ∈ T,
all: ∀x:A. B[x],
so_apply: x[s]
Rules used in proof :
equalitySymmetry,
equalityTransitivity,
instantiate,
minusEquality,
intEquality,
independent_isectElimination,
independent_functionElimination,
independent_pairFormation,
unionElimination,
dependent_functionElimination,
natural_numberEquality,
addEquality,
setEquality,
dependent_set_memberEquality,
voidEquality,
voidElimination,
isect_memberEquality,
hypothesisEquality,
independent_pairEquality,
productElimination,
lambdaEquality,
because_Cache,
applyEquality,
hypothesis,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
rename,
thin,
setElimination,
cut,
lambdaFormation,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution
Latex:
\mforall{}a:\mBbbR{}. \mforall{}b:\{b:\mBbbR{}| a \mleq{} b\} . \mforall{}f:[a, b] {}\mrightarrow{}\mBbbR{}. \mforall{}mc:f[x] continuous for x \mmember{} [a, b]. (\mint{} f[x] dx on [a, b] \mmember{} \mBbbR{}\000C)
Date html generated:
2016_07_08-PM-06_00_20
Last ObjectModification:
2016_07_05-PM-03_15_05
Theory : reals
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