Nuprl Lemma : Riemann-integral_functionality
∀[a:ℝ]. ∀[b:{b:ℝ| a ≤ b} ]. ∀[f,g:{f:[a, b] ⟶ℝ| ifun(f;[a, b])} ].
∫ f[x] dx on [a, b] = ∫ g[x] dx on [a, b] supposing ∀x:ℝ. (((a ≤ x) ∧ (x ≤ b)) ⇒ (f[x] = g[x]))
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],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s],
prop: ℙ,
ifun: ifun(f;I),
all: ∀x:A. B[x],
top: Top,
real-fun: real-fun(f;a;b),
implies: P ⇒ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
sq_stable: SqStable(P),
squash: ↓T,
cand: A c∧ B,
i-member: r ∈ I,
rccint: [l, u]
Lemmas referenced :
rleq_antisymmetry,
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,
Riemann-integral-rleq,
req_witness,
all_wf,
rleq_wf,
member_rccint_lemma,
rfun_wf,
sq_stable__rleq,
rleq_weakening_equal,
rleq_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
setElimination,
rename,
dependent_set_memberEquality,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesisEquality,
setEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaFormation,
independent_functionElimination,
independent_isectElimination,
productElimination,
functionEquality,
productEquality,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation
Latex:
\mforall{}[a:\mBbbR{}]. \mforall{}[b:\{b:\mBbbR{}| a \mleq{} b\} ]. \mforall{}[f,g:\{f:[a, b] {}\mrightarrow{}\mBbbR{}| ifun(f;[a, b])\} ].
\mint{} f[x] dx on [a, b] = \mint{} g[x] dx on [a, b] supposing \mforall{}x:\mBbbR{}. (((a \mleq{} x) \mwedge{} (x \mleq{} b)) {}\mRightarrow{} (f[x] = g[x]))
Date html generated:
2016_10_26-PM-00_05_54
Last ObjectModification:
2016_09_12-PM-05_38_23
Theory : reals_2
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