Nuprl Lemma : integral_functionality
∀[a,b:ℝ]. ∀[f,g:{f:[rmin(a;b), rmax(a;b)] ⟶ℝ| ifun(f;[rmin(a;b), rmax(a;b)])} ].
a_∫-b f[x] dx = a_∫-b g[x] dx supposing ∀x:ℝ. (((rmin(a;b) ≤ x) ∧ (x ≤ rmax(a;b))) ⇒ (f[x] = g[x]))
Proof
Definitions occuring in Statement :
integral: a_∫-b f[x] dx,
ifun: ifun(f;I),
rfun: I ⟶ℝ,
rccint: [l, u],
rleq: x ≤ y,
rmin: rmin(x;y),
rmax: rmax(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,
integral: a_∫-b f[x] dx,
rfun: I ⟶ℝ,
so_apply: x[s],
prop: ℙ,
squash: ↓T,
label: ...$L... t,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
guard: {T},
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
cand: A c∧ B,
top: Top,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y
Lemmas referenced :
rsub_functionality,
i-member_wf,
rccint_wf,
rmin_wf,
rmax_wf,
real_wf,
ifun_wf,
eta_conv,
rccint-icompact,
rmin-rleq-rmax,
iff_weakening_equal,
ifun_subtype_3,
rleq_weakening_equal,
rmin-rleq,
rleq-rmax,
Riemann-integral_wf,
rleq_wf,
Riemann-integral_functionality,
req_witness,
squash_wf,
icompact_wf,
rfun_wf,
interval_wf,
integral_wf,
all_wf,
req_wf,
member_rccint_lemma,
set_wf,
rleq_functionality_wrt_implies
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
dependent_set_memberEquality,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesisEquality,
hypothesis,
setEquality,
imageElimination,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
productElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
independent_pairFormation,
productEquality,
universeEquality,
functionEquality,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}[a,b:\mBbbR{}]. \mforall{}[f,g:\{f:[rmin(a;b), rmax(a;b)] {}\mrightarrow{}\mBbbR{}| ifun(f;[rmin(a;b), rmax(a;b)])\} ].
a\_\mint{}\msupminus{}b f[x] dx = a\_\mint{}\msupminus{}b g[x] dx
supposing \mforall{}x:\mBbbR{}. (((rmin(a;b) \mleq{} x) \mwedge{} (x \mleq{} rmax(a;b))) {}\mRightarrow{} (f[x] = g[x]))
Date html generated:
2016_10_26-PM-00_08_11
Last ObjectModification:
2016_09_12-PM-05_38_51
Theory : reals_2
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