Nuprl Lemma : integral-reverse
∀[a,c:ℝ]. ∀[f:{f:[rmin(a;c), rmax(a;c)] ⟶ℝ| ifun(f;[rmin(a;c), rmax(a;c)])} ]. (a_∫-c f[x] dx = -(c_∫-a f[x] dx))
Proof
Definitions occuring in Statement :
integral: a_∫-b f[x] dx,
ifun: ifun(f;I),
rfun: I ⟶ℝ,
rccint: [l, u],
rmin: rmin(x;y),
rmax: rmax(x;y),
req: x = y,
rminus: -(x),
real: ℝ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rfun: I ⟶ℝ,
so_apply: x[s],
prop: ℙ,
squash: ↓T,
uimplies: b supposing a,
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],
or: P ∨ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
req_witness,
i-member_wf,
rccint_wf,
rmin_wf,
rmax_wf,
real_wf,
ifun_wf,
squash_wf,
icompact_wf,
rfun_wf,
interval_wf,
eta_conv,
rccint-icompact,
rmin-rleq-rmax,
iff_weakening_equal,
integral_wf,
rminus_wf,
ifun_subtype_3,
rmin_ub,
rmin_lb,
rleq_weakening_equal,
rleq_wf,
rmax_lb,
rmax_ub,
set_wf,
integral-additive,
rleq-rmax,
int-to-real_wf,
radd_wf,
rmin-rleq,
req_functionality,
integral-single,
req_weakening,
radd-preserves-req,
req_wf,
rmul_wf,
req_inversion,
uiff_transitivity,
req_transitivity,
radd_functionality,
rminus-as-rmul,
rmul-identity1,
rmul-distrib2,
rmul_functionality,
radd-int,
rmul-zero-both,
radd_comm
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,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
dependent_functionElimination,
productElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
universeEquality,
independent_pairFormation,
inrFormation,
inlFormation,
isect_memberEquality,
natural_numberEquality,
minusEquality,
addEquality
Latex:
\mforall{}[a,c:\mBbbR{}]. \mforall{}[f:\{f:[rmin(a;c), rmax(a;c)] {}\mrightarrow{}\mBbbR{}| ifun(f;[rmin(a;c), rmax(a;c)])\} ].
(a\_\mint{}\msupminus{}c f[x] dx = -(c\_\mint{}\msupminus{}a f[x] dx))
Date html generated:
2016_10_26-PM-00_07_59
Last ObjectModification:
2016_09_12-PM-05_38_47
Theory : reals_2
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