Nuprl Lemma : integral-int-rdiv
∀[a,b:ℝ]. ∀[f:{f:[rmin(a;b), rmax(a;b)] ⟶ℝ| ifun(f;[rmin(a;b), rmax(a;b)])} ]. ∀[c:ℤ-o].
(a_∫-b (f[x])/c dx = (a_∫-b f[x] dx)/c)
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),
int-rdiv: (a)/k1,
req: x = y,
real: ℝ,
int_nzero: ℤ-o,
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,
int_nzero: ℤ-o,
implies: P ⇒ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
not: ¬A,
nequal: a ≠ b ∈ T ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
rfun: I ⟶ℝ,
ifun: ifun(f;I),
real-fun: real-fun(f;a;b),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
squash: ↓T,
label: ...$L... t,
true: True,
guard: {T},
rdiv: (x/y)
Lemmas referenced :
integral-rmul-const,
rinv_wf2,
int-to-real_wf,
rneq-int,
int_nzero_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformnot_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
set_subtype_base,
nequal_wf,
int_subtype_base,
req_witness,
int-rdiv_wf,
i-member_wf,
rccint_wf,
rmin_wf,
rmax_wf,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
req_functionality,
int-rdiv_functionality,
req_weakening,
req_wf,
ifun_wf,
rccint-icompact,
rmin-rleq-rmax,
integral_wf,
squash_wf,
icompact_wf,
rfun_wf,
interval_wf,
eta_conv,
real_wf,
equal_wf,
true_wf,
istype-universe,
iff_weakening_equal,
subtype_rel_self,
int_nzero_wf,
rdiv_wf,
rdiv_functionality,
integral_functionality,
int-rdiv-req,
member_rccint_lemma,
rleq_wf,
rmul_wf,
rmul_functionality,
rmul_comm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
because_Cache,
hypothesis,
independent_functionElimination,
dependent_functionElimination,
natural_numberEquality,
productElimination,
lambdaFormation_alt,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
equalityIstype,
applyEquality,
intEquality,
baseClosed,
sqequalBase,
equalitySymmetry,
dependent_set_memberEquality_alt,
setIsType,
inhabitedIsType,
equalityTransitivity,
imageElimination,
setEquality,
instantiate,
imageMemberEquality,
universeEquality,
isectIsTypeImplies,
productIsType
Latex:
\mforall{}[a,b:\mBbbR{}]. \mforall{}[f:\{f:[rmin(a;b), rmax(a;b)] {}\mrightarrow{}\mBbbR{}| ifun(f;[rmin(a;b), rmax(a;b)])\} ]. \mforall{}[c:\mBbbZ{}\msupminus{}\msupzero{}].
(a\_\mint{}\msupminus{}b (f[x])/c dx = (a\_\mint{}\msupminus{}b f[x] dx)/c)
Date html generated:
2019_10_30-AM-11_38_52
Last ObjectModification:
2019_01_01-PM-04_14_02
Theory : reals_2
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