Nuprl Lemma : integral-const
∀[a,b,c:ℝ]. (a_∫-b c dx = (c * (b - a)))
Proof
Definitions occuring in Statement :
integral: a_∫-b f[x] dx,
rsub: x - y,
req: x = y,
rmul: a * b,
real: ℝ,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
integral: a_∫-b f[x] dx,
subtype_rel: A ⊆r B,
rfun: I ⟶ℝ,
all: ∀x:A. B[x],
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
and: P ∧ Q,
prop: ℙ,
uimplies: b supposing a,
ifun: ifun(f;I),
real-fun: real-fun(f;a;b),
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rsub: x - y
Lemmas referenced :
req_witness,
top_wf,
member_rccint_lemma,
subtype_rel_dep_function,
real_wf,
rleq_wf,
rmin_wf,
rmax_wf,
subtype_rel_self,
set_wf,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
req_weakening,
req_wf,
i-member_wf,
rccint_wf,
ifun_wf,
rccint-icompact,
rmin-rleq-rmax,
integral_wf,
rmul_wf,
rsub_wf,
rmin-rleq,
Riemann-integral_wf,
radd_wf,
int-to-real_wf,
rminus_wf,
req_functionality,
rsub_functionality,
Riemann-integral-const,
uiff_transitivity,
radd_functionality,
rminus_functionality,
req_transitivity,
rmul-distrib,
rmul_over_rminus,
rmul_comm,
rminus-radd,
req_inversion,
radd-assoc,
radd-ac,
radd_comm,
rminus-as-rmul,
rmul_functionality,
rmul-identity1,
rmul-distrib2,
radd-int,
rmul-zero-both,
rminus-zero,
radd-zero-both
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality,
lambdaEquality,
hypothesisEquality,
hypothesis,
applyEquality,
sqequalRule,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
setEquality,
productEquality,
independent_isectElimination,
because_Cache,
lambdaFormation,
setElimination,
rename,
productElimination,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
minusEquality,
natural_numberEquality,
addEquality
Latex:
\mforall{}[a,b,c:\mBbbR{}]. (a\_\mint{}\msupminus{}b c dx = (c * (b - a)))
Date html generated:
2016_10_26-PM-00_07_38
Last ObjectModification:
2016_09_12-PM-05_38_42
Theory : reals_2
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