Nuprl Lemma : rsum-of-nonneg-zero-iff
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ].
uiff(Σ{x[i] | n≤i≤m} = r0;∀i:{n..m + 1-}. (x[i] = r0)) supposing ∀i:{n..m + 1-}. (r0 ≤ x[i])
Proof
Definitions occuring in Statement :
rsum: Σ{x[k] | n≤k≤m},
rleq: x ≤ y,
req: x = y,
int-to-real: r(n),
real: ℝ,
int_seg: {i..j-},
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
all: ∀x:A. B[x],
so_apply: x[s],
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
prop: ℙ,
not: ¬A,
rneq: x ≠ y,
or: P ∨ Q,
guard: {T},
false: False,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x]
Lemmas referenced :
int_seg_wf,
req_witness,
int-to-real_wf,
req_wf,
rsum_wf,
rleq_wf,
real_wf,
istype-int,
req-iff-not-rneq,
rneq_wf,
rless_transitivity1,
rless_irreflexivity,
rsum-of-nonneg-positive-iff,
rless_wf,
rleq_weakening,
rsum-zero-req
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
lambdaFormation_alt,
universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
addEquality,
natural_numberEquality,
hypothesis,
sqequalRule,
lambdaEquality_alt,
dependent_functionElimination,
applyEquality,
independent_functionElimination,
functionIsTypeImplies,
inhabitedIsType,
because_Cache,
functionIsType,
productElimination,
independent_pairEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
independent_isectElimination,
unionElimination,
voidElimination,
dependent_pairFormation_alt
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[x:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}].
uiff(\mSigma{}\{x[i] | n\mleq{}i\mleq{}m\} = r0;\mforall{}i:\{n..m + 1\msupminus{}\}. (x[i] = r0)) supposing \mforall{}i:\{n..m + 1\msupminus{}\}. (r0 \mleq{} x[i])
Date html generated:
2019_10_29-AM-10_12_37
Last ObjectModification:
2019_10_10-PM-10_00_27
Theory : reals
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