Nuprl Lemma : weighted-mean-properties_wf
∀[I:Interval]. ∀[F:{x:ℝ| x ∈ I} ⟶ {x:ℝ| x ∈ I} ⟶ r:{r:ℝ| r0 ≤ r} ⟶ {s:ℝ| (r0 ≤ s) ∧ (r0 < (r + s))} ⟶ {x:ℝ| x ∈ \000CI} ].
(weighted-mean-properties(I;F) ∈ ℙ)
Proof
Definitions occuring in Statement :
weighted-mean-properties: weighted-mean-properties(I;F),
i-member: r ∈ I,
interval: Interval,
rleq: x ≤ y,
rless: x < y,
radd: a + b,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
prop: ℙ,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
rge: x ≥ y,
true: True,
squash: ↓T,
less_than: a < b,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
so_apply: x[s],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
weighted-mean-properties: weighted-mean-properties(I;F),
or: P ∨ Q,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
uiff: uiff(P;Q),
all: ∀x:A. B[x],
prop: ℙ,
uimplies: b supposing a,
guard: {T},
uall: ∀[x:A]. B[x],
cand: A c∧ B,
and: P ∧ Q,
member: t ∈ T
Lemmas referenced :
req_weakening,
rless_functionality,
radd_functionality_wrt_rleq,
rless_functionality_wrt_implies,
interval_wf,
rless-int,
rleq_weakening_equal,
false_wf,
rleq-int,
req_wf,
i-member_wf,
real_wf,
all_wf,
rless_transitivity2,
rmul-is-positive,
radd-zero,
rmul-nonneg-case1,
rmul_wf,
trivial-rless-radd,
radd_wf,
rless_wf,
rleq_wf,
int-to-real_wf,
rleq_weakening_rless
Rules used in proof :
isect_memberEquality,
functionEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
baseClosed,
imageMemberEquality,
functionExtensionality,
applyEquality,
rename,
setElimination,
lambdaFormation,
lambdaEquality,
setEquality,
sqequalRule,
isect_memberFormation,
inlFormation,
independent_functionElimination,
productElimination,
dependent_functionElimination,
because_Cache,
productEquality,
independent_pairFormation,
independent_isectElimination,
natural_numberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
hypothesis,
cut,
hypothesisEquality,
dependent_set_memberEquality,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[I:Interval]. \mforall{}[F:\{x:\mBbbR{}| x \mmember{} I\}
{}\mrightarrow{} \{x:\mBbbR{}| x \mmember{} I\}
{}\mrightarrow{} r:\{r:\mBbbR{}| r0 \mleq{} r\}
{}\mrightarrow{} \{s:\mBbbR{}| (r0 \mleq{} s) \mwedge{} (r0 < (r + s))\}
{}\mrightarrow{} \{x:\mBbbR{}| x \mmember{} I\} ].
(weighted-mean-properties(I;F) \mmember{} \mBbbP{})
Date html generated:
2017_10_04-PM-11_10_46
Last ObjectModification:
2017_07_29-PM-09_35_52
Theory : reals_2
Home
Index