Nuprl Lemma : range_sup_functionality_wrt_subinterval
∀I:{I:Interval| icompact(I)} . ∀f:{x:ℝ| x ∈ I} ⟶ ℝ.
∀J:{J:Interval| icompact(J)} . (J ⊆ I ⇒ (sup{f[x] | x ∈ J} ≤ sup{f[x] | x ∈ I}))
supposing ∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f[x] = f[y]))
Proof
Definitions occuring in Statement :
range_sup: sup{f[x] | x ∈ I},
subinterval: I ⊆ J ,
icompact: icompact(I),
i-member: r ∈ I,
interval: Interval,
rleq: x ≤ y,
req: x = y,
real: ℝ,
uimplies: b supposing a,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
subinterval: I ⊆ J ,
prop: ℙ,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
and: P ∧ Q,
guard: {T}
Latex:
\mforall{}I:\{I:Interval| icompact(I)\} . \mforall{}f:\{x:\mBbbR{}| x \mmember{} I\} {}\mrightarrow{} \mBbbR{}.
\mforall{}J:\{J:Interval| icompact(J)\} . (J \msubseteq{} I {}\mRightarrow{} (sup\{f[x] | x \mmember{} J\} \mleq{} sup\{f[x] | x \mmember{} I\}))
supposing \mforall{}x,y:\{x:\mBbbR{}| x \mmember{} I\} . ((x = y) {}\mRightarrow{} (f[x] = f[y]))
Date html generated:
2020_05_20-PM-00_17_10
Last ObjectModification:
2020_01_03-PM-03_10_43
Theory : reals
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