Nuprl Lemma : I-norm_functionality_wrt_subinterval
∀[I:{I:Interval| icompact(I)} ]. ∀[f:{x:ℝ| x ∈ I} ⟶ ℝ].
∀[J:{J:Interval| icompact(J)} ]. ||f[x]||_x:J ≤ ||f[x]||_x:I supposing J ⊆ I
supposing ∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f[x] = f[y]))
Proof
Definitions occuring in Statement :
I-norm: ||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,
uall: ∀[x:A]. B[x],
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 :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
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)\} ]. ||f[x]||\_x:J \mleq{} ||f[x]||\_x:I supposing J \msubseteq{} 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_20_56
Last ObjectModification:
2020_01_03-PM-02_46_22
Theory : reals
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