Nuprl Lemma : I-norm-rleq
∀[I:{I:Interval| icompact(I)} ]. ∀[f:{x:ℝ| x ∈ I} ⟶ ℝ].
∀[c:ℝ]. ||f[x]||_x:I ≤ c supposing ∀[x:{r:ℝ| r ∈ I} ]. (|f[x]| ≤ c)
supposing ∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f[x] = f[y]))
Proof
Definitions occuring in Statement :
I-norm: ||f[x]||_x:I,
icompact: icompact(I),
i-member: r ∈ I,
interval: Interval,
rleq: x ≤ y,
rabs: |x|,
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,
I-norm: ||f[x]||_x:I,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
implies: P ⇒ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
guard: {T},
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A
Latex:
\mforall{}[I:\{I:Interval| icompact(I)\} ]. \mforall{}[f:\{x:\mBbbR{}| x \mmember{} I\} {}\mrightarrow{} \mBbbR{}].
\mforall{}[c:\mBbbR{}]. ||f[x]||\_x:I \mleq{} c supposing \mforall{}[x:\{r:\mBbbR{}| r \mmember{} I\} ]. (|f[x]| \mleq{} c)
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_34
Last ObjectModification:
2020_01_03-PM-02_38_06
Theory : reals
Home
Index