Nuprl Lemma : I-norm-non-neg
∀[I:{I:Interval| icompact(I)} ]. ∀[f:{x:ℝ| x ∈ I} ⟶ ℝ].
r0 ≤ ||f[x]||_x: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,
icompact: icompact(I),
i-member: r ∈ I,
interval: Interval,
rleq: x ≤ y,
req: x = y,
int-to-real: r(n),
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],
natural_number: $n
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),
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
guard: {T},
exists: ∃x:A. B[x],
cand: A c∧ B,
sq_stable: SqStable(P),
squash: ↓T,
icompact: icompact(I)
Latex:
\mforall{}[I:\{I:Interval| icompact(I)\} ]. \mforall{}[f:\{x:\mBbbR{}| x \mmember{} I\} {}\mrightarrow{} \mBbbR{}].
r0 \mleq{} ||f[x]||\_x: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_21_42
Last ObjectModification:
2020_01_03-PM-03_40_06
Theory : reals
Home
Index