Nuprl Lemma : I-norm_functionality
∀[I:{I:Interval| icompact(I)} ]. ∀[f:{x:ℝ| x ∈ I} ⟶ ℝ].
∀[g:{x:ℝ| x ∈ I} ⟶ ℝ]. ||f[x]||_x:I = ||g[x]||_x:I supposing ∀x:{x:ℝ| x ∈ I} . (f[x] = g[x])
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,
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],
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
so_apply: x[s],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
guard: {T},
prop: ℙ,
so_lambda: λ2x.t[x],
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B
Latex:
\mforall{}[I:\{I:Interval| icompact(I)\} ]. \mforall{}[f:\{x:\mBbbR{}| x \mmember{} I\} {}\mrightarrow{} \mBbbR{}].
\mforall{}[g:\{x:\mBbbR{}| x \mmember{} I\} {}\mrightarrow{} \mBbbR{}]. ||f[x]||\_x:I = ||g[x]||\_x:I supposing \mforall{}x:\{x:\mBbbR{}| x \mmember{} I\} . (f[x] = g[x])
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_19
Last ObjectModification:
2020_01_03-PM-02_40_12
Theory : reals
Home
Index