Nuprl Lemma : I-norm-positive-implies
∀I:{I:Interval| icompact(I)} . ∀f:{x:ℝ| x ∈ I} ⟶ ℝ.
(r0 < ||f[x]||_x:I) ⇒ (∃c:{c:ℝ| c ∈ I} . (r0 < |f[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,
rless: x < y,
rabs: |x|,
req: x = y,
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
I-norm: ||f[x]||_x:I,
guard: {T},
label: ...$L... t,
rfun: I ⟶ℝ,
sup: sup(A) = b,
rneq: x ≠ y,
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
exists: ∃x:A. B[x],
rrange: f[x](x∈I),
rset-member: x ∈ A,
rge: x ≥ y,
rgt: x > y
Latex:
\mforall{}I:\{I:Interval| icompact(I)\} . \mforall{}f:\{x:\mBbbR{}| x \mmember{} I\} {}\mrightarrow{} \mBbbR{}.
(r0 < ||f[x]||\_x:I) {}\mRightarrow{} (\mexists{}c:\{c:\mBbbR{}| c \mmember{} I\} . (r0 < |f[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_22_06
Last ObjectModification:
2020_01_03-PM-03_35_40
Theory : reals
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