Nuprl Lemma : real-fun-uniformly-less
∀a:ℝ. ∀b:{b:ℝ| a ≤ b} . ∀f:[a, b] ⟶ℝ.
(real-fun(f;a;b)
⇒ (∀c:ℝ. ((∀x:{x:ℝ| x ∈ [a, b]} . ((f x) < c)) ⇒ (∃c':{c':ℝ| c' < c} . ∀x:{x:ℝ| x ∈ [a, b]} . ((f x) ≤ c')))))
Proof
Definitions occuring in Statement :
real-fun: real-fun(f;a;b),
rfun: I ⟶ℝ,
rccint: [l, u],
i-member: r ∈ I,
rleq: x ≤ y,
rless: x < y,
real: ℝ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
rfun: I ⟶ℝ,
uall: ∀[x:A]. B[x],
real-fun: real-fun(f;a;b),
top: Top,
prop: ℙ,
and: P ∧ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uimplies: b supposing a,
guard: {T},
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
exists: ∃x:A. B[x],
sq_stable: SqStable(P),
squash: ↓T,
rge: x ≥ y
Latex:
\mforall{}a:\mBbbR{}. \mforall{}b:\{b:\mBbbR{}| a \mleq{} b\} . \mforall{}f:[a, b] {}\mrightarrow{}\mBbbR{}.
(real-fun(f;a;b)
{}\mRightarrow{} (\mforall{}c:\mBbbR{}
((\mforall{}x:\{x:\mBbbR{}| x \mmember{} [a, b]\} . ((f x) < c))
{}\mRightarrow{} (\mexists{}c':\{c':\mBbbR{}| c' < c\} . \mforall{}x:\{x:\mBbbR{}| x \mmember{} [a, b]\} . ((f x) \mleq{} c')))))
Date html generated:
2020_05_20-PM-00_23_40
Last ObjectModification:
2019_12_05-PM-04_32_08
Theory : reals
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