Nuprl Lemma : real-fun-uniformly-greater
∀a:ℝ. ∀b:{b:ℝ| a ≤ b} . ∀f:[a, b] ⟶ℝ.
(real-fun(f;a;b)
⇒ (∀c:ℝ. ((∀x:{x:ℝ| x ∈ [a, b]} . (c < (f x))) ⇒ (∃c':{c':ℝ| c < c'} . ∀x:{x:ℝ| x ∈ [a, b]} . (c' ≤ (f x))))))
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,
uimplies: b supposing a,
prop: ℙ,
and: P ∧ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
guard: {T},
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
exists: ∃x:A. B[x],
sq_stable: SqStable(P),
squash: ↓T,
rev_uimplies: rev_uimplies(P;Q),
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]\} . (c < (f x)))
{}\mRightarrow{} (\mexists{}c':\{c':\mBbbR{}| c < c'\} . \mforall{}x:\{x:\mBbbR{}| x \mmember{} [a, b]\} . (c' \mleq{} (f x))))))
Date html generated:
2020_05_20-PM-00_23_17
Last ObjectModification:
2019_12_05-PM-04_28_18
Theory : reals
Home
Index