Nuprl Lemma : real-fun-uniformly-positive
∀a:ℝ. ∀b:{b:ℝ| a ≤ b} . ∀f:[a, b] ⟶ℝ.
(real-fun(f;a;b) ⇒ (∀x:{x:ℝ| x ∈ [a, b]} . (r0 < (f x))) ⇒ (∃c:{c:ℝ| r0 < 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,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
rfun: I ⟶ℝ,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
real-fun: real-fun(f;a;b),
so_apply: x[s],
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
exists: ∃x:A. B[x],
nat_plus: ℕ+,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rless: x < y,
sq_exists: ∃x:A [B[x]],
cand: A c∧ B,
sq_stable: SqStable(P),
squash: ↓T,
le: A ≤ B,
less_than': less_than'(a;b),
rdiv: (x/y),
req_int_terms: t1 ≡ t2
Latex:
\mforall{}a:\mBbbR{}. \mforall{}b:\{b:\mBbbR{}| a \mleq{} b\} . \mforall{}f:[a, b] {}\mrightarrow{}\mBbbR{}.
(real-fun(f;a;b)
{}\mRightarrow{} (\mforall{}x:\{x:\mBbbR{}| x \mmember{} [a, b]\} . (r0 < (f x)))
{}\mRightarrow{} (\mexists{}c:\{c:\mBbbR{}| r0 < c\} . \mforall{}x:\{x:\mBbbR{}| x \mmember{} [a, b]\} . (c < (f x))))
Date html generated:
2020_05_20-PM-00_22_53
Last ObjectModification:
2020_01_08-AM-10_51_16
Theory : reals
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