Nuprl Lemma : classical-exists-implies-approx
∀I:{I:Interval| icompact(I)} . ∀f:{f:I ⟶ℝ| ifun(f;I)} .
((¬¬(∃x:{x:ℝ| x ∈ I} . (f[x] = r0))) ⇒ (∀e:{e:ℝ| r0 < e} . ∃x:{x:ℝ| x ∈ I} . (|f[x]| < e)))
Proof
Definitions occuring in Statement :
ifun: ifun(f;I),
icompact: icompact(I),
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
rless: x < y,
rabs: |x|,
req: x = y,
int-to-real: r(n),
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
implies: P ⇒ Q,
false: False,
uall: ∀[x:A]. B[x],
prop: ℙ,
sq_stable: SqStable(P),
squash: ↓T,
uimplies: b supposing a,
rfun: I ⟶ℝ,
interval-vec: I^n,
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
guard: {T},
top: Top,
req-vec: req-vec(n;x;y),
subinterval: I ⊆ J ,
cand: A c∧ B,
real-fun: real-fun(f;a;b),
ifun: ifun(f;I),
icompact: icompact(I),
so_apply: x[s],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
sq_exists: ∃x:A [B[x]],
rless: x < y
Latex:
\mforall{}I:\{I:Interval| icompact(I)\} . \mforall{}f:\{f:I {}\mrightarrow{}\mBbbR{}| ifun(f;I)\} .
((\mneg{}\mneg{}(\mexists{}x:\{x:\mBbbR{}| x \mmember{} I\} . (f[x] = r0))) {}\mRightarrow{} (\mforall{}e:\{e:\mBbbR{}| r0 < e\} . \mexists{}x:\{x:\mBbbR{}| x \mmember{} I\} . (|f[x]| < e)))
Date html generated:
2020_05_20-PM-00_41_02
Last ObjectModification:
2020_01_06-PM-00_15_38
Theory : reals
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