Nuprl Lemma : classical-exists-n-implies-approx
∀I:{I:Interval| icompact(I)} . ∀n:ℕ. ∀f:{f:I^n ⟶ ℝ| ∀a,b:I^n. (req-vec(n;a;b) ⇒ ((f a) = (f b)))} .
((¬¬(∃x:I^n. ((f x) = r0))) ⇒ (∀e:{e:ℝ| r0 < e} . ∃x:I^n. (|f x| < e)))
Proof
Definitions occuring in Statement :
interval-vec: I^n,
req-vec: req-vec(n;x;y),
icompact: icompact(I),
interval: Interval,
rless: x < y,
rabs: |x|,
req: x = y,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
not: ¬A,
exists: ∃x:A. B[x],
false: False,
interval-vec: I^n,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q
Lemmas referenced :
approx-zero,
real_wf,
rless_wf,
int-to-real_wf,
interval-vec_wf,
req_wf,
istype-void,
req-vec_wf,
istype-nat,
interval_wf,
icompact_wf,
rneq_wf,
rneq_irrefl,
rneq_functionality,
req_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
setIsType,
universeIsType,
isectElimination,
natural_numberEquality,
sqequalRule,
functionIsType,
productIsType,
setElimination,
rename,
applyEquality,
because_Cache,
productElimination,
voidElimination,
independent_isectElimination
Latex:
\mforall{}I:\{I:Interval| icompact(I)\} . \mforall{}n:\mBbbN{}. \mforall{}f:\{f:I\^{}n {}\mrightarrow{} \mBbbR{}| \mforall{}a,b:I\^{}n. (req-vec(n;a;b) {}\mRightarrow{} ((f a) = (f b)))\}\000C .
((\mneg{}\mneg{}(\mexists{}x:I\^{}n. ((f x) = r0))) {}\mRightarrow{} (\mforall{}e:\{e:\mBbbR{}| r0 < e\} . \mexists{}x:I\^{}n. (|f x| < e)))
Date html generated:
2019_10_30-AM-08_27_14
Last ObjectModification:
2019_05_29-AM-09_12_17
Theory : reals
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