Nuprl Lemma : proper-interval-to-int-bounded
∀a,b:ℝ.
∀f:{x:ℝ| x ∈ [a, b]} ⟶ ℤ. ∃B:ℕ. ∀x:{x:ℝ| x ∈ [a, b]} . ∃y:{x:ℝ| x ∈ [a, b]} . ((x = y) ∧ (|f y| ≤ B)) supposing a < \000Cb
Proof
Definitions occuring in Statement :
rccint: [l, u],
i-member: r ∈ I,
rless: x < y,
req: x = y,
real: ℝ,
absval: |i|,
nat: ℕ,
uimplies: b supposing a,
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uimplies: b supposing a,
implies: P ⇒ Q,
top: Top,
uall: ∀[x:A]. B[x],
sq_stable: SqStable(P),
and: P ∧ Q,
squash: ↓T,
prop: ℙ,
guard: {T},
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
cand: A c∧ B,
nat: ℕ
Lemmas referenced :
cantor-to-interval-onto-proper,
member_rccint_lemma,
istype-void,
sq_stable__rleq,
i-member_wf,
rccint_wf,
cantor-to-int-bounded,
cantor-to-interval_wf,
rleq_weakening_rless,
istype-nat,
bool_wf,
req_wf,
istype-le,
absval_wf,
istype-int,
rless_wf,
real_wf,
req_inversion
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isect_memberFormation_alt,
independent_isectElimination,
setElimination,
rename,
independent_functionElimination,
sqequalRule,
isect_memberEquality_alt,
voidElimination,
isectElimination,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
because_Cache,
setIsType,
inhabitedIsType,
universeIsType,
lambdaEquality_alt,
applyEquality,
functionIsType,
dependent_pairFormation_alt,
productIsType,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation
Latex:
\mforall{}a,b:\mBbbR{}.
\mforall{}f:\{x:\mBbbR{}| x \mmember{} [a, b]\} {}\mrightarrow{} \mBbbZ{}
\mexists{}B:\mBbbN{}. \mforall{}x:\{x:\mBbbR{}| x \mmember{} [a, b]\} . \mexists{}y:\{x:\mBbbR{}| x \mmember{} [a, b]\} . ((x = y) \mwedge{} (|f y| \mleq{} B))
supposing a < b
Date html generated:
2019_10_30-AM-07_42_34
Last ObjectModification:
2019_06_26-PM-03_10_43
Theory : reals
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