Nuprl Lemma : path-ss-point
∀[X:Top]
(Point(Path(X)) ~ {f:{x:ℝ| (r0 ≤ x) ∧ (x ≤ r1)} ⟶ Point(X)| ∀t,t':{x:ℝ| (r0 ≤ x) ∧ (x ≤ r1)} . (t ≡ t' ⇒ f t ≡ f t\000C')} )
Proof
Definitions occuring in Statement :
path-ss: Path(X),
unit-ss: 𝕀,
ss-eq: x ≡ y,
ss-point: Point(ss),
rleq: x ≤ y,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
top: Top,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
sqequal: s ~ t
Definitions unfolded in proof :
ss-point: Point(ss),
path-ss: Path(X),
ss-fun: X ⟶ Y,
ss-function: ss-function(X;Y;f),
fun-ss: A ⟶ ss,
set-ss: {x:ss | P[x]},
mk-ss: Point=P #=Sep cotrans=C,
all: ∀x:A. B[x],
member: t ∈ T,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
uall: ∀[x:A]. B[x]
Lemmas referenced :
unit_ss_point_lemma,
rec_select_update_lemma,
istype-top
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
Error :memTop,
isect_memberFormation_alt,
axiomSqEquality
Latex:
\mforall{}[X:Top]
(Point(Path(X)) \msim{} \{f:\{x:\mBbbR{}| (r0 \mleq{} x) \mwedge{} (x \mleq{} r1)\} {}\mrightarrow{} Point(X)|
\mforall{}t,t':\{x:\mBbbR{}| (r0 \mleq{} x) \mwedge{} (x \mleq{} r1)\} . (t \mequiv{} t' {}\mRightarrow{} f t \mequiv{} f t')\} )
Date html generated:
2020_05_20-PM-01_20_12
Last ObjectModification:
2020_02_08-AM-11_41_42
Theory : intuitionistic!topology
Home
Index