Nuprl Lemma : path-end-mem-open
∀X:SeparationSpace. ∀f:Point(Path(X)). ∀O:Open(X).
(f@r1 ∈ O ⇒ (∃z:{z:ℝ| z ∈ [r0, r1)} . ∀t:{t:ℝ| t ∈ [z, r1]} . f@t ∈ O))
Proof
Definitions occuring in Statement :
ss-mem-open: x ∈ O,
ss-open: Open(X),
path-at: p@t,
path-ss: Path(X),
ss-point: Point(ss),
separation-space: SeparationSpace,
rcoint: [l, u),
rccint: [l, u],
i-member: r ∈ I,
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]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
ss-mem-open: x ∈ O,
exists: ∃x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
cand: A c∧ B,
prop: ℙ,
ss-open: Open(X),
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
top: Top,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A
Lemmas referenced :
path-end-mem-basic,
ss-mem-basic_wf,
path-at_wf,
member_rccint_lemma,
member_rcoint_lemma,
rleq_transitivity,
int-to-real_wf,
rleq_wf,
real_wf,
i-member_wf,
rccint_wf,
all_wf,
ss-mem-open_wf,
rleq-int,
false_wf,
rleq_weakening_equal,
ss-open_wf,
ss-point_wf,
path-ss_wf,
separation-space_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
dependent_pairFormation,
because_Cache,
independent_pairFormation,
productEquality,
applyEquality,
sqequalRule,
isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
setElimination,
rename,
dependent_set_memberEquality,
natural_numberEquality,
independent_isectElimination,
setEquality,
lambdaEquality
Latex:
\mforall{}X:SeparationSpace. \mforall{}f:Point(Path(X)). \mforall{}O:Open(X).
(f@r1 \mmember{} O {}\mRightarrow{} (\mexists{}z:\{z:\mBbbR{}| z \mmember{} [r0, r1)\} . \mforall{}t:\{t:\mBbbR{}| t \mmember{} [z, r1]\} . f@t \mmember{} O))
Date html generated:
2020_05_20-PM-01_22_55
Last ObjectModification:
2018_07_06-PM-07_12_52
Theory : intuitionistic!topology
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