Nuprl Lemma : ss-homotopic_weakening
∀X:SeparationSpace
∀[x0,x1:Point(X)]. ∀a,b:Point(Path(X)). (ss-homotopic(X;x0;x1;a;b)) supposing (a ≡ b and b@r0 ≡ x0 and b@r1 ≡ x1)
Proof
Definitions occuring in Statement :
ss-homotopic: ss-homotopic(X;x0;x1;a;b),
path-at: p@t,
path-ss: Path(X),
ss-eq: x ≡ y,
ss-point: Point(ss),
separation-space: SeparationSpace,
int-to-real: r(n),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
ss-eq: x ≡ y,
not: ¬A,
implies: P ⇒ Q,
false: False,
top: Top,
and: P ∧ Q,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
ss-homotopic: ss-homotopic(X;x0;x1;a;b),
exists: ∃x:A. B[x],
path-at: p@t,
guard: {T}
Lemmas referenced :
ss-sep_wf,
path-at_wf,
member_rccint_lemma,
rleq-int,
false_wf,
rleq_weakening_equal,
int-to-real_wf,
rleq_wf,
path-ss_wf,
path-ss-point,
real_wf,
ss-eq_weakening,
ss-eq_wf,
unit-ss_wf,
unit_ss_point_lemma,
set_wf,
all_wf,
ss-point_wf,
separation-space_wf,
ss-eq_inversion,
i-member_wf,
rccint_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
voidElimination,
extract_by_obid,
isectElimination,
because_Cache,
isect_memberEquality,
voidEquality,
hypothesis,
natural_numberEquality,
productElimination,
independent_functionElimination,
independent_pairFormation,
independent_isectElimination,
dependent_set_memberEquality,
productEquality,
rename,
setEquality,
functionEquality,
applyEquality,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
independent_pairEquality,
setElimination
Latex:
\mforall{}X:SeparationSpace
\mforall{}[x0,x1:Point(X)].
\mforall{}a,b:Point(Path(X)). (ss-homotopic(X;x0;x1;a;b)) supposing (a \mequiv{} b and b@r0 \mequiv{} x0 and b@r1 \mequiv{} x1)
Date html generated:
2020_05_20-PM-01_20_35
Last ObjectModification:
2018_07_05-PM-03_52_53
Theory : intuitionistic!topology
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