Nuprl Lemma : path-comp-set
∀[A:SeparationSpace]. ∀[P:Point(A) ⟶ ℙ].
((∀a:Point(A). Stable{P[a]})
⇒ (∀a,b:Point(A). (a ≡ b ⇒ P[b] ⇒ P[a]))
⇒ path-comp-property(A)
⇒ path-comp-property({a:A | P[a]}))
Proof
Definitions occuring in Statement :
path-comp-property: path-comp-property(X),
set-ss: {x:ss | P[x]},
ss-eq: x ≡ y,
ss-point: Point(ss),
separation-space: SeparationSpace,
stable: Stable{P},
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
path-comp-property: path-comp-property(X),
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
and: P ∧ Q,
prop: ℙ,
exists: ∃x:A. B[x],
path-comp-rel: path-comp-rel(X;f;g;h),
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
cand: A c∧ B,
le: A ≤ B,
false: False,
not: ¬A,
stable: Stable{P},
satisfiable_int_formula: satisfiable_int_formula(fmla),
path-at: p@t,
sq_stable: SqStable(P),
rdiv: (x/y),
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+
Lemmas referenced :
path-ss-point,
set-ss-point,
set-ss-eq,
real_wf,
rleq_wf,
int-to-real_wf,
ss-eq_wf,
unit-ss_wf,
unit_ss_point_lemma,
i-member_wf,
rccint_wf,
rdiv_wf,
rless-int,
rless_wf,
path-comp-rel_wf,
set-ss_wf,
ss-point_wf,
path-at_wf,
member_rccint_lemma,
rleq-int,
istype-false,
rleq_weakening_equal,
path-ss_wf,
path-comp-property_wf,
subtype_rel_self,
stable_wf,
separation-space_wf,
false_wf,
or_wf,
not_wf,
istype-void,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
rneq-int,
full-omega-unsat,
intformeq_wf,
itermConstant_wf,
istype-int,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
rmul-nonneg-case1,
rmul_preserves_rleq2,
rmul_wf,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
rinv_wf2,
sq_stable__rleq,
rleq_functionality,
req_transitivity,
rmul-rinv,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
not-rless,
rleq_weakening_rless,
nat_plus_properties,
rsub_wf,
rleq-implies-rleq,
rmul-int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalHypSubstitution,
cut,
lambdaEquality_alt,
introduction,
extract_by_obid,
isectElimination,
thin,
Error :memTop,
hypothesis,
sqequalRule,
because_Cache,
setElimination,
rename,
dependent_set_memberEquality_alt,
functionExtensionality,
applyEquality,
hypothesisEquality,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
closedConclusion,
setEquality,
productEquality,
natural_numberEquality,
functionIsType,
setIsType,
universeIsType,
productIsType,
dependent_functionElimination,
independent_functionElimination,
productElimination,
dependent_pairFormation_alt,
independent_pairFormation,
promote_hyp,
independent_isectElimination,
inrFormation_alt,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
functionEquality,
unionIsType,
unionElimination,
voidElimination,
approximateComputation,
equalityIstype,
sqequalBase,
imageElimination,
int_eqEquality,
applyLambdaEquality
Latex:
\mforall{}[A:SeparationSpace]. \mforall{}[P:Point(A) {}\mrightarrow{} \mBbbP{}].
((\mforall{}a:Point(A). Stable\{P[a]\})
{}\mRightarrow{} (\mforall{}a,b:Point(A). (a \mequiv{} b {}\mRightarrow{} P[b] {}\mRightarrow{} P[a]))
{}\mRightarrow{} path-comp-property(A)
{}\mRightarrow{} path-comp-property(\{a:A | P[a]\}))
Date html generated:
2020_05_20-PM-01_21_43
Last ObjectModification:
2020_01_06-AM-11_22_58
Theory : intuitionistic!topology
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