Nuprl Lemma : path-comp-property_functionality
∀[X,Y:SeparationSpace]. (ss-homeo(X;Y) ⇒ (path-comp-property(X) ⇐⇒ path-comp-property(Y)))
Proof
Definitions occuring in Statement :
path-comp-property: path-comp-property(X),
ss-homeo: ss-homeo(X;Y),
separation-space: SeparationSpace,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
path-comp-property: path-comp-property(X),
all: ∀x:A. B[x],
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
guard: {T},
ss-homeo: ss-homeo(X;Y),
exists: ∃x:A. B[x],
path-ss: Path(X),
subtype_rel: A ⊆r B,
sq_stable: SqStable(P),
squash: ↓T,
ss-eq: x ≡ y,
ss-sep: x # y,
record-select: r.x,
path-at: p@t,
ss-comp: ss-comp(f;g),
compose: f o g,
ss-ap: f(x),
uimplies: b supposing a,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
path-comp-rel: path-comp-rel(X;f;g;h),
satisfiable_int_formula: satisfiable_int_formula(fmla),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
true: True,
i-member: r ∈ I,
rccint: [l, u],
rless: x < y,
sq_exists: ∃x:A [B[x]]
Lemmas referenced :
ss-eq_wf,
path-at_wf,
member_rccint_lemma,
rleq-int,
istype-false,
int-to-real_wf,
rleq_wf,
ss-point_wf,
path-ss_wf,
path-comp-property_wf,
ss-homeo_wf,
separation-space_wf,
ss-homeo_inversion,
ss-comp_wf,
unit-ss_wf,
subtype_rel_self,
sq_stable__ss-eq,
ss-ap_wf,
ss-fun_wf,
ss-eq_weakening,
ss-eq_functionality,
ss-ap_functionality,
path-comp-rel_wf,
real_wf,
i-member_wf,
rccint_wf,
rdiv_wf,
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_wf,
rmul-nonneg-case1,
rmul_preserves_rleq2,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
rinv_wf2,
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,
rless-int-fractions3,
decidable__lt,
intformnot_wf,
intformless_wf,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
istype-less_than,
rless_transitivity2,
rleq_weakening_rless,
rsub_wf,
rless-int-fractions2,
rless_transitivity1,
nat_plus_properties,
rleq-implies-rleq,
rmul-int
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalHypSubstitution,
universeIsType,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
dependent_functionElimination,
Error :memTop,
hypothesis,
natural_numberEquality,
productElimination,
independent_functionElimination,
independent_pairFormation,
because_Cache,
dependent_set_memberEquality_alt,
productIsType,
inhabitedIsType,
applyEquality,
imageMemberEquality,
baseClosed,
imageElimination,
independent_isectElimination,
dependent_pairFormation_alt,
promote_hyp,
setIsType,
closedConclusion,
approximateComputation,
lambdaEquality_alt,
voidElimination,
equalityIstype,
sqequalBase,
equalitySymmetry,
setElimination,
rename,
int_eqEquality,
unionElimination,
equalityTransitivity,
setEquality
Latex:
\mforall{}[X,Y:SeparationSpace]. (ss-homeo(X;Y) {}\mRightarrow{} (path-comp-property(X) \mLeftarrow{}{}\mRightarrow{} path-comp-property(Y)))
Date html generated:
2020_05_20-PM-01_20_52
Last ObjectModification:
2020_02_08-AM-11_43_46
Theory : intuitionistic!topology
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