Nuprl Lemma : path-comp-union
∀[A,B:SeparationSpace]. (path-comp-property(A) ⇒ path-comp-property(B) ⇒ path-comp-property(A + B))
Proof
Definitions occuring in Statement :
path-comp-property: path-comp-property(X),
union-ss: ss1 + ss2,
separation-space: SeparationSpace,
uall: ∀[x:A]. B[x],
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,
or: P ∨ Q,
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},
true: True,
record-select: r.x,
ss-sep: x # y,
union-sep: union-sep(ss1;ss2;p;q),
ss-eq: x ≡ y,
assert: ↑b,
outl: outl(x),
isl: isl(x),
btrue: tt,
bfalse: ff,
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
mk-ss: Point=P #=Sep cotrans=C,
ss-point: Point(ss),
union-ss: ss1 + ss2,
uimplies: b supposing a,
top: Top,
path-at: p@t,
exists: ∃x:A. B[x],
squash: ↓T,
less_than: a < b,
rneq: x ≠ y,
path-comp-rel: path-comp-rel(X;f;g;h),
rdiv: (x/y),
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
rccint: [l, u],
i-member: r ∈ I,
subtype_rel: A ⊆r B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
nat_plus: ℕ+,
isr: isr(x),
outr: outr(x)
Lemmas referenced :
path-in-union,
ss-eq_wf,
union-ss_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,
separation-space_wf,
istype-true,
rec_select_update_lemma,
rleq_weakening_equal,
istype-void,
path-comp-rel_wf,
unit_ss_point_lemma,
unit-ss_wf,
real_wf,
path-ss-point,
rless_wf,
rless-int,
rdiv_wf,
rccint_wf,
i-member_wf,
rinv_wf2,
itermConstant_wf,
itermVar_wf,
itermMultiply_wf,
itermSubtract_wf,
rmul_preserves_rleq2,
rmul-nonneg-case1,
rmul_wf,
rleq_functionality,
req_transitivity,
rmul-rinv,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
subtype_rel_self,
rleq_transitivity,
istype-less_than,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
intformless_wf,
intformnot_wf,
full-omega-unsat,
decidable__lt,
rleq-int-fractions3,
rleq-implies-rleq,
rsub_wf,
rmul-int,
rleq-int-fractions2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
because_Cache,
unionElimination,
universeIsType,
hypothesis,
sqequalRule,
Error :memTop,
natural_numberEquality,
productElimination,
independent_functionElimination,
independent_pairFormation,
dependent_set_memberEquality_alt,
productIsType,
inhabitedIsType,
equalitySymmetry,
equalityTransitivity,
equalityIstype,
functionIsTypeImplies,
lambdaEquality_alt,
independent_isectElimination,
voidElimination,
isect_memberEquality_alt,
dependent_pairFormation_alt,
applyEquality,
setIsType,
functionIsType,
rename,
setElimination,
inlEquality_alt,
baseClosed,
imageMemberEquality,
inrFormation_alt,
closedConclusion,
promote_hyp,
approximateComputation,
int_eqEquality,
setEquality,
inrEquality_alt
Latex:
\mforall{}[A,B:SeparationSpace].
(path-comp-property(A) {}\mRightarrow{} path-comp-property(B) {}\mRightarrow{} path-comp-property(A + B))
Date html generated:
2020_05_20-PM-01_21_30
Last ObjectModification:
2020_02_08-AM-11_39_59
Theory : intuitionistic!topology
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