Nuprl Lemma : path-comp-prod
∀[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),
prod-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,
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),
false: False,
not: ¬A,
prop: ℙ,
uimplies: b supposing a,
path-at: p@t,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
guard: {T},
pi2: snd(t),
pi1: fst(t),
ss-eq: x ≡ y,
or: P ∨ Q,
exists: ∃x:A. B[x],
ss-point: Point(ss),
record-select: r.x,
prod-ss: ss1 × ss2,
mk-ss: Point=P #=Sep cotrans=C,
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
rev_uimplies: rev_uimplies(P;Q),
path-comp-rel: path-comp-rel(X;f;g;h),
rneq: x ≠ y,
less_than: a < b,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
i-member: r ∈ I,
rccint: [l, u],
nat_plus: ℕ+,
satisfiable_int_formula: satisfiable_int_formula(fmla),
rdiv: (x/y),
req_int_terms: t1 ≡ t2
Lemmas referenced :
ss-eq_wf,
prod-ss_wf,
path-at_wf,
member_rccint_lemma,
rleq-int,
false_wf,
rleq_weakening_equal,
int-to-real_wf,
rleq_wf,
ss-point_wf,
path-ss_wf,
path-comp-property_wf,
separation-space_wf,
path-ss-point,
prod-ss-point,
pi1_wf_top,
equal_wf,
real_wf,
unit-ss_wf,
unit_ss_point_lemma,
set_wf,
all_wf,
pi2_wf,
prod-ss-eq,
top_wf,
prod-ss-sep,
ss-sep_wf,
path-at_functionality2,
path-comp-rel_wf,
rdiv_wf,
rless-int,
rless_wf,
subtype_rel_sets,
i-member_wf,
rccint_wf,
rleq-int-fractions3,
less_than_wf,
rleq_transitivity,
rneq-int,
full-omega-unsat,
intformeq_wf,
itermConstant_wf,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
rmul_wf,
rmul-nonneg-case1,
rmul_preserves_rleq2,
rmul_comm,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
req-iff-rsub-is-0,
rleq_functionality,
req_transitivity,
rmul-rinv,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
rsub_wf,
rleq-implies-rleq,
rmul-int,
rleq-int-fractions2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
natural_numberEquality,
productElimination,
independent_functionElimination,
independent_pairFormation,
because_Cache,
independent_isectElimination,
dependent_set_memberEquality,
productEquality,
setElimination,
rename,
lambdaEquality,
applyEquality,
independent_pairEquality,
equalityTransitivity,
equalitySymmetry,
setEquality,
functionEquality,
inlFormation,
inrFormation,
dependent_pairFormation,
imageMemberEquality,
baseClosed,
multiplyEquality,
approximateComputation,
intEquality,
promote_hyp,
int_eqEquality
Latex:
\mforall{}[A,B:SeparationSpace].
(path-comp-property(A) {}\mRightarrow{} path-comp-property(B) {}\mRightarrow{} path-comp-property(A \mtimes{} B))
Date html generated:
2020_05_20-PM-01_21_05
Last ObjectModification:
2018_07_04-AM-11_34_50
Theory : intuitionistic!topology
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