Nuprl Lemma : unit-balls-homeomorphic+-2
∀n:ℕ+. homeomorphic+(B(n;r1);rn-metric(n);[r(-1), r1]^n;max-metric(n))
Proof
Definitions occuring in Statement :
real-ball: B(n;r),
max-metric: max-metric(n),
rn-metric: rn-metric(n),
interval-vec: I^n,
homeomorphic+: homeomorphic+(X;dX;Y;dY),
rccint: [l, u],
int-to-real: r(n),
nat_plus: ℕ+,
all: ∀x:A. B[x],
minus: -n,
natural_number: $n
Definitions unfolded in proof :
mfun: FUN(X ⟶ Y),
is-mfun: f:FUN(X;Y),
homeomorphic+: homeomorphic+(X;dX;Y;dY),
mdist: mdist(d;x;y),
rn-metric: rn-metric(n),
rev_uimplies: rev_uimplies(P;Q),
true: True,
squash: ↓T,
cand: A c∧ B,
uiff: uiff(P;Q),
less_than': less_than'(a;b),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
interval-vec: I^n,
prop: ℙ,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
nat: ℕ,
guard: {T},
real-ball: B(n;r),
subtype_rel: A ⊆r B,
ext-eq: A ≡ B,
nat_plus: ℕ+,
le: A ≤ B,
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
member: t ∈ T,
real-vec: ℝ^n,
all: ∀x:A. B[x]
Lemmas referenced :
mfun_wf,
rmul_wf,
meq_wf,
is-mfun_wf,
metric-on-subtype,
unit-balls-homeomorphic+,
req_weakening,
mdist-symm,
req_functionality,
real-vec-dist-from-zero,
nat_plus_wf,
interval-vec_wf,
max-metric_wf,
rccint_wf,
i-member_wf,
iff_weakening_equal,
subtype_rel_self,
rminus-int,
real_wf,
true_wf,
squash_wf,
member_rccint_lemma,
nat_plus_subtype_nat,
istype-false,
rleq-int,
max-metric-mdist-from-zero-2,
real-ball_wf,
rleq_wf,
rleq_weakening,
rleq_transitivity,
real-vec-norm_wf,
rn-metric_wf,
istype-le,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_plus_properties,
real-vec_wf,
mdist_wf,
req_inversion,
int_seg_wf,
int-to-real_wf
Rules used in proof :
setEquality,
productIsType,
functionExtensionality,
closedConclusion,
minusEquality,
functionIsType,
universeEquality,
instantiate,
baseClosed,
imageMemberEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
applyEquality,
inhabitedIsType,
setIsType,
voidElimination,
isect_memberEquality_alt,
int_eqEquality,
dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
unionElimination,
dependent_functionElimination,
because_Cache,
dependent_set_memberEquality_alt,
independent_pairFormation,
hypothesisEquality,
natural_numberEquality,
universeIsType,
hypothesis,
productElimination,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
lambdaEquality_alt,
sqequalRule,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}n:\mBbbN{}\msupplus{}. homeomorphic+(B(n;r1);rn-metric(n);[r(-1), r1]\^{}n;max-metric(n))
Date html generated:
2019_10_30-AM-11_26_26
Last ObjectModification:
2019_10_29-PM-01_06_58
Theory : real!vectors
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