Nuprl Lemma : unit-balls-homeomorphic+
∀n:ℕ+. homeomorphic+({q:ℝ^n| mdist(rn-metric(n);λi.r0;q) ≤ r1} rn-metric(n);{q:ℝ^n| mdist(max-metric(n);λi.r0;q) ≤ r1} \000C;max-metric(n))
Proof
Definitions occuring in Statement : 
max-metric: max-metric(n)
, 
rn-metric: rn-metric(n)
, 
real-vec: ℝ^n
, 
homeomorphic+: homeomorphic+(X;dX;Y;dY)
, 
mdist: mdist(d;x;y)
, 
rleq: x ≤ y
, 
int-to-real: r(n)
, 
nat_plus: ℕ+
, 
all: ∀x:A. B[x]
, 
set: {x:A| B[x]} 
, 
lambda: λx.A[x]
, 
natural_number: $n
Definitions unfolded in proof : 
rge: x ≥ y
, 
less_than': less_than'(a;b)
, 
meq: x ≡ y
, 
metric: metric(X)
, 
rtermConstant: "const"
, 
req-vec: req-vec(n;x;y)
, 
real-vec-mul: a*X
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
rat_term_to_real: rat_term_to_real(f;t)
, 
rtermDivide: num "/" denom
, 
rat_term_ind: rat_term_ind, 
rtermVar: rtermVar(var)
, 
pi1: fst(t)
, 
rtermMultiply: left "*" right
, 
pi2: snd(t)
, 
true: True
, 
label: ...$L... t
, 
sq_stable: SqStable(P)
, 
squash: ↓T
, 
rdiv: (x/y)
, 
rneq: x ≠ y
, 
stable: Stable{P}
, 
req_int_terms: t1 ≡ t2
, 
rev_uimplies: rev_uimplies(P;Q)
, 
uiff: uiff(P;Q)
, 
sq_exists: ∃x:A [B[x]]
, 
rless: x < y
, 
scale-metric: c*d
, 
rev_implies: P 
⇐ Q
, 
iff: P 
⇐⇒ Q
, 
is-mfun: f:FUN(X;Y)
, 
mfun: FUN(X ⟶ Y)
, 
homeomorphic+: homeomorphic+(X;dX;Y;dY)
, 
mdist: mdist(d;x;y)
, 
rn-metric: rn-metric(n)
, 
metric-leq: d1 ≤ d2
, 
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}
, 
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 : 
itermAdd_wf, 
int_term_value_add_lemma, 
int_term_value_mul_lemma, 
istype-less_than, 
rleq_functionality_wrt_implies, 
rleq_weakening_equal, 
real-vec-norm-0, 
rmul-rinv, 
rn-metric-meq, 
req-vec_inversion, 
mdist-same, 
not-rless, 
rleq_antisymmetry, 
real-vec-norm-nonneg, 
real-vec-norm-is-0, 
istype-false, 
meq_inversion, 
meq-max-metric-iff-meq-rn-metric, 
meq_weakening, 
meq_functionality, 
iff_weakening_uiff, 
req_witness, 
meq-max-metric, 
real-vec-mul-mul, 
req-vec_weakening, 
real-vec-mul_functionality, 
req-vec_functionality, 
rtermConstant_wf, 
rmul-identity1, 
rdiv_functionality, 
rneq-by-function, 
rneq-symmetry, 
req_wf, 
assert-rat-term-eq2, 
rtermDivide_wf, 
rtermMultiply_wf, 
rtermVar_wf, 
rinv-mul-as-rdiv, 
mdist-max-metric-mul, 
iff_weakening_equal, 
squash_wf, 
true_wf, 
real_wf, 
subtype_rel_self, 
equal_functionality_wrt_subtype_rel2, 
rmul_functionality, 
rabs-of-nonneg, 
sq_stable__rless, 
rless-implies-rless, 
rleq_weakening_rless, 
rsub_wf, 
real-vec-norm-mul, 
rabs_wf, 
rmul-rinv3, 
req_transitivity, 
rinv_wf2, 
real-vec-mul_wf, 
rdiv_wf, 
minimal-not-not-excluded-middle, 
minimal-double-negation-hyp-elim, 
stable__meq, 
false_wf, 
not_wf, 
unit-cube-to-unit-ball, 
real_term_value_var_lemma, 
real_term_value_const_lemma, 
real_term_value_mul_lemma, 
real_term_value_sub_lemma, 
real_polynomial_null, 
req-iff-rsub-is-0, 
rless_functionality, 
rleq_functionality, 
real-vec-dist-symmetry, 
req_weakening, 
mdist-symm, 
req_functionality, 
itermMultiply_wf, 
itermSubtract_wf, 
rless_transitivity1, 
rless_wf, 
decidable__lt, 
rless-int, 
rmul_preserves_rless, 
rleq-int, 
scale-metric_wf, 
real-vec-dist_wf, 
real-vec-dist-from-zero, 
nat_plus_wf, 
rmul_wf, 
meq_wf, 
mfun_wf, 
metric-on-subtype, 
max-metric_wf, 
is-mfun_wf, 
unit-ball-to-unit-cube, 
rn-metric-leq-max-metric, 
nat_plus_subtype_nat, 
max-metric-leq-rn-metric, 
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 : 
addEquality, 
multiplyEquality, 
hyp_replacement, 
applyLambdaEquality, 
inlFormation_alt, 
instantiate, 
universeEquality, 
imageMemberEquality, 
baseClosed, 
imageElimination, 
equalityIstype, 
inrFormation_alt, 
unionEquality, 
functionEquality, 
unionIsType, 
functionIsType, 
productIsType, 
closedConclusion, 
setEquality, 
equalitySymmetry, 
equalityTransitivity, 
functionExtensionality, 
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+(\{q:\mBbbR{}\^{}n|  mdist(rn-metric(n);\mlambda{}i.r0;q)  \mleq{}  r1\}  ;rn-metric(n);\{q:\mBbbR{}\^{}n|  mdist(max-metri\000Cc(n);\mlambda{}i.r0;q)  \mleq{}  r1\}  ;max-metric(n))
Date html generated:
2019_10_30-AM-11_26_21
Last ObjectModification:
2019_10_29-PM-01_08_07
Theory : real!vectors
Home
Index