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
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